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Electronics has been my profession for well over a               excused for thinking this or that particular textbook
quarter of a century and my hobby for even                       to be mainly an ego trip for the author.
longer. Over that whole period, I have been an                      Now make no mistake, maths is an essential tool
avid collector of knowledge of the subject, so that              in electrical engineering in general and in electro-
by now my card index system contains references                  nics in particular. Indeed, the research laboratories
to hundreds of articles published during that time.              of all the large electronics companies employ at
Now references are all very well, but one often                  least one 'tame mathematician' to help out when-
needs information in a hurry, so it has been my                  ever an engineer finds himself grappling with the
practice, more often than not, also to save the                  mathematical aspects of a problem where his own
article itself. Thus I now have, stored in many                  maths is too rusty. For the practising electronic
bulging files, an invaluable hoard of articles,                  engineer (unless also a born mathematician) can no
photocopies and originals, from dozens of maga-                  more expect to be fluent in all the mathematical
zines, books and learned journals. For some years                techniques he may ever need, or indeed may have
the feeling has been growing that I should not sit               learnt in the past, than the mathematician can
on all this information, but should share it around.             expect to be abreast of all the latest developments
Of course, it is all freely available already, in the            in electronics (it takes the engineer all his time to do
various publications in which it originally ap-                  that!). It seems particularly appropriate therefore
peared, but that makes it a very diffuse body of                 to attempt to explain analog electronic circuits as
knowledge and consequently very elusive. In this                 simply as possible, appealing as far as possible to
book I have tried to bring some of it together,                  nothing more complicated than basic algebra and
concentrating on what I have found over the years                trigonometry, with which I assume the reader of
to be the most useful, and seeking to explain it as              this book to be familiar. This has been done
simply as possible. Whether or not I have suc-                   successfully in the past. Older readers may recall
ceeded, the reader must judge for himself.                       the articles by 'Cathode Ray', the pen-name of a
   This book is not a textbook, but I hope never-                well known writer of yesteryear on electronics,
theless that you will learn a good deal from it.                 which appeared over many years in the magazine
Textbooks have traditionally presented a great                    Wireless WorM. The approach adopted in this
deal of information compressed within a relatively               book is not essentially different. The pace is more
confined s p a c e - a format which is appropriate in            leisurely and discursive than in a typical textbook,
conjunction with a course including lessons or                   the aim being to take the reader 'inside' electronic
lectures, at a school, polytechnic or university.                circuits so that he can see what makes them t i c k -
However, it makes life very difficult for the student,           how and why exactly they do what they do. To this
however keen, who is working on his own with no                  end, vector diagrams are particularly useful; they
one to consult when something is not clear. It must              illustrate very graphically what is going on, en-
be said also that some textbooks seem to delight in              abling one to grasp exactly how the circuit works
the most abstract treatment of the subject, dragging             rather than simply accepting that if one slogs
in degree-level maths at every turn, even when a                 through the maths, the circuit does indeed behave
more concrete a p p r o a c h - using simple vector              as the textbooks say. There will of course be those
diagrams, for e x a m p l e - w o u l d be perfectly satisfac-   whose minds work in a more academic, mathe-
tory and much more readily comprehensible to                     matical way, and these may well find their needs
normal mortals. On occasions even, one might be                  served better by conventional textbooks.
x   Preface

   With this brief apology for a style which some       my colleague and friend of more than a quarter of
will undoubtedly find leisurely to the point of         a century's standing, Mick G. Thanks also to Dave
boredom, but which will I hope materially assist        Watson who produced the 'three-dimensional wire
others, it only remains to mention two minor            grid' illustrations of poles and zeros in Appendix 4
points before passing on to the main body of the        and elsewhere. For permission to reproduce circuit
book. First, I must apologize to British and many       diagrams or other material, supplied or originally
other non-US readers for spelling 'analog'              published by them, my thanks are also due to all
throughout in the North American manner: they           the following:
will in any case be used to seeing it spelt thus,
whereas 'analogue' looks very quaint to North           C. Barmaper Ltd
American eyes. Second, the following pages can          EDN
be read at different levels. The technically minded     Electronic Design
adolescent, already interested in electronics in the    Electronic Engineering
early years of secondary or high school, will find      Electronic Product Design
much of practical interest, even if the theory is not   Electronics Worm (formerly Wireless World)
appreciated until later. Technicians and students       ETI
at technical colleges and polytechnics will all find    Ever Ready Company (Great Britain) Ltd
the book useful, as also will electronics under-        Hewlett-Packard Journal
graduates. Indeed, many graduates and even post-        Maplin Electronic Supplies Ltd
graduates will find the book very handy, especially     Maxim Integrated Products UK Ltd
those who come into electronics from a different        Microwave Journal
background, such as a physics degree.                   Microwaves & RF
   Writing the following pages has turned out to be     Motorola Inc.
a not inconsiderable task. My sincere thanks are        New Electronics
due first to my ever-loving (and long-suffering)        Philips Components Ltd (formerly Mullard Ltd)
wife, who shared the typing load, and also to           Practical Electronics
those who have kindly vetted the work. In par-          Practical Wireless
ticular, for checking the manuscript for howlers
and for many helpful suggestions, I must thank my                                             Ian Hickman
colleagues Pete C., Dave F., Tim S. and especially                                                Eur. Ing

    1                    Passive components

The passive components used in electronic circuits           by
all make use of one of the three fundamental
phenomena of resistance, capacitance and induc-
tance. Just occasionally, two may be involved, for
example delay cable depends for its operation on             where P (lower-case Greek letter rho) is a property
both capacitance and inductance. Some com-                   of the material of the wire, called resistivity. In the
ponents depend on the interaction between an                 case of copper the value of P is 1.55 • 10 - 8 0 m in
electrical property and, say, a mechanical prop-             other words, the resistance between opposite faces
erty; thus a piezoelectric sounder operates by               of a solid cube of copper of 1 m side is 0.0155 ~f~.
virtue of the small change in dimension of certain           The term (//A)p is called the resistance of the wire,
types of ceramic dielectric when a voltage is                denoted by R. So one may write
applied. But most passive components are simply                                                  l
resistors, capacitors or inductors. In some ways                                            R -~p                         (1.2)
inductance is the most subtle effect of the three,
since with its aid one can make transformers,                Combining (1.1) and (1.2) gives I = E/R, the form
which will be described later in this chapter.               in which most people are familiar with Ohm's law
                                                             (see Figure 1.1). As mentioned earlier, when
                                                             current flows through a resistance, energy is dis-
                                                             sipated as heat. The rate at which energy is

                                                                             I (amperes)
Some substances, for example metals (particularly                                    1.0
copper and a l u m i n i u m - also gold, but that's a bit
expensive for everyday use), conduct electricity
                                                                                 ~51 0.5-
well; these substances are called conductors.
They are distinct from m a n y others called
insulators, such as glass, polystyrene, wax, P T F E          -r        jl
                                                                                    t /
                                                                               __0w.5 ~          ,       ,
                                                                                                          i      v ,~---E (volts)
                                                             -1.5                               0.5      1      1.5
etc., which in practical terms do not conduct
electricity at all. In fact, their resistivity is about
1018 or a million million million times that of
metals. Even though copper, say, conducts elec-                                     -1.0
tricity well, it exhibits some resistance to the flow
of electricity and consequently it does not conduct                 The slope of the line is given by gl/gE. In this illustration
                                                                    gl = 1 A and gE = 1 V, so the conductance G = 1 S. The S
perfectly; energy is lost in the process, appearing in              stands for siemens, the unit of conductance, formerly called
the form of heat. In the case of a wire of length 1                 the mho. G = 1/R.
metres and cross-sectional area A square metres,
                                                             Figure 1.1 Current through a resistor of R ohms as a
the current I in amperes which flows when an                 function of the applied voltage. The relation is linear,
electrical supply with an electromotive force                as shown, for a perfect resistor. At DC and low frequen-
(EMF) of E volts is connected across it is given             cies, most resistors are perfect for practical purposes.
2   Analog Electronics


                R                                                                             Rb

                    -_       2R                  R-         R/2
                                                                              A                            C      A. . . .            C
                                                                                      Star or wye ,h,               Delta or mesh A

                                                                                         AtoA                            AtoA,
                                                                    1                              RbRc                    R2R 3
               R1                                               1       1
                                                                                  R 1= R b+ R c+    Ra            Ra= R 1 + R 2+ R 3
                    =                            R2 =
                                                                 R1R 2
               R2            1 + R2                             R1 + R2                            RaR c                 R1R 3
                                                                                  R2=Ra+ Rc+        R"---~        Rb= R1+ R2 + R 3

                                                                                                    RaR b                R1R 2
                                                                                  R3=Ra+ Rb+            R---"-~   Rc= R1+ R2 + R 3
      For resistors in series, total         For resistors in parallel,
      resistance is                          1      1       1       1
      n, =     +         +                   n_ =       +       +

Figure 1.2 Resistors in combination.
  (a) Series parallel (also works for impedances).
  (b) The star-delta transformation (also works for impedances, enabling negative values of resistance effctively to
be produced).

dissipated is measured in watts, where one watt                             1 Mf~ (one million ohms) were available, but were
equals one joule per second. If a current of I                              expensive owing to the vast number of turns of
amperes flows through a resistance of R ohms,                               very fine wire needed to achieve this resistance.
the power dissipated is given by W = IZR. Using                             Nichrome (an alloy of chromium and nickel) is
Ohm's law it also follows that W = E I = E Z / R ,                          used for high-power resistors designed to dissipate
where E is the E M F necessary to cause the current                         several or many watts, whilst precision wirewound
I to flow through the resistance R. Clearly from                            resistors may use constantan or manganin (alloys
(1.2), if a second identical wire is connected in                           of copper with nickel or manganese respectively).
series with the first (doubling l) the resistance is                        Such resistors have an extremely low temperature
doubled, whilst if it is connected in parallel (dou-                        coefficient of resistance; they are available manu-
bling A) the resistance is halved (Figure 1.2 also                          factured to a tolerance of better than 0.05% and
shows the useful 'star-delta' equivalence).                                 are stable to within one part per million (1 PPM)
    Electronic engineers use resistors from a frac-                         per year. Such resistors are used as reference
tion of an ohm up to millions of ohms. Low-value                            standard resistors in measurements and standards
resistors up to a few thousand ohms are often                               laboratories. In many electronic circuits, resistors
wirewound, although pure copper wire is seldom                              with a tolerance of 1, 2 or 5% are entirely
used owing to its high temperature coefficient of                           satisfactory; indeed, in the era of thermionic
resistance, namely +0.4% per degree centigrade.                             valves 20% was the norm.
At one time, wirewound resistors with values up to                              In the interests of economy, most low-power
                                                                                      Passive components       3

resistors up to 1 W rating are not wirewound, and          Resistors are mass produced in certain preferred
indeed the resistive element is frequently non-         values, though specialist manufacturers will supply
metallic. Carbon composition resistors have a           resistors of any nominal value, at a premium.
cylindrical resistance element made of an insulat-      Appendix 1 shows the various E series, from E6
ing compound loaded with carbon, usually                which is appropriate to 20% tolerance resistors, to
protected by a moulded phenolic covering. Such          E96 for 1%.
resistors were universally used at one time and are        Resistors of 1% tolerance are readily available
still widely employed in the USA. The resistance        in metal film and metal glaze construction. Metal
tends to rise as the resistor ages, owing to the        glaze resistors use a film of glass frit and metal
absorption of moisture: the effect is less              powder, fused onto a ceramic core, resulting in a
pronounced where the resistor is run at or near         resistor with good surge and short-term overload
its rated dissipation and operates for long periods.    capability and good stability even in very low and
Carbon composition resistors not only are in-           very high resistance values. Metal film resistors
expensive but also behave very well at radio            have a conducting film made entirely of metal
frequencies, unlike wirewound resistors and to a        throughout and consequently offer a very low
lesser extent spiralled film resistors.                 noise level and a low voltage coefficient.
   The next big improvement in resistor technology         The latter can be a very important consideration
was the carbon film resistor, popularly known in        in critical measurement or very low-distortion
the early days as a Histab resistor owing to its        applications. Ohm's law indicates that the current
improved ageing characteristic. It was available in     through a resistor is directly proportional to the
5, 2 and 1% tolerances, and the 5% variety is still     voltage across it; in other words, if the current is
widely used in the U K and Europe as a general          plotted against the voltage as in Figure 1.1, the
purpose low-wattage resistor. Manufacture is            result should be a perfectly straight line, at least if
highly automated, resulting in a low-cost resistor      the rated dissipation is not exceeded. Hence a
that is very reliable when used within its rated        resistor is described as a 'linear' component. It
voltage and power limits. (Note that for resistance     can more accurately be described by a power series
values much above 100 kf~, it is not possible in the    for current as follows:
case of a carbon film resistor to dissipate its rated
                                                              I - (E + 0~E2 + ]3E 3 + 7E 4 + - . . )
power without exceeding its rated working                                                                  (1.3)
voltage.) The carbon film is deposited pyrolytically                            R
on a ceramic rod, to a thickness giving an end-to-      If at, [3, 7 and the coefficients of higher powers of
end resistance of a few per cent of the required        E are all zero, the item is a perfectly linear resistor.
final value. End caps and leads are then fitted and     In practice, 0t is usually immeasurably small.
a spiral groove is automatically machined in the        Coefficient [3 will also be very small, but not
carbon film. The machine terminates the cut when        necessarily zero. For instance, the contact resis-
the required resistance is reached, and a protective    tance between individual grains of carbon in a
insulating lacquer is applied over the film and end     carbon composition resistor can vary slightly
caps. Finally the resistance and tolerance are          with the current flowing, i.e. with the applied
marked on the body, usually by means of the             voltage, whilst with film resistors the very small
standard code of coloured bands shown in                contact resistance between the film and the end
Appendix 1.                                             caps can vary likewise. A quality control check
   Metal oxide resistors are manufactured in much       used in resistor manufacture is to apply a pure
the same way as carbon film, except that the            sinusoidal voltage of large amplitude across
resistive film is tin oxide. They exhibit a higher      sample resistors and check the size of any third-
power rating, size for size, than carbon film, and      harmonic component g e n e r a t e d - indicating a
when derated to 50 or 25% of maximum they               measurable value of ]3. Contact resistance varia-
exhibit a degree of stability comparable to Histab      tion can also be responsible for the generation of
or semiprecision types respectively.                    an excess level of random noise in a resistor, as can
4   Analog Electronics

ragged edges of the spiral adjustment cut in a film     deciding whether it is worth achieving a particular
resistor.                                               nominal value by the above means.
   It is sometimes convenient to connect two or              Variable resistors are available in various
more resistors in series or parallel, particularly      technologies: wirewound, carbon film, conductive
when a very low or very high resistance is required.    plastic, cermet etc. Both ends of the resistive track
It has already been noted that when two equal           are brought out to contacts, in addition to the
resistors are connected in series, the resultant        'slider' or 'wiper'. When the component is used
resistance is twice that of either resistor alone,      purely as a variable resistor, connections are made
and if they are connected in parallel it is half. In    to one end of the track and the wiper. It may be
the general case of several resistors of different      useful to connect the other end of the track to the
values, the results of series and parallel combina-     wiper since then, in the event of the wiper going
tions are summarized in Figure 1.2a. So, for            open-circuit for any reason, the in-circuit resis-
example, to obtain a resistance of 0.33 ft (often       tance will only rise to that of the track rather than
written as 0R33) three 1 f~ (1R0) resistors in          go completely open-circuit. When the component
parallel may be used. Not only does this arrange-       is used as a potentiometer, the wiper provides a
ment provide three times the power rating of a          signal which varies between the voltage at one end
single resistor, it also offers a closer initial        of the track and that at the o t h e r - usually
tolerance. In values down to 1R0, resistors are         maximum and zero respectively (Figure 1.3).
available with a 1% selection tolerance; whereas        Thus the voltage at the output depends upon the
for values below 1R0, 5 or 10% is standard. This        position of the wiper. But what about the effect of
would be an inconveniently large tolerance in           the resistance of any circuit we may wish to
many applications, for example the current sensing      connect to the wiper? Well, this is as convenient
shunt in a linear laboratory power supply. The          a point as any for a digression to look at some of
parallel resistor solution may, however, involve a      the corollaries of Ohm's law when connecting
cost penalty, for although three IR0 resistors          sources of electricity to loads of one sort or
will usually be cheaper than a higher-power             another, e.g. batteries to bulbs or whatever.
0R33 resistor, the assembly cost in production is           Figure 1.4a shows an ideal battery or voltage
higher.                                                 source, and Figure 1.4b a more realistic one with a
    Series resistors may be used likewise either to     finite 'internal resistance'. It would clearly be
obtain a value not otherwise readily available (e.g.    imprudent to short-circuit the ideal battery, since
200M); or to obtain a closer tolerance (e.g. two        Ohm's law indicates that with a resistance of zero
 1% 750K resistors where a 1M5 resistor is only         ohms between its terminals the resultant current
available in 5% tolerance); or to gain twice the        would be infinite- smoke and sparks the order of
working voltage obtainable with a single resistor.      the day. To be more precise, the foregoing scenario
Unequal value resistors may be combined to give a       must be fictional: for if the voltage source really
value not otherwise readily obtainable. For ex-         has zero internal resistance there must always be E
ample, E96 values are usually restricted to resistors   volts between its terminals, however much current
above 100R. Thus a 40R resistance may be                it supplies; whereas if the short-circuit really has
produced by a 39R resistance in series with 1R0,        zero resistance there can be no voltage between the
a cheaper solution than three 120R resistors in         source's terminals, however much current flows.
parallel. Likewise, a 39R 1% resistor in parallel       Shades of the irresistible force and the immovable
with 1K0 is a cheaper solution for 37R5 at 2%           object! In practice a source, be it battery or power
than two 75R 1% resistors in parallel, as the 1K0       supply, will always have some internal or source
resistor may be 5 or 10% tolerance. If you don't        resistance, say Rs. In principle one can measure Rs
believe it, do the sums! In addition to its initial     by noting the open-circuit voltage E and measur-
selection tolerance, a resistor's value changes with    ing the short-circuit current Isc through an am-
ageing, especially if used at its maximum dissipa-      meter. Then Rs = E/Isc. In practice this only works
tion rating. This must be borne in mind when            approximately, for the ammeter itself will have a
                                                                                                                             Passive c o m p o n e n t s   5

                               Variable resistors                                                               Potentiometer used as
                                                                    Potentiometer                               a variable resistor
                                                                         i           i   =   _
                                    Y or

                       Ril l
                                                                                             ~ s s - - - " tl
                                                                               jf    11~'~

                                                         c.""                            /             /i
                                                         ,,"                        /                  /I                                  Attenuated
                                                                                                                                 ~-1       signal
                     .~                      //                    i/                            /,"        I
                      ,,                I/                     /                         I,/                I

                                    /                /                         ,,"                          i
                                   Ii                                    //                                 I
                    Rm/5 - 4 - - -
                        l ,
                                        __Z. . . . . . .
                                        /              ,,/f
                                                                        "//"                                I

                               0                              50                                         100
                                                  Percentage rotation of wiper

  (A) Linear law.
  (B) Log law (20% log shown; some potentiometers have a 10% log law). Used for volume controls.
  (C) Reverse log law.

Figure 1.3   Variable resistors and potentiometers.

small but finite resistance: nevertheless you can, in                               common or garden primary (i.e. non-rechargeable)
the case of a dry (Leclanch8 primary type) battery,                                 battery such as the zinc-carbon (Leclanch6) variety
get a reasonable estimate of its source resistance.                                 is set by the rise in internal resistance rather than
(It is best not to try this with batteries having a low                             by any fall in the battery's EMF as measured off
internal resistance, such as lead-acid or Ni-Cd                                     load. (Measuring the open-circuit voltage and the
types.) Naturally it pays to short-circuit the bat-                                 short-circuit current to determine the internal
tery through the ammeter for no longer than is                                      resistance is even less successful in the case of a
absolutely necessary to note the reading, as the                                    laboratory stabilized power supply, where Rs may
procedure will rapidly discharge the battery.                                       be zero or even negative, but only up to a certain
Furthermore, the current will in all probability                                    rated output current.)
be gradually falling, since with most types of                                         The observant reader will not fail to notice that
battery the internal resistance rises as the battery                                the current flowing in the load resistance in Figure
is discharged. In fact, the end of the useful life of a                             1.4c must also be responsible for dissipating
6   Analog Electronics



                                             O                                          o
                                  (a)                                          (b)

                                        v=2Rs+R      L

                              J                E

                    1~                  I = Rs ~ RL

               +                    Load
               ,.~2V                RL

                             0V                             0                                  1"-
                                                                     0.5      1 9     1.5      2   V (volts)
Figure 1.4 The maximum power theorem.
   (a) Ideal voltage source.
   (b) Generator or source with internal resistance Rs.
   (c) Connected to a load RL.
   (d) E = 2 V, Rs = I f~. Maximum power in the load occurs when RL = Rs and V = E/2 (the matched condition),
but only half the power is supplied to the load. On short-circuit, four times the matched load power is supplied,
all dissipated in the battery's internal resistance Rs.

energy in the internal resistance of the source itself.    50%. This result is usually dignified with the title
Figure 1.4d shows the power (rate of energy)               of the maximum power theorem. The matched
dissipation in the source resistance and the load          condition gives the greatest possible power in the
for values of load resistance from zero to infinity.       load, but only at the expense of wasting as much
   It can be seen that the maximum power in the            again in the internal resistance of the source. In
load occurs when its resistance is equal to the            many cases, therefore, the source is restricted to
internal resistance of the source, that the terminal       load resistances much higher than its own internal
voltage V is then equal to half the source E M F E,        resistance, thus ensuring that nearly all of the
and that the same power is then dissipated in the          power finishes up where it is really wanted - in
source's internal resistance as in the load. This is       the load. G o o d examples of this are a radio
called the matched condition, wherein the effi-            transmitter and a hand flashlamp; an even more
ciency, defined as the power in the load divided           telling example is a 660 M W three-phase turbo-
by the total power supplied by the source, is just         alternator!
                                                                                      Passive components      7

    Now Ohm's law relates the current through a           won't in fact correspond to midtravel, as a volume
resistor to the applied E M F at any instant and          control is designed with a non-linear (approxi-
consequently, like the maximum power theorem,             mately logarithmic) variation of track resistance.
applies to both AC and DC. The AC waveform                This gives better control at low volumes, as the ear
shown in Figure 1.5 is called a sinusoidal wave-          does not perceive changes of loudness linearly.
form, or more simply a sine wave.                             Preset potentiometers for circuit adjustment on
   It is the waveform generated across the ends of a      test, on the other hand, almost invariably have
loop of wire rotating in a uniform magnetic field,        linear tracks, often with multiturn leadscrew op-
such as the earth's field may be considered to be, at     eration to enable very fine adjustments to be made
least over a localized area. Its frequency is meas-       easily. Potentiometers for user operation, e.g. tone
ured in cycles per second or hertz (Hz), which is         and volume controls, are designed for continued
the modern term. As a necessary result of Ohm's           use and are rated at greater than 100000 opera-
law, not only is the current waveform in a resistive      tions, whereas preset controls are only rated for a
circuit the same shape as the voltage waveform,           few hundred operations.
but also its peaks and troughs line up with the
voltage waveform as shown in Figure 1.5. The sine
wave shown contains alternating energy at one
frequency only, and is the only waveshape with            Capacitors are the next item on any shopping list
this important property. An audio-frequency sine          of passive components. The conduction of elec-
wave reproduced through a loudspeaker has a               tricity, at least in metals, is due to the movement of
characteristically round dull sound, like the flue        electrons. A current of one ampere means that
pipes of a flute stop on an organ. In contrast, a         approximately 6242 x 1014 electrons are flowing
sawtooth waveform or an organ-reed stop con-              past any given point in the conductor each
tains many overtones or harmonics.                        second. This number of electrons constitutes one
   Returning to the potentiometer, which might be         coulomb of electrical charge, so a current of one
the volume control in a hi-fi reproducing organ           ampere is alternatively expressed as a rate of
music or whatever, to any circuitry connected to          charge movement of one coulomb per second.
the wiper of the potentiometer it will appear as a           In a piece of metal an outer electron of each
source of an alternating EMF, having some inter-          atom is free to move about in the atomic lattice.
nal resistance. When the wiper is at the zero             Under the action of an applied EMF, e.g. from a
potential (ground or earth) end of the track, this        battery, electrons flow through the conductors
source resistance is zero. At the other end of the        forming the circuit towards the positive pole of
track, the source resistance seen 'looking back'          the battery (i.e. in the opposite direction to the
into the wiper circuit is equal to the resistance of      conventional flow of current), to be replaced by
the track itself in parallel with the source resistance   other electrons flowing from the battery's negative
of whatever circuit is supplying the signal to the        pole. If a capacitor forms part of the circuit, a
volume control. If this source resistance is very         continuous current cannot flow, since a capacitor
much higher than the resistance of the track, then        consists of two plates of metal separated by a non-
the resistance looking back into the wiper simply         conducting m e d i u m - even a vacuum, for example
increases from zero up to very nearly the track           (Figure 1.6a). If a battery is connected across the
resistance of the potentiometer as the volume is          plates, its E M F causes some electrons to leave the
turned up to maximum. In the more likely case             plate connected to its positive pole or terminal and
where the source resistance is much lower than the        an equal number to flow onto the negative plate,
track resistance - let's assume it is zero - then the     as indicated in Figure 1.6c. A capacitor is said to
highest resistance seen at the wiper occurs at            have a capacitance C of one farad (1 F) if an
midtrack and is equal to one-quarter of the end-          applied E M F of one volt stores one coulomb
to-end track resistance. If the potentiometer is          (1 C) of charge. The capacitance is proportional
indeed a volume control, then midtrack position           to A, the area of the plates in Figure 1.6a, and
8     Analog Electronics





                                                  Generator                          Load

                  v (volts)                                            (a)

                                                                     1--                  One cycle                ._1
                                                                                                                   "] v m sin(tot)
                  Vm - -      --                              2~~~-'~3~
                                                                                                                   4 ~ / 0 (radians)
                                                                                                                     Is     I,.


                 i (amperes)

                  lm - - -                                                                                         Im tsin(tot)
                                                              2~                             37r                   4~ Ir
                     0                                                                                                      "-

                 -I m -


                 w (watts)                                                                         Vm sin (tot) I m sin(tot)
                                   Area 1                                                               m
                                                                                                   - Vml 1/2 (1 - cos(2tot))
               Vmlml          ~             Area2 ~

                                                                 1                           I             "         I     "-


    One cycle corresponds to 3 6 0 ~ (or 2r~ radians), e.g. I revolution of a loop of wire in a magnetic field. If the wave-
    form has a frequency o f f Hz then each cycle lasts I / f seconds. Thus there are to = 2~f radians per second. Note
    that there are two power peaks per cycle of the applied voltage, so the angular frequency of the power waveform
    is 203 t radians per second.
    Peak power load = Vmlm--Vm2/RL--Im2RL, occurs at 0=rc/2,3rc/2 radians etc. Power in load at
    0 = tot = 0,1t, 2rc etc. is zero. Since area 1 equals 2, average power in load is (1/2)(vmZ/RL) = VZ/RL, where
    V = Vm/v/2. V is called the effective or root m e a n square (RMS)voltage.
Figure 1.5     Alternating voltage and power in a resistive circuit.
                                                                                                                                                      Passive components              9

                                                                                                                               (-) indicates electrons which
                                                                                                                               have flowed away from the
                                                                                                                             + positive metal plate
                                       Area A
                       I           .I      _ _                      ,
                                                                         I    ....

          Vacuum d

                   ,           t           _-i ,e.!ectric                                                                   ,       .       . r   .   ::_

                                                                                                                  _     _       [
          --I-                                                           I
                   (a)                                                  (b)                                                 (c)

                                         Switch closed                                                                                                             Switch closed
                 + s v ~ _~=0                                                                                                                          5f~
                                                                                     o~                                                 +

                 T             .....                   f~at t = O
                                                                                     ~" 1.8                                                                               T3F
                               OV                                                                   0   15 Time (seconds)                                            OV
                                   (d)                                                                  (e)                                                  (f)

                       5                         //I            "

                 3.15 . . . . . . . .                                                                                                                                connection
                                   #                    I

                                                                                                                                                                    All foil strips
                           0                           15                            Time (seconds)               Polystyrene
                                                         (g)                                                      films
                                                                                                                  (dielectric)          (h)
Figure 1.6       Capacitors.

inversely proportional to their separation d, so                                                              As mentioned earlier, an insulator or dielectric
that C = k(A/d) (provided that d 2 is much smaller                                                        is a substance such as air, polystyrene, ceramic etc.
than A). In vacuo the value of the constant k is                                                          which does not conduct electricity. This is because,
8.85 x 10 -12 F/m, and it is known as the permit-                                                         in an insulator, all of the electrons are closely
tivity of free space ~o. Thus in vacuo C = eo(A/d).                                                       bound to the respective atoms of which they
More commonly, the plates of a capacitor are                                                              form part. But although they cannot be completely
separated by material of some k i n d - air or a                                                          detached from their parent atoms (except by an
solid s u b s t a n c e - rather than the vacuum of free                                                  electrical force so great as to rupture and damage
space. The permittivity of air is for practical                                                           the dielectric), they can and do 'give' a little (as in
purposes the same as that of free space.                                                                  Figure 1.6c), the amount being directly propor-
10   Analog Electronics

tional to the applied voltage. This net displace-         4.67V. After 15 s the charge would be all gone,
ment of charge in the dielectric enables a larger         there would be zero voltage across the capacitor,
charge to be stored by the capacitor at a given           as indicated by the dashed line in Figure 1.6e. But
voltage than if the plates were in vacuo. The ratio       of course as the voltage across the capacitor falls,
by which the stored charge is increased is known as       so too must the current through the 5 f~ resistance
the relative permittivity t~r . Thus C = e,O~r(A/d). So   as shown by the full line. In fact, after a time
when a battery is connected to a capacitor there is       T= CR seconds (15 s in this case) the current will
a transient electrical current round the circuit, as      only have fallen to 37% of its original value. But
electrons flow from the positive plate of the             to come back to that point in a minute, though;
capacitor to the positive terminal of the battery,        meanwhile concentrate for the moment on work-
and to the negative plate from the negative               ing out the stored energy. At any moment the
terminal.                                                 power being dissipated in the resistor is IZR, so
   It was stated earlier that if the total transient      initially it is 5 W or 5 joules per second. Suppose a
flow of current needed to charge a capacitor to one       5 f~ variable resistor is used, and its value linearly
volt amounts to a total charge of one coulomb, the        reduced to zero over 15 s. Then the initial 1 A will
capacitor is said to have a capacitance of one            be maintained constant for 15 s, by the end of
farad. More generally, the charge stored on a             which time the capacitor will be discharged. The
capacitor is proportional to both the size of the         initial heat dissipation in the resistor will be 5 J per
capacitor and the applied EMF; so Q - CV, where           second, falling linearly to zero, just like the
Q coulombs is the charge stored when a voltage V          resistance, since I 2 is constant. So the average
exists between the terminals of a capacitor of C          power is 2.5 W maintained for 15 s, or a total of
farads. In electronics capacitors as small as 10 -12      37.5 J.
farad (called one picofarad and written 1 pF) up to          Starting with twice the capacitance, the initial
a few thousand microfarads or more are used. You          rate of voltage drop would only have been 0.167 V
will also encounter nanofarads (1 n F - 10 -9 F) and      per second; and, reducing the resistance to zero
microfarads (1 g F - 10 -6 F). Capacitors as large        over 30 s to maintain the current constant at 1 A as
as 500000 gF are found in computer power sup-             before, the average power of 2.5W would have
plies, where it is necessary to store considerable        been maintained for twice as long. So the energy
energy, whilst small capacitors up to several farads      stored by a capacitor at a given voltage is directly
are now readily available for memory back-up              proportional to its capacitance. Suppose, however,
purposes.                                                 that the 3 F capacitor had initially been charged to
    So just how much energy can a capacitor store?         10 V; then the initial charge would be 30 C and the
This can be answered by connecting a resistor             initial current through the 5 f~ resistor would be
across a charged capacitor and finding out how            2A. The initial power dissipation I2R would be
much heat the electrical energy has been converted        2 0 W and the discharge time 15 s (reducing the
into by the time the capacitor is completely dis-         resistance steadily to zero over that period, as
charged. Imagine a 3 F capacitor charged up to 5 V        before). So with an average power of 10 W, the
(Figure 1.6d). The stored charge Q is given by CV,        stored energy appearing as heat in the resistor is
in this case 15 coulombs or 15C. (It is just              now 150 J or four times as much. Thus the energy
unfortunate that we use C F to mean a capacitor           stored in a capacitor is proportional to the square
of value C measured in units of farads, and Q C for       of the voltage. In fact, quite simply the stored
a charge of value Q measured in units of cou-             energy is given by
lombs!) Well then, imagine a 5 ohm resistor con-                                J -- 1CV2
nected across the capacitor and see what happens.
Initially, the current I will of course be just 1 A, so   You may wonder about that 89 shouldn't there be
the capacitor is being discharged at a rate of 1 C        another 89 2 lurking about somewhere? Well,
per second. At that rate, after 1 s there would be        certainly the sums agree with the formula. Going
 14C left, so the voltage would be V = Q / C =            back to the 3 F capacitor charged to 5V, the
                                                                                          Passive components       II

formula gives 37.5 J -     and that is indeed what it         same voltage. Another disadvantage of the capa-
was.                                                          citor as an energy store is leakage. The dielectric of
    Suppose that, instead of discharging the capa-            a capacitor is ideally a perfect insulator. In prac-
citor, it is charged up to 5V from an initially               tice its resistivity, though exceedingly high, will not
discharged state (Figure 1.6f). If it is charged via          be infinite. The result is that a discharge resistor is
a 5 Ft resistor the initial current will be 1 A, and if       effectively built into the capacitor, so that the
the resistance is linearly reduced to zero over 15 s,         stored charge slowly dies away as the positive
a total charge of 15C will be stored in the                   charge on one plate is neutralized by the leakage
capacitor (Figure 1.6g). With a constant current              of electrons from the other. This tendency to self-
of 1 A and an average resistance of 2.5 f~, the               discharge is called the shunt loss of a capacitor.
heat dissipation in the resistor will be 37.5J.                  Figure 1.6e shows how the voltage falls when a
Furthermore at the end of 15s there will be                   capacitor is discharged: rapidly at first, but ever
37.5 J of energy stored in the capacitor, so the              more slowly as time advances. The charge on the
5V battery must have supplied 75 J, as indeed it              capacitor at any instant is proportional to the
has: 5 V at 1 A for 15 s equals 75 J. So one must             voltage, and the rate of discharge (the current
expend C V joules of energy to store just half that           through the resistor) is likewise proportional to
amount in a capacitor.                                        the voltage. So at each instant, the rate of dis-
   If a fixed 5 f~ resistor is used, the voltage across       charge is proportional to the charge remaining at
the capacitor will have reached only 63% of its               that instant. This is an example of the exponential
final value in a time CR seconds, as shown by the            function, a fundamental concept in many branches
solid line in Figure 1.6g. In theory it will take an          of engineering, which may be briefly explained as
infinite time to reach 5 V, since the nearer it gets,         follows.
the less the potential difference across the resistor            Suppose you invest s at 100% compound
and hence the lower the current available to supply           interest per annum. At that very favourable rate,
the remaining charge. However, after a period of              you would have s at the end of the first year, s at
5 CR seconds the voltage will be within 1% of its             the end of the second and so on, since the yearly
final value, and after 12 CR within one part in a             rate of increase is equal to the present value.
million. But this doesn't alter the fact that of a            Suppose, however, that the 100% annual interest
total energy C V 2 joules provided by the battery,            were added as 50% at the end of each six months;
only half is stored in the capacitor and the other            then after one year you would have s           If (100/
half is dissipated as heat in the resistor. Of course         12)% were added each month, the year-end total
one could charge the capacitor directly from the              would be s             If the interest were added not
battery without putting a resistor in series. How-            monthly, daily or even by the minute, but con-
ever, the only result is to charge the capacitor and          tinuously, then at the end of a year you would have
store the 1CV2 joules more quickly, whilst dis-               approximately s                  The rate of interest
sipating 1CV2 joules as before but this time in              would always be 100% per annum, and at the
the internal resistance of the battery. This makes a          start of the year this would correspond to s
capacitor rather inconvenient as an energy storage            per annum. But as the sum increased, so would the
device. Not only is charging it from a fixed voltage          rate of increase in terms of pounds per annum,
source such as a battery only 50% efficient, but it           reaching s          28 ... per annum at the year end.
is only possible to recover the stored energy com-            The number 2.718 28 ... is called exponential e.
pletely if one is not fussy about the voltage at              The value of an investment of s at n% compound
which it is a c c e p t e d - for example, when turning it    continuous interest after t years is s e (n/lOO)t, often
into heat in a resistor. Contrast this with a                written s exp((n/100)t).
secondary (rechargeable) battery such as a lead-                 Now, going back to the resistor and capacitor
acid accumulator or a Ni-Cd (nickel-cadmium)                 circuit: the rate of discharge is proportional to the
battery, which can accept energy at a very nearly            charge (and hence voltage) remaining; so this is
constant voltage and return up to 90% of it at the            simply compound interest o f - 100% per annum!
12   Analog Electronics

 So V - V0 e -it, where t is measured in units of         corresponds to 360 ~ rotation of the loop of wire
time of CR seconds, not years, and V0 is the              in a magnetic field, as mentioned earlier. The next
capacitor voltage at time to, the arbitrary start         rotation is 360 ~ to 720 ~, and so on; the frequency
of time at the left of the graph in Figure 1.6c. To       is simply the number of rotations or cycles per
get from units of CR seconds to seconds simply            second (Hz). The voltage v at any instant t is given
write V - Vo e -t/cR. Thus if V0 - 1 V, then after a      by v = Vm sin(c0t) (assuming v equalled zero at to,
time t - CR seconds, it will have discharged to           the point from which t is measured), where Vm is
e -1 - 1/2.718 28 - 0.37 V approximately.                 the voltage at the positive peak of the sine wave
   Remember that in a circuit with a direct current       and co is the angular frequency, expressed in
(DC) source such as a battery, and containing a           radians per second (co is the lower-case Greek
capacitor, only a transient charging current flows;       letter omega). There are 2re radians in one com-
this ceases entirely when the capacitor is fully          plete cycle, so co = 2~f radians per second.
charged. So current cannot flow continuously in              One can represent the phase relationship be-
one direction through a capacitor. But if first a         tween the voltage and current in a capacitor with-
positive supply is connected to the capacitor, then       out needing to show the repetitive sinusoidal
a negative one, and so on alternately, charging           waveforms of Figure 1.7, by using a vector diagram
current will always be flowing one way or the             (Figure 1.8a). The instantaneous value of the
other. Thus an alternating voltage will cause an          voltage or current is given by the projection of
alternating current apparently to flow through a          the appropriate vector onto the horizontal axis.
capacitor (see Figure 1.7). At each and every             Thus at the instant shown, corresponding to just
instant, the stored charge Q in the capacitor             before 0 = r~/2 or t = 1/4f, the voltage is almost
must equal CV. So the waveform of charge                  at its maximum positive value, whilst the current is
versus time is identical to that of the applied           almost zero. Imagine the voltage and current
voltage, whatever shape the voltage waveform              vectors rotating anticlockwise at co/2rc revolutions
may be (provided that C is constant, which is             per second; then the projection of the voltage and
usually the case). A current is simply the rate of        current vectors onto the horizontal axis will
movement of charge; so the current must be zero           change in sympathy with the waveforms shown
at the voltage peaks, where the amount of charge          in Figure 1.7b and d. Constantly rotating vectors
is momentarily not changing, and a maximum                are a little difficult to follow, but if you imagine the
when the voltage is zero, where the charge is             paper rotating in the opposite direction at the
changing most rapidly. In fact for the sinusoidal         same speed, the diagram will appear frozen in
applied voltage shown, the current waveform has           the position shown. Of course this only works if
exactly the same shape as the voltage and charge          all the vectors on the diagram are rotating at the
waveforms; however, unlike the resistive circuit          same speed, i.e. all represent things happening at
(see earlier, Figure 1.5), it is moved to the l e f t -   the same frequency. So one can't conveniently
advanced in time - by one-quarter of a cycle or 90 ~      show the power and energy waveforms of Figure
or re/2 radians. The current waveform is in fact a        1.7e on the vector diagram.
cosine wave, this being the waveform which charts            Now the peak value of the capacitor current Im
the rate of change of a sine wave. The sine wave is       is proportional to the peak value of the voltage
a waveform whose present rate of change is equal           Vm, even though they do not occur at the same
to the value of the waveform at some point in the         instant. So one may write Im = Vm/Xc, which
future, namely one-quarter of a cycle later (see          looks like Ohm's law but with R replaced by Xc.
Figure 1.7). This sounds reminiscent of the ex-           Xc is called the reactance of the capacitor, and it
ponential function, whose present rate of change is       differs from the resistance of a resistor in two
equal to its present value: you might therefore           important respects. First, the sinusoidal current
expect there to be a mathematical relation between        through a capacitor resulting from an applied
the two, and indeed there is.                             sinusoidal voltage is advanced by a quarter of a
   One complete cycle of a sinusoidal voltage             cycle. Second, the reactance of a capacitor varies
                                                                                                                 Passive components               13

                                                                        v = Vm sin(o)t)
                                                                         O-           ~ /= lm cos(cot)
                                                                                          C farads


                v (volts)
                                                                                                                       v = Vm sin(co/)

                                                                                                                4          ~

               VmI -    I/4~12    '     ~            f                                               '   '       2If              t   (seConds)


             q (coulombs)

                                            .   ._       .......... 2      ~    n                               4~.,,.~..--


              i (amperes)
                   .                                                                                                i = I m cos(cot)

                .tm-I ~/4 ~                                         llf           ~                             2}f                     --
               - |
            Energy J (joules)
            Power W (watts)                                                                              Stored energy

                  89 2 = 1C[Vm sin(c0t)] 2 = 89
  Stored energy J =                                  289 -cos(2c0t)]
              vi = Vm sin(c0t)/m cos(c0t) = 1 Vm/m sin(2c0t)watts.
  Energy flow =
  The maximum energy stored is 89          C(-
                                 CVm 2 = 89 Vm) 2 joules; the energy stored is zero at 0 - cot = 0, ~, 2~ etc.
  The rate of energy flows is iv watts. This is positive (into the capacitor) over the first quarter-cycle, 0 to ~z/2, as r a n d i
  are both positive. It peaks at n/4, where W = 1 Vmlm, the point at which the stored energy is increasing most
  rapidly. By zt/2, where energy flow W is zero again, 1 CV 2 joules of energy have been stored. The energy flow
  W = vi now becomes negative as the capacitor gives up its stored energy over the remainder of the first half-cycle
  of voltage. The zero net energy means that no power is dissipated, unlike the resistive case of Figure 1.5.

Figure I. 7 Alternating voltage and power in a capacitive circuit.
14   Analog Electronics

                          A                           \          Paperrotating
                               I                          \\     at--coradians
                                                               \\ per second
                                                                 \x                              I
                                          ICE                     \
                      /q                                           ,x A
                                                                       ~                   v
                      /90~ ]"                                          ~                                 T
                   - 0 ~                    ,,iv


                                    co                                                               I

                                                                                            v                L


Figure 1.8 Phase of voltage and current in reactive components.
  (a) ICE: the current I leads the applied EMF E (here V) in a capacitor. The origin 0 represents zero volts, often
referred to as ground.
  (b) ELI: the applied EMF E (here V) across an inductor L leads the current I.

with frequency. For if the voltage and charge                              and simply write V for the maximum or peak
waveforms in Figure 1.7 were of twice the fre-                             voltage Vm.
quency, the rate of change of charge (the current)                            Capacitors are used for a number of different
at t - 0 would be twice as great. Thus the reac-                           purposes, one of which has already been men-
tance is inversely proportional to the frequency,                          tioned: energy storage. They are also used to
and in fact Xo - 1/coC. Recalling that the instan-                         pass on alternating voltage signals to a following
taneous    voltage     v = Vmsin(cot), note that                           circuit whilst blocking any associated constant
i = Vm sin(cot) / (1/coC) - Vm sin(cot)coC. However,                       voltage level, or to bypass unwanted AC signals
that can't be right, since i is not in fact in phase                       to ground. It has always been a problem to obtain
with v, see Figure 1.7, but in advance or 'leading'                        a very high value of capacitance in a reasonably
by 90 ~ A way round this difficulty is to write                            small package, and a number of different construc-
i = Vm sin(c0t)jcoC, where j is an 'operator' and                          tions are used to meet different requirements. If
indicates a 90 ~ anticlockwise rotation of a vector.                       the two plates of the capacitor in Figure 1.6(a)
This makes the formula for i agree with the vector                         each have an area of A square metres and are
diagram of Figure 1.8a. So from now on, just tack                          separated by d metres in vacuo, the capacitance, it
the j onto the reactance to give Xc = 1/jcoC, and                          was noted earlier, is given by C = ~o(A/d) farads
you will find that the 90 ~ displacement between                           (where ~ is the lower-case Greek letter epsilon)
current and voltage looks after itself. Also, for                          approximately, if d 2 is small compared with A. So
convenience, from now on, drop the subscript m                             if A = l m 2 and d = l m m ,     the capacitance is
                                                                                          Passive components      15

8.89 x 10 - 9 farads or 889 pF in a vacuum and just           oxide is a non-conductor, so when anodization is
0.06% higher in air. Capacitance values much                  complete no more current flows, or at least only a
larger than this are frequently required; so how              very small current called the leakage current or
can they be achieved? Simply by using a solid                 shunt loss. For low-voltage electrolytics, when
dielectric with a relative permittivity of ~r as              new, the leakage current at 20~ is usually less
shown earlier. The dielectric can be a thin film of           than 0.01 ~tA per ~tF, times the applied voltage,
plastic, and ~r is then typically in the range 2 to 5.        usually quoted as 0.01CV ~tA. The forming voltage
The resultant increase in capacitance is thus not             is typically some 20% higher than the capacitor's
large but, with a solid film to hold the plates apart,        rated maximum working voltage. The higher the
a much smaller value of d is practicable.                     working voltage required, the thicker the oxide
   The plates may be long narrow strips of alum#              film must be to withstand it. Hence the lower the
niurn foil separated by the d i e l e c t r i c - say poly-   capacitance obtainable in a given size of capacitor;
styrene film 0.02 mm t h i c k - and the sandwich is          the electrolyte is a good conductor, so it is the
rolled up into a cylinder as in Figure 1.6h. Note             thickness of the oxide which determines the capa-
that unlike in Figure 1.6a, both sides of each plate          citance. Working voltages up to 450 V or so are the
now contribute to the capacitance, with the excep-            highest practically obtainable. In use, the working
tion of the outer plate's outermost turn. The foil is         voltage should never be exceeded; nor should the
often replaced by a layer of metal evaporated onto            capacitor's polarity be reversed, i.e. the red or +
the film (this is done in a vacuum), resulting in an          terminal must always be more positive than the
even more compact capacitor; using this tech-                 black or - terminal. (This restriction does not
nique, 10 ~tF capacitors fitting four to a matchbox           apply to the 'reversible electrolytic', which consists
can be produced. The dielectric in a foil capacitor           of two electrolytic capacitors connected in series
being so thin, the electric stress in the dielectric-         back to back and mounted in a single case.)
measured in volts per m e t r e - can be very high. If           Before winding the aluminium foils and paper
the voltage applied across the capacitor exceeds              separators together, it is usual to etch the surface
the rated working voltage, the dielectric may be              of the foils with a chemical; this process can
punctured and the capacitor becomes a short-                  increase the surface area by a factor of ten or
circuit. Some foil capacitors are 'self-healing'. If          even thirty times. The resultant large surface area
the dielectric fails, say at a pinhole, the resultant         and the extreme thinness of the oxide layer enable
current will burn out the metallization in the                a very high capacitance to be produced in a small
immediate vicinity and thus clear the short-circuit.          volume: a 7V working 500000 ~tF capacitor as
The external circuitry must have a certain mini-              used in mains power supplies for large computers
mum resistance to limit the energy input during               may be only 5 to 8cm diameter by 10 to 15cm
the clearing process to a safe value. Conversely, if          long. The increase in surface area of the electrodes
the circuit resistance is too high the current may be         produced by etching is not without its disadvan-
too low to clear the short-circuit.                           tages. With heavy etching, the aluminium strip
   For values greater than 10 ~tF, electrolytic capa-         develops a surface like a lace curtain, so that the
citors are usually employed. Aluminium foil elec-             current flows through many narrow necks of
trolytic capacitors are constructed much like                 metal. This represents a resistive component inter-
Figure 1.6h; the film separating the plates is                nal to the capacitor, called the series loss, resulting
porous (e.g. paper) and the completed capacitor               in energy dissipation when an alternating or ripple
is enclosed within a waterproof casing. The paper             current flows. So while heavy etching increases the
is impregnated with an electrolyte during manu-               capacitance obtainable in a given size of capacitor,
facture, the last stage of which is 'forming'. A DC           it does not increase pro rata the rated ripple
forming voltage is applied to the capacitor and a             current that the capacitor can support. The
current flows through the electrolyte. The resultant          plates of an aluminium electrolytic capacitor are
chemical action oxidizes the surface of the positive          manufactured from extremely pure aluminium,
plate, a process called anodization. Aluminium                better than 'four nines pure', i.e. 99.99%, as any
16   Analog Electronics

impurity can be a centre for erosion leading to           becomes doughnut shaped as in Figure 1.9c. A
increased leakage current and eventual failure. The       series of closely spaced turns form a solenoid- the
higher the purity of the aluminium foil the higher        resultant field is then concentrated as represented
its cost, and this is reflected in the price of the       in Figure 1.9d. The form and strength of the
capacitor. Whilst a cheaper electrolytic may save         magnetic field is determined by the strength of
the manufacturer a penny or two, a resultant early        the current causing it and the way the current
equipment failure may cost the user dear.                 flows, as is clear from Figure 1.9a to d. In the
   In radio-frequency circuits working at frequen-        case of a solenoid of length l, if one ampere flows
cies up to hundreds of megahertz and beyond,              in the wire and there are N turns, the magneto-
capacitors in the range 1 to 1000pF are widely            motive force (MMF), denoted by F, is just N
employed. These commonly use a thin ceramic               amperes.
disc, plate, tube or multilayer structure as the             This is often written as N ampere turns, but the
dielectric, with metallized electrodes. The lower         number of turns is really irrelevant: the solenoid
values, up to 100pF or so, may have an NPO                could equally well consist of a single turn of
dielectric, i.e. one with a low, nominally zero,          copper tape of width l carrying N amperes. The
temperature coefficient of capacitance. In the            effect would be the same if it were possible to
larger values, N750, N4700 or N15000 dielectrics          ensure that the current flow of N amperes was
may be used (N750 indicates a temperature coeffi-         equally distributed across the width of the tape.
cient o f - 7 5 0 parts per million per degree C), and    Dividing the tape up into N turns in series, each
disc ceramics for decoupling purposes with capa-          carrying one Nth of the total current, is a con-
citances up to 470nF (0.47 ~tF) are commonly              venient subterfuge for ensuring this.
available. As with resistors, many circuits nowa-            If a long thin solenoid is bent round into a toroid
days use capacitors in surface mount packaging.           (Figure 1.9e), then instead of returning round the
   Variable capacitors are available in a variety of      outside of the solenoid the magnetic lines of force
styles. Preset variable capacitors, often called          are closed upon themselves entirely within the
trimmers, are used in production for the setting-         solenoid and there is no external field. The
up adjustments to tuned circuits. They may use            strength of the magnetic field H A/m within the
solid or air dielectric, according to the application,    toroid depends upon the strength of the magneto-
as also do the variable capacitors used in radio sets     motive force per unit length, in fact H = I / l
for tuning.                                               amperes per metre, where l is the length of the
                                                          toroid's mean circumference and I is the effective
                                                          c u r r e n t - the current per turn times the number of
Inductors and transformers
                                                          turns. The magnetic field causes a uniform mag-
The third type of passive component mentioned at          netic flux density, of B webers per square metre,
the beginning of the chapter is the inductor. This is     within the toroidal winding. The ratio B/H is
designed to exploit the magnetic field which sur-                                                        o,
                                                          called the permeability of free space ~t and its
rounds any flow of current, such as in a wire - or        value is 4~ x 10 -7. So B = ~t0H in a vacuum or air,
indeed in a stroke of lightning. This is illustrated in   or even if the toroid is wound on a solid core,
Figure 1.9a, which shows lines of magnetic force          provided the core material is non-magnetic. If the
surrounding the current in a wire. The lines form         cross-sectional area of the core is A m 2 , the total
closed loops and are shown more closely packed            magnetic flux 9 Wb (webers) is simply given by
together near the wire than further out, to indicate      ~ = BA.
that the magnetic field is strongest near the wire.          If the toroid is provided with a ferromagnetic
However, the lines are merely a crude representa-         c o r e - Faraday experimented with a toroid wound
tion of the magnetic field, which is actually             on an anchor r i n g - it is found that the flux density
continuous throughout space: some writers talk            and hence the total flux is greatly increased. The
of tubes rather than lines, to indicate this. If the      ratio is called the relative permeability of the
wire is bent into a circular loop, the magnetic field     material ~r" Thus in general B = ].t0~trH, where
                                                                                                        Passivecomponents       17


             (a)                         (b)                            ~1~                                  (d)

         Cross-sectional                                       Inductance of 1 H
         area A m 2


                                                                                                        ~    1~          E


                                                                                                      0.999 V
                         4VPD                  2 V PD                      +1 V                       EMF
                         /R     \ 1A                   \0.5A
                                                                      J                                         9
                                               /g                  1V ~               -      -    -                 Typical
                                                                                                                    flux line
                                       3~VI_             T
                                                        IV -       EMF~I                  (say)-----~   ~
         3VT -                  1T

                                                                            - -   v

                          (g)                    (h)                                            (i)
Figure1.9      The magnetic field.
   (a) End view of a conductor. The cross indicates current flowing into the paper (a point indicates flow out). By con-
vection, the lines of flux surrounding the conductor are as shown, namely clockwise viewed in the direction of current
flow (the corkscrew rule).
   (b) The flux density is greatest near the conductor; note that the lines form complete loops, the path length of a
loop being greater the further from the wire.
   (c) Doughnut-shaped (toroidal) field around a single-turn coil.
   (d) A long thin solenoid produces a 'tubular doughnut', of constant flux density within the central part of the coil.
   (e) A toroidal winding has no external field. The flux density B within the tube is uniform over area A at all point
around the toroid, if the diameter of the solenoid is much smaller than that of the toroid.
   (f) Changing current in single-turn coil.
   (g) EMF and PD: sources combining.
   (h) EMF and PD: sources opposing, and energy storage.
   (i) Energy storage in inductor field.
18   Analog Electronics

gr--1 for a vacuum, air and non-magnetic                 which does not affect the principle of the thing.)
materials. In the cases shown in Figure 1.9a to d        Having assumed the coil to have negligible resis-
the flux density (indicated by the closeness of          tance, the current will ultimately become very
spacing of the flux lines) varies from place to          large; so why isn't it already huge after just one
place, but at each point B - ~t0H. B - la0~trH can       second? The reason is that the steadily increasing
be expressed in terms of total flux 9 and M M F F        flux induces an E M F in the coil, in opposition to
as follows:                                              the applied EMF: this is known as Lenz~ law. If
                             F                           the flux 9 increases by a small amount d~ in a
                    -2 = u~ 7                            fraction of a second dt, so that the rate of increase
                                                         is d~b/dt, then the back EMF induced in the single-
so that
                             F                           turn coil is
                   ~=                                                                 d~
                        ///(~t0~trA)                                          EB --    dt                 (1.4)

The term l/~to~trA is called the reluctance S, of the    or - N d~b/dt in the case of an N-turn coil. But
magnetic path, and it has the units of amperes per
                                                                                    MMF              F
weber. Thus, in a magnetic circuit, flux equals
M M F divided by reluctance; this is a similar sort                      ~        reluctance         S
of result to current equals E M F divided by             So for a one-turn coil, 9 I/S. As the current and
resistance in an electric circuit. Just as the indi-     flux are both increasing, for this to remain true
vidual resistances around an electric circuit can be     their rates of change must also be equal, i.e.
added up when working out the total E M F needed
                                                                                  d~     1 dI
to cause a current I to flow, so in a magnetic circuit                            d--7 = ~ d t                       (1.5)
(e.g. a core of magnetic material with a perme-
ability ~tr, having an airgap) the reluctances can be    Substituting     (1.4)       in     (1.5)           gives   EB =
added up to find the total M M F needed to cause a       -(1/S)(dI/dt) for a one-turn coil. In the more
given total flux.                                        general case where the M M F F = NI, then 9 = NIl
    So far the field produced by a constant current      S and so d~b/dt=(N/S)(dI/dt). This rate of flux
of 1 amperes has been considered; but what               increase will induce a voltage EB = - d r b / d t              in
happens when the current changes? Indeed, how            series in each of the N turns of the coil, so
does the current come to flow in the first place?                        d~                NdI       -N2dI
(Figure 1.9c rather begs the question by assuming             EB--N--~--N~dt=                            S    dt     (1.6)
that the current is already flowing.) Consider what
happens when an E M F of one volt is connected to        The term N2/S, which determines the induced
a large single-turn coil, as in Figure 1.9f. Assume      voltage resulting from a unit rate of change of
for the moment that the coil has negligible resis-       current, is called the inductance L and is measured
tance. Nothing in this universe (except a politi-        in henrys: that is,
cian's promises) changes instantaneously, so the                                  N2
moment after connecting the supply the current                                L - ~ henrys
must be the same as the moment before, i.e. zero.
Clearly one can expect the current to increase           You must keep the difference between an electro-
thereafter, but how fast? Assume that the current        motive force (EMF) and a potential drop or
increases at one ampere per second, so that after        difference (PD) very clearly in mind, to understand
 one second the M M F F is just one ampere turn,         the minus sign in equation (1.4). To illustrate this,
 and that the reluctance S = 1, so that the resulting    consider two secondary batteries and a resistor
flux 9 is one weber. (In fact, for this to be so, the    connected as in Figure 1.9g. The total E M F
coil would have to be very large indeed or im-           round the circuit, counting clockwise, is 3 + 1
mersed in a magnetic medium with a huge relative         volts, and this is balanced by the PD of IR volts
permeability, but that is a minor practical point        across the resistor. The batteries supply a total of
                                                                                            Passive components      19

4 W of power, all of which is dissipated in the                 the resistor, and E = - L dI/dt (where the induc-
resistor. If now the polarity of the 1 V battery is             tance is unity in this case). Of course dI/dt is itself
reversed, as in Figure 1.9h, the total E M F acting is          now negative (current decreasing), as the polarity
3 - 1 V, so the current is 0.5 A. The 3 V battery is            reversal witnesses. After a fraction of a second, the
now supplying 3 x 0.5 = 1.5 W, but the dissipa-                 current being now less than one ampere, the
tion in the resistor I/R is only 1 W. The other 0.5             voltage across the resistor will have fallen likewise;
watts is disappearing into the 1 V battery; but it is           so the rate of decrease of current will also be lower.
not being dissipated, it is being stored as chemical            The fall of current in the coil will look just like the
energy. The situation in Figure 1.9i is just the                fall of voltage across a discharging capacitor
same; the applied E M F of the battery is opposed               (Figure 1.6e, solid line). Suppose, however, that
by the back E M F of the inductor (which in turn is             the resistor is a variable resistor and its resistance
determined by the inductance and the rate of                    increases, keeping the value inversely proportional
increase of the current), whilst energy from the                to the current. Then IR will be constant at 1 V and
battery is being stored in the steadily increasing              the current will fall linearly to zero in 1 second,
magnetic field. If the internal resistance of the               just like the dashed line in Figure 1.6e.
battery and the resistance of the inductor are                     Since the induced voltage across the resistor has,
vanishingly small, the current will continue to                 by this dodge, been maintained constant at 1 V, the
increase indefinitely; if not, the current will reach           energy dissipated in it is easily calculated. On
a limit set by the applied E M F and the total                  opening the switch the dissipation is 1 V x 1 A,
resistance in the circuit.                                      and this falls linearly to zero over one second. So
   Returning now to Figure 1.9f, if the switch is               the average power is 0.5W maintained for one
closed one second after connecting the battery, at              second, giving a stored energy of 0.5J. If the
which time the current has risen to 1 A, then there             inductance had been 2 H and the current 1 A
is no voltage across the ends of the coil. No back              when the switch was opened, the initial rate of
E M F means that d~/dt must be zero, so dI/dt is                fall would have been 0.5 A per second and the
also zero. Hence the current now circulates indefi-             discharge would have lasted 2 s, dissipating 1 J in
nitely, its value frozen at 1 A - provided our coil             the resistor (assuming its value was adjusted to
really has zero resistance. (In the meantime, dis-              maintain 1 V across it as before). Thus the stored
connect the battery and replace it with a 1 9~                  energy is proportional to the inductance L. On the
resistor; you will see why in a moment.) Thus                   other hand, if the current was 2 A when the switch
energy stored in the magnetic field is preserved                was opened, the voltage across the 1 ~ resistor
by a short-circuit, just as the energy stored in a              would have been 2 V, so the rate of fall would need
capacitor is preserved by an open-circuit. Now                  to be 2 A/s (assuming 1 H inductance). Thus the
consider what happens on opening the switch in                  initial dissipation would have been 4 W, falling to
Figure 1.9f, thus substituting the 1 ~ resistor in              zero over 1 s, giving a stored energy of 2 J. So the
place of the short-circuit. At the moment the                   stored energy is proportional to the square of the
switch opens the current will still be 1A; it                   current. In fact, the stored energy is given by
cannot change its value instantaneously. This will                                    j - 1LI2
establish a 1 V PD across the resistor, of the
opposite polarity to the (now disconnected) bat-                This result is reminiscent of J - CV2/2 for a
tery; that is, the top end of the resistor will be              capacitor.
negative with respect to the lower end. The coil is                An inductor can be and often is used as an
now acting as a generator, feeding its stored energy            energy store, as will appear in a later chapter in the
into the r e s i s t o r - initially at a rate of 1 joule per   context of power supplies, where certain limita-
second, i.e. 1 W. How much energy is there stored               tions to the inductor's power storing ability will
in the field, and how long before it is all dissipated          become apparent. In particular, the energy stored
as heat in the resistor? Initially the current must be          in the magnetic field of a short-circuited inductor
falling at 1 A per second, since there is 1 V across            is rapidly lost due to dissipation in the resistance of
20   Analog Electronics

its windings. The ratio of inductance to series loss      anticlockwise displacement of a vector, multiply-
L/r, where r is the resistance of the inductor's          ing by j again will result in a further 90 ~ anticlock-
winding, is much lower than the ratio C/R,                wise rotation. This is equivalent to changing the
where R is the shunt loss, for a high-quality             sign of the original vector. Thus j x j = - 1 , a
capacitor. At very low temperatures, however,             result which will be used extensively later.
the electrical resistivity of certain alloys and com-        In the meantime, imagine two identical lengths
pounds vanishes e n t i r e l y - a phenomenon known      of insulated wire, glued together and bent into a
as superconductivity. Under these conditions an           loop as in Figure 1.9f. Virtually all of the flux
inductor can store energy indefinitely in its mag-        surrounding one wire, due to the current it is
netic field, as none is dissipated in the conductor.      carrying, will also surround the other wire. Now
   In addition to use as energy storage devices,          connect a battery to one loop - c a l l e d the pri-
inductors have several other applications. For            m a r y - and see what happens to the other l o o p -
example, inductors with cores of magnetic material        called the secondary. Suppose the self-inductance
are used to pass the direct current output of a           of the primary is 1 H. Then an applied E M F of 1 V
rectifier to later circuitry whilst attenuating the       will cause the current to increase at the rate of 1 A
alternating (hum) components. Air or ferrite              per second, or conversely the rate of change of 1 A
cored inductors (RF chokes) are used to supply            per second will induce a back E M F of 1 V in the
power to radio-frequency amplifier stages whilst          primary; it comes to the same thing. But all the
preventing RF power leaking from one stage to             flux produced by the primary also links with the
another via the power supply leads. This applica-         secondary, so an EMF identical to the back E M F
tion and others make use of the AC behaviour of           of the primary will be induced in the secondary.
an inductor. Just as the reactance of a capacitor         Since a dI/dt of 1 A/s (a rate of increase of 1 A per
depends upon frequency, so too does that of an            second) in the primary induces an E M F of 1 V in
inductor. Since the back E M F EB = N d ~ / d t =         the secondary, the two windings are said to have a
- L di/dt, it follows that the higher the frequency,      mutual inductance M of 1 H. If the two coils were
the smaller the alternating current required to give      placed slightly apart so that only a fraction c (a
a back E M F balancing the applied alternating            half, say) of the flux caused by the primary current
EMF. In fact, the reactance XL of an inductor is          linked with the secondary, then only 0.5 V would
given by XL = 2rcfL = coL where f is the frequency        be induced in the secondary and the mutual
in hertz, co is the angular velocity in radians per       inductance would be only 0.5 H.
second, and L is the inductance in henrys. This              In the above example the two coils were iden-
may be represented vectorially as in Figure 1.8b,         tical, so that the self-inductance of the secondary
from which it can be seen that when the voltage is        was also 1 H. In the general case, the maximum
at its positive peak, the current is zero but increas-    value of mutual inductance M between two un-
ing. If you draw the waveforms for an inductor            equal coupled inductors L1 and L2 is given by
corresponding to those of Figure 1.7 for a capa-          M = x/(L1L2), whilst if only some of the flux of
citor, you will find that the current is increasing (or   one winding links with the other winding then
becoming less negative) all the time that the             M = kv/(L1L2 ), where k is less than unity. As a
applied voltage is positive and vice versa, and           matter of interest k = Cv/(S1S2 ), where c is as
that the net energy flow is zero. Again, you can          before the fraction of the primary flux linking
look after the 90 ~ phase shift between the voltage       the secondary, and S1, $2 are the reluctances of
and lagging current by using the j operator and           the primary and secondary magnetic circuits re-
writing XL = jcoL, thus keeping the sums right.           spectively.
With a capacitor, the voltage produces a leading             Coupling between coils by means of mutual
current; the exploitation of this difference is the       inductance is used in band-pass tuned circuits,
basis of a particularly important application,            which are briefly mentioned in the chapter cover-
namely tuned circuits, which will be considered           ing r.f. In this application, quite small values of
later. Note that if multiplying by j signifies a 90 ~     coupling coefficient k are used. Right now it is time
                                                                                       Passive components       21

to look at coupled circuits where k is as large as          reactance) connected to one winding of a transfor-
possible, i. e. where c is unity, so that all the flux of   mer appears at the other transformed by the
the primary links the secondary and vice versa.             square of the turns ratio.
   Figure 1.10a shows a t r a n s f o r m e r - two coils      So far an almost perfect transformer has been
wound on a ferromagnetic core, which results in             considered, where Ealp = Esls, ignoring the small
much more flux per ampere turn, owing to the                magnetizing current Io, which flows in the primary
lower reluctance of the magnetic path. The resul-           when the transformer is off load. The term 'mag-
tant high value of inductance means that only a             netizing current' is often used loosely to mean Ipol.
small 'magnetizing current' Im flows. This is 90 ~          The difference is not large since Ic is usually much
out of phase with the alternating voltage, Ea at 50         smaller than Im. In a perfect transformer, the
or 60 Hz say, applied to the primary winding.               magnetizing inductance would be infinity, so that
(There is also a small in-phase current I~ due to           no primary current at all would flow when the
the core loss. This together with Im makes up the           transformer was off load. In practice, increasing
primary off-load current Ipol.) Since EB --                 the primary inductance beyond what is necessary
-LdI/dt--Nd~b/dt,            and all of the flux (b         makes it more difficult to ensure that virtually all
links both windings, it follows that the ratio of           the primary flux links the secondary, resulting in
secondary voltage Es to the primary back EMF                undesirable leakage reactance. Further, the extra
EpB is equal to the turns ratio:                            primary and secondary turns increase the winding
                                                            resistance, reducing efficiency. Nevertheless the
                       Es       Ns                          'ideal transformer', with its infinite primary induc-
                             = --                  (1.7)
                       gpB     Np                           tance, zero leakage inductance and zero winding
If a resistive load R is now connected to the               resistance, if an unachievable goal, is useful as a
secondary, a current Es/R will flow, since Es               touchstone.
appears to the load like a source of EMF. By                    Figure 1.10d shows the equivalent circuit of a
itself, this current would produce a large flux in the      practical power transformer, warts and all. Rc
core. However, the flux cannot change, since the            represents the core loss, which is caused by hyster-
resultant primary back EMF EpB must balance the             esis and eddy currents in the magnetic core. Eddy
fixed applied EMF Ea (see Figure 1.10b and c).              currents are minimized by building the core up
Consequently an additional current Ip flows in the          from thin stampings insulated from each other,
primary to provide an M M F which cancels out the           whilst hysteresis is minimized by stamping the
M M F due to the secondary current. Hence                   laminations from special transformer-grade steel.
                                                            The core loss Rc and the magnetizing inductance
IpNp -- IsNs so
                        Is     Up                           Lm are responsible for the current Ipol in Figure
                             = --                  (1.8)    1.10c. They are shown connected downstream of
                                                            the primary winding resistance Rwp and leakage
If, for example, Ns/Np - 0.1, i.e. there is a ten-to-       inductance Lip since the magnetizing current and
one step-down turns ratio, then from (1.7) the              core loss actually reduce slightly on full load. This
secondary voltage will only be one-tenth of the             is because of the extra voltage drop across Rwp and
primary voltage. Further, from (1.8) Ip will only be        Lip due to Ip. A useful simplification, usually valid,
one-tenth of Is. The power delivered to the load is         is to refer the secondary leakage inductance and
I~R and the power input to the transformer                  winding resistance across to the primary, pro rata
primary is I2R ~ where R ~ is that resistor which,          to the square of the turns ratio (see Figure 1.10e,
connected directly to Ea, would draw the same               L1 and Rw). Using this simplification, Figure 1.10f
power as R draws via the transformer (assuming              shows the vector diagram for a transformer with
for the moment that the transformer is perfectly            full-rated resistive load: for simplicity a unity turns
efficient). Since in this example Ip is only a tenth of     is depicted so that Es = Ea approximately. Strictly,
Is and Es is only a tenth of Ea then R p must equal         the simplification is only correct if the ratio of
(R • 100) ohms. Hence a resistance (or indeed a             leakage inductance to total inductance and the 'per
22   Analog Electronics
                                                                                        /.                                  Power in = power out

                    _.7               ~Iux V--                      'p+lm         ~ ' ]                                       R
                                                                                                              's=Es/R I2R,= 12s

                                                                o                        i l:                       U                        2
                            turns         turns
                                                                            100 turns              10 turns                 So R' =          R
                                    Ca)                                                  (b)

                                                                Lip            Rwp                                       Lls       Rws
            Ea~                                                                                       r-----i

                  /                                        OF                            ii1~-~l     Perfect

                  Im                                                                                 transformer

                       EpB = - N ~dt

                                                                                     evs~f 19
     Ip      LI             Rw            rp               Is                                             I
                                                r------1                                                   I
                                                                                                          I I1~I
      o--                             Lm


                                                                                                        & - It. - E ; R L

                                                                                             Es = EpB
                                                                                                            Passive components   23

                                                     Primary and secondary
                                                     layer windings                                Primary     Secondary
                          Insulating bobbin          interleaved with        _       _

                                                     insulation e.g. paper

                                                                                             - -      . .

                                              .'E and r
          Secondary -                          laminations                       ,       ,

                                                                                     ;         t
                                                                             Interwinding earthed screen



Figure 1.10 Transformers.

unit' resistance (the ratio of winding resistance to         and eddy current) loss roughly equalled the full-
Erated/Irated) is the same for both windings. The            load copper loss (winding resistance loss). The
transformer designer will of course know the                 increasingly popular toroidal transformer (Figure
approximate value of magnetizing inductance,                 1.10h) exhibits a very low core loss, so that at full
since he will have chosen a suitable core and                load the copper loss markedly predominates. In
number of turns for the application, making                  addition, the stray magnetic field is much lower
allowance for tolerances on the core permeability,           than with three-limb cores and there is less ten-
hysteresis and eddy current losses. The precise              dency to emit annoying audible hum. Originally
value of magnetizing inductance is then unimpor-             commanding a premium over conventional trans-
tant, but it can be measured if required on an               formers on account of these desirable properties,
inductance bridge or meter, with the secondary               toroidal transformers are now produced at such a
open-circuit. The total leakage inductance referred          volume and level of automation that there is little
to the primary can be found by repeating the                 price differential. Both types are built down to a
measurement with the secondary short-circuited.              price, which means minimizing the core size and
   In the case of a power transformer the answers            number of turns per volt, leading to a high peak
will only be approximate, since the magnetizing              flux density.
inductance, core loss and leakage reactance vary                In small transformers of, say, 50 to 100W
somewhat with the peak flux level and hence with             rating, Rw (referred to the secondary) is often
the applied voltage.                                         nearly one-tenth of the rated load resistance. So
   Small transformers with laminated three-limb              the full-load output voltage is only 90% of the off-
cores as in Figure 1.10g, designed to run most if            load v a l u e - described as 10% regulation. Taking
not all of the time at maximum rated load, were              account of core loss as well, the full-load efficiency
traditionally designed so that the core (hysteresis          of such transformers barely reaches 90%. For very
24   Analog Electronics

small transformers with a rating of just a few          the secondary short-circuited, corresponding to no
watts, the regulation may be as poor as 30% and         voltage drop across the primary. The secondary is
the efficiency less than 70%, whereas for a large       designed with enough inductance to support the
mains distribution transformer the corresponding        maximum voltage drop across the measuring
figures might be 2% and 98%.                            circuit, called the burden, with a magnetizing
    Note that in Figure 1.10e, if the rated Is flowed   current which, referred to the primary, is less
in a purely capacitive or inductive load instead of     than 1% of the primary current. It is instructive
RL, the losses in the transformer would be just as      to draw out the vector diagram for a current
great. Therefore the rated secondary load for a         transformer on load, corresponding to the voltage
transformer is always quoted in terms of the rated      transformer case of Figure 1.10e and f.
secondary volt-ampere product (VA) rather than
in watts. Furthermore, the secondary current
rating is strictly root mean square (RMS) or
effective current. Thus with a non-linear load,
e.g. a capacitor input rectifier circuit (see Chapter
10), the transformer must be derated appropriately       1. A 7.5V source is connected across a 3.14 f~
to avoid overheating, since the RMS value of the            resistor. How many joules are dissipated per
current will be much greater than that of a                 second?
sinusoidal current of the same mean value.               2. What type of wire is used for (a) high wattage
   In some ways, power transformers are easy to             wirewound resistors, (b) precision wirewound
design, at least in the sense that they are only            resistors?
required to work over a very limited range of            3. A circuit design requires a resistance of value
frequencies, say 45-65Hz or sometimes 45-                   509 f~ + 2%, but only El2 value resistors, in
440 Hz. Signal transformers, on the other hand,             1%, 2% and 5% tolerance, are available.
may be required to cover a 1000:1 bandwidth or              What value resistor, in parallel with a 560 ft
more, say 20 Hz to 20 kHz or 1 to 1000 MHz. For             resistor, is needed to give the required value?
these, special techniques and construction meth-            Which of the three tolerance values are
ods, such as sectionalized interleaved windings,            suitable?
may be used. By these means it is possible to            4. What type of capacitor would usually be used
produce a transformer covering 30 Hz to 2 MHz,              where the required capacitance is (a) 1.5 pF,
almost a 100 000:1 bandwidth.                               (b) 4700 gF?
   An interesting example of a signal transformer is     5. A 1 gF capacitor is charged to 2.2 V and a
the current transformer. This is designed with a            2.5 gF capacitor to 1.35 V. What is the stored
primary which is connected in series with a heavy           energy in each? What is the stored energy after
current circuit, and a secondary which feeds an             they have been connected in parallel? Explain
AC milliammeter or ammeter with a replica (say              the difference.
0.1%) of the primary current, for measurement            6. A 1 Mf~ resistor is connected between the two
purposes.                                                   terminals of the capacitors, each charged to
   With the power transformers considered so far,           the voltage in Question 5 above. How long
designed for a specified rated primary voltage, the         before the voltages across each capacitor are
safe off-load condition is with the secondary open-         the same to within (a) 10%, (b) 0.1%?
circuit; the secondary load connected then defines       7. A black box with three terminals contains
the secondary current actually drawn. In the case           three star-connected capacitors, two of
of a current transformer, designed for a specified          15pF, one of 30pF. Another black box,
maximum primary current, where the current is               identical to all measurements from the outside,
determined by the external primary circuit and not          contains three delta-connected capacitors.
by the load, we are in the topsy-turvy constant             What are their values? (This can be done by
current world. The safe off-load condition is with          mental arithmetic.)
                                                                               Passive components   25

8. Define the reluctance   S of a magnetic circuit.     10. An ideal transformer with ten times as many
   Define the inductance   of a coil with N turns, in       primary turns as secondary is connected to
   terms of S.                                              240V AC mains. What primary current
9. The reactance of 2.5    cm of a particular piece         flows when (a) a 56ft resistor or (b) a
   of wire is 1 6 ~ at     100MHz. What is its              10gF    capacitor   is connected    to the
   inductance?                                              secondary?

    2                            Passive circuits

Chapter 1 looked at passive c o m p o n e n t s - resis-       Figure 2.1a shows such a top-cut or treble-cut
tors, capacitors and inductors - individually, on           circuit, which, for the sake of simplicity, will be
the theoretical side exploring their characteristics,       driven from a source of zero output impedance
and on the practical side noting some of their uses         (i.e. a constant voltage source) and to fed into a
and limitations. It also showed how, whether we             load of infinitely high-input impedance, as indi-
like it or not, resistance always turns up to some          cated. What is Vo, the signal voltage passed to the
extent in capacitors, inductors and transformers.           load at any given frequency, for a given alternating
Now it is time to look at what goes on in circuits          input voltage Vi9. Since both resistors and capaci-
when resistors, capacitors and inductors are delib-         tors are linear components, i.e. the alternating
erately combined in various arrangements. The               current flowing through them is proportional to
results will figure importantly in the following            the applied alternating voltage, the ratio Vo/Vi is
chapters.                                                   independent of vi; it is a constant at any given
                                                            frequency. This ratio is known as the transfer
                                                           function of the network. At any given frequency
CR and LR circuits
                                                            it clearly has the same value as the output voltage
It was shown earlier that whilst the resistance of a        Vo obtained for an input voltage Vi of unity. One
resistor is (ideally, and to a large extent in practice)    can tell by inspection what Vo will be at zero and
independent of frequency, the reactance of capa-            infinite frequency, since the magnitude of the
citors and inductors is not. So a resistor combined         capacitor's reactance Xc = 1/2nfC will be infinite
with a capacitor or an inductor can provide a               and zero respectively. So at these frequencies, the
network whose behaviour depends on frequency.               circuit is equivalently as shown in Figure 2.1b,
This can be very useful when handling signals, for          offering no attenuation at 0 Hz and total attenua-
example music reproduction from a disc or gra-              tion at c~ Hz (infinite frequency) respectively.
mophone record. On early 7 8 R P M records,                 Since the same current i flows through both the
Caruso, Dame Nellie Melba or whoever was                    resistor and the capacitor,          Vo = iXc and
invariably accompanied by an annoying high-                 vi = i(R + Xc). A term such as R + Xr containing
frequency hiss (worse on a worn record), not to             both resistance and reactance is called an im-
mention the clicks due to scratches. In the case of        pedance Z. Recalling that X~ = 1/jcoC, then
an acoustic gramophone, the hiss could be tamed
                                                                      Vo      1/jcoC             1
somewhat by stuffing a cloth or small cushion up
                                                                      vi   R + (1/jo~C)     1 + jcoCR
the horn (the origin of 'putting a sock in it',
perhaps). But middle and bass response was also            Thus Vo/Vi is a function ofjco, i.e. the value of Vo/vi
unfortunately muffled, as the attenuation was not          depends upon jr The shorthand for function ofjr
very frequency selective. When electric pick-ups,          is F(jr so, in the case of the circuit of Figure 2.1 a,
amplifiers and loudspeakers replaced sound boxes           F(jr = 1/(1 + jeoCR). Figure 2.1c shows what is
and horns, an adjustable tone control or 'scratch          going on for the particular case where 1/r        - R,
filter' could be provided, enabling the listener           i.e. at the frequency f = 1/2nCR. Since the same
selectively to reduce the high-frequency response,         current i flows through both R and C, it is a
and with it the hiss, pops and clicks.                     convenient starting point for the vector diagram.
                                                                                                                                                                                    Passive circuits   27

                                                                                                                                      Rs=0              R
                    R s= 0                PD          = iR

             ~)           vi

                                        If v . = l V ,
                                                      i"~                   PD =
                                                                              C                   Vo
                                                                                                            0                        Rs=0              R
                                                          jox7                                    1
                                        %- ----W- = ---7--R
                                                      R + Jtt~S'                              1
                  (a)                   If T = CR, Vo=
                                                                                              T       1                                                 (b)
                                                                                     jco+ Y
                                                                                                                                                                                    = ~ vi = 1 L 0

        v o = iX c = i/jcoC               o \\\\                   vi
                                                                                X                                                                              -j0.5
                               vo                                                   11,
                  (c)               A                     iR                        "-a                                         (d)          vo = 0.5 - j0.5 = 0.707 < 45 ~
                                          f = 2--~Hz
                                                                                                      vi                        oo
           t.0 -- ** .          .         .               .                                           ~'0~ i~               100"         -
                          Vo at 030                    coo                  iR at too                      (e)                         (f)                 1
         Ivol = M                                                                                                           M (dB)
             1                                                                                                              o ~ ~             ~ , ,
         0.8                                                                                                               -3 . . . .
                                                                                                                           -6 . . .            .
                                                                                                                          - 1 0 I-
                                                                                                                                               I     -- 1               ~ l
           0                                                    t                     I                     I                        MlOO fo/lO            fo                lOfo             100fo
       arg;o              1/T            2/T                  3/T                    4/T                   5/T   f

       _45o~(=                          1 radian )
                                                                                                                          _45 ~
       _57.3~ . . . . .     ~       -     ~           ~
        _.., ,I             ,, .;
                            I ",I             ,                    ,
                                                                                          I                 !    "-       _90 ~
                          1/T 1.5_.__7                        3/7"                   4/T                   5/T        f
                             (= x/2T)                          (g)                                                                                                     (h)

Figure 2.1        C R l o w - p a s s (top-cut) lag circuit.
28        Analog Electronics

Next you can mark in the potential drop iXc across                    f = 1/2rcCR, F(jm) evaluates to 0.707/-45 ~ the
the capacitor. The PD iR is added to this as shown,                   same answer obtained by the vector diagram. But
giving vi. Note that iXc = i(1/jmC) = -ji(1/mC).                      note that, unlike the voltage vectors CA and AB in
Recalling that j indicates a 90 ~ anticlockwise                       Figure 2.1c, you will not find voltages of 0.5 and
rotation, -ji(1/mC) will simply be rotated in the                     -j0.5 at any point in the circuit (Figure 2. l a). As is
opposite direction to ji(1/mC), i.e. downwards.                       often the case, the geometric (vector) solution ties
The PD iR, on the other hand, is in phase with i,                     up more directly with reality than the algebraic. In
so the two PDs must be added vectorially as shown                     general, a quantity x + jy in cartesian form can be
to obtain vi. CAB is a right-angled triangle, so by                   converted to the magnitude and phase angle polar
Pythagoras's theorem the magnitude of vi is                           form Ms as follows:
iv/(X 2 + R2). So for the case illustrated, where
                                                                      M-    v/(x 2 + y2)   ( ~ _ t a n - l ( y / x ) , i.e. tan d~=y/x
the reactance of the capacitor in ohms is numeric-
ally equal to R,                                                      To convert back again,
     Vo            iXc                                                           x - M cos 4)          y - M sin 4)
     Vi       iv/(X 2 + R 2)        v'[(XclX )2 + (RlXc)q             Thus in Figure 2.1d, Vo = 0.707cos (-45 ~ +
                  1                                                   0.707 sin(-45~
          =                = 0.707
              v/(1 + 1)                                                   If the top-cut or low-pass circuit of Figure 2.1a
                                                                      were connected to another similar circuit via a
with the phase angle shown.                                           buffer amplifier- one with infinite input impe-
   You can get the same result by algebra from the                    dance, zero output impedance and a gain of
transfer function rather than by geometry from                        unity at all frequencies- the input to the second
the vector diagram. Starting with F(jm) =                             circuit would simply be the output of the first. The
1/(1 + jmCR), the trick is to get rid of the awk-                     transfer function of the second circuit being iden-
ward term in the denominator by multiplying top                       tical to that of the first, its output would also be
and bottom by the 'complex conjugate' of                              0.707/-45 ~ relative to its input at f = 1/2rcCR,
1 +jmCR. The complex conjugate of P + j Q is                          which is itself 0.707/-45 ~ for an input of 1/0 ~ to
simply P - jQ, or in this case 1 - jmCR. So                           the first circuit. So the output of the second circuit
               1(1 - jmCR)                                 1 - jmCR   would lag the input of the first by 90 ~ and its
 F(jm) - (1 + jmCR)(1 - jmCR)                          1 + o}2C2R 2   magnitude would be 0.707 x 0.707 = 0.5 V. In gen-
                                                                      eral, when two circuits with transfer functions
remembering that j2 _ _ 1. When 1/mC = R                              F1 (jm) and Fz(jm) are connected in cascade- the
                                                                      output of the first driving the input of the second
                                1   D

                                        jl = ! _ j ~   1
                      F .(m)-
                          i     1+ 1           2                      (assuming no interaction) - the combined transfer
                                                                      function Fc(jm) is given by Fc(jm) = Fl (jm)Fz(jco).
This expresses the output voltage for unity refer-                       At any frequency where F1 (jm) and F2(jm) have
ence input voltage, in cartesian or x + jy form                       the values MIs 1 and M2/_~2 say, Fc(jm) simply
(Figure 2.1d). The terms x and y are called the                       has the value M1M2/_(~I + ~ 2 ) . This result is
in-phase and q u a d r a t u r e - or sometimes (mislead-             much more convenient when multiplying two
ingly) the real and imaginary - parts of the answer,                  complex numbers than the corresponding car-
which can alternatively be expressed in polar                         tesian form, where (a + jb)(c + jd) = (ac - bd) +
coordinates. These express the same thing but in                      j(ad + bc). On the other hand, the cartesian form
terms of the magnitude M and the phase angle ~I,                      is much more convenient than the polar when
(or modulus and argument) of Vo relative to vi,                       adding complex numbers. Returning to Figure
written M / ~ . You can see from Figure 2.1d that                     2.1c, it should be remembered that this is shown
the magnitude of V o = v / ( 0 . 5 2 + 0 . 5 2 ) = 0 . 7 0 7          for the particular case of that frequency at which
(Pythagoras again) and that the phase angle of Vo                     the reactance of the capacitor equals R ohms; call
relative to vi is - 4 5 ~ or -0.785 radians. So at                    this frequency fo and let 2rcfo = COo.In Figure 2.1e,
                                                                                                                Passive circuits   29

                          L                                                                               R

                                                                                                  O r ........... ~ L   ~o

                                                 R                             R                                jco
          Vo                     R       ~                      F (j03) =         1               F(j03) =          1' T -
                                                                                                              j03+ ~-
          v---?"= F(jr        R + j03~ = jco + R                            R + jtoC
            1                                        T                       j03CR
                                           1                           = 1 +j03CR
                          L                "I"
                If T =   ---~, F(j03)=       I                                J03
                                         J03+T                         =         1
                                                                            J03+ T
                               (a)                                           (b)

Figure 2.2 CR and L R circuits.
  (a) L R low-pass circuit.
  (b) CR high-pass (bass-cut) lead circuit. We can normalize the frequencyf to the break, corner or 3 dB frequency f0
where f0 = 1/2rcCR (i.e. where 030 = 1/CR = l / T ) , by using 03n = 03/030 instead of 03. Then when 03 = l I T = 03o, the
normalized radian frequency COn= 1. F(j03) then simplifies to
                                                                    j03/030              j03n
                                                 F(j03n) =
                                                             (j03/030) + (030/030)     j03n + 1
or, more generally, normalized F(s) = s/(s + 1).
   (c) L R high-pass circuit. Again, if cois normalized as above, F(s) - s~ (s + 1).

Vo (which also represents the transfer function if                          Figure 2.1f. As co increases from 0 to infinity,
v i - 1/0 ~ is shown for various values of co from                          decreases from 0 ~ to - 9 0 ~ whilst Vo decreases from
zero to infinity. Since the vectors iXc and iR in                           unity to zero. You now have a universal picture
Figure 2.1 c are always at right angles whatever the                        which applies to any low-pass circuit like
frequency, it follows that the locus or line joining                        Figure 2.1a. Simply multiply the co values in
the tips of the Vo vectors for all frequencies is a                         Figure 2.1f by 1/2rcCR to get its actual frequency
semicircle, due to a theorem worked out a long                              response.
time ago by a gentleman called Euclid. Figure 2.1e                              One can alternatively plot these changes in Ivol
is a simple example of a circle diagram - a very                            (signifying the modulus of Vo, i.e. the M part of
useful way of looking at a circuit, as will appear in                       M / G ) and arg Vo (signifying the angle 9 part of M)
later chapters. It is even more useful if you normal-                       against a linear base of frequency, as in Figure
ize co, the angular frequency, vi has already been                          2.1g. However, for most purposes it is better tO
normalized to unity, i.e. 1/0 ~ or 1/0 radians,                             plot Vo against a logarithmic baseline of frequency,
making the transfer function evaluate directly to                           as this enables you to see more clearly what is
Vo at any frequency. In particular, for the exam-                           happening at frequencies many octaves above and
ple of Figure 2.1a, F(jco) - ( 1 / v ~ ) / _ - 45 ~ =                       below f0, the frequency where 2 r c f 0 = c o 0 =
0.707/_ - 0.785 radians whenl/2rcfC - R, i.e.                               1 / C R = 1/ T, the frequency where Vo/Vi turned
when c o - 1/CR. It is useful to give the particular                        out to be 0 . 7 0 7 / - 4 5 ~ (Note that f0 depends
value of co where c o - 1/CR the title coo. Dividing                        only on the product C R = T, not on either C or
the values of co in Figure 2.1e by coo, they simply                         R separately.) At the same time, it is convenient to
become 0, 0.1, 0.2, 0.5,1, 10 etc. up to oc, as in                          plot the magnitude on a logarithmic scale of
30    Analog Electronics

 decibels (unit symbol dB*). This again compresses           sistance R at 0 Hz to an open-circuit at infinite
 the extremes of the range, enabling one to see very         frequency. By contrast, the input impedance of
large and very small values clearly. So rather than          Figure 2.1a falls from a capacitive open-circuit to
plotting Ivo/vil one can plot 20 loglolVo/Vi I instead.      R as we move from 0 Hz to c~ Hz. Similarly, the
This is called a Bode plot, after the American               source impedances seen by the load, looking back
author H. W. Bode 1, and is shown in Figure                  into the output terminals of Figures 2. l a and 2.2a,
2.1h. Note that multiplying two numbers is equiva-           also differ.
lent to adding their logarithms. So if the value at a           Figure 2.2b and c show bass-cut or high-pass
particular frequency of a transfer function M / ~            circuits, with their response. Here the response
is expressed as M d / ~ , where Md is the ratio              rises (with increasing frequency) at low frequen-
M expressed in decibels, then the product                    cies, at + 6dB per octave, becoming fiat at high
Mdl//tI)l • MdE/tI)2 is simply expressed as                  frequencies. Clearly, with a circuit containing only
 (Mdl "q'-Md2)/(~l 4- ~2).                                   one resistance and one reactance driven from a
    At very low frequencies, the reactance of C is           constant voltage source and feeding into an open-
very high compared with R, so i is small.                    circuit load, Vo can never exceed vi so the two
Consequently there is very little PD across R and            responses    shown exhaust the possibilities.
 the output is virtually equal to vi, i.e. independent       However, if we consider other cases where the
of frequency o r - in the jargon - ' f l a t ' (see Figure   source is a constant current generator or the load
 2.1h). At very high frequencies, the reactance of C         is a current sink (i.e. a short-circuit), or both, we
is very low compared with R, thus the current is             find arrangements where the output rises indefi-
virtually determined solely by R. So each time the           nitely as the frequency rises (or falls). These cases
frequency is doubled, Xc halves and so does the              are all shown in Figure 2.3 2.
PD across it. Now 201og100.5 comes to - 6
(almost exactly), so the output is said to be falling
                                                             Time domain and frequency domain analysis
by 6dB per octave as the frequency increases.
Exactly at f0 , the response is falling by 3 dB per          There is yet another, more recent representation of
octave and the phase shift is then - 4 5 ~ as shown.         circuit behaviour, which has been deservedly pop-
The slope increases to - 6 d B / o c t a v e and falls to    ular for nearly half a century. By way of introduc-
 0dB/octave as we move further above and below               tion recall that, for a resistor, at any instant the
f0 respectively, the phase shift tending to - 9 0 ~ and      current is uniquely defined by the applied voltage.
 0 ~ likewise.                                               However, when examining the behaviour of capa-
    The L R top-cut circuit of Figure 2.2a gives             citors and inductors, it turned out to be necessary
 exactly the same frequency response as the CR               to take account not only of the voltage and
 top-cut circuit of Figure 2.1a, i.e. it has the same        current, but also of their rate of change. Thus
 transfer function. However, its input impedance             the analysis involves currents and voltages which
 behaves quite differently, rising from a pure re-           vary in some particular manner with the passage of
                                                             time. In many cases the variation can be described
* Decibels, denoted by 'dB', indicate the ratio of the       by a mathematical formula, and the voltage is then
power at two points, e.g. the input and output on an         said to be a determinate function of time. The
amplifier. For example, if an amplifier has a power gain     formula enables us to predict the value of the
of one hundred times (100 mW output for 1 mW input,
say), then its gain is 20dB (or 2 Bels or •            In    voltage at any time in the future, given its present
general, if the power at point B is 10n times that at        value. Some varying voltages are indeterminate,
point A, the power at B is + 1ON dB with respect to A. If    i.e. they cannot be so described, an example is the
the impedances at A and B are the same, then a power         hiss-like signal from a radio receiver with the aerial
gain of • 100 or + 20 dB corresponds to a voltage gain of    disconnected. In this section, interest focuses on
• 10 (since W = E2/R). In practice, voltage ratios are
often referred to in dB (dB= 20 x loglo(Vl/V2)), even        determinate functions of time, some of which have
when the impedance levels at two points are not the          appeared in the analysis already. One example was
same.                                                        the exponential function, where vt (the voltage at
                                                                                                                                                                              Passive circuits                   31

          Characteristic curves                                    Constant voltage input                                                       Constant current input
                                       Curve             Voltageoutput              Current output                                   Voltage output       ]      Current output
                                       no.               into open circuit          into short circuit                               into open circuit     i     into) short circuit
                                                                     -                                                                                '                       I                                  '

                                             1                                          vi                          io                                            v~                   ii         i~

                                                                                        o-                          o            o                            -       o               o--~   -----o
                                                                                                                                             jc0T                                                  jeT
                                                               j~T                              1             j~T            ,            R~ ;~T                                                 1 + j~T

                   3       4                                                            ~                                    I                                                '
                                                     ~               ~--,O1~o                                                    o----                        -   o               ~
dB 0                                         2                                          vi                          i~
                                                     5z_       o'                                                                "                                                                         z_5
                                   Y                             ~              l
                                                                                        ~--I                  1     ~                     R                                                            1

                                             3                                          vi                          io           ii           ~[~                     v~
                                                                                        O                           O            ~            Y                       O
                                                                                               1 1 +jmT                                    R 1 +jmT
                                       ~..       |                              !
                                                                                               R jtoT
                                                                                                   ,    . _              i
                                                                                                                                               je0r       ~                   !

                                             4                                          vi                          i~           ii                               v~
                                                                                        o--                         o            o-~.                                 o
 +90~                  5                                                                       1
                                                                                               -~ (1 + jc0T)                              a (1 + .i~T)
               1               4
                                                                                        ~------I|                   o            Or               -               ~

    0                                        5                                          vi                          io               ii           I                   v~
                                                                                        o---                                     O --~            -                   O
                                                                                                       jt0C                                   jo)L
                                                                                    ,                                    _/

 _90 ~t   J              ~ 2       _ ,                                                                                           ~            i                       ~

                                                                                        vi                          io               ii       T                       v~

                                                                                        o~                          o                o            -                       o

Figure 2.3    All combinations of one resistance and one reactance, and of one reactance only, and their frequency
characteristics (magnitude and phase) and transfer functions (reproduced by courtesy of Electronics and Wireless

any instant t) is given by vt = vo e at, describing a                                           time, already mentioned, is the sinusoidal func-
voltage which increases indefinitely or dies away to                                            tion, e.g. the output voltage of an AC generator.
zero (according to whether Qtt is positive or nega-                                             Here vt = V sin(cot), where the radian frequency co
tive) from its initial value of v0 at the instant t = 0.                                        equals 2n times the frequency f in hertz, and V is
Another example of a determinate function of                                                    the value of the voltage at its positive peak.
32    Analog Electronics

                                    +1                                                                     (tot)

                        -           /j~"t             =0         1/2f~~,,,~j/"l/f       ................ 3/2f
                        - - ~ -1-

                                         +1                                                            Cos (COt)

                                         /                 ~'t
                            J            -1
                                                           I              (b)

Figure 2.4        Initial transients.
  (a) Sine wave c o n n e c t e d to a circuit at to.
  (b) C o s i n e wave c o n n e c t e d to a circuit at to.

Considering a voltage or a circuit response                                 where T = l/f, the applied voltage (defined by
specifically as a function of time is described as                          reference to its phase at t = to) would be a cosine
time domain analysis.                                                       wave (Figure 2.4b). In the first case we have an
   The sinusoidal waveform is particularly impor-                           input voltage exhibiting a change of slope but no
tant owing to its unique properties. Already men-                           change of value at to, whereas in the second we
tioned is the fact that its rate of change is described                     have an abrupt change of value of the voltage at to
by the cosine wave, which has exactly the same                              itself, but the slope or rate of change of voltage is
shape but is advanced in time by a quarter of a                             zero both just before and just after to. It: is not
cycle or 90 ~ (see Figure 1.7). Now it turns out that                       surprising that the response of the circuit in these
all repetitive waveforms can be analysed into the                           two cases differs, at least in the short term.
sum of a number of sine and cosine waves of                                 However, the fine detail of the initial conditions
related frequencies, so it is exceedingly useful to                         when the signal was first applied become less
know how a circuit responds to sine wave inputs of                          important with the passage of time, and after a
different frequencies. You can find out by connect-                         sufficiently long period become completely irrele-
ing a sine wave obtained from a signal generator,                           vant. The response of the circuit is then said to be
for example, to the circuit's input and seeing what                         in the steady state. The difference between this and
comes out of the output, at various frequencies. In                         the initial response is called the transient, and its
the real world, the waveform will be connected to                           form will depend upon the initial conditions at to.
the circuit under test at some specific instant, say                        For many purposes it is sufficient to know the
when its value is zero and rising towards its                               steady state response of a circuit over a range of
positive peak. The applied voltage is thus a sine                           frequencies, i.e. the nature of the circuit's response
wave as in Figure 2.4a, whereas had the connec-                             as a function of frequency. This is known as
tion been made at a to which was T/4 seconds later,                        frequency domain analysis, since the independent
                                                                                                         Passive circuits    33

variable used to describe the circuit function is                    cy equals zero. The law of indices states that to
frequency rather than time. The method is based                      multiply together two powers of the same number
upon the Laplace transform, an operational                           it is only necessary to add the indices: 4 x 8 =
method that in the 1950s gradually replaced the                      22 • 23 = 25 = 32,         for   example.   Similarly,
operational     calculus   introduced    by Oliver                   e ~'t eJmt= e (~+jm)', thus expressing compactly in a
Heaviside many years earlier 3. The transform                        single term the frequency and rate of growth (or
method eases the solution of integral/differential                   decline) of a sinusoid. It is usual to use s as
equations by substituting algebraic manipulation                     shorthand for cy + jco; s is called the complex
in the frequency domain for the classical methods                    frequency variable. A familiarity with the value
of solution in the time domain. It can provide the                   of F(s), plotted graphically for all values of cy and
full solution for any given input signal, i.e. the                   co, provides a very useful insight into the behaviour
transient as well as the steady state response,                      of circuits, especially of those embodying a
though for reasons of space only the latter will                     number of different CR and/or L R time constants.
be dealt with here.                                                  The behaviour of such circuits is often more
   In the frequency domain the independent vari-                     difficult to envisage by other methods.
able is co - 2rcf, with units of radians per second.
The transfer function, expressed as a function of
                                                                     Frequency analysis: pole-zero diagrams
frequency F(jco) has already been mentioned. You
will recall that j, the square root o f - 1 ,    was                 To start with a simple example, for the circuit of
originally introduced to indicate the 90 ~ rotation                  Figure 2.1a it was found that F(jco) = 1/(1 + jcot),
of the vector representing the voltage drop across a                 giving a response of 0 . 7 0 7 / - 4 5 ~ at coo = l I T
reactive component, relative to the current                          = 1/CR. Taking the more general case using
through it. However, j also possesses another                        cy + jco,
significance. You may recall (in connection with                                            1            1/T
Figure 1.7) that on seeing that the differential (the                             F(s) = 1 + s-----~= s + ( l / T )
rate of change) of a sine wave was another wave-
form of exactly the same shape, it seemed likely                     Figure 2.5a shows a pair of axes, the vertical one
that the sinusoidal function was somehow con-                        labelled jco, the horizontal one cy. Plotting the
nected with the exponential function. Well, it turns                 point cy = - 1 / T (marked with a cross) and draw-
out that                                                             ing a line joining it to the point co = 1 / T on the
                                                                     vertical axis, gives you a triangle. This is labelled
                       e j~ - cos 0 + j sin 0                        CAB to show that it is the same as the triangle in
                   e -j~ - cos 0 - j sin 0                           Figure 2.1c, where Vo/Vi = CA/CB. As co increases
                                                                     from zero to infinity, the reactance of the capacitor
This is known as Euler's identity. So sinusoidal                     will fall from infinity to zero; so in both diagrams
voltage waveforms like V sin cot can be represented                  the angle BCA will increase from zero to 90 ~ So in
in exponential form since using (2.1) you can write                  Figure 2.5a, ,I, represents the angle by which
             eJo~t _ e-jCot                       eJcot + e - J c~   Vo lags Vi, reaching - 9 0 ~ as co approaches oc.
 sin cot -                     and    cos cot -                      Likewise, since Vo/Vi = CA/CB, the magnitude of
                  2j                                     2
                                                                     the transfer function is proportional to 1/CB.
One can also allow for sine waves of increasing or                   Expressed in polar ( M / G ) form, the transfer
decreasing amplitude by multiplying by an expo-                      function has evaluated to ( C A / C B ) / t a n - I ( A B /
nential term, say e c~t, where cy is the lower-case                  AC), as you move up the jco axis above the
Greek letter sigma. As noted earlier, if cy is positive              origin, where cy = z e r o . In fact, if you plot the
t h e n e ~t increases indefinitely, whilst if cy is nega-           magnitude of F(s), for co from 0 to oc (with cy = 0),
tive then the term dies away to zero. So e~te jc~                    on a third axis at right angles to the cy and jco axes,
represents a sinusoidal function which is increas-                   (Figure 2.5b), you simply get back to the magni-
ing, d e c r e a s i n g - or staying the same amplitude if          tude plot of Figure 2.1g. This may sound a
34    Analog Electronics

                                                                                                           M-'IF(s)I ,~=0                                                            jo



                                               co=T=     '('   - ~ - = co   o)
                      c                        A                              -0

                      -1/7"                        Origin


                                                                             -jco                                   (b)

                                                                                    M (dB)                                          Asymptotes
                                                                                     .. ~,                 ,              ~         M1   ,M2                         --6 dB/octave
                                         jco                                         o]                    ~_ t---!                            -,~:..~-/\-~-
                                                                                   -127                    i --'"~,~I
                                                                  o)0say           -18 -5                  i                  ~            ~         /
                                                                                       !                   I                  I                  !               ~               -
                                                                                                           i                  i                  I
                          / / , ,                                                      |                    ,                 !                  !                      /                  l o g co
                                                                                       L _ I          ,     l                 i        ,       , |       .....         IL:            :~

                                     /                                                                -   1/ T1               -1               -l/T2                 -12 dB/octave

  -I/T2    = c r o a - 1/T1 = cro/a            0                                     ,-, i" ' ~                                , / ' -.~a,
                           -1                                                      -90~       ~                                I --~]~'--                               '-       -
                ~     7(r~ T9                                                    -18o~                         I               I                  ,,,~."                     ~             log ~o
                                                                                           t _ ....            i ....             , ........... , . . . . . . . .
Figure 2 . 5 Analysis of circuit of Figure 2.1, giving m a g n i t u d e a n d p h a s e as functions of c o m p l e x f r e q u e n c y cr + jc0.
((c) a n d (d) r e p r o d u c e d by c o u r t e s y of Maxim I n t e g r a t e d P r o d u c t s UK Ltd).
                                                                                                Passive circuits    35

complicated way to get there, but the light at the          you surmount, and the more rapidly the phase
end of the tunnel is worth waiting for, so stick with       twizzles round from 0 ~ to - 1 8 0 ~ in the vicinity of
it. See what happens if you plot the value of F(s) as       the peak. Keep this picture in mind, as you will
in Figure 2.5b for values of s where cy is not zero.        meet it again later.
Remember, you are plotting the value of                        The infinitely high mountain is called a pole
                                                            and occurs, in the low-pass case of F(s)=
                      lIT                      mo
     F(s) --                                                (1/T)/[s + (l/T)] at s = - l i T + j0. This is the
               (a + jm) + l i t          (cy + jm) -+- mo   complex frequency which is the solution of." de-
not the ratio CA/CB. (The vector diagram of                 nominator of F(s) = 0. If there were two low-pass
Figure 2.1c only ties up with the cy +jm plot for           circuits like Figure 2.1a in cascade (but assuming a
the particular case where cy = 0, i.e. for a sine wave      buffer amplifier to prevent any interaction between
of constant amplitude.) Consider first the case             them) with different critical frequencies fl and f2,
where cy = 1/T. For very large values of m this             then you would have
really makes very little difference, but, at c0 = 0,
                                                                            1/T1             1/T2
                            1/T            =0.5/00              F(s) -                •
                                                                         s + (l/T1)       s + (l/T2)
        F(s) - ( l / T + j0) + l I T
Conversely, for small negative values of cy, F(s) (at
                                                                         [(s + (1/T1)][(s + (1/T2)]
c o - 0 ) is greater than unity: and, as cy reaches
-1/T,                                                                                     1/T1T2
                                1/T                                      ,2 +             + (l/V2)] + (1/v    r2)
             F(s) -
                      ( - l I T + j0) + l I T               and the s-plane plot - which is known as a pole-
i.e. it explodes to infinity. This is shown in three-       zero diagram- would have two poles on the - ~
dimensional representation in Figure 2.5c. But,             axis, at distances l/T1 and 1~Tz, to the left of the
you may object, for values of cy more negative              origin. At any frequency corresponding to m on
than - 1 / T the picture shows F(s) falling again but       the vertical jm axis, the response M/__G- F(jm)
still positive; whereas at cy = - 2 I T ,                   would be (1/dld2)/(~l + ~2), as shown in Figure
                                                            2.5d and e. This is called a second-order system as
         F(s) -    (-2/T + j0) + 1/T = - 1                  F(s) has a term in s 2 in the denominator. As with
                                                            the first-order system of Figure 2.1, the overall
Remember, however, that F(s) is a vector quantity           response falls towards zero (or - e ~ dB) as m tends
with a magnitude M (always positive) and a phase            to infinity, but at 12dB per octave, as shown in
9 . The minus sign indicates that the phase has             Figure 2.5f. This also shows the phase and ampli-
switched suddenly to - 1 8 0 ~ To make this clearer,        tude responses of the two terms separately, and
consider the value of F(s) as cy becomes progres-           what is not too obvious from the Bode plot is that
sively more negative, for a value of m constant at          at a frequency m - 1/v/(T1T2) the output lags 90 ~
0.1 (1/T) instead of zero. The value of                     relative to the input. This can, however, be de-
                                                            duced from the pole-zero diagram of Figure 2.5e
             F(s) -                                         using secondary- or high-school geometry. Call
                      (cy + j 0 . 1 / T ) + l I T
                                                            1/v/(T1T2) by the name m0 for short. Then con-
increases,     reaching     maximum value of
                             a                              sider the two right-angled triangles outlined in
(1/T)/(jO.1/T) = 1 0 / - 90 ~    when     cy=-l/T,          bold lines in Figure 2.5e. The right angle at the
where you surmount the north face of the infinitely         origin is common to both. Also (1/T1)/mo
high F(s) mountain. Descending the western slope,           = co0(1/T2); therefore the two triangles are similar.
the amplitude M falls as 9 increases to - 1 8 0 ~           Therefore the two angles marked ~2 are indeed
Clearly the smaller the constant positive value of m        equal. Therefore ~1 + ~2 - 90 ~ measured anti-
during your westward journey the higher the slope           clockwise round the respective poles, making Vo
36    Analog Electronics

lag Vi by 90 ~ at 03o, mostly owing to the lower-              round the back of the world, where the inter-
frequency lag circuit represented by T1. The point             national dateline crosses the equator.
Cro = - 1 / v / ( T 1 T 2 ) , to which the poles are related      By contrast, the high-pass lead circuit of Figure
through the parameter at by - 1 / T 2 = ~cr0 and               2.2b has a pole-zero diagram (not shown) with a
- 1 / T 1 = cr0/a, is therefore significant. If a = 1          finite zero, located at the origin, as well as a pole.
the two poles are coincident, as also are the two              The phase contribution to F(j03) of a zero on the
- 6 dB/octave asymptotes and the two 0 ~ to - 9 0 ~            -or axis is the opposite to that of a pole, starting at
phase curves in Figure 2.5f. As at increases, so that          zero and going to +90 ~ as you travel north from
 - l/T1 moves towards zero and - 1/T2 moves                    the origin along the j03 axis to infinity. However, in
towards infinity, the corresponding asymptotes in              this case the phase is stuck at +90 ~ from the
Figure 2.5f move further and further apart on their            outset, as the zero is actually right at the origin.
log scale of 03. There is then an extensive region             The zero is due to the s term in the numerator of
either side of O3o where the phase shift dwells at             F(s), so the magnitude contribution of this term at
around - 9 0 ~ and the gain falls at 6dB/octave.               any frequency co is directly (not inversely) propor-
However, even if a = 1 so that - l/T1 = - 1/T2,                tional to the distance from the point co to the
you can never get a very sharp transition from the             z e r o - again the opposite to what you find with a
flat region at low frequencies to the - 1 2 dB/octave          pole.
regime at high frequencies. Setting a less than 1                 Note that the low-pass circuit of Figure 2.1 and
just does not help, it simply interchanges -1/T1               the high-pass circuits of Figure 2.2b and c each
and -1/T2. Although one cannot achieve a sharp                 have both a pole and a zero. A crucial point
transition with cascaded passive CR (or LR) low-               always to be borne in mind is that however
pass circuits, it is possible with circuits using also         simple or complicated F(s) and the corresponding
transistors or operational amplifiers (as will                 pole-zero diagram, the number of poles must
appear in later chapters) and also with circuits               always equal the number of zeros. If you can see
containing R, L and C.                                         more poles than zeros, there must be one or more
    I have been talking about pole-zero diagrams,              zeros at infinity, and vice versa. This follows from
and you have indeed seen poles (marked with                    the fact that F ( j 0 3 ) = Vo/Vi = M / ~ . Now 9 has
crosses), though only on the -or axis so far; but              units of radians, and a radian is simply the ratio of
where are the zeros? They are there all right but off          two lengths. Likewise, M is just a pure dimension-
the edge of the paper. In Figure 2.5a there is                 less ratio. So if the highest power of s in the
clearly a zero of F(s) at 03= cr since                         denominator of F(s) is s 3 (a third-order circuit
                                                               with three poles), then implicitly the numerator
                              030       = 0
            r(s)~     = (0 + jc~) + 030                        must have terms up to s 3, even if their coefficients
                                                               are zero: i.e. F(s) = 1/(as 3 + bs 2 -+-cs -+-d) is really
Incidentally, F(s) also becomes zero if you head               F(s) = (As 3+Bs 2 + Cs + D)/(as 3 + bs 2 + cs + d),
infinitely far south down the -j03 axis, or for that           where A = B = C = 0 and D = 1. So poetic justice
matter east or west as cy tends to +c~ or - c r                and the theory of dimensions are satisfied and, just
Indeed the same thing results heading northeast,               as Adam had Eve, so every pole has its zero. As a
letting both co and cy go to + cr or in any other              further example, now look at another first-order
direction. So the zero of F(s) completely sur-                 circuit, often called a transitional lag circuit. This
rounds the diagram, off the edge of the map at                 has a finite pole and a finite zero both on the -cr
infinity in any direction. This 'infinite z e r o ' - if you   axis, with the pole nearer the origin. The transi-
don't find the term too c o n f u s i n g - can alterna-       tional lag circuit enables us to get rid of (say) 20 dB
tively be considered as a single point, if you                 of loop gain, in a feedback amplifier (see Chapter
imagine the origin in Figure 2.5a to be on the                 3) without the phase shift ever reaching 90 ~ and
equator, with the j03 axis at longitude 0 ~ Then, if           with neligible phase shift at very high frequencies.
you head off the F(s) map in any direction, you                Figure 2.6a shows the circuit while Figure 2.6b to f
will always arrive eventually at the same point,               illustrate the response from several different points
                                                                                               Passive circuits     37

of view. Note that K is the value of the transfer           much larger than the input voltage. Consequently,
function at infinitely high frequencies, where the           (1/R)v/(L/C) is often called the magnification
reactance of C is zero, as can be seen by replacing         factor or quality factor Q of the circuit, and
C by a short-circuit. When s = O,F(s) gives the              can alternatively, be written as Q - C O o L / R -
steady state transfer function at zero frequency,            X L o / R - - X c o / R . If, on the other hand, R is
where the reactance of C is infinite, so                     much larger than cooL and 1/cooC, the output
                                                             will start to fall at 6dB per octave when 1/
                            1/ T2 Tz l / T2
                  F(s) - K . . . . .             1           c o C - R and at 12 dB per octave at some higher
                            1 / T1 T1 1 / T1
                                                             frequency where c o L - R. These results can be
as can be seen by replacing C by an open-circuit.            derived more formally, referring to Figure 2.7a, as
You may recall meeting - 1 v/(T1T2) in Figure 2.5,           follows:
and it turns out to be significant again here. At
                                                            F(jco) - v--2~         iXc
co - - V/(co01co02), r + (~z --- 9 0 ~ but the actual
phase shift for this transitional lag circuit is only
                                                                      vi     i(XL + R + Xc)
r   -- r z.                                                                 1/ jcoC                     1
  In Figure 2.2b, by normalizing the frequency,                       jcoL + R + 1/jcoC        (jco)ZLC+jcoCR +1
one finished up with the delightfully simple form
F(s) = s/(s + 1) - not a time constant in sight.            or more generally,
However, this is only convenient for a simple first-                                       1
order high-pass circuit, or a higher-order one                               F(s) = LC 2 + CRs + 1                (2.2)
where all the corner frequencies are identical.
                                                            This    equation       can        be     factorized  into
With two or more different time constants, it is
                                                            1/[(s + a)(s + b)], where a and b are the roots of
best not to try normalizing, though in Figure 2.6
                                                            the equation
you could normalize by setting v/(co01co02 = 1 if                           L C s 2 -k- C R s -+- 1 - 0         (2.3)
you felt so inclined. Then T1 becomes r (say) and
T2:  1/a.                                                   When s - - a     or - b , the denominator of (2.2)
                                                            equals zero, so F(s) will be infinite, i.e. - a and
                                                            - b are the positions of the poles on the pole-zero
Resonant circuits                                           diagram. As any algebra textbook will confirm, the
                                                            two roots o f a x 2 + b x + c - 0 are given by
Figure 2.7 shows an important example of a two-
                                                            x - { - b + v/(b 2 - 4ac)}/Za, so applying this for-
pole (second-order) circuit. At some frequency the
                                                            mula to (2.3) we get
circuit will be resonant, i.e. IjcoLJ = I1/jcoCI. At this
frequency, the PD across the inductor will be equal                           -CR   + v/(C2R   2 -   4LC)         (2.4)
in magnitude and opposite in sign to the PD across                      s-              2LC
the capacitor, so that the net PD across L and C
                                                            Now algebra is an indispensable tool in circuit
together (but not across each separately) will be
                                                            analysis, but a result like (2.4) by itself doesn't give
zero. The current i will then exhibit a maxi-
                                                            one much feel for what is actually going on. So
mum of i=vi/R. At resonance, then, i[jcoL+
                                                            consider a normalized case where L - 1 H and
(1/jcoC)] = 0, so co2LC = 1 and co = 1/v/(LC).
                                                            C - I F , so that c o o - l/v/(1 x 1 ) - 1 radian per
Give this value of co the label coo. At coo the
                                                            second, and consider first the case where
output voltage Vo = iXc = (vi/R)(1/jcoC), so
                                                            R - 10 ohms.         From       (2.4),      s - { - 10 +
        Vo           -j       _jl   /-{L'~       .Xco       v/(100-4)}/2--5+4.9.             Now we can draw
                                                            the poles in as in Figure 2.7b, which also shows
         Z    =           =                  -
                                                            the corresponding 45 ~ break frequencies on the jco
where Xco is the reactance of the capacitor at coo.         axis col and co2, at which each pole contributes 45 ~
Clearly, for given values of L and C, as R becomes          phase lag and 3 dB of attenuation. As there are no
very small the output voltage at resonance will be          terms in s in the numerator of (2.2), no zeros are
38   Analog Electronics

                                                                                                                      ! t~o2
                      R1                                                                                        •)Z
                                                                                                                        co = 4 - ( ~    01 ~ 09

                 vi                             Vo

                                     R2                             Cz (po
                                                                        -1      -1      -1
                 o-                             o
                               (a)                                      T 2 -~'-(r 1T2) T1


                                                                        r          Comer frequency                                .,,ioc'~ ""

                                                                              0j       --'0)01       '1           C002't'/~               ........
                                                               -:Olo~/;|                -;~.~ i                          ',
                                                              = -10 log K["               I-u ~ r            ~--
                                                                 -~Olo~l                  !       :i:~-~;:-----                             .       -
                                                                          ,              -1                             T
                                                                        +90"             T1 ~-(rlr:~)
                                                                            0 o-          i,         i          -~--I               . . . .
                                                                        -90 . . . .
                                                                                           i          I                  I
                                                                                                      !                  I
                                                                                           I          I                  Io

                         (c)                                                                      (d)

                  K                                           vi= 1                (v o) ~ = K                                             vi = 1

                                                     Locus ':'f Vo =J
                  "-.~                                         /,
                                                                                                                                       co increasing
           V/-(c0 01o~02)                                                             ~= V-(~ 01~o9
                                          (e)                                                             (f)
                                                                                                                                 Passive circuits   39

visible on the diagram: they are there, but both at                              v/(4-4)}/2-        1 + 0 , so that the two poles are
infinity. The Bode diagram (Figure 2.7c) shows                                   indeed coincident at s - - 1 or at s - 1 + j0 to be
amplitude and phase approximated by straight-                                    precise, since s is after all the complex frequency
line asymptotes rather than the true curves: the                                 variable. The roots a and b are both equal to
pole at o r - -0.1 will cause 3dB of attenuation                                 - R / 2 L (i.e. - 1 in the numerical example), so
and - 4 5 ~ phase shift at o3-0.1. F r o m 0)1 to 0)2 is
just under seven octaves, i.e. 9.9/0.1 ~ 27 so the                                                                           1
phase r will settle down to - 9 0 ~ over most of the                                          F(s) - [s + (R/ZL)][s + (R/ZL)]
middle of this range, whilst at 6dB/octave the                                                                   1                     1
attenuation at o32, will have risen to just under                                                      Is + (R/2L)] Is + (R/ZL)]
7 x 6 - 42 dB (actually 39.9 dB) plus another 3 dB
due to the pole at c y - - 9 . 9 . At o3-9.9, r will                             that is, the same as a cascade of two identical first-
have risen to 90 + 45 ~ heading for 180 ~ at higher                              order lags. Figure 2.7e shows how we can use this
frequencies still.                                                               fact to construct points on the vector locus of Vo
   The picture looks in fact remarkably like the                                 for three frequencies o3-0.1, o 3 - 1 and c o - 1 0 .
two cascaded lags of Figure 2.5d to f. I think the                               For example, a single lag will provide a normalized
circle diagram goes like Figure 2.7d, almost a                                   output of 0 . 7 0 7 / - 4 5 ~ at o3- 1, and a second circle
complete semicircle before finally diving into the                               diagram erected on this base gives the final output
origin at - 1 8 0 ~ but I haven't actually tried                                 as 0.707 • 0 . 7 0 7 / - 4 5 0 - 4 5 0 - 0 . 5 / - 9 0 ~ At very
drawing it to scale. If you substitute lower and                                 high frequencies, the phase shift is clearly tending
lower values of R in (2.4), you will find that the                               to - 1 8 0 ~ This all agrees with the corresponding
two poles move towards each other, so it looks as                                Bode plot, which would show a horizontal at-
though they must eventually meet.                                                tenuation asymptote changing straight away to
   N o w look at the case where R - 2            ohms.                           - 1 2 d B / o c t a v e at o 3 - 1 , at which frequency the
Substituting this in (2.4) gives s - { - 2 +                                     phase shift @ will already have reached 90 ~

Figure 2.6    Three ways of representing frequency response for transitional lag circuit.
  (a) Transitional lag circuit.
                  F(j0)) - Vo =         (1/j0)C) + R2                      1 + j0)CR2          1 .1. j0)T2      T2[(1/T2) + j0)]
                              i     ( 1 / j 0 ) C ) "1" R 2 .1. R1     1 -t- j0)C(R1 -9 R2)    1 .1. j0)T1      T 1 [ ( l / T 1) %- j0)]
Mor generally,
                      F(s) = K s ,1, (1 / T2)         where                    /)___2= ) _ _ 1
                                                                     K = T2 = 100___2 00 _ 1=         CR2            __      R~
                                  s .1. (l/T1)                           T1 1/0)01 0)02          C ( R 1 at-   R2)        R1 .1. R2
   (b) Pole-zero plot. Up tOv/(0)01 0)02) the lag (Dp i n c r e a s e s faster than the lead qbz.Thereafter Cz catches up, so the net
lag falls again. The zero is shown by a circle.
   (c) Pole-zero plot in three dimensions. The effect of the zero depresses the value of F(s) at infinity to K. The further
the zero at - I / T 2 is along the -or axis (the smaller R2 is), the smaller is K (the response at infinity) (reproduced by
courtesy of Maxim Integrated Products UK Ltd).
  (d) Bode plot, showing the use of asymptotes. The - 6 dB/octave asymptote is cancelled beyond 0)o2by the +6 dB/
octave asymptote. They crudely approximate the actual amplitude shown dashed. Likewise, phase asymptotes run-
ning from 0 ~below the corner frequency to - 9 0 ~ or +90 ~ above, indicate the phase tendency if 0)02 >> 0)01.
  (e) Circle diagram. For any phase angle less than Cm there are two different possible values of Vo, the larger at a fre-
quency below v/[(l / TI)(I / T2)], the smaller at a frequency above.
  (f) Using the circle diagram.lf you are better at gometry than at calculus or complex numbers, the circle diagram en-
ables you to find the attenuation at the frquency where the phase shift qbis maximum. The bold triangles with bases
(Vo)~ = K and vi = I are similar, as they have three equal angles, qb is common, each has an angle a, and each has
an angle 90 ~ + a. Therefore when the output vector is tangential to the circle, at 0 ) v / ( 0 ) o 1 0 ) o 2 ) , K / v o = vo/l, whence
Vo = v/-K. Expressed in decibels, this amounts to half the maximum attenuation, agreeing with the Bode plot.
40   AnalogElectronics

                               R             L

                                                               vo       load
                                         i             --[-             resistance
                                                        _t ~o ....I
       Ideal voltage generator
       (zero internal resistance)
                                                                                                                          .co 1
                                                                                           -9.9              -5


                 0                                |
                                                   !                            /     -6 dB/oct
                                                  Ii     "                  I
              --40                                I
                                                  I                                                         -12 dB/oct

                                                  !                         I                      I
                                                  I                         I                      I
                                                  J,                        I                      i                       log co
                           0.01 0.620.05         0.1     0.2    ()15        1               5     10   20   5'0     100
                                   I              I                 I
                                   I              I                 I
                               -                  a                 I

            --90 ~ -

           .180 ~. . . .

                                                                                                                                         Passive circuits   41

                                                   0                                                           vi

                                               1                                                                            =0.01


                                         V o

                                          c0=10                                                                     vi

                                                                    ---- ..,.. , . . . . . , ,
                                                       ia) = I0                                  ill .......                     = o.1
                                                                                                                        (co =0.1)



Figure 2.7      Resonant LC circuit analysis.

Figure 2.7e shows the circle diagram only: the                                             point on the jco axis, is proportional to the product
pole-zero and Bode plots are easy to visualize                                             of the reciprocals of the distances from the point to
for this case.                                                                             each of the poles. Consider the frequency co = 0.1,
    Now consider what happens to a normalized                                              i.e. one-tenth of the resonant frequency. Com-
1 H, 1 F tuned circuit when R is less than 2 ohms,                                         pared with the situation at co--0, the distance 11
say 1.4ohms. From (2.4) we need to find the                                                to the upper pole is slightly less whilst the distance
square root of (1.42 - 4 ) , i.e. v/(-2.04). When                                          12 to the lower is slightly greater; this is shown in
you first started learning algebra you would                                               Figure 2.8a. So will Vo at co = 0.1 be greater or less
simply have been told that the equation didn't                                             than vi - or is perhaps the magnitude of the
factorize. However, j solves the problem:                                                  transfer function still unity, as at c0=0? Imagine
v / ( - 2 . 0 4 ) - v / ( - 1 ) v / ( 2 . 0 4 ) - jl.428 286. So                           that each pole is a thumbtack and that a tight loop
                                                                                           of cotton encloses these and a pencil point at the
         - 1 . 4 • v / ( 1 . 9 6 - 4)
   s-                                   = - 0 . 7 + j0.714 143                             origin. As you move the pencil upwards and
                      2                                                                    round, it will trace out an ellipse around the
and these roots are the poles shown in Figure 2.8a.                                        poles and entirely to the left of the jc0 axis,
Now you will recall that the magnitude of the                                              except where it touches it at the origin. Clearly
transfer function at any frequency co, denoted by a                                        then the sum 11 -%/2 increases as you move up the
42   Analog Electronics


                                                                  + 0.714 143

                                        -0.71                     0
                                             J ~
                                                                  -0.714 143

                                                                                When co = 0, l 1 = l2 = 1

                                                             M (dB)j
                 Decreasing R

                                                                                                            = 1.0
                                                                   +6-                           ,, ~ / ' / = 1.414
                                                                    o.                            ~9 / / = ~ 2.0 ~              log   o

                                                                  R -,/2A 1 (dB-at 3 = 1) 1 l.zz                   ~~"~,
                                                                                                                                ~ 1 2 dB/

                                                                   0~ ~      R                      =0                          octave


                                                                 Constant                  "'t ii
                                - J-c           m                current                    -             -                o
                        L         T             m

                                                                 AC signal             |
                                                                                       Z                 ~
                                                                 source                T     c                L            Vo

                                                                                                          v   ..           O

Figure 2.8
  (a - b) Analysis of resonant LC circuit with complex poles.
  (c) Shunt current fed parallel tuned circuit.
                                                                                                             Passive circuits     43

jco axis; so you might be tempted to think that the                    F(jco) =
magnitude of Vo which is proportional to                                            (jco)2(1/co~) + jco(1/LcoZ)R + 1
 (1/ll)(1/12) will decrease. But not so: for in                                                       1
percentage terms, ll initially decreases faster than
                                                                                    (jco/co0) 2 + j(co/coo)(R/Lcoo)+ 1
/2 increases, so the magnitude of the transfer
function IF(jco)l = IVo/Vil actually increases. At                 So now, using con to denote the ratio of the actual
 co= 0.1,(1/ll)(1/12) = 1.000 15, i.e. there is a                  co to the resonant frequency coo,
 'peak' of minuscule dimensions at frequencies                                                        1
well      below    the    resonant    frequency.     At                 F(jcon) - (jcon) 2 -+- j c o n ( R / c o o L ) -+- 1    (2.6)
 co= 0.21F(jco)] is already back to unity.
   As R decreases, the positions of the poles                      It was noted earlier that cooL~R, the ratio of the
 migrate away from each other around a semicircle                  reactance of the inductor to the circuit resistance,
 of radius equal to unity centred on the origin. For               is the voltage magnification factor Q, so (2.6) can
 normalized values of R much smaller than unity,                   be written as
 the poles approach ever closer to the jco axis, as in                                                  1
 Figure 2.8b. As the Bode plot shows, the output                               F(jcon)                                          (2.7)
                                                                                              _co2 + jcon(1/Q)+ 1
 voltage at coo increases indefinitely as R falls, but
 IF(jeo)l can never exceed x2 (or + 6 d B ) at 0. 707co0           From this it is very clear that at resonance
 and 1.22co0 since v/-~.5 and v/1.5 are the values of co           (COn = l ) , Vo/Vi --- - j Q = Q / - 90 ~
 which make                                                           Q has another significance that is less well
                                                                   known. Consider a sinusoidal current of I amperes
                                                                   RMS (i.e. Ix/~ amperes peak) and frequency f Hz
               (jco)2LC + jcoCR + 1                                flowing in an inductor of inductance L henrys
                                                                   and resistance R ohms. The energy stored in the
when L = C = 1, as R tends to zero. Likewise as                    inductor at the peak of the current is
you shave past the pole travelling north up the jco                (1/2)L(Iv/2) 2 = LI 2 joules. The power dissipation
axis, the phase flips from zero to - 1 8 0 ~ more                  in the resistor is I2R watts or I2R joules per
rapidly for smaller values of R (poles nearer the                  second, i.e. I2R/2rcf joules per radian. So the
axis).                                                             ratio of peak energy stored to energy dissipated
  You can see how easily numerical answers drop                    per radian is LIZ/(IZR/2rcf)= mL/R. The term
out of the expression                                              coL/R is called the quality factor Q of the inductor
                                                                   at the frequency co, and is simply the ratio of
                               1                                   reactance to resistance. It looks as though the Q
            F(jco) - (jco)ZLC + jcoCR + 1                (2.5)     of an inductor is simply proportional to frequency,
                                                                   but, as noted earlier, the effective resistance of an
for the particular case of a tuned circuit where                   inductor tends to rise with frequency, so the Q
L = 1 H and C = 1 F, which is of course resonant                   increases with frequency rather less rapidly. In the
at coo = 1 / v / ( L C ) = 1 rad/s. You won't often be             case of a tuned circuit, of particular interest is the
using a tuned circuit with those particular values,                value of Q at c00, so assume for the moment that R
but one can simplify the maths for any old tuned                   is a constant which includes the coil's loss resis-
circuit, using the t e c h n i q u e - mentioned e a r l i e r -   tance, the series loss (if any) of the capacitor, and
of normalization. The trick is to normalize to coo,                even the source resistance of vi, in case it's not
i.e. to express any radian frequency under con-                    quite an ideal voltage source. In the case of an
sideration not as its actual value, but by its ratio to            inductor on its own, Q was defined in terms of the
the resonant frequency coo. Since coo = 1/x/(LC),                  energy dissipated per radian and the peak energy
just replace LC in the denominator of (2.5) with                   stored, but in the case of a loss-free tuned circuit
 1/co~" likewise C - 1/Lco'~). With these substitu-                the energy stored at coo is constant. It simply
tions, (2.5) becomes                                               circulates back and forth from LI2/2 in the
44    Analog Electronics

inductor to CV2/2 in the capacitor a quarter of a                Design. H. W. Bode, D. Van Nostrand
cycle later, then L(-I)2/2 in the inductor, and so                Company Inc., New York 1945.
on. With losses, energy is dissipated and so the               2. Transfer Functions, Cathode Ray, p. 177.
amplitude decreases, u n l e s s - in the steady s t a t e -      Wireless World, April 1962.
it is made up by energy drawn from a source.                   3. State Variables for Engineers (Chapter 3), Da
    You may see equation (2.6) expressed in yet                   Russo, Roy, Close; John Wiley and Sons Inc.
another form. The reciprocal of Q is sometimes                    1965.
denoted by D, the energy dissipated per radian
divided          by        the  stored    energy.   Then
F ( j c 0 n ) - 1/{ (jc0 n + jc%D + 1} or, more generally,
                      "  2                                     Questions
normalized F(s) = 1/{s 2 + Ds + 1}. This general
                                                                1. Define a linear component. Give examples.
form applies equally to the voltage across the
                                                                2. Define impedance. How does it differ from
capacitor of a series voltage fed tuned circuit like
Figure 2.7a, to a second-order low-pass LC filter
                                                                3. What is the complex conjugate of x + jy?
section, and to second-order active low-pass filters
                                                                   Express the cartesian value x + jy in polar
(which are covered in a later chapter). In the high-
                                                                   (M/G) form.
Q case it also applies to the shunt current fed
                                                                4. Prove that the Q of an inductor is equal to the
parallel tuned circuit of Figure 2.8 (c), and it
                                                                   ratio of the energy stored, to the energy
would also apply in the low-Q case if the series
                                                                   dissipated per radian.
loss resistance was equally divided between the
inductor and capacitor branches or was due                      5. An audio equipment includes a switch-select-
solely to the finite resistance of an imperfect                    able scratch filter (top-cut circuit), fed from a
current source. In the case shown where the loss                   voltage source. The filter consists of a 10 K
is entirely associated with the inductor, the fre-                 series resistor feeding a 3300pF shunt capa-
quency at which the circuit appears purely resistive               citor, and connected to a load resistance of
is not quite the same as the frequency of maximum                  1 Mf~. What is the frequency where the at-
impedance.                                                         tenuation has risen to 3 dB? What is the rate of
    A detailed study could be made of many more                    cut-off, in dB per octave, at much higher
circuit arrangements and their representations as                  frequencies?
pole-zero diagrams, Bode plots and so on.                       6. If, in question 5, the voltage source were
However, to save you wading through acres of                       replaced by an ideal 1 mA current source,
text, the results are summarized in Appendix 4.                    what would be the voltage across the capacitor
Here I have shown for each circuit the circle                      at (a) 5 kHz, (b) 5 Hz?
diagram (vector diagram as a function of fre-                   7. As a result of careful retuning, the power
quency), the pole-zero diagrams for F(s) in plan                   output of an RF amplifier was increased
and isometric view, and the Bode plot, all together                from 70mW to 105mW. What was the in-
for comparison. I have not seen this done in any                   crease, expressed in dB?
other book, but I think you will find it helpful.               8. A 1 V AC. generator with zero output impe-
You will in any case be meeting more pole-zero                     dance is connected through a series 159mH
plots, circle diagrams etc. in later chapters. But in              inductor (with a Q of 100) to a lossless
the meantime, let's look at some active devices-                   0.159 ~tF shunt capacitor. What is the max-
transistors, opamps and the like, including those                  imum voltage which can occur across the
buffer amplifiers which have already been men-                     capacitor? At what frequency is this maximum
tioned.                                                            observed?
                                                                9. In the previous question, what would have
                                                                   been the maximum voltage across the capaci-
                                                                   tor if the dissipation constant D of both the
1. Network Analysis and Feedback Amplifier                         inductor and the capacitor had been 0.005?
                                                                               Passive circuits   45

    How much does it differ from the previous        resistor and the capacitor, and the input of a
    value, and why?                                  third such circuit connected to the resistor/
10. The output of a non-inverting opamp with a       capacitor of the second. What is the phase-
    gain of unity is connected through a 100 K       shift between the input voltage at the first
    resistor to a 4.7 nF capacitor whose other end   opamp and the voltage across the third capa-
    is grounded. The input of a second, identical,   citor, at 340 Hz? At what frequency will these
    circuit is connected to the junction of the      two voltages be in antiphase?

    3                    Active components

If passive components are the cogs and pinions of        Turning then to semiconductors, the simplest of
a circuit, an active component is the mainspring.        these is the diode.
The analogy is not quite exact perhaps, for the
mainspring stores and releases the energy to drive
                                                         The semiconductor diode
the clockwork, whereas an active component
drives a circuit by controlling the release of           This, like its predecessor the thermionic diode,
energy from a battery or power supply in a par-          conducts current in one direction only. It is
ticular manner. In this sense, active devices have       arguable that diodes in general are not really
existed since the days of the first practical applica-   active devices at all, but simply non-linear passive
tions of electricity for communications. For             devices. However, they are usually considered
although the receiving apparatus of the earliest         along with other active devices such as transistors
electric telegraphs may have been passive in the         and triacs, and the same plan is followed here.
sense that the indicators were operated solely by           The earliest semiconductor diode was a point
the received signal, the sending key at the originat-    contact device and was already in use before the
ing end represented an active device of sorts. The       First World War, being quite possibly contempor-
sensitivity of the receiving end was soon improved       ary with the earliest thermionicdiodes. It consisted
by using a very sensitive relay to control the flow      of a sharp pointed piece of springy wire (the 'cat's
of power from a local receiving end b a t t e r y -      whisker') pressed against a lump of mineral (the
probably the earliest form of amplifier, and a           'crystal'), usually g a l e n a - an ore containing
clear example of an active device.                       sulphide of lead. The crystal detector was widely
   Relays are still widely used and the more             employed as the detector in the crystal sets which
sensitive varieties can provide an enormous gain,        were popular in the early days of broadcasting.
defined as the ratio of the power in the circuit         Given a long aerial and a good earth, the crystal
controlled by the contacts to the coil power             set produced an adequate output for use with
required to operate the r e l a y - typically 30 to      sensitive headphones, whilst with so few stations
60dB. Nowadays we are more likely to think of            on the air in those days the limited selectivity of
discrete bipolar transistors and MOSFETs or              the crystal set was not too serious a problem. The
linear integrated circuits like opamps as typical        crystal and cat's whisker variety of point contact
active devices, but generations of electronic en-        diode was a very hit and miss affair, with the
gineers were brought up on thermionic valves.            listener probing the surface of the crystal to find
    Everybody still uses valves every day, for the       a good spot. Later, new techniques and materials
cathode ray tube (CRT) in a television set is just a     were developed, enabling robust preadjusted point
special type of valve, and the picture it displays       contact diodes useful at microwave frequencies to
was in all probability broadcast by a valved             be produced. These were used in radar sets such as
transmitter. However, this book does not cover           AI Mk.10, an airborne interceptor radar which
valve circuits and applications since, CRTs apart,       was in service during (and long after!) the Second
they are nowadays restricted to special purpose          World War. Germanium point contact diodes are
high-power uses such as broadcast transmitters,          still manufactured and are useful where a diode
industrial RF heating, medical diathermy etc.            with a low forward voltage drop at modest current
                                                                                                          Active components      47

                                            Small-signal . . . . .OlOOe
                                            alOaeS || l~OWer/
                                         ImA u m ~
                                    20     Germani

                                    15-1         ""x/ l/                                    Anode

                                                                                           I .I-++++++++++++
                                                 0.2   0.4   0.6      0.8              N

               /I   Small.signal-,v-~
           dH~e vOltage            -20 1
                                               ~tA                                                (b)

Figure 3.1 Semiconductor diodes.
  (a) I/Vcharacteristics.
  (b) Diagrammatic representation of PN diode, showing majority carriers and depletion region.

(a milliampere or so) combined with very low                                five valence electrons in its outer shell, unlike
reverse capacitance is required. However, for the                           quadravalent silicon which has four outer elec-
last twenty years silicon has predominated as the                           trons), then there are electrons 'going begging',
preferred semiconductor material for both diode                             with no corresponding electron in a neighbouring
and transistor manufacture, whilst point contact                            atom with which to form a bond pair. These spare
construction gave way to junction technology even                           electrons can move around in the semiconductor
earlier.                                                                    lattice, rather like the electrons in a metallic con-
   Figure 3.1 a shows the I/V characteristics of prac-                      ductor. The higher the doping level, the more free
tical diodes. Silicon diodes are manufactured hun-                          electrons and the higher the conductivity of the
dreds or thousands at a time, commencing with a                             silicon, which is described as N type. This simply
thin wafer of single crystal silicon several inches in                      indicates that the current flow is by means of nega-
diameter, which is later scribed and separated to                           tive charge carriers, i.e. electrons. P type silicon is
obtain the individual diodes. Silicon is one of those                       obtained by doping the monocrystalline lattice with
substances which crystallize in a cubic lattice struc-                      a sprinkling of trivalent atoms, such as boron.
ture; another is sodium c h l o r i d e - c o m m o n s a l t - but         Where one of these is substituted in the lattice next
that is a compound, not an element like silicon.                            to a silicon atom, the latter has one of its electrons
Silicon, in the form of silicon dioxide, is one of the                      'unpaired', a state of affairs described as a hole. If
most abundant elements in the carth's crust, occur-                         this is filled by an electron from a silicon atom to the
ring as quartz, in sandstone etc. When reduced to                           right, then whilst that electron has moved to the left,
elementary silicon, purified and grown from the                             the hole has effectively moved to the right. It turns
melt as a single crystal, it is called intrinsic silicon                    out that the spare electrons in N type silicon are
and is a poor conductor of electricity, at least at                         more mobile than the holes in P type, which explains
room temperature. However, if a few of the silicon                          why very high-frequency transistors are more easy
atoms in the atomic lattice are replaced with atoms                         to produce in N P N than PNP t y p e s - but that is
of the pentavalent element phosphorus (which has                            jumping ahead.
48   Analog Electronics

    To return to the silicon junction diode: the         solely for reverse biased use. A special doping
construction of this is as in Figure 3.1b, which         profile giving an abrupt or 'hyperabrupt' junction
indicates the lack of carriers (called a depletion       is used. This results in a diode whose reverse
region) in the immediate neighbourhood of the            capacitance varies widely according to the magni-
junction. Here, the electrons from the N region          tude of the reverse bias. The capacitance is speci-
have been attracted across to the P region to fill       fied at two voltages, e.g. 1 V and 15 V, and may
holes. This disturbance of the charge pattern that       provide a capacitance ratio of 2 : 1 or 3 : 1 for types
one would expect to find throughout N type and P         intended for use at U H F , up to 30:1 for types
type material represents a potential barrier which       designed for tuning in AM radio sets. In these
prevents further migration of carriers across the        applications the peak-to-peak amplitude of the R F
junction. When the diode is reverse biased, the          voltage applied to the diode is small compared
depletion region simply becomes more extensive.          with the reverse bias voltage, even at minimum
    The associated redistribution of charge repre-       bias (where the capacitance is maximum). So the
 sents a transient charging current, so that a           varactor behaves like a normal mechanical air-
reversed biased diode is inherently capacitive. If       spaced tuning capacitor except that it is adjusted
a forward bias voltage large enough to overcome          by a DC voltage rather than a rotary shaft. Tuning
the potential barrier is applied to the junction,        varactors are designed to have a low series loss rs
about 0.6 V in silicon, then current will flow; in the   so that they exhibit a high quality factor Q, defined
case of a large-area power diode, even a current of      as Xc/rs , over the range of frequencies for which
 several amperes will only result in a small further     they are designed.
increase in the voltage drop across the diode, as           Another use for varactors is as frequency multi-
indicated in Figure 3.1a. The incremental or slope       pliers. If an R F voltage with a peak-to-peak
 resistance rd of a forward biased diode at room         amplitude of several volts is applied to a reverse
 temperature is given approximately by 25/Ia ohms.       biased diode, its capacitance will vary in sympathy
 where the current through the diode Ia is in            with the R F voltage. Thus the device is behaving
 milliamperes. Hence the incremental resistance at       as a non-linear (voltage dependent) capacitor, and
 10 ~tA is 2K5, at 100 ~tA is 250R, and so on, but it    as a result the RF current will contain harmonic
 bottoms out at a few ohms at high currents, where       components which can be extracted by suitable
 the bulk resistance of the semiconductor material       filtering.
 and the resistance of the leads and bonding pads            The P type/intrinsic/N type diode or PIN diode
 etc. come to predominate. This figure would apply       is a PN junction diode, but fabricated so as to have
 to a small-signal diode: the minimum slope resis-       a third region of intrinsic (undoped) silicon be-
 tance of a high-current rectifier diode would be in     tween the P and N regions. When forward biased
 the milliohm region.                                    by a direct current it can pass radio-frequency
    Power diodes are used in power supply rectifier      signals without distortion, down to some mini-
 circuits and similar applications, whilst small         mum frequency set by the lifetime of the current
 signal diodes are widely used as detectors in           c a r r i e r s - holes or e l e c t r o n s - in the intrinsic
 radio-frequency circuits and for general purpose        region. As the forward current is reduced, the
 signal processing, as will appear in later chapters.    resistance to the flow of the R F signal increases,
 Also worth mentioning are special purpose small-        but it does not vary over an individual cycle of the
 signal diodes such as the tunnel diode,/ backward        RF current. As the direct current is reduced to
 diode, varactor diode, PIN diode, snap-off diode,       zero, the resistance rises towards infinity; when the
 Zener diode and Schottky diode; the last of these is    diode is reverse biased, only a very small amount
 also used as a rectifier in power circuits. However,     of RF current can flow, via the diode's reverse
 the tunnel diode and its degenerate cousin the          biased capacitance. The construction ensures that
 backward diode are only used in a few very              this is very small, so the PIN diode can be used as
 specialized applications nowadays.                      an electronically controlled RF switch or relay, on
    The varactor diode or varicap is a diode designed    when forward biased and off when reverse biased.
                                                                                        Active components     49

It can also be used as a variable resistor or               voltage. Higher-voltage diodes, rated at 6 V and up,
attenuator by adjusting the amount of forward               operate by a different mechanism called avalanche
current. An ordinary PN diode can also be used as           breakdown, which exhibits a small positive tempera-
an R F switch, but it is necessary to ensure that the       ture coefficient. In diodes rated at about 5 V both
peak R F current when 'on' is much smaller than             mechanisms occur, resulting in a very low or zero
the direct current, otherwise waveform distortion           temperature coefficient of voltage. However, the
will occur. It is the long lifetime of carriers in the      lowest slope resistance occurs in diodes of about
intrinsic region (long compared with a single cycle         7 V breakdown voltage.
of the RF) which enables the PIN diode to operate
as an adjustable linear resistor, even when the
                                                            Bipolar transistors
peaks of the RF current exceed the direct current.
   When a PN junction diode which has been                   Unlike semiconductor diodes, transistors did not
carrying current in the forward direction is sud-            see active service in the Second World War; they
denly reverse biased, the current does not cease            were born several years too late. In 1948 it was
instantaneously. The charge has first to redistri-          discovered that if a point contact diode detector
bute itself to re-establish the depletion layer. Thus       were equipped with two cat's whiskers rather than
for a very brief period, the reverse current flow is        the usual one, spaced very close together, the
much greater than the small steady-state reverse            current through one of them could be influenced
leakage current. The more rapidly the diode is              by a current through the other. The crystal used
reverse biased, the larger the transient current            was germanium, one of the rare earths, and the
and the more rapidly the charge is extracted.               device had to be prepared by discharging a capa-
Snap-off diodes are designed so that the end of             citor through each of the cat's whiskers in turn to
the reverse current recovery pulse is very abrupt,          'form' a junction. Over the following years, the
rather than the tailing off observed in ordinary PN         theory of conduction via junctions was elaborated
junction diodes. It is thus possible to produce a           as the physical processes were unravelled, and the
very short sharp current pulse by rapidly reverse           more reproducible junction transistor replaced
biasing a snap-off diode. This can be used for a            point contact transistors.
number of applications, such as high-order har-                However, the point contact transistor survives
monic generation (turning a V H F or U H F drive            to this day in the form of the standard graphical
current into a microwave signal) or operating the           symbol denoting a bipolar junction transistor
sampling gate in a digital sampling oscilloscope.           (Figure 3.2a). This has three separate regions, as
   Small-signal Schottky diodes operate by a funda-         in Figure 3.2b, which shows (purely diagrammati-
mentally different form of forward conduction. As a         cally and not to scale) an N P N junction transistor.
result of this, there is virtually no stored charge to be   With the base (another term dating from point
recovered when they are reverse biased, enabling            contact days) short-circuited to the emitter, no
them to operate efficiently as detectors or rectifiers      collector current can flow since the collector/base
at very high frequencies. Zener diodes conduct in the       junction is a reverse biased diode, complete with
forward direction like any other silicon diode, but         depletion layer as shown. The higher the reverse
they also conduct in the reverse direction, and this is     bias voltage, the wider the depletion layer, which is
how they are normally used. At low reverse vol-             found mainly on the collector side of the junction
tages, a Zener diode conducts only a very small             since the collector is more lightly doped than the
leakage current like any other diode. But when the          base. In fact, the pentavalent atoms which make
voltage reaches the nominal Zener voltage, the              the collector N type are found also in the base
diode current increases rapidly, exhibiting a low           region. The base is a layer which has been con-
incremental resistance. Diodes with a low break-            verted to P type by substituting so many trivalent
down v o l t a g e - up to about 4 V - operate in true      (hole donating) atoms into the silicon lattice, e.g.
Zener breakdown: this conduction mechanism ex-              by diffusion or ion bombardment, as to swamp the
hibits a small negative temperature coefficient of          effect of the pentavalent atoms. So holes are the
50      Analog Electronics

                                                                                                                                Ic Collector
                                                       +10V                                                            ..___._.~.I. +6 V (say)
                                                  _._..~.l Collector

        Base - - ~ Collector                                             N
                       Emitter                                           Collector
                                                                     _   _   o

                  PNP                                                    depletion
     C o l l e c t o ~ Base                       +++++++++++        --'region
                                     Base         +++++++++++
                                                  +++++++++++        P
       Emitter NPN                                +++++++++++
                                                                         m   ~

               (a)                                                   --region

                                                                                                                       l 0v+ t 1~=Ib+~c
                                          _.____•             Emitte~r
                                                         (b)                                                                        (c)

                   150 --                                                                                                                         BDY 20
                            73   =   25~
                                                                                                             I b =0.7 A
                                                                                      I0-                                             =0.6A
                                                                                                                                      =0.5 A
                   100-                                                                                                               =0.4 A
                                                                                                                                      =0.3 A
                                                                                 I c (A)
             I c (mA)                                                                                                               -- =0.2A
                                                                                                                                      = 50 mA
                            ~    ~   1        0     0       gA
                            ~                           50 gA
                                                                                                -        I     I           I
                                              I                  I                          0            1     2           3
                                              2                 4                                                  Vce (v)
                                          v~ (v)
                                              (d)                                                                                                 min. typ. max.

                            Vce=5 V
                            73 = 25~
                                                                                                                       102 -
                            typxcal values                                             ~ BC 108C
                                                                                           BC 109C
                                                                                                                           10-            /// t           I

                                                                                                                    I c (mA)
                                                                                           BC 107B
                                                                                           BC 108B
                                                                                           BC 109B
                                                                                                                                          I           !

                                                                                                                                        I / I Base-emitter
                                                                                                                       10-1--          I ] tt voltage versus
            200-                                                                            BC 107A                                   I [ / collector current
                                                                                            BC 108A                                   ~li             Vce -'-- 5 W

               0                      I                     I                    I                  "1                 10 -2
                                                                                                                                    ~I!I              ~ =25~
                                                                                                                                                  I           I       I
                   10-2              10--1                  1                    10                                             400           600    800             1000
                                                        I c (mA)                                                                                V~ (mY)
                                                                                          Active components      51

majority carriers in the base region, just as elec-          reason, the bipolar junction transistor is often
trons are the majority carriers in the collector and         described as having a 'pentode-like' output char-
emitter regions. The collector 'junction' turns out          acteristic, by an analogy dating from the days of
then to be largely notional; it is simply that plane         valves. This is a fair analogy as far as the collector
for which on one side (the base) holes or P type             characteristic is concerned, but there the similarity
donor atoms predominate, whilst on the other (the            ends. The pentode's anode current is controlled by
collector) electrons or N type donor atoms pre-              the gl (control grid) voltage, but there is, at least
dominate, albeit at a much lower concentration.              for negative values of control grid voltage, negli-
   Figure 3.2c shows what happens when the base/             gible grid current. By contrast, the base/emitter
emitter junction is forward biased. Electrons flow           input circuit of a transistor looks very much like a
from the emitter into the base region and, simul-            diode, and the collector current is more linearly
taneously, holes flow from the base into the                 related to the base current than to the base/emitter
emitter. The latter play no useful part in transistor        voltage (Figure 3.2f and g). For a silicon N P N
action: they contribute to the base current but not          transistor, little current flows in either the base or
to the collector current. Their effect is minimized          collector circuit until the base/emitter voltage Vbe
by making the N type emitter doping a hundred                reaches about + 0 . 6 V , the corresponding figure
times or more heavier than the base doping, so               for a germanium N P N transistor being about
that the vast majority of current flow across the             +0.3V. For both types, the Vbe corresponding
emitter/base junction consists of electrons flowing          to a given collector current falls by about 2 mV for
into the base from the emitter. Some of these                each degree centigrade of temperature rise,
electrons flow out of the base, forming the greater          whether this is due to the ambient temperature
part of the base current. But most of them, being            increasing or due to the collector dissipation
minority carriers (electrons in what should be a P           warming the transistor up. The reduction in Vbe
type region) are swept across the collector junction         may well cause an increase in collector current and
by the electric field gradient existing across the           dissipation, heating the transistor further and
depletion layer. This is illustrated (in diagram-            resulting in a further fall in Vbe. It thus behoves
matic form) in Figure 3.2c, while Figure 3.2d                the circuit designer, especially when dealing with
shows the collector characteristics of a small-              power transistors, to ensure that this process
signal N P N transistor and Figure 3.2e those of             cannot lead to thermal runaway and destruction
an N P N power transistor. It can be seen that,              of the transistor.
except at very low values, the collector voltage has            Although the base/emitter junction behaves like
comparatively little effect upon the collector cur-          a diode, exhibiting an incremental resistance of
rent, for a given constant base current. For this            25/Ie at the emitter, most of the emitter current

Figure 3.2 The bipolar transistor.
   (a) Bipoar transistor symbols.
   (b) NPN junction transistor, cut-off condition. Only majority carriers are shown. The emitter depletion region is
very much narrower than the collector depletion region because of reverse bias and higher doping levels. Only a
very small collector leakage current Icb flows.
   (c) NPN small signal silicon junction transistor, conducting. Only minority carriers are shown. The DC common
emitter current gain is hFE = Iclb, roughly constant and typically around 100. The AC small signal current gain is
hfe -- dlc/dIb = ic/ib.
   (d) Collection current versus collector/emitter voltage, for an NPN small signal transistor (BC 107/8/9).
   (e) Collector current versus collector/emitter voltage, for an NPN power transistor.
   (f) hvz versus collector current for an NPN small signal transistor.
   (g) Collector current versus base/emitter voltage for an NPN small signal transistor.
(Parts (d) to (g) reproduced by courtesy of Philips Components Ltd.)
52   Analog Electronics

      1000      Tj = 25~
          q     f= 1 kHz                                   Vce = i0 V                                                                  i2
      hfe I     typicalvalues                                             ,,,                                                          _    ~r   .[=

                                                                                  Base                . r                         rc I Collector
       3oo-lBc lO9C~f                      ~         ~                                 Vl                                    r         r~ v2
          IBC107B} ~                                           ""   10V ___                                                            ][
       200-~BC 108B ~
          [ B C 109BI ~                v       ~                     5V                                                      _           itter
                                                                                                                                  re~_ Em_

       100    [BC   108Al                                                        Em~ter                          (b)
                                   i                       l                I
             10-2                10-1                      1               10
                                           I c (mA)

                            __     -           +                                   A                  A
                                                                            100 I.tA~                      3k3
                                                                                56k11 lmA "
             AC                    + ic                                               :2n~ing~                         tl0 I.tF-- --
             signal ib + ib - ,-lie + ib + ic
             ib               Y                                                                                                   Output
                                 /z-h7 o v                           AF input 33k~
                                                                         .                    .2k7 .              2 100 I.tF
                                                                                                                  T              ~'0V
                                                                    (c)                     /r/rz

                                                                                                                                           +9 V

                        T                                                                       5k6
                                                         Input from
              Input                                      band II dipole 560R                                      10 nF
                                                                                        10 nF                          10 nF
                                                                                                    3k3                            Output
                                                                                                                 10 nF
                                                                                                   Active components    53

                                                                     ~]         i          0+9 V

          Input                                      10 nF      _    [~ lOOk~ J BCI09C


                                                                      _   /~    _         o0   V

Figure 3.3 Small-signal amplifiers.
   (a) hfe versus collector current for an NPN small-signal transistor of same type as in Figure 3.5f (reproduced by
courtesy of Philips Components Ltd.).
   (b) Common emitter equivalent circuit.
   (c) Common emitter audio amplifier. Ib =base bias or standing current; Ic =collector standing current;
ic = useful signal current in load.
   (d) Common base RF amplifier.
   (e) Common collector high-input-impedance audio amplifier.

appears in the collector circuit, as has been                       linear hFE axis in one and the logarithmic hfe axis
described above.                                                    in the other).
   The ratio Ic/Ib is denoted by the svmbol hFE and                      Once the AC performance of a transistor is
is colloquially called the DC current gain or static                considered, it is essential to allow for the effects
forward current transfer ratio. Thus if a base                      of reactance. Just as there is capacitance between
current of 10 ~tA results in a collector current of                 the various electrodes of a valve, so too there are
3 m A - typically the case for a high-gain general                  unavoidable capacitances associated with the
purpose audio-frequency N P N transistor such as a                  three electrodes of a transistor. The collector/
B C 1 0 9 - then hFE -- 300. As Figure 3.2f shows, the              base capacitance, though usually not the largest
value found for hFE will vary somewhat according                    of these, is particularly important as it provides
to the conditions (collector current and voltage) at                a path for AC signals from the collector circuit
which it is measured. When designing a transistor                   back to the base circuit. In this respect, the
amplifier stage, it is necessary to ensure that any                 transistor is more like a triode than a pentode,
transistor of the type to be used, regardless of its                and as such the Miller effect will reduce the
current gain, its Vbe etc., will work reliably over a               high-frequency gain of a transistor amplifier
wide range of temperatures: the no-signal DC                        stage, and may even cause an RF stage to
conditions must be stable and well defined. The                     oscillate due to feedback of in-phase energy
DC current gain hvE is the appropriate gain                         from the collector to the base circuit.
parameter to use for this purpose. When working                         One sees many different theoretical models for
out the stage gain or AC small-signal amplification                 the bipolar transistor, and almost as many differ-
provided by the stage, hfe is the appropriate                       ent sets of parameters to describe it: z, g, y, hybrid,
parameter, this is the AC current gain dlc/dlb.                     s, etc. Some equivalent circuits are thought to be
Usefully, for many modern small-signal transistors                  particularly appropriate to a particular configura-
there is little difference in the value of hFE and hfe              tion, e.g. grounded base, whilst others try to model
over a considerable range of current, as can be                     the transistor in a way that is independent of how
seen from Figures 3.5f and 3.6a (allowing for the                   it is connected. Over the years numerous workers
54   Analog Electronics

have elaborated such models, each proclaiming the         appearing at the input port as the result of the
advantages of his particular equivalent circuit.          voltage variations at the output port (again this
   Just one particular set of parameters will be          will be a complex number at high frequencies).
mentioned here, because they have been widely             Finally, h22 is the output admittance, m e a s u r e d -
used and because they have given rise to the              like h i : - with the input port open-circuit to
symbol commonly used to denote a transistor's             signals. These parameters are called hybrid be-
current gain. These are the hybrid parameters             cause of the mixture of units: impedance, admit-
which are generally applicable to any two-port            tance and pure ratios.
network, i.e. one with an input circuit and a                  In equation (3.4) the input voltage vl is shown
separate output circuit. Figure 3.7a shows such a         as being the result of the potential drop due to il
two-port, with all the detail of its internal circuitry   flowing through the input impedance plus a term
hidden inside a b o x - the well-known 'black box'        representing the influence of any output voltage
of electronics. The voltages and currents at the two      variation v2 on the input circuit. When considering
ports are as defined in Figure 3.4a, and I have used      only small signals, to which the transistor responds
v and i rather than V and I to indicate small-signal      in a linear manner, it is valid simply to add the two
alternating currents, not the standing DC bias            effects as shown. In fact the hybrid parameters are
conditions. Now vl, il, v2, and i2 are all variables      examples of partial differentials: these describe
and their interrelation can be described in terms of      how a function of two variables reacts when first
four h parameters as follows:                             one variable is changed whilst the other is held
                                                          constant, and then vice versa. Here, v~ is a function
                  Vl = h11il + h12v2              (3.4)   of both il and V 2 - SO hll --OVl/Oil with V2 held
                  /2 = h21 il -+- h22 v2          (3.5)   constant (short-circuited), and h12 = OVl/OV2 with
                                                          the other parameter il held constant at zero (open-
Each of the h parameters is defined in terms of two       circuit). Likewise, i2 is a function of both il and v2;
of the four variables by applying either of the two       the relevant parameters h21 and h22 are defined by
conditions il = 0 or v2 = 0:                              (3.7) and (3.9), and i2 is as defined in equation
                                                          (3.5). Of course the interrelation of Vl, il, v2 and i2
                              v~]                 (3.6)   could be specified in other ways: the above scheme
                    h~l   =   tl    v2=0
                                                          is simply the one used with h parameters.
                          i2                                  The particular utility of h parameters for speci-
                    h21 = --                      (3.7)
                              ll v2=0                     fying transistors arises from the ease of determing
                                                          hll and h21 with the output circuit short-circuited
                              V1                          to signal currents. Having defined h parameters,
                    h~2 = - -                     (3.8)
                              1:2 i1=0                    they can be shown connected as in Figure 3.4b.
                                                          Since a transistor has only three electrodes, the
                    h22 = i2[                     (3.9)
                                                          dashed line has been added to show that one of
                              1~2 Ii1=0
                                                          them must be common to both the input and the
Thus hll is the input impedance with the output           output ports. The common electrode may be the
port short-circuited as far as AC signals are             base or the collector, but particularly important is
concerned. At least at low frequencies, this im-          the case where the input and output circuits have a
pedance will be resistive and its units will be ohms.     common emitter.
Next, h21 is the current transfer ratio, again with             Armed with the model of Figure 3.4b and
the output circuit short-circuited so that no output      knowing the source and load impedance, you can
voltage variations result: being a pure ratio, h21        now proceed to calculate the gain of a transistor
has no units. Like hll it will be a complex quantity      s t a g e - provided you know the relevant values of
at high frequencies, i.e. the output current will not     the four h parameters (see Figure 3.4c). For ex-
be exactly in phase with the input current. Third,        ample, for a common emitter stage you will need hie
h12 is the voltage feedback ratio, i.e. the voltage       (the input impedance hll in the common emitter
                                                                                                                                                         Active components                 55

                                                                                                          il                                                                    i2
                                                               ~ +                                    +                                +                                             +

     Port 1--~ Vl o _ ~                                              l v ~---Port 2
                                                                                                                                                                  h21 il             v2

                                                                                                          .....                  I .


                                                          l                                                                                /
       10 3 -           f= 1 kHz                      1~ BC 108C                          103- f = 1 kHz         I BC 108C
                        /3 = 25oC                        BC 109C                                7~ = 25~       1 ~ BC 109C
     (k~)                typical values                { BC 107B                                typical values   { BC 107B
                                                      2~ BC 108B                                               2 ~ BC 108B
        102-                                             BC 109B                          102 -
                                                                                                     1 .~        ( BC 109B
                                                      _{ BC 107A                         hre
                                                      3~ B C 108A                      (10--4)       3                108A
            I0-                                       ~                    Vce= 10V              10-
                            Vce=5V                                                                                Vce =   1 0 V ~ ~ , j

             1    L
                                                                           -                          1
                            !                     I                            I       i                                   1                             I ....            !

                 10-2     10--1                   1                    10             102                 102             10-1                           1                 10             102
                                            I c (mA)                                                                                   I c (mA)

                                     103 -
                                                              Vce = 5 to 10V
                                                              f=lkHz                                                        BC108C
                                                                =  25oc                                                     BC 109C
                                                                                                                          I BC 107B
                                                               ypical values                                         ~-- 2 Be 108B
                                      102 -
                                                                                                                            BC 109B
                                   hoo                                                                                ~31              Be 107A
                                (I.t~ -1)                                                                                              BC 108A

                                            1                              I                I     '                  I                               I
                                                10-2                 I0-I                   I                        I0                         102
                                                                                      Ic (mA)

Figure 3.4 h parameters.
  (a) Generalized two-port black box. v and i are small-signal alternating qualities. At both ports, the current is
shown as in phase with the voltage (at least at low frequencies), i.e. both ports are considered as resistances (impe-
  (b) Transistor model using hybrid parameters.
  (c) h parameters of a typical small-signal transistor family (see also Figure 3.6a) (reproduced by courtesy of
Philips Components Ltd.).
56   Analog Electronics

configuration), hfe (the common emitter forward         are probably used more often in the examination
current transfer ratio or current gain correspond-      hall than in the laboratory. The notable exception
ing to h21), hre (the common emitter voltage feed-      are the scattering parameters s, which are widely
back ratio corresponding to h12) and hoe (the           used in radio-frequency and microwave circuit
common emitter output admittance corresponding          design (see Appendix 5). Not only are many
to h22). You will generally find that the data sheet    U H F and microwave devices (bipolar transistors,
for the transistor you are using quotes maximum         silicon and gallium arsenide field effect transistors)
and minimum values for hfe at a given collector         specified on the data sheet in s parameters, but s
current and voltage, and may well also include a        parameter test sets are commonplace in RF and
graph showing how the typical or normalized             microwave development laboratories. This means
value of hfe varies with the standing collector         that if it is necessary to use a device at a different
current I~. Sometimes, particularly with power          supply voltage and current from that at which the
transistors, only hFE is quoted: this is simply the     data sheet parameters are specified, they can be
ratio Ic/Ib, often called the DC current gain or        checked at those actual operating conditions.
static forward current transfer ratio. As mentioned        The h parameters for a given transistor config-
earlier, for most transistors this can often be taken   uration, say grounded emitter, can be compared
as a fair guide or approximation for hfe, for           with the elements of an equivalent circuit designed
example compare Figures 3.2f and 3.3a. From             to mimic the operation of the device. In Figure
these it can be seen that over the range 0.1 to         3.3b re is the incremental slope resistance of the
10 mA collector current the typical value of hFE is     base/emitter diode; it was shown earlier that this is
slightly greater than that of hFE, SO the latter can    approximately equal to 25/Ie where Ie is the
be taken as a guide to hfe, with a little in hand for   standing      emitter   current     in milliamperes.
safety. Less commonly you may find hoe quoted on        Resistance rc is the collector slope or incremental
the data sheet, whilst hie and hre are often simply     resistance, which is high. (For a small-signal
not quoted at all. Sometimes a mixture of param-        transistor in a common emitter circuit, say a
eters is quoted; for example, data for the silicon      BC109 at 2 m A collector current, 15 K would be
N P N transistor type 2N930 quote hFE at five           a typical value: see Figure 3.4c.) The base input
different values of collector current, and low-         resistance rb is much higher than 25/Ie, since most
frequency (1 kHz) values for hib, hrb, hfe and          of the emitter current flows into the collector
h o b - all at 5V, 1 mA. The only data given to         circuit, a useful approximation being hfe x 25/Ie.
assist the designer in predicting the device's per-     The ideal voltage generator ~tbc represents the
formance at high frequency are fT and Cobo. The         voltage feedback h12 (hre in this case), whilst the
transition frequency fT is the notional frequency at    constant current generator CXcbrepresents h21 or
which Ihfe[ has fallen to unity, projected at - 6 dB    hfe, the ratio of collector current to base current.
per octave from a measurement at some lower             Comparing Figures 3.3b and 3.4b, you can see
frequency. For example, fT (min.) for a 2N918           that hll = rb + re, hi2 -- [.tbc, h21 -- 0~cb and h22 =
N P N transistor is 600 MHz measured at 100 MHz,        1/(re+rc).
i.e. its common emitter current gain hfe at 100            Not the least confusing aspect of electronics is
MHz is at least 6. Cobo is the common base output       the range of different symbols used to represent
capacitance measured at I~ = 0, at the stated Vcb       this or that parameter, so it will be worth clearing
and test frequency (10 V and 140 kHz in the case of     up some of this right here. The small-signal
the 2N918).                                             common emitter current gain is, as has already
   If you were designing a common base or common        been seen, sometimes called 0tcb but more often hfe;
collector stage, then you would need the corre-         the symbol [3 is also used. The symbol 0tce or just at
sponding set of h parameters, namely hib, hfb, hrb      is used to denote the common base forward
and hob or hie, hfc, hrc and hoc respectively. These    current transfer ratio hfb: the term 'gain' is perhaps
are seldom a v a i l a b l e - in fact, h parameters    less appropriate here, as ic is actually slightly less
together with z, v, g and transmission parameters       than ie, the difference being the base current ib.
                                                                                                            Active components   57

                                                                                        R l typ. 8 kf~
                                                                                        R2 typ. 100 D.
                  b                           b                                                              BDX63
                                                                           r- . . . .                                    I

                                                    PNP                u-!-                  J                           ici
                                                                         !                                                i
                                                                           L .          .     .         .            J
                                e'                         ] C'

                      Complementary               Complementary                                   (b)

Figure 3.5
  (a) Darlington connected discrete transistors.
  (b) Typical monolithic NPN Darlington power transistor (reproduced by courtesy of Philips Components Ltd).

It follows from this that [3= a / ( 1 - ~). The sym-              one is interested in switching the transistor on or
bols a and [3 have largely fallen into disuse,                    off as quickly as possible, it can more usefully be
probably because it is not immediately obvious                    considered as a charge-controlled rather than a
whether they refer to small-signal or DC gain: with               current-controlled device. Here again, although
hfe and hFE - or heo and hFB - you know at once                   sophisticated theoretical models of switching per-
exactly where you stand.                                          formance exist, they often involve parameters
   When h parameters for a given device are                       ( s u c h as rbb' , the extrinsic or ohmic part of the
available, their utility is limited by two factors:               base resistance) for which data sheets frequently
first, usually typical values only are given (except              fail to provide even a typical value. Thus one is
in the case of hfe) and second, they are measured at              usually forced to adopt a more pragmatic ap-
a frequency in the audio range, such as 1 kHz. At                 proach, based upon such data sheet values as are
higher frequencies the performance is limited by                  available, plus the manufacturer's application
two factors: the inherent capacitances associated                 notes if any, backed up by practical in-circuit
with the transistor structure, and the reduction of               measurements.
current gain at high frequencies. The effect of these                  Returning for the moment to small-signal am-
factors is covered in a later chapter, which deals                plifiers, Figure 3.3c, d and e shows the three
with radio-frequency (RF) circuits.                               possible configurations of a single-transistor am-
   In addition to their use as small-signal ampli-                plifier and indicates the salient performance feat-
fiers, transistors are also used as switches. In this             ures of each. Since the majority of applications
mode they are either reverse biased at the base, so               nowadays tend to use N P N devices, this type has
that no collector current flows or conducting                     been illustrated. Most early transistors were PNP
heavily so that the magnitude of the voltage drop                 types; these required a radical readjustment of the
across the collector load approaches that of the                  thought processes of electronic engineers brought
collector supply rail. The transistor is then said to             up on valve circuits, since with PNP transistors the
be bottomed, its Vce being equal to or even less                  'supply rail' was negative with respect to ground.
than Vbe. For this type of large-signal application,              The confusion was greatest in switching (logic)
the small-signal parameters mentioned earlier are                 circuits, where one was used to the anode of a
of little if any use. In fact, if (as is usually the case)        cut-off valve rising to the (positive) HT rail, this
58   Analog Electronics

being usually the logic 1 state. Almost overnight,          Figure 3.6a shows the symbols and Figure 3.6b
engineers had to get used to collectors flying up to     and c the construction and operation of the first
- 6 V when cut off, and vice versa. Then NPN             type introduced, the depletion mode junction F E T
devices became more and more readily available,          or JFET. In this device, in contrast to the bipolar
and eventually came to predominate. Thus the             transistor, conduction is by means of majority
modern circuit engineer has the great advantage          carriers which flow through the channel between
of being able to employ either NPN or PNP                the source (analogous to an emitter or cathode)
devices in a circuit, whichever proves most con-         and the drain (collector or anode). The gate is a
venient- and not infrequently both types are used        region of silicon of opposite polarity to the source
together. The modern valve circuit engineer, by          cum channel cum drain. When the gate is at the
contrast, still has to make do without a thermionic      same potential as the source and drain, its deple-
equivalent of the PNP transistor.                        tion region is shallow and current carriers (elec-
   A constant grumble of the circuit designer was        trons in the case of the N channel FET shown in
for many years that the current gain hFE of power        Figure 3.6c) can flow between the source and the
transistors, especially at high currents, was too        drain. The FET is thus a unipolar device, minority
low. The transistor manufacturers' answer to this        carriers play no part in its action. As the gate is
complaint was the Darlington, which is now avail-        made progressively more negative, the depletion
able in a wide variety of case styles and voltage        layer extends across the channel, depleting it of
(and current) ratings in both NPN and PNP                carriers and eventually pinching off the conducting
versions. The circuit designer had already for           path entirely when Vgs reaches -Vp, the pinch-off
years been using the emitter current of one tran-        voltage. Thus for zero (or only very small) voltages
sistor to supply the base current of another, as in      of either polarity between the drain and the source,
Figure 3.5a. The Darlington compound transistor,         the device can be used as a passive voltage-
now simply called the Darlington, integrates both        controlled resistor. The JFET is, however, more
transistors, two resistors to assist in rapid turn-off   normally employed in the active mode as an
in switching applications, and usually (as in the        amplifier (Figure 3.6d) with a positive supply rail
case of the ubiquitous TIP120 series from Texas          (for an N channel JFET), much like an NPN
Instruments) an antiparallel diode between collec-       transistor stage. Figure 3.6e shows a typical
tor and emitter. Despite the great convenience of a      drain characteristic. Provided that the gate is
power transistor with a value of hFE in excess of        reversed biased (as it normally will be) it draws
1000, the one fly in the ointment is the saturation      no current.
or bottoming voltage. In a small-signal transistor          The positive excursions of gate voltage of an N
(and even some power transistors) this may be as         channel JFET, or the negative excursions in the
low as 200 mV, though usually one or two volts,          case of a P channel JFET, must be limited to less
but in a power Darlington it is often as much as 2       than about 0.5V to avoid turn-on of the gate/
to 4 V. The reason is apparent from Figure 3.5b:         source junction; otherwise the benefit of a high
the Vcesat of the compound transistor cannot be          input impedance is lost.
less than the Vcesat of the first transistor plus the       In the metal-oxide semiconductor field effect
 Vbe of the second.                                      transistor (MOSFET) the gate is isolated from
                                                         the channel by a thin layer of silicon dioxide,
                                                         which is a non-conductor: thus the gate circuit
Field effect transistors
                                                         never conducts regardless of its polarity relative to
An important milestone in the development of             the channel. The channel is a thin layer formed
modern active semiconductor devices was the              between the substrate and the oxide. In the en-
field effect transistor, or FET for short. These         hancement (normally off) MOSFET, a channel of
did not become generally available until the             semiconductor of the same polarity as the source
1960s, although they were described in detail and        and drain is induced in the substrate by the voltage
analysed as early as 1952.                               applied to the gate (Figure 3.7b). In the depletion
                                                                                                                                                 Active components          59

                             Drain                        d

                Gate~                      or g - ~
                             Source        s
                                 N channel
                               d           d
                        g~               or g _ ~

                                  s                        s
                                        P channel                                                                                                         =+9 V Vad
                                           (a)                                                                                              2k2

                                                                                                                  0.1 gtF
                                                                                                                                 -     ~J2N3819 ~"

          Vos                          vo____~s_                                                                       I
                ]            ~
                                                                                                      Input                1M
                                             ,It /
                                                                                                                                 L) II ~10012F
                    i            L--                                                                          T                  1470Ry           1       = 0 V Vss

                                 T                   7ZS$S66
                                                                     Id ( m A )                ~Pinch-off limit

                                                                                        9    "ql

                                                                                I I~ I! ; I Pinch-offregion
                                                                                                              i             [i[i"     ~               /          -

                                                                         ,o/J~//~iKnee- ~v II
                                                                           I /I //t" i
                                                                                                                                                     =                ins
                                                           Vds     15V _1 / I I /              voltage 3 v l                           ~         %,
                                                               Vds= 2                                  4V

                                                         Vgs (V)     [ -5          0]              I
                                                                                               5 plV 10           15       2'0         Vds (V)
                                                                     '      V           "~
                                                                    Vp          ds(p)        ivy, t                             v~,~,,

Figure 3.6 Depletion mode junction field effect transistors.
  (a) Symbols.
  (b) Structure of an N channel JFET.
  (c) Sectional view of an N channel JFET. The P+ upper and lower gate regions should be imagined to be con-
nected in front of the plane of the paper, so that the N channel is surrounded by an annular gate region.
  (d) JFET audio-frequency amplifier.
  (e) Characteristics of N channel JFET: pinch-off voltage I/p = - 6 V.
(Parts (b), (c) and (e) reproduced by courtesy of Philips Components Ltd.)
60     Analog Electronics

           M0 S-type              Circuit symbol          'O ,.I O
                                                                                                                        Vos>Vp                                       [
           Normally-on                                            bO                                       s                       g                                     ;Io
           (depletion type)
                                                                            VGS     oxide layer (Si Oz)            vnet~i                A t)
                                                                                     ~,\\\\\'~                      ~ * x , \ \ \ \ \ \ \ \ \ ~ \ ' ~ ' %."q\\\\\~
           Normally--off                                          >0
           (enhancementtypd                                                                                                                                  inversion layer
                                                                             Vos.                                                     P                        n-channel
                                                                                                                                 s u bstrat 9

                                                                                                                                   --1       -                                    1Z$6S14
                                                                       "Io                                                               b
           Nmmotly-on I)                                          <0
           (depletion type)                                                                                                       (b)

                                                                                            depletion ~               enhancement
           (enhancementtype}                                                                 n -channel ~ Io                             IoI                 ~                   =
                                                                                                                                                                               V6s 4V
                                                                                              O -E
                                                                                             M SF TI                         /
                                                                                                                            4~                                                 3V
            1)Cannot be mode so for
                                          (a)                                               ~176176
                                                                                                               6 O,          vo~                 d                                Vo~
                        -   -             .::}+                                              n-channel
                                                                                                               9                         1o                                    V6s =2V

                       ']                  ~
                                                                                        b             " /~~~ ' nor mafly         on                                            0I V
                                                                                               Vp              0             Vos                 0                                    Vos
                                                                                             n - c h a n n e l ~ l~                      I~ l        ~                         vos=O
                                                                                                   ~ / I .or~o,y on                              i       /       ~             -I V

                                                                                                                                                 0                   ~z sssTz         VoS

                                         (0                                                                                           (d)

Figure 3.7 Metal-oxide semiconductor field effect transistors.
  (a) MOSFET types. Substrate terminal b (bulk) is generally connected to the source, often internally.
  (b) Cross-section through an N channel enhancement (normally off) MOSFET.
  (c) Cross-section through an N channel depletion (normally on) MOSFET.
  (d) Examples of FET characteristics: (i) normally off (enhancement); (ii) normally on (depletion and enhance-
ment); (iii) pure depletion (JFETs only).
(Parts (a) to (d) reproduced by courtesy of Philips Components Ltd.).

(normally on) MOSFET, a gate voltage is effec-                                      much easier to arrange for positive ions to be
tively built in by ions trapped in the gate oxide                                   trapped in the gate oxide than negative ions or
(Figure 3.7c). Figure 3.7a shows symbols for the                                    electrons, P channel depletion MOSFETs are not
four possible types, and Figure 3.7d summarizes                                     generally available. Indeed, for both JFETs and
the characteristics of N channel types. Since it is                                 MOSFETs of all types, N channel devices far
                                                                                                                                   Active components                              61

           Gate 2 ~,                                                                              28
           Gate 1 ~-                              Source and                                      24        i ~[ ' - ' ~ - }       t         vGls'--i.0~              ~   .....     :
                                                  substrate (bulk)                                22

                                   (a)                                                                          r f       "--                           §        v
                                                                                            ~     16
                                                                                            ""    14
                                                                                            z     1:~

                                   9 rut                                                    og:
                                                                                               10 ._h~_. f                                              "       OV
                                                                                                         ....                                               '    t    "     t,.   ....
           VGG                                    30                                        = 6.0
                                                                                                        lr..~.r-        ......                          :,o

                                                                                                        o     ,.o       ,.o    ~.o     ~.o     ,0  ,~           ,~   ,~   ,~         2
                                                                                                                        ,,o~. o,,~,,.,'o-so,.,,c,~ ,,o~,A~E Ivo~T~



                               s            ?g,                ?g2            ?a

                                                                                                                         . upper' MOS-FET

                                                                                                                          . lower' MOS-FET

                                                                     7z.'$.                                         S    ?z~6sr
Figure 3.8 Dual-gateMOSFETS.
  (a) Dual-gate N channel MOSFET symbol. Gate protection diodes, not shown, are fabricated on the chip in
many device types. These limit the gate/source voltage excursion in either polarity, to protect the thin gate oxide
layer from excessive voltages, e.g. static charges.
  (b) Drain characteristics (3N203/MPF203).
  (c) Amplifier with AGC applied to gate 2.50 f~ source and load (3N203/MPF203).
(Parts (b) and (c) reproduced by courtesy of Motorola Inc.)
  (d) Construction and discrete equivalent of a dual-gate N channel MOSFET (reproduced by courtesy of Philips
Components Ltd).

outnumber P channel devices. In consequence, one                                   the input signal may be applied to the gate and the
only chooses a P channel device where it notably                                   local oscillator signal to the substrate. In high-
simplifies the circuitry or where it is required to                                power MOSFETS, whether designed for switching
operate with an N channel device as a comple-                                      applications or as H F / V H F / U H F power ampli-
mentary pair. (These are further described in the                                  tiers, the substrate is always internally connected
chapter covering audio amplifiers.)                                                to the source.
   Note that whilst the source and substrate are                                      In the N channel dual-gate M O S F E T (Figure
internally connected in many MOSFETS, in some                                      3.8) there is a second gate between gate 1 and the
(such as the Motorola 2N351) the substrate con-                                    drain. Gate 2 is typically operated at +4 V with
nection is brought out on a separate lead. In some                                 respect to the source and serves the same purpose
instances it is possible to use the substrate, where                               as the screen grid in a tetrode or pentode.
brought out separately, as another input terminal.                                 Consequently the reverse transfer capacitance
For example, in a frequency changer application,                                   Crss between drain and gate 1 is only about
62   Analog Electronics

            ~   Drain                        Drain               Source   ' 9


Gate:                        Gate
                                                           P+               Drain (anode)

            ~ Source                        l, Source
Figure 3.9 The gain enhanced MOSFET (GEMFET).
  (a) Symbols for GEMFET, COMFET (conductivity modulated FET) and other similar devices.
  (b) Structure and equivalent circuit of the GEMFET (reproduced by courtesy of Motorola Inc.).

0.01 pF, against 1 pF or thereabouts for small-            electrons and reducing the drain region on voltage
signal JFETS, single-gate MOSFETs and most                 drop. However, nothing comes for free in this
bipolar transistors designed for RF applications.          world, and the price paid here is a slower switch-
   N channel power MOSFETs for switching ap-               off than a pure MOSFET; this is a characteristic of
plications are available with drain voltage ratings        devices like bipolar transistors which use minority
up to 500 V or more and are capable of passing             carrier conduction. An interesting result of the
20 A with a drain/source voltage drop of only a            additional drain P layer is that the antiparallel
few volts, corresponding to a drain/source resis-          diode inherent in a normal power M O S F E T - and
tance in the fully on condition of rds on on of just a     in D a r l i n g t o n s - is no longer connected to the
few hundred milliohms. Other devices with lower            drain. Consequently COMFETS, GEMFETs and
drain voltage ratings exhibit r d s o n resistances as     similar devices will actually block reverse drain
low as 0.010ohms, and improved devices are                 voltages, i.e. N channel types will not conduct
constantly being developed and introduced.                 when the drain voltage is negative with respect to
Consequently these figures will already doubtless          source. Indeed, the structure has much in common
be out of date by the time you read this. A very           with an insulated gate silicon controlled rectifier
high drain voltage rating in a power MOSFET                ( S C R ) - and SCRs together with other members of
requires the use of a high-resistivity drain region,       the thyristor family are the next subject in this
so that very low levels of rds on cannot be achieved       necessarily brief review of active devices.
in high-voltage MOSFETS.
   A development which provides a lower drain/
source voltage drop in the fully on condition
utilizes an additional P type layer at the drain           The name thyristor, given because the action is
connection. This is indicated by the arrowhead on          analogous to that of the thyratron tube (a gas-
the symbol for this type of device (Figure 3.9a).          filled triggered discharge valve), applies to a whole
The device is variously known as a conductivity            family of semiconductors having three or more
modulated power MOSFET or COMFET (trade-                   junctions (four layers or more). These devices,
mark of GE/RCA), as a gain enhanced MOSFET                 which can have two, three or four external term-
or G E M F E T (trademark of Motorola), and so on.         inals, are bistable (they are either off or on) in
Like the basic MOSFET these are all varieties of           operation and may be unidirectional or bidirec-
insulated gate field effect transistors (IGFETs).          tional. This extensive family of devices is summar-
The additional heavily doped P type drain region           ized in Figure 3.10, which also indicates the
results in the injection of minority carriers (holes)      maximum voltage and current ratings typically
into the main N type drain region when the device          available in each type of device.
switches on, supplementing the majority carrier                The best-known member of this family is the
                                                                                                                                    Active components            63

                                                                                       SCR           Silicon controlled rectifier
                                                                                                           4 KV - 4 KA
 t/PUT      [ Pr. rom~O~li@i~u~c~oAn.....
                                   t                           (             _                                                            (.....-           --

                                                                                    ASCR         Assymetricatsilicon controled
 ]                         trigger DIAC NPN
                Bidirectional                                                                              rectifier
                                                                                                          2 KV -    1,5 KA                      ,           v
                                                                                                                                    l.)             F
                                                                                             t   '

     DIAC            50 V -     1 0 0 mA

                                                                                       GTO                 Gate turn off
                                                                                                           4 KV - 2 KA

                                                                                    TRIAC                Triode AC switch
                                                                                                          600 V - 40 A
                                                                                                                                          -r-       - r I   _

      SUS          Silicon unilateral switch
                         30V-      1A
                                                   ~       7
                                                           {                                                                        !!               i i

                                                                                                                                          V- i
                                                                                       MOS                  MOS thyristor
                                                       u       i__..--   !   --        Thy                 5 0 0 V - 20 A

                    Silicon bilateral switch
                          30 V - 1 A

      SCS          Silicon controled switch
                        IOOV-      I"A

                                                       o" r
Figure 3.10    Four-layer (three-junction) device family, with typical characteristics.
  The UJT is a single-junction device but is shown here as its characteristic resembles that of the three-junction
PUT. Likewise, the characteristic of the two-junction diac resembels that of the three-junction bidirectional diode
thyristor. (Reproduced by courtesy of Motorola Inc.)

silicon-controlled rectifier, also known as a reverse                             lector current of the PNP section exceeds the
blocking triode thyristor: it is a unidirectional                                 externally supplied trigger current, both transistors
conducting device having three terminals (anode,                                  conduct heavily and continue to do so even if the
cathode and gate). This is a four-layer device                                    external base current is removed. This type of SCR
(PNPN), the operation of which can be repre-                                      will only turn off again when the current through
sented by two complementary transistors so inter-                                 the device falls below some minimum holding
connected that the collector current of each                                      current. This is a current which is so small that
supplies the base current of the other (Figure                                    the loop gain of the two devices no longer exceeds
3.11a). If a current is supplied to the base of the                               unity, so that one or other transistor no longer
N P N section of the SCR from an external source,                                 receives enough base current to keep it switched
it will start to conduct, its collector supplying base                            on. Thus the SCR makes good use of the fact that
current to the PNP section. If the resultant col-                                 as the current through a transistor is reduced, the
64    Analog Electronics




 L    .,         I


                           .   .   .    .      .                                           (b)

           22~ r                                   _

                     Ct~               ~/Oiac          ,

     C~ ......         ~,~
Figure 3.11
  (a) The SCR modelled as a latching complementary DC coupled bipolar transistor pair with positive feedback.
  (b) The diac.
  (c) Typical application. (Reproduced by courtesy of Motorola Inc.)

current gain falls. The gate of an SCR should not                 rapidly, owing to the need to charge the effective
be forward biased whilst the SCR is blocking a                    gate input capacitance quickly.
negative voltage at its anode, since this results in                 The triode AC semiconductor switch or triac is a
increased leakage current, causing higher dissipa-                bilateral-controlled switch and can thus be used to
tion. SCRs are the heavyweights of the active                     control power in AC circuits. It can be triggered
semiconductor world, being capable of controlling                 on by either a positive or negative input current at
tens of kilowatts of power. A more recent devel-                  the gate relative to main terminal 1 (MT1), what-
opment is the gate turn-off (GTO) thyristor, which                ever the polarity of the voltage at MT2. MT1
is capable of being switched off again by injecting               corresponds to the cathode of SCR and MT2 to
at the gate a pulse of current of the opposite                    the anode; clearly the terms anode and cathode are
polarity from the trigger current. The structure                  inappropriate to a device which can conduct in
of the MOS thyristor is similar to a standard MOS                 either direction. As with all members of the
power FET with the addition of a P type layer in                  thyristor family, a substantial pulse of trigger
series with the drain. This results in a four-layer               current is desirable in order rapidly to switch the
NPNP structure which can be triggered on by a                     device on fully. This ensures that the full supply
positive voltage at the gate of the MOSFET                        voltage appears rapidly across the load, minimiz-
structure. The latter is brought out as the gate                  ing the turn-on period during which the device is
terminal, in place of the P base region of the NPN                passing current whilst the voltage across it is still
section. This results in a high-voltage, high-current             large. This is particularly important in high-fre-
device capable of controlling kilowatts of power.                 quency applications where switching losses repre-
Whilst the steady-state gate current is zero, a low-              sent a large proportion of the device dissipation.
impedance drive is required to turn the device on                 Rapid injection of trigger current can be ensured
                                                                                        Active components       65

by using a two-terminal trigger device such as a            with respect to the other input terminal, the output
diode AC switch or diac (Figure 3.11b). This                will move in the inverse or negative direction.
symmetrical device remains non-conducting, until            Ideally, the operational amplifier has no offset
the voltage across it - of either polarity - exceeds        voltage: that is to say that if its input terminals are
some breakover value (typically 20 to 30 V), when           at the same voltage, the output terminal will be at
it switches to a low-resistance state. This dis-            zero volts, as shown in Figure 3.12b. Furthermore,
charges the capacitor Ct via the triac's gate               the output should respond only to differential
terminal, ensuring rapid turn-on.                           inputs, i.e. voltage differences between the two
                                                            input terminals, so as the wiper of the potentiometer
                                                            in Figure 3.12b is moved away from the central
Operational amplifiers
                                                            position towards either supply rail, the output
The final few pages of this chapter are devoted to          voltage should remain unaffected.
operational amplifiers (opamps) and comparators.               A voltage variation common to both inputs, as
Both are linear integrated circuits (ICs) composed          in Figure 3.12b, is called a common mode input. If a
of many components, active and passive, formed               1 V common mode input results in a 1 mV change
in a single die of silicon. Their manufacture uses          in output voltage, the common mode rejection ratio
the same range of processes used in the production          (CMRR) is described as 60dB or 1000: 1; this
of discrete semiconductor devices, namely photo-            would be a very poor performance for a modern
lithography, diffusion, ion implantation by bom-            opamp. An opamp's rejection of common mode
bardment,      passivation and oxide growth,                AC signals is poorer than for DC voltage changes,
metallization and so on. Although an opamp or               and is progressively worse the higher the fre-
a comparator may contain dozens of components,              quency. Ideally, the common mode input r a n g e -
mainly active ones, it is here considered as a single       the range of common mode voltage for which the
active component. At one time this might have               C M R R remains h i g h - would extend from the
been considered a cavalier attitude, but opamps             negative to the positive supply rail voltage. In
are now so cheap - often cheaper than a single              practice, it often only extends to a point 2 or 3
small-signal bipolar transistor or F E T - that such        volts short of each rail voltage, though with many
an approach is entirely justified. Indeed, in the           opamps, including the CA3140 and the ubiquitous
frequency range up to one or two hundred kilo-              LM324, it does extend to and include the negative
hertz they are much simpler and more convenient             supply rail. This feature is becoming more
to employ than discrete devices, which they                 common as many recently introduced opamps
furthermore outperform. The need for them ex-               are designed for single-rail working, i.e. using
isted long before the IC opamp was available.               just a positive supply and 0V ground. A few
They were therefore produced first with discrete            opamps have a CM input range extending right
semiconductors and passive components, and later            up to the positive rail, the LF355 being one ex-
in hybrid f o r m - transistor chips and miniature          ample, though this is only a typical, not a guaran-
passive components mounted on an insulating                 teed, characteristic.
substrate encapsulated in a metal or plastic multi-             The earliest commonly available integrated cir-
lead housing. And even before that there were               cuit opamp, the 709, could exhibit an unfortunate
operational amplifiers built with thermionic                behaviour known as latch-up. If the common mode
valves.                                                     input voltage exceeded the limits of the specified
   Figure 3.12a shows the usual symbol for an               range, the sense of the amplifier's gain could
opamp. It has two input terminals and one output            reverse. This effectively interchanged the ! and
terminal. The non-inverting input terminal (NI or +)        NI input terminals, turning negative feedback
is so called because if it is taken positive with respect   into positive feedback and locking the circuit up
to the other input terminal, the output voltage will        with the output stuck 'high' or 'low' as the case
also move in a positive direction. Conversely, if the       may be. Naturally, the circuit designer arranged
inverting input terminal (I or - ) is taken positive        for the input normally to remain within the
66       Analog Electronics

                                                                                                                                         + 15 V (V+ or Vr )
                 Inputs ~ ~ ~ ~ ~ - - - ~                      Output


                                                              R2           i2


                                                                                                                                         -15 v-W-or Ve~)
                          vi                                                           A
                                                             (c/                                                   e=l                   ~-e

                                                                    I                                                                     vo =A
                                                              VS ='_+IGV
     : loe                                                    RL = 2.0 k         -~        ......

     z 104                                                                                                                         (d)
     Ig lo3

     9 ' 1o2
                          \                                                            "|
                                     Phm'Sh~             ~         ~ \
               1.0   10        100     1.0 k    10 k  100 k        1.0 M        10 M
                                         f, FREQUENCY(Hz}


Figure 3.12 Theopamp.
  (a)    Symbol.
  (b)    Ideal opamp before the application of feedback (see text).
  (c)    The inverting connection.
  (d)    The non-inverting connection.
  (e)    Device TL081 gain and phase versus frequency, open loop (reproduced by courtesy of Motorola Inc.).

common mode range, but an unforeseen voltage                                                        with frequency. Modern opamps, like their ther-
spike capacitively coupled into the opamp's input                                                   mionic forebears, are specified as having a certain
circuit, or some other interference mechanism,                                                      minimum voltage gain at DC, though as with the
could kick the circuit into the latched-up state.                                                   other parameters, the gain falls with increasing
All modern opamps are arranged so that the gain                                                     frequency. Voltage gain is a particularly appro-
does not reverse its sign when the CM range is                                                      priate parameter for opamps with JFET or MOS
exceeded.                                                                                           input stages, as the input current for such types is
   Another important opamp parameter is the                                                         negligible. However, bipolar opamps and their
power supply rejection ratio (PSRR). A typical                                                      discrete semiconductor ancestors draw a small
modern opamp would provide 100dB PSRR at                                                            but finite input c u r r e n t - namely the base current
DC, but as with CMRR, this figure deteriorates                                                      of the transistors forming the input stage. In some
                                                                                                Active components   67

cases this could be more significant than the input          selecting the appropriate ratio of R2/R1, providing
voltage, so the gain of discrete semiconductor               it falls well short of the open loop gain A of the
opamps was often quoted as a transresistance,                opamp, i.e. the gain measured with R 2 - - o c and
i.e. as so many volts change at the output per               R1 = 0 .
microampere change of input current. Modern                     Remember that the opamp manufacturer
bipolar opamps draw so little input bias current             usually quotes a typical value for A and also a
that they too are specified in terms of voltage gain.        minimum value, say one-third of the typical, but
   Just as the input CM range would ideally extend           no maximum value. You can easily see what this
from rail to rail, so ideally should the output              means in terms of gain accuracy. Take the non-
range. Again, in practice, when driving a load               inverting connection of Figure 3.12d, for example.
connected to ground as in Figure 3.12b, most                    The voltage at the junction of R2 and R1 equals
opamps will run out of steam when the output                 voR1/(R1 + R2). Denote the fraction R1/(R1 + R2)
voltage is within one volt or so of the supply rail.         of the output voltage fed back to the input by the
Opamps with complementary MOS (CMOS)                         symbol 13. Also, assume that the output voltage is
output stages can better this, reaching ever closer          numerically equal to A, so that the opamp's input
to the supply rail the higher the resistance of the          voltage e is just unity, as indicated in Figure 3.12d.
load. Other characteristics of the ideal opamp of            Then by inspection,
Figure 3.12b are very high voltage gain (even the
                                                                          (R1)A+I                    ARI+RI+R2
venerable 709 managed 12 000 minimum, 45 000
                                                                  Vi --    R1 + R2               -     R1 + R2
typical), a low output resistance (50ohms open
loop is typical, reduced in practice by negative             and Vo- A. So
feedback), and a gain maintained constant up to as
                                                                                    "Vo     N(R1 -'t-R2)
high a frequency as possible.
    The very high gain of the opamp is the key to its                               vi    AR1 + (R1 + R2)
usefulness. Imagine it connected as in Ficure 3.12c,         and dividing top and bottom by R1 + R2 gives
where for clarity the necessary power supplies and
                                                                      Vo               A                     A
highly advisable local decoupling capacitors have                                                     m

                                                                      Vi        AR1/(R1 + R2) 4- 1        1 + A[3
been omitted. Whatever the output voltage Vo, the
differential input voltage between the two input             This is the gain with negative feedback, i.e. when
terminals will be exceedingly small owing to the             the fraction 13of the output is subtracted from the
amplifier's enormous gain. Therefore the two inputs          input. If A I3 is very much greater than unity then
must be at virtually the same input voltage, namely          the denominator approximately equals AI3, and so
0 V. Since the NI terminal is grounded: the inverting        the gain simply equals 1/13. Suppose you want a
terminal can thus be described as a 'virtual earth'. If      stage gain of • 100 or 40 dB, then you might make
it is further assumed that the amplifier's transresis-       R1 = 100 R and R2 = 9 K9, whence [3=0.01. If
tance is very high, i.e. the input current it draws is       the type of opamp being used had a minimum
negligible, then il q--/2 -- 0, SO il = - i 2 and actually   open loop gain (often called the large-signal open
flows in the direction shown. So Vo = i2R2 = -ilR2           loop voltage gain Avol or the large-signal differ-
whence Vo/Vi = - R z / R 1 . So the gain of the com-         ential voltage amplification Avo and quoted in
plete circuit is defined by the ratio of the two             volts per millivolt in manufacturers' data sheets)
resistors, provided only that the opamp's voltage            of 10 000 (or 10 V/mV), then the worst case gain
gain is exceedingly high and that the input current it       will be
draws is negligible compared with il. Naturally                            Vo               10 000
enough, Figure 3.12c is known as the inverting                                  =                         =   99
                                                                           Vi        1 + (0.01 • 10000)
connection and, bearing in mind that in this case
the inverting terminal is a virtual earth, the input         or 1% low. Clearly the larger the gain required, the
resistance of the circuit is just R1. You can have any       smaller 13must be, and the larger A must then be to
gain you like, from less than unity upwards, by              ensure that A I3 >> 1.
68   Analog Electronics

   In the inverting connection of Figure 3.12c one            unity. Beyond this unity-gain bandwidth, typically
is not feeding back a fraction of the output                  3 MHz in the case of the TLO81, the rate of roll-
voltage, but rather feeding back a current propor-            off of most opamps increases to - 1 2 dB per octave
tional to Vo and balancing this against a current             or beyond, associated with a phase shift of 180 ~ or
proportional to the input vi, so a different ap-              more. At these frequencies the negative feedback
proach is appropriate. Equating the currents                  provided via R1 and R2 will have become positive.
through R1 and R2,                                            However, as the amplifier's gain is by then less
                                                              than unity the circuit will be stable even in the case
                    l~i- |    1   -(-A)
                                                              where [31 = 1 (i.e. 100% feedback: R1 = infinity in
                      R1            R2
                                                              Figure 3.12d).
whence                                                           When building or modifying existing circuits or
                        RI(1 + A )
                vi-                      + 1                  developing new ones, it not infrequently happens
                                                              that they don't work as expected. When trying to
and of course Vo - - A .          Hence, after a little re-   diagnose the problem, there is no real substitute
arrangement,                                                  for a thorough understanding of how they should
                Vo            -A                              work. So it's worth looking at the inverting opamp
                                                              connection in more detail, remembering that only
                vi       1 + (A + 1)/G
                                                              at very low frequencies is Avol a real number with
where G - R2/R1. If (A + 1)/G >> 1 then                       no associated phase shift.
              Vo             -A                                   Figure 3.13b shows the response of a typical
              --=                     ~ -G                    opamp in the inverting mode with the feedback
               Vi      (A + 1)/G
                                                              resistor Rf in Figure 3.13a open-circuit. If Ri is
so G is simply the demanded gain, as derived                  short-circuited (or the input impedance of the
earlier. Interestingly, in this inverting case, where         opamp is very high) then the voltage e at the
one can choose R1/R2 so as to give a gain of less             inverting input is simply vi. Taking this to be
than unity, the condition (A + 1)/G >> 1 may not              unity, then at very low frequencies the output will
be sufficient. In fact, A must be much greater than           be numerically equal to Avol which is typically
G or unity, whichever is the larger.                          200000 in the case of the opamp whose char-
   This analysis of opamp circuits has been in                acteristics are illustrated in Figure 3.12e. Figure
terms of ratios such as A and G and of symbols                3.13b presents the same information as the Bode
for variables such as vi and Vo. Apart from stating           plot of Figure 3.12e, but represented as an output
vi to be the input voltage, it has not been defined           vector whose magnitude and phase relative to the
exactly, so it could apply equally well to a change           unity input vector varies with frequency in the
of input voltage from one steady value to another,            way illustrated. Figures 3.15e and 3.16b both
or to a continuously varying signal like a sine               illustrate the open loop response of the amplifier,
wave. In the latter case, however, the simple anal-           i.e. the response in the absence of feedback. In
ysis above only applies exactly up to the highest             the vector diagram of Figure 3.13b, the unit input
frequency at which A is a real number, having no              has been drawn to the left of the diagram's origin
associated phase shift to provide a j component. In           or ground reference point, so that Vo extends to
fact, for many opamps A begins to fall at quite a             the right at low frequencies, giving us the sort of
low frequency. This is due to a top-cut or low-pass           circle diagram met earlier. Here, though, as
characteristic deliberately built in to the opamp.            already noted, at frequencies beyond the unity-
Figure 3.12e shows characteristics typical of an              gain frequency the phase shift increases to 180 ~,
opamp of this type, known as internally compen-               corresponding to a 12dB per octave roll-off,
sated. The open loop gain of such an opamp starts             resulting in the little pothook on the locus of v0
to roll off at a frequency in the order of 10 Hz and          as co tends to infinity. Figure 3.13c shows the
continues to fall at a rate of 6dB per octave,                closed loop vector diagram for the inverting
associated with a 90 ~ phase lag, until it reaches            unity-gain connection, i.e. where A = 1. At
                                                                                                                                      Active components          69

                                       Rf           i                                                                                      200 000
                                                                                                                                             ~- co=O

                                                                                                       co= 20x 106 i f                          co= 10
               1            e                           ,v    _
                                                                                                         = 10 x 106                             co= 20
                                                                                                                                      co increasing
                                       Let R i = Rf = R
                                                                                                                                            i =       10.2
                                               0                                           co=l                                                   I
                          VRi = 200 001        .'
                                                                             4 VRf= vo
                                                                                  2000011                                 vRl9= 10.0'"        I
                                                                                                                                            ~/[-~11.3 o

     Iv i = 2 0 0 0 0 2
                                        e = 1I
                                                                                                                                                      "~. 5"7o

                                                                                                                  03-2x       10 6          ~e=l

                                                                                                                            VRf   = 1 0 . 0 5 / % = 10

                                                                              03= 60 x 10 6                                                   t .5.7 ~

                                vRi                                            vRf            vo


         co increasing                                        co

         03=0t       vi
                                        ~~--             vi             -1
                                                                                                             //                       Vo

                                                                                                                                     co increasing

                                                                        (f)      ~ ~

Figure  3.13   O p a m p inverting connection.
  (a) Circuit.
  (b) Typical open-loop gain/frequency plot.
  (c-e) Closed loop vector diagrams for 03 = I (0.16 Hz), 03 = 2 x 10 6 (~ 300 kHz), 03 = 60 x 10 6 (~ 10 MHz).
  (f) Closed loop vector diagrams combined, showing frequency locus.
70   Analog Electronics

frequencies well below 10 Hz, Avol has zero phase      the demanded gain +1, then the actual gain is
angle, so the vectors are all in line, in Figure       only 1 in 200000 parts less than unity, since the
3.13c therefore they have been offset vertically for   loop gain = Avol.
clarity. Note that for an Avol of 200000 the gain         Opamps which are internally compensated for
is just short of unity by one part in 100 000, since   use at gains of unity upwards, such as the TL081,
vRi (the voltage across the input resistor Ri) and     make useful general purpose opamps. If they are
vRf both equal 200001, as shown. Figure 3.13d          to be used for audio-frequency operations where
shows the situation at 300 kHz, where the open         any small DC offset is unimportant, then offset
loop gain has fallen to 20 dB or • 10, again taking    adjustment connections are unnecessary. Conse-
the voltage at the opamp input e as unity. The         quently only three pins per opamp are required,
output voltage Vo lags - e by 90 ~ but, as there is    namely NI and I inputs and the output. So a
still 20 dB of loop gain in hand, Vo only lags vi by   standard 14-pin dual in-line (DIL) plastic pack-
an additional 11.3~ over the expected 180 ~. By        age can accommodate four opamps, with the two
loop gain is meant the excess of the opamp's gain      remaining pins providing connections for the
at any given frequency over the demanded gain at       positive and negative supply rails (e.g. TL084,
that frequency. It can be seen from the vector         LM324 and MC3404). However, internally com-
diagram that although the open loop gain of the        pensated opamps are not the best choice for
TL081 at 300 kHz is only • 10, the gain is only        applications requiring a gain substantially in
2% short of unity. This is due to the 90 ~ phase       excess of unity; the higher the required gain, the
shift between e and Vo: without the phase shift,       less optimum they prove. Figure 3.14c shows that
the vector diagram would look like Figure 3.13c        at a gain with feedback (GWF) or demanded gain
and the gain shortfall would be 20%. Figure            of •        or 60dB, the internally compensated
3.13e shows the situation at 10MHz, well               TL081 family will provide a frequency response
beyond the unity-gain frequency. Vo now has            which is flat to only a little over 1 kHz. For this
almost 180 ~ of excess phase shift, bringing it        application the uncompensated TL080 would be a
almost back in phase with e and vi. Figure 3.13f       much more appropriate choice. Indeed, as Figure
summarizes the closed loop cases of Figure 3.13c,      3.14d shows, even the externally compensated
d and e, showing the loci of vi and Vo relative to     ~tA709 opamp, dating from the 1960s, can pro-
the inverting terminal input voltage e of unity.       vide 60dB gain at a small-signal bandwidth of
Note that the behaviour shown beyond the unity-        300 kHz when appropriately compensated. Note
gain frequency is a bit conjectural as it assumes      that the rate of change of output voltage of an
that the opamp's output impedance is zero. In          opamp is limited to some maximum rate called
fact, in the case of the TL08I it is nearer            the slew rate, determined by the IC process used
200ohms, although at low frequencies this is           and the compensation components, internal or
reduced to a very low effective value by the           external. This slew rate limit results in the full-
application of negative feedback. However, at          power bandwidth being substantially less than the
10 MHz and beyond there is no loop gain to             small-signal bandwidth.
speak of, so the locus of Vo will not finish up           There is a half-way house between opamps
actually at the origin as c0 tends to infinity, but    internally compensated for unity gain and those
somewhere else close by. Figure 3.14a shows the        completely uncompensated. This is the decompen-
general non-inverting connection. Figure 3.14b         sated opamp, which is partially internally compen-
shows the corresponding zero-frequency vector          sated and can be used without further external
diagram for a demanded gain of +2. If you              compensation, down to a specified gain. For ex-
compare it with Figure 3.13c you will find that        ample, the Motorola MC34085 quad JFET input
it is identical except for the location of the zero-   opamp is compensated for Avcl (closed loop gain
voltage or ground reference point, and the gain        or gain with feedback) of 2 or more, whilst the
shortfall is likewise one part in 100000 for an        Signetics 5534 bipolar single low-noise opamp is
Avol of 200000. If, however, R2 =zero, making          compensated for gains of 3 or more.
                                                                                                                          Active components         71

                                                      L                             i..!

                                     2                i
                                                      L                                                                                ~'O
                                                                                v   I
                                                      0   %/2



                                                          ,o    80

         ~100                                             (.~
                                                          ,"-   40
                         I                                o                                    3
         =   60                                           9     20--        '
                                                          4(                                   2


                         I                                            lOO       1.0 k              10 k               100 k    1.0 M         10 M
                      3 kHz               Log frequency                                            f, F R E Q U E N C Y (Hz)

                              (c)                                                                            (d)

Figure 3.14 More about opamps.
  (a) Non-inverting connection. Demand gain = 1/13 -- (RI q- R2)/RI.
  (b) For the case where 13= 0.5 (R2 = RI).
  (c) Gain/bandwidth plot for gain = 1000 (60 dB), for internally compensated opamp (see Figure 3.12e).
  (d) G = 60, 40,20 and 0dB for externally compensated opamp MC1709, with appropriate compensation com-
ponents for each gain setting (reproduced by courtesy of Motorola Inc.).

    A long-standing limitation on the IC opamp            input impedance at both the inverting and non-
 manufacturer's art has been the difficulty of pro-       inverting terminals. However, when an opamp is
 ducing high-performance on-chip PNP transistors.         used in the inverting mode, where the non-invert-
 The traditional method was to use lateral PNP            ing terminal is tied to ground, the input impedance
 transistors, but these have a transition frequency       is simply equal to R1 whilst when it is used in the
fT of only about 2MHz, severely limiting the              non-inverting mode, the inverting input terminal is
 obtainable performance of the complete opamp.            internal to the circuit, the input 'seeing' only the
 But some time ago now, National Semiconductor            non-inverting input. The conclusion is that there is
 introduced the vertically integrated PNP (VIP)           really no need in practice for a high input im-
 structure, leading to opamps such as the                 pedance at the device's inverting input.
 LM6161/64/65 family with a 300 V/gs slew rate,              Current feedback opamps capitalize on this
 a 4.5MHz power bandwidth and drawing only                relaxation and use a low-impedance inverting
4.8 mA supply current. The LM6161 is unity-gain           input. Without going into the details, this permits
 stable, whilst the 6165 is stable for gains of • or      the opamp designer effectively to remove a pole
more.                                                     in the frequency response. The result is that
    Another exciting development is the current           whereas the bandwidth of a regular opamp falls
feedback opamp. Standard opamps have a high               as you demand a higher set gain, owing to its
72   Analog Electronics

                                                        .A   -                             I

                                                ,.3k             ,oo
                                                                           ooiisool                                                                  .....
                                                                                                                                                                   vcc   'r:


                                  I 1,              '                  I 1 1 I I                                                                     [       I      " " ' ,~,.5.ov
                                  l s0:-v~                             I I 1 ,1 I                                    5.0
                o                                                                                                                      in            i       I
                > 4.0
                ~ 3.o
                                20 v...,,,. ,,'~,~"
                                                                                                                                     .,~ |! /~.o =~
                                                                                                                                        |Hr    I I
                                                                                                                                                                   _- L "
                o 2.0                                                                                                2.0
                ~ 1.0
                                        ,r          2:o..v                                         /o ,              1.0
                                                                                                                                        I!~      '-i.o''v           i     I        I    1   1

                                      .LfO'                                                                            o !                  .R
                                            9                          l     I !       ~       l
                                                                           Vcc = +15 V
                    100 w                                                  VEE = -15 V
                                                                                                                                                                         V'cc -',I 5 ~
                o   50                                                     TA = +25oc   - - - -                      -50
                                                                                                                                                                         VEE = -15 V
                                  ,     ,
                                                                             ii                                 ~   -10o                                                 TA = +25oc

                            0           0.1       0.2      0.3    0.4             0.5      0.6                                   0.1          0.2            0.3   0.4             05       0.6
                                                 ITLH. RESPONSETIME Ips}                                                                    ITHL. RESPONSETIME (~s)


Figure 3.15
  (a) Typical comparator.
  (b) Performance (response time) for various input overdrives (LMl11/211/311)(reproduced by courtesy of
Motorola Inc.).

limited gain-bandwidth product, that of a current                                                                   Finally, in this section on opamps, mention
feedback opamp does n o t - at least to a first-                                                                must be made of Norton opamps such as the
order approximation. Current feedback opamps                                                                    LM3900 and the MC3301. These are generally
can therefore offer wider bandwidth at high gain,                                                               similar to conventional opamps but, although
faster slew rates and shorter settling times than                                                               they have a differential input, it is ground refer-
standard opamps, even decompensated ones. Nor                                                                   enced rather than floating, as in a conventional
is this premium AC performance achieved at the                                                                  opamp. Consequently they are mainly used in
cost of sacrificing the DC characteristics. Input                                                               single supply rail AC coupled circuits and less
offset voltages and temperature coefficients as low                                                             demanding DC comparator or logic applications.
as 300 ~tV and 5 ~tV/~ (typical) are available in a
device with a 100 MHz bandwidth, making
current feedback opamps an excellent choice for
critical applications such as 12-bit two-pass sub-                                                              To finish this review of active components, a word
ranging flash analog-to-digital converters.                                                                     about comparators. The sole purpose in life of a
   At the time of writing, current feedback opamps                                                              comparator is to signal as quickly and as accu-
are available from five American manufacturers,                                                                 rately as possible whether the voltage at one of its
including Analog Devices. We can expect to see                                                                  inputs is positive or negative with respect to the
more manufacturers offering such devices in the                                                                 other input. This represents one bit of information,
future.                                                                                                         and a comparator's output is designed to be
                                                                                   Active components      73

compatible with transistor-transistor logic (TTL)       Reference
and other logic families. One of the earliest
comparators is the 710, and this is still one of        1. Current-feedback opamps ease high-speed cir-
the fastest general purpose comparators, only              cuit design. P. Harold, EDN, July 1988.
bettered by special purpose emitter coupled logic
(ECL) compatible types. Later comparators such
as the LM111 are suitable for use with + 15 V and
-15 V supplies as used by opamps, rather than the
less convenient + 12 V and - 6 V rails required by       1. Define intrinsic silicon. What type of diode
the 710.                                                    depends upon it for its action?
   Although comparators, like opamps, have a             2. Which has the greater mobility in silicon, holes
differential input, high values of gain, common             or electrons? What is the consequence for RF
mode and power supply rejection, and even in
some cases (see Figure 3.15a) input offset adjust-
                                                         3. What is the incremental or slope resistance Rd
ment facilities, they are very different in both
                                                            of a diode, in terms of the current I m A that it
purpose and design. A comparator's response
time is quoted as the time between a change of              is passing?
polarity at its input (e.g. from - 9 5 mV to +5 mV       4. In normal use, the peak-to-peak RF voltage
or +95 mV to - 5 mV, i.e. a 100mV change with               across a varicap diode is kept small relative to
5 mV overdrive) and the resultant change of logic           the bias voltage. In what application is this not
state at its output (measured at half-swing), as            the case?
illustrated in Figure 3.15b. Given suitable circuitry    5. How does the forward base emitter voltage of
to interface between its rail-to-rail swing and logic       a transistor, at a given collector current, vary
levels, an opamp can be used as a comparator.               with temperature? How is a transistor's fT
However, its response time will generally be much           measured?
longer than that of a purpose designed compara-          6. What is the approximate value of the low-
tor. For the opamp is only optimized for linear             frequency input resistance of a common emit-
operation, wherein the input terminals are at               ter transistor when Ic = 2 mA?
virtually the same voltage and none of the internal
                                                         7. What is rbb' and what is its significance for the
stages is saturated or cut off. Its speed of recovery
                                                            rise and fall times of a switching transistor?
from overdrive at the input is unspecified, whereas
the comparator is designed with precisely this           8. What restrictions are there on the operating
parameter in mind. Furthermore, whilst an                   Vgs of a junction FET? What does FET action
opamp may be specified to survive a differential            depend upon; majority or minority carriers?
input voltage of +30 V, it is not designed to be         9. Why is an opamp such as the TL080 preferred
operated for long periods toggled, i.e. with any            over an internally compensated type, such as
substantial voltage difference between the NI and I         the TL081, for an audio frequency amplifier
terminals. Extended periods of operation in the             with 60 dB gain?
toggled state, especially at high temperatures, can     10. What is the minimum gain which can be
lead to a change in input offset voltage in certain         provided by an opamp operating in the non-
opamps, e.g. BIMOS types.                                   inverting connection?

    4              Audio-frequency signals
                      and reproduction
Audio frequencies are those within the range of         1. The device's current gain is very large, so that
h e a r i n g - which is a definition vague enough to      the collector current virtually equals the emitter
mean different things to different people. To the          current.
line communication engineer, everything has to fit      2. Base/emitter voltage variations are due solely to
within a 4 kHz wide telephony channel, whilst to           the emitter current variations flowing through
the radio telephone engineer, audio frequencies are        the transistor's internal re, where re -25/Ie, i.e.
those between 300Hz and 3kHz. To the hi-fi                 the device's mutual conductance gm - -
enthusiast the definition means 20 Hz to 20 kHz,           (40 • Ie)mA/V.
no less, even though only the young can hear            3. Collector voltage variations have negligible
sound waves of 15 kHz and above.                           effect on the collector current.
   This chapter looks at the amplification of audio-
frequency electrical signals and their transforma-      Then at frequencies where the reactance of the
tion into sound waves. It also looks at the inverse     100 ~tF emitter decoupling capacitor is very low, a
process of turning sound waves into electrical          1 mV input at the base in Figure 3.3c will cause a
signals and of recording and playing them back,         40 ~tA change in emitter (collector) current, from
and so will adopt the hi-fi definition of audio as      approximation 2, since in this case r e - 25 ~ and
20 Hz to 20 kHz.                                        so gm =40mA/V. This will cause a collector
                                                        voltage change of 40 ~tA • 3 K3 = 133 mV, or in
                                                        other words
Audio amplifiers
                                                                              collector load resistance Rc
Figure 3.3c shows a perfectly serviceable, if simple,    voltage gain Av - i n t e r n a l emitter resistance r e
audio-frequency (AF) amplifier. This is a small
signal single-ended class A amplifier, meaning that     Owing to the approximations, Av = 133 is an
it employs a single transistor biased so as to          optimistic upper bound, even assuming the input
conduct all the time, with equal positive and           is provided from a zero-impedance source and the
negative current swings about the mean current.         output feeds into a following stage with infinite
Of the 9.9 mW of power it draws from the supply,        input resistance. Now the input resistance of the
most is wasted, even when it is producing the           circuit shown is approximately hfere, in parallel
maximum output voltage swing of which it is             with the 33 K and 56 K bias resistors, say typically
capable. But this is generally of little consequence    3 K3 if hfe = 150. So if the stage were the middle
since efficiency is only really important in a later    one in a long string of identical stages, then the
power output stage, where one may well be dealing       gain per stage would simply be hf~/2, since the
with many watts rather than a few milliwatts. The       collector current of one stage would be equally
gain of the circuit of Figure 3.6c can be estimated     divided between its collector load resistance and
without even putting pencil to paper, by making         the input of the following stage. This gives a gain
certain approximations:                                 for the stage of 150/2 and represents a more
                                                          Audio-frequency signals and reproduction      75

practical approximation than the figure of 133         current, which will produce a flux in the core and
derived above. The output impedance of the             could even saturate it. Therefore an airgap is
stage is approximately 3300 ohms, assuming the         introduced into the magnetic path, such as to
collector slope resistance is very high. Therefore     increase the reluctance and set the DC component
the maximum power the stage can deliver will be        of flux at just one-half of the maximum usable
obtained when the load resistance connected to the     value. But designing the choke is a little previous.
output, downstream of the 10 gF DC block, is also      First one must decide the value of RE which in
3300 f~ (see maximum power t h e o r e m - Chapter     turn is determined by the available supply voltage
1). With its l mA standing bias current, the           Vs and the output power Po which is required.
maximum current swing that the stage can               Given Vs and Po, then an approximate value for
handle without cutting off completely for a part       RE can be simply calculated. From Figure 4. l a the
of the half-cycle is 2 mA peak-to-peak (p-p). Of       maximum peak-to-peak current swing ipp max
this, 1 mA p-p will flow in the load and 1 mA p-p      cannot exceed 2Ia, for Ia cannot fall below zero
corresponds (assuming a sine wave and no distor-       and, to avoid distortion, the positive and negative
tion) to 1/(2x/~) mA RMS. The power in a 3K3 load      swings must be symmetrical. Likewise the voltage
is then W = I2R = {(1/2x/~) x 1 0 - 3 } 2 3 3 0 0      at the active device's output Vpp max cannot fall
0.4 mW, giving an efficiency at maximum output         below zero and is thus limited to +2Vs on the
of 0.4/9.9 ~ 0. 04 or 4%.                              other half-cycle. Since the peak-to-peak current or
   Even if one discounts the power wasted in the       voltage excursion of a sine wave is 2x/~ times the
bias resistors and the 2K7 emitter resistor, it is     RMS value, the maximum output power available
clear that a class A stage with collector current      cannot exceed P o - (Vpp/2X/~)(ipp/2X/~), given a
supplied via a resistor would result in a very         load resistance RL=Vpp/ipp---Vs/Ia; whence
inefficient power amplifier stage. The traditional                              "2
                                                       Pomax -- Vpp/(8 X RL) --IppRL/8.      This    power
solution to the problem of efficiency in single-       would actually be obtainable with an ideal device,
ended class A output stages is transformer             as can be seen in Figure 4.1b, where a load line
coupling: in certain circumstances a choke feed        representing RE has been superimposed on an
could be used instead (Figure 4.1a). Here, the         idealized I/V plot. Note that the values of the
circle represents an idealized power amplifier de-     parameter vi shown are arbitrary incremental
vice, be it bipolar power transistor or power FET.     values relative to the quiescent no-signal con-
When the alternating input signal Vi is zero, a        dition. Note also that on the negative-going swings
constant standing current Ia flOWS through the         of vi, when the current i is less than Ia, the stored
device, defined by suitable bias arrangements          energy in the choke causes the output voltage to
(not shown). If the DC winding resistance of the       rise above Vs, forcing a current IL = ia-Ia
choke Lc is much lower than the load resistance        through RE.
RE, the DC current will all flow through the choke,       In practice, the power device will have a finite
as shown. However, any AC signal current ia            bottoming voltage and its output characteristic
through the device will flow through RE in pre-        will only approximate the ideal shown, as in
ference to Lc, provided that the reactance c0Lc of     Figure 4.1c. This will have two effects: first, the
the latter is much greater than RE, where ia repre-    maximum available output power will be less than
sents the instantaneous value of the signal current,   V2p/8RL; and second, the uneven spacing of the Vi
not its RMS value. This can be arranged to apply       lines will introduce distortion. For a given Vs a
right down to the lower audio frequencies, less        value of Ia and RL would be chosen so as to
than 100Hz say, by designing Lc to have a              maximize both the peak-to-peak voltage and cur-
reactance XL much greater than 2rcl00 x RE - no        rent swings: this means choosing a load line to
simple task, as it happens. To obtain the required     extend into the knee of the highest practicable ia
inductance coupled with a winding resistance           curve as shown. The maximum and minimum
much less than RL a ferromagnetic core will be         values of current Iamax and Iamin and of voltage
required. However, the choke is carrying a DC           Vmax and Vmin (as indicated in Figure 4.1c) then
76   AnalogElectronics

                                                                                                                                              -v i = +3
                       RL [              Lc
                                                                                                                                             ~- +2
                   ia+la                                                                                 ,,
                  vi g (
                             -- ~ .           0V                                                                                              -~1

                                   (a)                                                                        i
                                                                                                                               /--,.         ~.._-3
                                                                                                              Vs         /             2Vs    Va
                                                                                                                    aia             1
                                                                                                  Load line slope - Ava -           RL



     lamax     . . . . . .

        21a              fl
                                                                                                              ~~]            Loudspeaker
      Ia min
                                                                                                         - f/~7              -0V
                               t                I                     y        Va

                              Vmin              Vs          Vmax                                                             (d)

                                                                  ~                    RL


                                                              .   .       .        .         g

                                                     ia, t . . + - - - - ~ ,           _d        ia~

                                                                          .7.          (e)
                                                                         Audio-frequency signals and reproduction             77

                                                               * Matched pair
                 stages        ~                   N3k9                                                                   1
                                         ]LT44 U                                                  Loudspeaker

                From           111
                                                           (          22o:       i   -
                                                                             LT700                9V              I
                 ~1       OlaF I~''~ I 1"-"-- tF    681q
                                                                                                  (PP3, O06P etc.)I

                                                                                     [    L ........                 1+
                                         - [-                  -4 o                  -r
Figure 4.1   Audio amplifiers.
  (a) Choke fed single-ended class A stage.
  (b) Ideal output characteristics.
  (c) Practical output characteristics.
  (d) Transformer coupled class A output stage.
  (e) Transformer coupled push-pull class A amplifier.
  (f) Practical form of (e) as used in early transistor radios (operates in class B for battery economy).A temperature
dependent resistor RT may be included to compensate for variation of the output transistors' Vbe with
temperature. This avoids crossover distortion at low temperatures, and excessive battery drain when hot. The
modest amount of negative feedback improves fidelity from shocking to poor.

define the load line and hence RE and also the                      the efficiency of a single-ended class A output stage
maximum power output. A little more power                           is readily derived from Figure In the no-
would be obtainable by increasing the peak-to-                      signal condition, dissipation in the power device is
peak input vi , but only at the expense of rapidly                  Vsla watts; the output power is zero, and so is the
worsening distortion. Conversely distortion at full                 efficiency. At full output power, Vpp = 2 Vs, so the
rated output could be reduced, at the cost of                       power in the load is (2Vs)Z/8RL - V2s/2RL -
poorer efficiency, by designing the stage to have                    Vsla/2. Meanwhile, the supply delivers a sinusoi-
an actual maximum output considerably in excess                     dal current varying between zero and 2ia at a
of its nominal rated output. However, distortion                    constant voltage of Vs. So the average power
can be more economically reduced by the applica-                    drawn from the supply is still Vsla watts. Thus
tion of negative feedback, as will appear later.                    the efficiency rl (lower-case Greek letter eta) is
   The choke feed arrangement of Figure 4.1a is                     given by
inconvenient in that usually RE cannot be freely
                                                                                useful power in load     Vsia/2
chosen: for example, in an audio power amplifier
                                                                         rl - t o t a l power supplied = gsia = 50~176
RE would be a loudspeaker with a resistance of 3, 8
or 15 ~ typically. Thus the transformer-coupled                     The power in the load thus equals the power
arrangement of Figure 4.1d is more practical,                       dissipated in the active device, and this seems to
enabling the speaker resistance to be transformed                   agree with the maximum power theorem, but does
to whatever value of RE is convenient, by suitable                  it? It was shown earlier that the maximum power
choice of transformer ratio. An upper bound on                      theorem states that the maximum power will be
78   Analog Electronics

obtained in the load when the load resistance equals       standing current Ia is reduced to a very low
the internal resistance of the source. Now the             value, ideally zero. The stage now draws no
output slope resistance of the ideal active device         current when v i - 0 , giving rise to the alternative
in Figure 4.1b (the source resistance as far as RL is      name of quiescent push-pull. If you assume the
concerned) is infinitely high, and still much higher       same peak current through each transistor, namely
than RL even in the practical case of Figure 4.1c.         2Ia then the same value of RL' is required for the
The solution is quite simple: the value of RL chosen       class B stage as for the class A stage. The max-
in practice is that which will extract the maximum         imum output power for the class B stage then
power from the source within the limits of the             turns out to be the same as for the class A stage,
available voltage and current s w i n g s - for the        but Vs supplies a pulsating current, namely a full-
output device is not an ideal generator in its own         wave rectified sine wave of peak value 2Ia. Now
right, but only a device controlling the flow of           the average value of half a cycle of a sine wave is
power from the ideal voltage source Vs. As applied         2/re of the peak (a handy result to remember), so
to the output of an amplifier, the maximum power           the power drawn from the supply Vs works out at
theorem is in effect a 'maximum gain theorem'. If          (8/g)(V2s/RL ') compared with 4V2s/RL ' for the
the drive voltage vi is so small that the output           class A stage. Thus the maximum efficiency at
voltage swing will not exceed 2Vs even with                full output for a push-pull class B stage works
RL = ec, then it is indeed true that maximum               out at 1"1= ~/4 or 78% against 50% for the class A
power will be obtained in a load RL equal to the           stage. Of course, with practical devices, more like
output slope resistance of the device.                     Figure 4.1 c than b, the achievable efficiency will be
   Figure 4.1e shows a transformer coupled push-           less, say 70% at best; this gives rise to the rule of
pull class A amplifier, sometimes described as class       thumb that a class B output stage can supply an
A parallel push-pull. The load resistance RL is            output power of up to five times the dissipation
transformed to an equivalent collector-to-collector        rating of each output transistor.
load resistance RL t, where R L t = 2Vs/Ia. As tran-           Matched devices are just as essential for a class
sistor Q1 swings from bottomed to cut-off, point A         B power amplifier as they are for a class A
swings from 0 V (2 Vs negative with respect to point       amplifier. However, the distortion in a class A
B) to +2Vs (2V~ volts positive with respect to             amplifier will be low at small signal levels, since
B). Hence Vpp = 4 Vs and Ipp = 2Ia = 2(2 Vs/RL').          each transistor is operating in the most linear
Maximum power in the load is thus 2V2s/RL ~                middle part of its current range. In a class B
watts, whilst power drawn from the supply is               amplifier, on the other hand, each transistor is
4V2/RL ~ giving rl = 50%, the same result as for           biased at one end of its current swing, i.e. is almost
the single-ended class A stage. However, there are         cut off. It is therefore difficult to avoid some
several advantages compared with a single-ended            crossover distortion (discussed later) and conse-
stage. First, for a given rating of transistor, twice      quently some overall negative feedback (NFB) is
the output power can be obtained, but most                 always used with class B amplifiers. In the case of
importantly there is now no net DC flux on the             early transistor radios, this usually succeeded in
transformer core, so no airgap is needed. This             improving the performance from shocking to
makes it e a s i e r - and more e c o n o m i c a l - to   poor.
provide a high primary inductance and low wind-               A desirable economy in any audio power am-
ing resistance. On the debit side, the two transis-        plifier is the omission of the heavy, bulky and
tors should be a well-matched pair, and the circuit        expensive output transformer. Figure 4.2 shows
requires a balanced drive voltage vi to provide the        various ways in which this has been achieved. In
inputs needed by Q1 and Q2.                                Figure 4.2a a transformer is still employed but
   With a push-pull circuit like Figure 4.1e, effi-        only for the drive signal, not the output. This
ciency can be increased by operating the devices in        arrangement is known as series push-pull, or
 class B - an option not available with a single-          e v e n - c o n f u s i n g l y - single-ended push-pull. A
 ended stage as at Figure 4.1a. In class B the             little later the arrangement of Figure 4.2b,
                      Audio-frequency signals and reproduction   79

                                    -v S


                   * Matched pair

                                           n 1)or
                                           ,n 2)


 In                                        C7


80   Analog Electronics
                          .I. ~:
                                                                    >           i
                                                               ¢',q e¢~
                              ,     Jl   il    7
                                                     ~o                   ~o   i~            ~r-,-
                                                                                    zl.>      ¢-.q
                      Audio-frequency signals and reproduction   81
r-                                              "7
                           ~i ~                   I
                             II                   I
82   Analog Electronics

                                                 Power supply
                      CI, 2 = FF32K                                  BR1
                           Orange          Red                       (WQ58N)
                                                                                               _ +50V
                          240V                                                   k         J - DC
                          AC                                                   C2
                                                                               10,000   I.tF
                           Orange        T1 Grey        !                                      _ -50 V
                                                                                               - DC
                                                       C1            .[.             FS2
                                                       63 v
                                                       10,000 gF T
                                 (d)ii                          ,,                             ~ov


Figure 4.2     Audio power amplifiers.
  (a) Series push-pull. Dot on coils indicates start of winding. All windings in same sense. Therefore all dot ends
go positive together; dot voltages are in phase.
  (b) Quasi-complementary push-pull.
  (c) Full complementary push-pull amplifier. The dotted components (680pF, 1.5kf~) can be added if
electrostatic speakers are used.
(Parts (b) and (c) reproduced by courtesy of Electronics and Wireless World.)
  (d) Complementary MOSFET high-power hi-fi audio amplifier. Power 150 W. Frequency response 15 Hz to
40 kHz. Total harmonic distortion 0.01% at I kHz. Input sensitivity 850 mV RMS for rated output. Damping
factor 200 (8 f~ load). Main heatsink not shown. (Reproduced by courtesy of Maplin Electronic Supplies.)
                                                            Audio-frequency signals and reproduction        83

known as quasi-complementary push-pull, became           A k V 2 sinZ(cot) = 1AkV2 + 89       2 sin(2cot) (4.2)
popular. This was because, at the time, silicon
PNP high-power transistors were readily available       The term (1/2)AkV 2 is a small DC offset in the
whereas similarly rated NPN types with matching         output, which is not audible and so need not
characteristics were unobtainable or very expen-        concern us. However, if co = 2765 radians per
sive, although lower-power NPN devices were             second, making A V sin(c0t) the note 440 Hz (con-
common and cheap enough. Nowadays matched               cert pitch A above middle C), then the amplifier
complementary pairs of bipolar power transistors        output will also include a component of amplitude
are readily available (Figure 4.2c), whilst matched     (1/2)AkV 2 and frequency 880Hz. Thus the
N and P channel audio power FETs are now                second order distortion has introduced a second-
commonly used in high-power audio amplifiers            harmonic component in the output which wasn't
(Figure 4.2d). FETs have the advantage of a             in the input. Figure 4.3b(iii) shows two sine waves,
drain-current gate-voltage characteristic that pro-     one twice the frequency of the other. You can see
vides a more gradual transition from the cut-off        that if they are added, one peak will indeed be
state to conducting than the corresponding collec-      flattened whilst the other becomes even peakier, as
tor-current/base-voltage characteristic of a bipolar    shown in Figure 4.3b(i).
transistor (compare Figures 4.3a and 3.2g). The            Assuming that k - 0.1 and that V = 1 corre-
trick with either bipolar or FET devices is to          sponds to full rated-output, it follows from
choose a small quiescent current such that the          equations (4.1) and (4.2) that at full output the
input/output characteristic of the stage is as linear   amplitude      of   the    second     harmonic    is
as possible, this is known as class AB operation.       (1/2)kV 2 = 0.05 times the wanted fundamental,
   Practical circuits invariably incorporate meas-      i.e. there is 5% distortion. At one-tenth of full
ures to stabilize the quiescent current against tem-    output, where V-- 0.1, the second-harmonic com-
perature and supply voltage variations. Before          ponent is (1/2)(0.1)(0.1) 2 or 0.0005, representing
looking at how NFB reduces the remaining distor-        only 0.5% distortion, or one-tenth of the pre-
tion, here's an overview of what different sorts of     vious figure. Expressing the same thing logarith-
distortion there are.                                   mically this means that, for every 1 dB decrease
                                                        or increase in vi, the fundamental component of
                                                        Vo falls or increases by 1 dB but the second-
Distortion                                              harmonic distortion component falls or increases
The most noticeable and therefore the most objec-       by 2dB, which explains why the distortion is
tionable type of distortion is due to non-linearity.    worse in louder passages of music.
Figure 4.3b shows, in exaggerated form, the sort of        Now 5% of distortion on a single sine wave does
distortion which might be expected in a single-         not in fact sound at all objectionable, but you
ended class A amplifier. Linear enough at very          don't normally listen to isolated sine waves. The
small signal levels, at maximum output the peak of      trouble begins when vi includes many different
the sine wave is more compressed on one half cycle      frequencies at once, as for example in music. To
than on the other. This type of distortion is called    get some idea of the problem, consider what
second order, because the transfer characteristic       happens when vi consists of just two equal ampli-
can be decomposed into two parts as shown: an           tude sine waves of different frequencies, say
ideal linear part and a parabolic component. Thus       1000 Hz and 1100 Hz. Then
Vo -- A(vi -+-kv2), so if Vi : V sin (cot) then
                                                                    vo - A -~ M + k -~ M
      Vo = a{Vsin(cot) +k[Vsin(cot)] 2}         (4.1)
A V sin (cot) is the wanted amplified output, but we    where M stands for 'music'; in this case
also get a term Ak{ V sin(cot)} 2. As the chapter on    M - sin(col t) + sin(co2t), which has a maximum
trigonometry in any maths textbook will tell you,       value of 2, giving a peak output voltage vo of V as
sin20 = (1/2)(1 + sin 20). So the unwanted term is      before, with c O l - - 2nlO00, co2 - 2n1100 radians per
84   Analog Electronics

                                                                                                  y = x + kx2
             < 16
              ~ 12-
             "a       8-                                                      . ~ _ _ ~_ u_u ~,mo

                               I       I      i      I     I
                              2 4 6 8 10
               (a)                            s
                           Gate/sourcevoltageVg (V)

              (ii)                                                                 (iii) ~ .

                                                                                                              ,    ~Second
                  /                                                                   --~or
                                                                                        I     !     ~l
                                                                                                              Ii   [harmonic

                                                                                        i I          %!            i
                               vo = A(v i +      )
                                              kvi2                                      I I



             y=x-kx        3                                   U component
                           I        ~             ....


                                                                                Audio-frequency signals and reproduction      85

            Time]          V
                                         ' [sin (cot)]3 = sin3 (cot)   vl
                 '~.~f~~~-.               - 1/4 sin (3c0t)

                                        9 3/4 sin (cot)                YO                                          !    y

                                                                            0                           2       3 kHz
                t=O                                                                            (e)

Figure 4.3     Even-order and odd-order distortion.
  (a)   Typical N channel MOSFET Id/Vgs characteristic.
  (b)   Second-order distortion, typical of a single-ended class A amplifier.
  (c)   Third-order distortion, typical of a push-pull amplifier.
  (d)   Third-order distortion analysed.
  (e)   Third-order intermodulation distortion with two tones.

second. The distortion comes from the k[( V/2)M] 2                      teristic is virtually linear at very small amplitudes,
term, i.e. k{(V/z)[sin(colt ) + sin(co2t)]} 2. Writing                  but at larger excursions the gain falls, being
(e01t) - A and (cozt) - B for short, the expansion                      reduced by an amount proportional to the cube
for                                                                     of the voltage swing. At some higher input ampli-
                                                                        tude than shown, the output will cease to increase
 (sin A + sin B) 2 = sinZA + sinZB + 2 sin A sin B
                                                                        altogether as the output devices bottom: but
                                                             (4.3)      assume that over the range shown the distortion
                                                                        is purely third order. Then for a single tone as
is needed. The sin 2 term has cropped up earlier,
                                                                        shown in Figure 4.3c,
so you can tell from (4.3) that there will be
components at the second harmonic of each
                                                                                    Vo -- A(vi - kv~)
tone, i.e. 2000 Hz and 2200 Hz. But there is also
the 2 sin A sin B term. Now any handy maths                                            = A{ V sin(cot) - k[V sin(cot)] 3}   (4.4)
textbook will tell you that c o s ( A + B ) -
cos A cos B - sin A sin B     and    cos(A - B) -                       Now sin 3 0 = (3/4) sin 0 - (1/4) sin30, so there is
cos A cos B + sin A sin B. Subtracting the first                        a third-harmonic distortion term, but also a dis-
from the second gives cos(A - B) - c o s ( A + B) -                     tortion component at the fundamental. The com-
2 sin A sin B. Thus Vo contains not only harmo-                         plete distortion term has been shown in Figure
nics but also sum and difference frequencies of                         4.3c, it represents the shortfall of Vo compared with
2100 Hz and 100 Hz. These are not harmonically                          what it would have been without the curvature of
related and will therefore sound discordant.                            the transfer characteristic caused by the third-
Clearly with the complex sounds of many differ-                         order term. This is shown to a larger scale in
ent frequencies typical of music, be it pop or                          Figure 4.3d and also decomposed into its two
classical, there will be a host of discordant sum                       constituent sine waves. Bearing in mind that it is
and difference products, giving the typically                           multiplied by - k in (4.4), the sin(~ot) distortion
'woolly' or 'muddy' sound of an early 78 RPM                            term indicates that the curvature of the transfer
record or the soundtrack of a prewar film.                              characteristic has reduced the fundamental output,
   Figure 4.3c shows (exaggerated) the effect of the                    as well as introducing a third-harmonic distortion
third-order distortion typical of a well-balanced                       term. Again, a few per cent of 1320 Hz tone when
class A push-pull amplifier. The transfer charac-                       listening to the note A above middle C is not
86   Analo9 Electronics

objectionable or even noticeable: it is the effect on           On the other hand, many push-pull amplifiers
complex sounds like music that is of concern.                do indeed leave something to be desired, it is true,
Substituting the two-tone signal previously de-              for the above reasoning only applies to class A
noted by M into (4.4) and turning the handle on              push-pull amplifiers, whereas the majority are class
the maths, you will find no less than half a dozen           B designs. The problem here is crossover distortion.
distortion terms, including components at the two            Figure 4.4a shows two stages with a curvature in
input frequencies of 1000 Hz and 1100 Hz. These              one direction only (like Figure 4.3b), overlapped
have been shown (exaggerated) in Figure 4.3e in              so as to provide an almost perfectly linear average
the frequency domain rather than, as up to now, in           characteristic giving an undistorted output (i).
the time domain. In this form the diagram indi-              Next to it, (ii) shows the effect of reducing the
cates frequency and RMS amplitude but not rela-              small class AB standing current towards pure class
tive phases of the components. The output                    B, i.e. moving the two halves of the transfer
contains not only the two input tones at 1000                characteristic in the negative direction: a region
and 1100 Hz and the third harmonic of each, but              of low gain appears at mid swing. Excess forward
also third order intermodulation products. These             bias is no better, as a + movement in the char-
are 2fl - f 2 and 2f: - f l, in this case at 900 Hz and      acteristic shows. The increase in gain results in a
1200 Hz. You will not be surprised to learn that (at         larger output, due entirely to a region of excess
least at modest distortion levels) 1 dB change in vi         gain at midswing, as shown at (iii). Now the
results in a 1 dB change in the fundamental level in         distortion at full output in (ii) or (iii) may amount
Vo but a 3 dB change in the distortion products.             to only a fraction of a per cent and sound quite
The intermodulation products, being close in                 acceptable. But the same amount of crossover
frequency to the input tones from which they                 distortion will still be present at smaller outputs,
result, are very noticeable- even if not in principle        representing a much higher, very objectionable,
more discordant than the sum and difference terms            percentage distortion as shown in (iv). Thus, un-
found in the case of second-harmonic distortion.             like class A amplifiers with primarily second- or
Some writers have therefore claimed that a single-           third-harmonic distortion, with a class AB or B
ended amplifier is preferable for the highest-fidelity       push-pull amplifier the percentage distortion in the
applications to any push-pull amplifier. However,            output actually gets worse rather than better at
the reasoning is hardly convincing. It is true that if       lower output levels. Even though the quiescent
two identical single-ended stages are harnessed to           bias may be optimized, there is no guarantee that
work in push-pull, the remaining distortion will be          the shape of the curvature at the origin, at the foot
purely odd order (mainly third). But an amplifier is         of each half of the characteristic, will be such as to
not immune from third-harmonic distortion                    give a linear average (shown dashed) through the
simply by virtue of being single ended: after all,           origin. Even if it does, the quiescent setting may
if it is overdriven it will run out of gain on both          vary with temperature or ageing of the compo-
positive- and negative-going peaks. At lower                 nents; so, as mentioned earlier, NFB is employed
levels, the second-harmonic components of the                to ameliorate matters. If efficiency were of no
two halves of the push-pull amplifier cancel out             importance, the amplifier could be biased up to
whilst the remaining third-harmonic distortion is            class A by moving the two halves of the character-
simply that which is found in the two single-ended           istic in the positive direction so far that points A
stages originally. It may be true that a single-ended        coincide with points B, eliminating crossover dis-
amplifier with 0.2% distortion, all second order, is         tortion entirely. Crossover distortion results in a
preferable to a similarly rated push-pull amplifier          comparatively sharp little wrinkle on each flank of
with 0.2% distortion, all third order, but that is           the waveform, as in (ii). Being symmetrical there
not usually the choice. Given two of the class A             are no even-order distortion products; nor is the
single-ended stages, it is possible to design them           third-order component particularly large. How-
into a push-pull amplifier of twice the power rating         ever, many other odd-order products - fifth,
with virtually zero distortion, even at the higher rating.   seventh, ninth, eleventh and so o n - are all present,
                                                                          Audio-frequency signals and reproduction 87

                                                         A A                                       A
                                                                                                   9 V

                                                             (i)           (ii)        (iii)          (iv)

                Io   Oaa"


                                          .~0~           ~I              ~ ~_         ~f .....                ~~_
                vi = . J - - - L   -L-vo~                          ~j
                                                   Topcut            Top lift               Basscut                Basslift
                A = 100            -       V RMS
                                                             ~ + 1 0             _
              vi = 0.1V                                     vi=~'~'+0~l ~     -~k                        -    1
              RMS~....~~                       RL= 8R0          Dl     ~'-100 B~                              _[
                                                                     l            .            ~910R fi 8R0
                                                                                            9o9~             T
              +1.41~                    ]a ~         14.1279

                     ~ ~           E ~              -~B910R~I

                                       1.27151                H RL

                                       (iii)                                          (c)

Figure 4.4
  (a) Crossover distortion.
  (b) Effect of tone controls (frequencydistortion).
  (c) Effect of negative feedback on distortion.
88   Analog Electronics

resulting in a multitude of intermodulation prod-         taking a look at the use of negative feedback to
ucts as well as third, fifth, seventh etc. harmonics,     reduce distortion.
which explains why even quite small levels of
crossover distortion sound so objectionable.
   The previous section has dealt at some length
                                                          Negative feedback
with non-linearity, which gives rise to waveform          NFB is applied around the main amplifier of, for
distortion: the next variety of distortion can be         example, a hi-fi system, principally to reduce the
disposed of relatively easily. In fact we are so used     amplifier's distortion to a very low level. Strange
to it and find so little to object to in it that it is    as it may seem, there is no generally agreed figure-
frequently not thought of as distortion at all. I         like 0.1%, for example - which may be taken as an
refer to frequency distortion. An amplifier may be        adequate specification for all hi-fi amplifiers. It
perfectly linear in the sense that the ratio of output    appears that the human ear is so discerning that
to input voltage is independent of volume, whilst         distortion levels lower than this are noticeable,
being quite non-linear in terms of its frequency          under favourable listening conditions, at least in
response. Indeed the whole purpose of tone con-           the case of classical music. (In the case of pop,
trols is to enable the listener to adjust the degree of   rock etc., limiters, volume compressors, fuzz boxes
frequency non-linearity to suit his taste, especially     and other non-linear electronic musical 'aids'
as regards high and low notes using treble and bass       frequently render the question of distortion at
controls, or in a more complex manner with a              the reproducing end of the chain quite immaterial.)
'graphic equalizer'. These enable the ratio Vo/Vi,             To see how NFB reduces distortion, take a
i.e. the gain A to be set to exhibit different values     concrete example: an audio power amplifier with a
for different frequencies. Frequency distortion in a      non-inverting open loop voltage gain Avol = 100
power amplifier is very small indeed: it is intro-        and a 'shortfall' (Figure 4.3c) of 1% (without
duced, if and when required, in the early small-          feedback) when supplying an output of 10 V RMS
signal stages of the amplifier. A square wave is a        into an 8 ohm load, i.e. 12.5W. If the load is a
convenient signal to show the effect of frequency         reasonably efficient loudspeaker in a domestic
distortion, since it is a signal containing many          setting, this should provide enough sound to sat-
frequencies. However, as they are all harmonically        isfy anybody, and probably too much for the
related to its lowest or fundamental frequency, it        neighbour's liking. Asume further that it is a
has a constant determinate shape, unlike the              class A amplifier, or a very well set up class B
constantly changing waveform of a piece of                one, so that the distortion is entirely third harmo-
music. Figure 4.4b shows the effect of frequency          nic, as in Figure 4.3c. Figure 4.4c summarizes the
distortion on a square wave. What is not so               plot so far. Now add a voltage divider at the
obvious is that the frequency distortion is accom-        output to provide a signal whose amplitude is
panied by phase distortion. For example, not only         9% of the output, i.e. a feedback fraction
does top cut reduce the amplitude of the higher           13=0.09, and a network at the input which sub-
frequencies in Vo compared with their relative level      tracts the feedback voltage from the input voltage,
in vi, it also retards them in phase. Like frequency      now applied at point C (Figure 4.4c(ii)). The
distortion, phase distortion is not objectionable or      voltage levels have all been marked in and, just
even noticeable, at least in mono reproduction. To        to check that they all tie up, note that the gain-
prevent it altering the spatial sound image in stereo     with-negative-feedback or closed loop gain
reproduction, the tone controls are arranged to
                                                                         Avol            + 100
operate equally on both channels.                              Avc~ - 1 + [3Avo~    1 + (0.9 • 100)    10
   There are also other forms of distortion. One of
these, transient intermodulation distortion (TID),        as in the diagram. But this isn't fair, you may
was for long unrecognized, although it has at-            argue, and I agree entirely; it assumes the amplifier
tracted considerable attention for some years             is perfectly linear, whereas it has already been
now. However, that topic is best left until after         assumed to have 1% open loop shortfall at
                                                             Audio-frequency signals and reproduction      89

10 V RMS output. It applies well enough at low            is x 10 or 20 dB: not so very different. The heavier
levels where the percentage distortion is negligible,     the overall negative feedback, the closer the loop
but at 10 V RMS output it is no longer a case of          gain approaches the gain reduction due to feed-
pure sine waves. Consider the situation at the            back, and the two terms are often used loosely as
positive peak of the output waveform: it should           though they were synonymous. Either way, you
be 10 V RMS or 14.1421 V at the peak, when vi             can see that the NFB has reduced the amplifier's
instantaneously equals 1.414 21 V. But assuming a         third-harmonic distortion at 10 V RMS output
1% shortfall (relative to a distortion-free charac-       from 0.25% to 0.025%, a very worthwhile im-
teristic, as defined in Figure 4.3c) then the voltages    provement, whilst the 1% gain shortfall has been
at various points round the circuit are actually as       reduced in the same ratio. As noted earlier, the
shown in Figure 4.4c(iii). If you want to derive          amplifier's open loop distortion appears, inverted,
them for yourself to check, take my tip and assume        on the waveform at point E. It is often possible to
a nominal voltage at E of unity, so that the voltage      disconnect an amplifier's NFB line and examine its
at A is 99. It is then easy to fill in the rest and,      open loop distortion directly. In some designs,
finally, to proportion all the answers to a vi of         however, the NFB is DC coupled and used in
1.414 21 V, i.e. 1 V RMS. Comparing (iii) with (ii),      setting the amplifier's operating point; in this
there is an actual output of 14.1279 V peak instead       case the NFB cannot conveniently be disabled by
of x/2(10) = 14.1421 or 99.9%; at E there is 0.142        just disconnecting it. The amplifier's open loop
706V peak instead of 0.141421V peak or                    distortion can still be measured, though, simply by
100.91%. In other words the 1% open loop                  examining the signal in one of the early low-level
'distortion' of the amplifier has been reduced by         stages. Assuming that these are very linear and
a factor of 10, the amount of gain sacrificed when        that virtually all of the distortion occurs in the
the NFB was incorporated. Nine-tenths of the              output stage, then one has only to examine the
open loop 'distortion' can still be seen on the           signal in a stage following the combination of
waveform at E; it is in the opposite sense, however.      the input and feedback signals. This can be very
It thus predistorts the drive to the main amplifier,      illuminating, not to say alarming. Consider an
to push it relatively harder at the peaks than            amplifier with 0.2% distortion at full rated output:
elsewhere.                                                doesn't sound too bad, does it? Suppose it has
   Note that shortfall and distortion are not the         40dB of loop gain, though: one can deduce
same thing in normal parlance. Figure 4.3d shows          straight away that there is a massive 20% of
that three-quarters of the shortfall is due to a          open loop distortion, and confirm this by examin-
component at the fundamental frequency which              ing the signal in one of the early stages!
reduces the net fundamental, and only one-quarter            Heavy negative feedback can, then, considerably
is due to the third-harmonic component. Now a             reduce the level of distortion of a test signal
distortion meter measures the power at harmonic           consisting of a single sine wave of amplitude
frequencies only and compares it with the output          within the amplifier's rating. However, NFB is
at the fundamental; it does not know about or             far from the universal panacea for all hi-fi ills.
measure the slight reduction in fundamental out-          Consider the amplifier response of Figure 4.3c. If
put. So one must not talk of an open loop                 the input is increased beyond that shown, the
distortion of 1% at 10 V RMS, when what is                crushing of the peaks will gradually become
really meant is a shortfall of 1%.                        more severe until they reach the flat dashed
   If vi in Figure 4.4c(ii) is set to zero, the link AB   portion of the characteristic where no further
removed and a signal of 1 V RMS applied at B, the         increase is possible; the output devices have run
output will be - 9 V RMS, the minus sign indicat-         out of supply rail voltage. Note that as Figure 4.3d
ing that the output is inverted, since D is a subtract    shows, for every 1% increase in third-harmonic
or inverting input. Thus the loop gain or 'gain           distortion there is a 3% reduction in gain. The
within the loop' is x9 or 19dB, whilst the gain           result is a gentle sort of limiting, with gradually
reduction due to feedback (Avd compared with Avol         increasing distortion on overdrive. Imagine, how-
90   Analog Electronics

ever, that the same amplifier is improved with           modest amount of NFB, used well within its
heavy negative feedback: the transfer characteris-       rating.
tic will be considerably linearized, as shown long          A course commonly adopted to ease the design
dashed in Figure 4.3c. As the input is increased,        problems of applying NFB is to limit the fre-
the output will rise pro rata with no gain reduction     quency range over which the full loop gain is
or distortion until the maximum possible ampli-          applied. For example, suppose that the gain of
tude is reached. Beyond this point the peaks are         the amplifier block in Figure 4.4c(ii) is rolled off by
simply sliced off as with a sharp knife. Hard            20 dB, from x 100 to x 10, over the frequency
limiting of this sort involves high orders of distor-    range of 1-10kHz. Over the same frequency
tion and, on programme material, severe intermo-         range, the overall gain with feedback Avcl will
dulation, resulting in a very nasty sound indeed.        fall a little, but this can be easily corrected if
Nor is the problem limited to the output stage.          desired with pre-emphasis in the preamplifier
Once the output clips, the feedback signal can no        supplying vi. At frequencies of 10 kHz and above
longer increase pro rata with the input signal; the      the loop gain is now less than unity, so the
loop gain has momentarily fallen to zero. With the       opportunities for TID due to feedback are largely
amplifier now open loop, not only is the output          removed. It is, of course, true that harmonic
stage overdriven, but in all probability earlier         distortion of signals at 10 kHz and above will not
stages as well. This may shift their DC operating        be reduced by feedback, but the harmonics will all
point, resulting in distortion even on smaller           be at 20 kHz or higher anyway, i.e. above the limit
inputs following an overdriven transient, until          of audible frequencies. But why roll off the gain at
they recover. This is one form of transient inter-       all? The answer is that it will roll off of its own
modulation distortion (TID).                             accord anyway, at some sufficiently high fre-
   In fact, heavy NFB can make severe demands            quency, simply because practical amplifying de-
upon the earlier stages even when the amplifier is       vices do not have an infinite bandwidth. It is up to
not overdriven. As noted above, they have to             the designer to ensure that the gain rolls off in a
handle an inverted version of the distortion caused      controlled, reproducible manner, so that there is
by the output stage, superimposed upon the               no excessive gain peak, or worse still, possibility of
normal signal. The distortion includes harmonics         oscillation. The same criterion applies at the low-
at two, three, four etc. times the input frequency as    frequency end, if the amplifier has AC coupled
well as sum frequency and higher-order compo-            stages, e.g. RC interstage coupling and an output
nents when the input consists of more than just          transformer in a valved amplifier. Transistor am-
one frequency. Consider a cymbal clash in a              plifiers are usually DC coupled throughout in the
symphony, for example: it includes frequency com-        forward path, and frequently the feedback path is
ponents at up to 20 kHz and usually occurs when          also DC coupled, so stability considerations are
the rest of the orchestra is in full spate to boot.      limited to the high-frequency end of the spectrum.
Thus the inverted distortion components which               It was noted in the previous chapter that an
must be handled linearly by the earlier stages will      internally compensated opamp is designed with a
include components up to 100 kHz or more, riding         dominant lag which rolls off all of the open loop gain
on top of an already full amplitude swing. This is a     before the phase shift due to other stages becomes
severe test for the penultimate stage which drives       significant. If, however, in a high-power audio
the output devices; if it fails the test the result is   amplifier you wish to retain say 20 dB of loop gain
transient intermodulation distortion. This can re-       up to as high a frequency as possible, despite the
sult in discordant difference frequency components       earlier remarks about TID, then you might try using
appearing which would not have been produced if          a higher rate of roll-off. Figure 4.5 illustrates this
a more modest degree of NFB had been employed.           case, where the gain rolls off at 12 dB per octave. In
In fact, it is now generally agreed that the best        Figure 4.5a the basic amplifier with an open loop
fidelity is obtained from an amplifier with as little    gain A of x 100 has been shown as inverting so that
open loop distortion as possible, combined with a        the feedback voltage A [3at D can be simply added to
                                                                         Audio-frequency signals and reproduction           91

                                       [~_                                         1
                     C             E         100)               vi = C     ]   I
                                                                           '   ' D     =   13%
                Vi                                                  ,~-        ,                                   ,...._

                                                           v~       ~ 0                                            v

                                                                                                               B =v o
                         D                          R1              '    Ig'2C+D

                 D= ~         B = ~3vo        R2
                    "'1 " "'2                  ~         (a)

                     E=C+D         ID                                                      0

                         (c)                                                                   (d)

Figure 4.5

the input voltage Vi applied at point C. In Figure                 relative to b and c. At the particular radian fre-
4.5b the vector diagram for low frequencies (before                quency c0 shown in Figure 4.5d (call it COp), the
the frequency response has started to roll off) shows              magnitude of the feedback voltage at D is the
that the amplifier input at E is simply the sum of vi              same as the magnitude of the amplifier's input at
and the inverted feedback voltage: it is also of course            E. As can be seen, vi at point C is now very small
in antiphase with vo at B and just one-hundredth of                compared with the voltage at D and hence also
the amplitude. Assuming 13= 0.1, then Avcl =                       compared with the output; owing to the (almost)
100/[1 + (100 x 0.1)] = 9.09, so Vo =9.09vi. In                    180 ~ phase shift, the negative feedback has become
Figure 4.5c the two coincident lags are contributing               positive, resulting in a substantial gain peak at O3p.
between them a total of 90 ~ phase shift between the                  In Figure 4.6a the locus of the feedback voltage
voltages at E and B, and of course a corresponding                 has been plotted as a function of 0~, from zero to
6 dB fall in the open loop gain Avol. But despite the              infinity. For convenience, Vo is not shown; it is just
small phase shift between vi and Vo (at points C and               the same only 1/13 times larger. Figure 4.6b shows
B) the gain is virtually unaffected - as was also found            the open loop gain (from vi to the junction of R1
in the previous chapter in connection with internally              and R2) and the phase shift ~, i.e. with the
compensated opamps. Figure 4.5d shows the situa-                   feedback connection broken. Points 1, 2 and 3
tion at a much higher frequency, where the two lags                indicate the zero frequency, 90 ~ corner frequency
have been running for some octaves and hence not                   and unity loop gain frequency e0p, corresponding
only is the amplifier's phase shift very nearly 180 ~              to the vector diagrams of Figure 4.5b, c and d.
but its gain is greatly reduced. Note that, for clarity,           Figure 4.6b also shows the closed loop gain Vo/Vi
Figure 4.5d has been drawn to an expanded scale                    with its peak at the critical unity loop gain (0 dB
92   Analog Electronics

                                                                                 E=-I [0

                                                                          %               \
                                           E ---1                                             l
                                                         s ~'. ~
                                                                   "~" Enlargedview /
                                                                                                                         1] -0


                     dB                                                                                           k   +jo~
               A                                                                                       x-,
                              ~''--..                   -6 dB/octave
                                  -12 dB/octave                       log f

                                                                      logf                        II      I

             _90 ~
                          ~   T      w
                                                                   lags                                 !


                1/~ [

                                                                   ~ 2        dB/octave                       (c)


Figure 4.6

loop gain) point 3. Figure 4.6c shows the corre-                          being the sum of the angles subtended at point 3
sponding open and closed loop pole-zero dia-                              by each of the two closed loop poles, i.e. nearly
grams. For open loop, there are two coincident                            180 ~ Since Figure 4.6a is incomplete- showing
poles (indicated by the roman numeral II) on the                          the voltage at points D and E but not vi at C - it
axis, each contributing 45 ~ phase shift at point 2                       can do duty for both the open and closed loop
on the jc0 axis. For closed loop, point 3 indicates                       cases. Note that the closer the open loop phase
the frequency of maximum gain, the phase shift                            shift to 180 ~ at fDp, the smaller the voltage Vi at C
                                                             Audio-frequency signals and reproduction      93

in Figure 4.5d, until at 180 ~ the amplifier is           frequency with a lag of only 90 ~ A similar effect
capable of supplying its own input without any            can be achieved by leaving both of the lags in the
need for vi at all! This corresponds to an infinite       forward gain A to run indefinitely, but bridging a
closed loop peak in Figure 4.6b and to closed loop        capacitor C across R1 in Figure 4.5a such that
poles actually on the + and -j00 axis in Figure           CR1--1/o04. This arrangement, which modifies
4.6c. The result is a self-sustaining oscillation at      both the frequency response and the phase re-
COp; the amplifier is unstable. If the phase shift        sponse of the feedback voltage, is a simple example
exceeds 180 ~ so that the locus in Figure 4.6 (a)         of a beta network. In high-performance wide band
passes to the north of the point - 1 +j0, the             linear amplifiers such as are used as submerged
amplitude will rapidly increase until limiting oc-        repeaters in frequency division multiplex (FDM)
curs, resulting in a non-sinusoidal oscillation.          telephony via submarine cables, quite complex 13
Since at each peak, positive or negative, the             networks are called for. Once the amplifier de-
amplifier has to recover from overload, the fre-          signer had done his bit and produced the best
quency of the non-sinusoidal oscillation is lower         possible amplifier, the specialist 13 network de-
than COp;this is called a relaxation oscillation. For     signer would take over and produce the optimum
stability, the locus must not pass through or             13 network, taking into account the frequency
enclose the point - 1 + j0: this is called the Nyquist    response, phase, linearity and noise requirements.
stability criterion.                                         An alternative way of ensuring amplifier stabil-
   The upshot of this exercise is the conclusion that     ity is to roll the gain off at 10dB per octave,
two coincident lags is perhaps not the ideal way to       corresponding to a maximum loop phase shift of
control the high-frequency performance of a high-         150 ~ The Nyquist and open loop Bode and pole-
power audio amplifier. Of course one could argue          zero diagrams are shown in Figure 4.8, from which
that if the gain peak comes well above 20 kHz,            it can be seen that the 10dB per octave is
what does it matter? In the case of an audio-             approximated by alternating between 6 and 12
frequency amplifier there will be no components           dB per octave. The result is a 30 ~ phase margin
of vi at that frequency anyway: you could even add        at 00p: constructing a vector diagram like Figure
a simple low-pass filter ahead of the amplifier to        4.5d with 0 = 30 ~ will quickly convince you that
make sure. Now if the amplifier were perfectly            the peak is limited to approximately twice the
linear without NFB, there might be something in           amplitude of the low-frequency response, or just
this a r g u m e n t - but then if it were we would not   +6dB.
need NFB anyway. However, it was shown earlier               I have tended to concentrate on the use of NFB
that when an amplifier has significant open loop          to reduce distortion, but it should be clear that it
distortion, the net input at E includes harmonic          will tend to reduce any difference between the
components, representing the difference between           input and the feedback sample of the output
the input voltage and the feedback sample of the          voltage, due to whatever cause. Thus, for example,
output voltage. So even if vi were band limited to        if any noise, hum or other extraneous signal is
20 kHz, the input to the amplifier at E is not: one       picked up in the earlier stages of the amplifier, its
had better think again.                                   level at the amplifier's output, relative to the
   Figure 4.7 shows one way of reducing the gain          wanted signal, will be reduced in proportion to
peak and thus enhancing stability: it is a method         the loop gain. In addition, the NFB will reduce the
frequently used in linear regulated power supplies.       amplifier's output impedance, again in proportion
The forward gain A has one of the two lags                to the loop gain.
cancelled at a frequency lower than 00p, i.e. it is a        Having looked at class A, AB and B audio
transitional lag (see Figure 2.6). The effect on the      power amplifiers, a word now about class D
open loop Nyquist, Bode and pole-zero diagrams            amplifiers (which enjoyed a brief period of
is shown in Figure 4.7. It increases the phase            popularity) might not come amiss. The earlier
margin at the critical unity loop gain frequency          analysis of amplifier efficiency showed that the
(Figure 4.7a), with the locus approaching infinite        class B amplifier was considerably more efficient
94   Analog Electronics

                                                              dB                   2          ~B/octave

      .   .     .   .
                        -1.o                     A~
                                                                         -12 dB/octave "~,
                                                                         _             -   ....   ~

                                                               0 ~

                           i                                 _180~


               Zero                    ....              ~
                                                         r    +~


Figure 4.7

than a class A amplifier, mainly because the                  effect of the inductor is to integrate or smooth the
transistors only conducted heavily when the vol-              current pulses, resulting in a sinusoidal current at
tage across them was lowest. The class D amplifier            the signal frequency flowing through the loud-
pushes this philosophy to the limit: each output              speaker's voice coil, together of course with a
transistor is either bottomed or cut off. Figure 4.9          small supersonic ripple component.
shows the scheme. With no input signal, the out-                 The output stage of Figure 4.9b has the attrac-
put transistors conduct alternately, applying a               tion of simplicity, but there will be some switching
supersonic square wave to the inductor: the result,           loss as, with bipolar devices, the turn-on of one
if the inductor has a high reactance at the switch-           transistor will be quicker than the turn-off of the
ing frequency, is that only a small triangular                other. In Figure 4.9a the on drive to one device can
magnetizing current flows. When an input signal               be delayed relative to the off drive to the other,
is present, the mark/space ratio is modified so that          thus positively avoiding any conduction overlap.
the upper transistor conducts proportionately                 Of course, with this arrangement, as soon as one
longer than the lower, for up to 100% of the                  device turns off the output voltage will fly off in the
time at the positive peak of a full output amplitude          direction of the other rail, so the catching diodes
sine wave, and vice versa for negative peaks. The             shown will be necessary. In fact, they are necessary
                                                                            Audio-frequency signals and reproduction       95

                                                 -1.0                                                      A~



                      m13               '~~~            I"~-"-            ~-~      '
                                                                                   1' ' ' ~ - "
                                        I                                          I

                                        I               I           I
                                        t               I,          ,             l
                       o                i               1           I              s
                                                        I           I              I              ......
                     _90~                               i           I              t

                                  (c)                        +jco

                                            r\               ,p~-
                                                                                              --J~.- +13

Figure 4.8

anyway since the magnetizing current is out of                          hysteresis. Overall N F B is applied from the output
phase with the voltage, reversing sign half-way                         via R8 to the input stage Tr~ which operates as an
through each half-cycle under no signal con-                            integrator due to C2. The collector voltage of Trl
ditions, as shown in Figure 4.9d.                                       always moves in such a direction as to drive the latch
   The full circuit of Figure 4.9c operates as follows.                 via R5 to switch back in the opposite direction. The
Weak positive feedback via R12 round Tr2, Tr3, Tr4                      switching frequency is determined by the slew rate
and Tr5 causes them to act as a latch with a small                      of the integrator and the amount of hysteresis of the
96       AnalogElectronics

                                                                                o V+                                                       o V+


            LV1 ,                                   ~                            Loudspeaker
                                        .                               x       oOV                                       (b)                  V

                                  i ]                        .     .        .        .                   Tr4      "                 -     - [~L +34V
                              o             10.,. I          5k   Rgll Tc,     /d,                                                             i,
                                            T /                      -'-"Jl00g ]ju~
                                                                  3"9k                                                                               C7

                   R2                             12k                           Tr
                             C~00p                                                    R14
    C1 R, /        ~-~~Tr,                  I                i    R 1 0 ~ ! C42"7k'~'1                  Irr~""
                                                                                                             ~            ~               ]1~176176
                   V~__~             c~4 ]          3R.~kU 2 ~ ~ ~                                     ,~C6 2712
                                                                                                          MM' '                      I
                                                                                                                                B~Y641 *I             ~

                   j        luK                                    .....                 MM2711                                                           0V
o    ~                                                             R8

      0 I" . . . . . . . . . ~i~_.f. 28RL                0                                            28RL      0
                                                                                                                                  ..........         28RL
     -h                          _                      -h                                                     -h -- -
             Trl       D2                                         ,D1 Trl D2 Tr2                                         D1        Tr2
             conductsconducts                                        Conducts                                          conducts conducts
                                                                                                                      ..... VoltageacrossRL

Figure  4.9 ClassD audio amplifiers.
  (a) Commonemitter class D output stage.
  (b) Commoncollector class D output stage.
  (c) Completeclass D audio amplifier.
  (d) Output current and voltage waveformsfor positive, zero and negative output levels.
(Parts (c)and (d)courtesyof Electronics and Wireless World.)
                                                            Audio-frequency signals and reproduction       97

latch: about 50 kHz sounds plausible. The inductor                                 Rf
L is 100 laH or more, the higher the inductance the
higher the efficiency, but if it is too large the
amplifier's response will no longer be reasonably
flat up to 20 kHz. The circuit shown provided 5 W
output with less than 0.25 % distortion at all volume
                                                                        NI[                  Loudspeaker

     One problem with this type of amplifier is the
generation of audible difference tones due to
intermodulation between the input signal and the
switching frequency under certain circumstances.
This could be minimized by raising the switching
                                                                    o          A


frequency (a simple matter with modern high-
speed semiconductors) and using a different type
                                                         Figure 4.10 Use of positive feedback to achieve zero
of pulse width modulator, maybe the scheme is            output impedance.
worth resurrecting after all. That is more than can
be said for the sliding bias amplifier, which was
used in very early models of transistorized car          reduce the output impedance to a very low value,
radio. A single output transistor was used, for          it can never actually make it zero. In the world of
reasons of economy: not only were power transis-         hi-fi marketing, superlatives are the order of the
tors expensive but, being germanium, they had a          day, so it would look impressive to be able to claim
top junction temperature of around 75~             Re-   an output impedance of zero, making the amplifier
calling that in a class A amplifier the dissipation      appear to the speaker as an ideal voltage source.
in the transistor(s) is lowest at full output, in the    This can be achieved by the judicious application
single-ended class A sliding bias amplifier, the         of a little current derived positive feedback (PFB).
transistor was normally biased at a fraction of          Figure 4.10 shows the scheme. If the loudspeaker
the standing current required to cope with full          is disconnected, there is no current through Rp and
output. A part of its drive signal was rectified         hence no positive feedback. On connecting the
and smoothed, and used to increase the bias in           loudspeaker, the output voltage (in the absence
sympathy with the size of the signal. Thus the           of PFB) would fall, even if only fractionally
device always draws just enough current to cope          (thanks to the NFB), owing to the amplifier's
with the present size of the signal without distor-      residual output impedance. However, in Figure
t i o n - or at least that's the theory!                 4.10 the current through the very low-value re-
                                                         sistor Rp causes a voltage drop which, fed back to
                                                         the amplifier's non-inverting terminal, causes the
Loudspeakers                                             output voltage to rise slightly. This completely
The reduction in amplifier output impedance re-          offsets the internal voltage drop across the ampli-
sulting from the employment of NFB was men-              fier's output impedance, effectively providing a
tioned earlier. In amplifiers employing heavy            zero output impedance. Indeed, one could increase
negative feedback, e.g. 40 to 50dB of loop gain,         the PFB even further and cause the amplifier to
the output impedance is reduced to a very low            exhibit a modest negative output impedance, if
level- a fraction of an ohm1,2. This is generally        desired, say - 8 f~, cancelling out the voice coil
reckoned to be a good thing since it increases the       resistance and resulting in a near infinite damping
damping effect on a moving coil loudspeaker, thus        factor, at least at low frequencies where the
discouraging the cone from flapping around at            reactance of the voice coil is negligible.
frequencies where it has a mechanical resonance.            Audio amplifiers, whether hi-fi or otherwise, are
The result should be a smoother, more level              usually rated to drive a nominal load impedance,
frequency response. However, NFB can only                such as a 4, 8 or 15 ohm loudspeaker. The vast
98   Analog Electronics




                Wire connecting top __._/
                of coil to terminal.                                      l~ Outer pole piece
                A similar wire                                               Central pole piece
                connects the bottom
                of the coil to a                                             Filling
                second terminal                                           ~Ma~net
                             Magnet housing
                                               Moving coil


Figure 4.11
  (a) Vital elements of a moving coil loudspeaker unit.
  (b) Goodmans loudspeaker unit (cutaway), showing the moving coil, the magnet and pole pieces, and the
(Reproduced from Beginners' Guide to Radio, G. J. King. Heinemann Newnes 1984.)

majority of reproducers are moving coil speakers             principles and construction of a moving coil
whose impedance, at least at low frequency, is               loudspeaker. The voice coil is located in a power-
resistive and equal to one of the three common               ful magnetic field. Current flowing in the voice coil
values just mentioned. Figure 4.11 shows the                 causes the latter to be subject to a mechanical force
                                                                Audio-frequency signals and reproduction       99

 at right angles to both the current and the                not capable of radiating a satisfactory level of
 magnetic field, i.e. along the axis of the coil.           distortion-free sound at low frequencies, despite
 Thus the alternating audio-frequency currents              various palliatives.
 cause the voice coil and cone assembly to vibrate             One popular palliative is the loudspeaker with a
 in sympathy, turning the electrical signal into an         long-throw voice coil and very flexible roll surround
 acoustic one. A lightweight paper cone of small            supporting the outer edge of the cone. The theory is
 size results in a relatively high speaker efficiency at    that greater bass radiation can be achieved in a
middle and high frequencies, but with negligible            smallish box by enabling the cone to move back
 output at lower frequencies.                               and forth at low frequencies with a relatively long
    Such a speaker is appropriate in a small tran-          throw. Clearly if the magnetic gap and the voice coil
sistor portable radio. For hi-fi work a much larger         were the same length (in the axial direction) the coil
speaker with an extended bass response and a                would quit the flux entirely at each extreme of
much stronger, stiffer cone is preferred. The stiff-        movement, resulting in gross distortion. But if the
ness of the cone reduces its tendency to vibrate in         coil is much longer than the magnet's airgap (or vice
various 'bell' modes at different frequencies, result-      versa, but the former is the cheaper), distortion due
ing in a smoother frequency response with less              to this cause is avoided. Further, if the cone is made
coloration. However, the much greater weight of             very strong (i.e. heavy), not only will break-up
the larger, stronger cone results in a less efficient       resonances be avoided, but its efficiency at mid-
s p e a k e r - except of course at the lower frequencies   frequencies will be depressed to match that at low
that the smaller speaker was incapable of reprodu-          frequencies. With modern amplifiers capable of
cing. Considerable ingenuity has been exercised             delivering tens or hundreds of watts, the low
over the years to develop loudspeaker systems               efficiency is of little consequence.
capable of fairly realistic reproduction of all sorts          Long-throw loudspeakers can produce some
of music, from symphony orchestra to cathedral              unpleasant effects. Consider, for example, organ
organ. For the latter, a bass response right down           music. The high notes are radiated from a cone
to 32Hz or better still 16Hz would be ideal,                which is travelling back and forth an inch or more
although few domestic living-rooms are large                if a powerful low note is also sounding. This
enough to do justice to such a speaker svstem, or           results in frequency modulation of the high note
even to accommodate it conveniently, whilst the             by the low note, owing to the Doppler effect. By
frequency range of broadcast or recorded music              using separate speakers- a woofer and a tweeter-
(CDs excepted) extends only down to 50 or 30 Hz             to reproduce the low and the high notes, the effect
respectively. Details of bass reflex enclosures can         can be diminished. It can be reduced even further
be found in many standard works 3. The non-                 with a three-speaker system of woofer, barker and
resonant acoustic line is another design capable            tweeter, or bass, middle and treble reproducer.
of providing extended bass response, as is the              Figure 4.12 shows a very simple two-way crossover
exponential horn loaded loudspeaker. The latter             network. The low-pass filter for the woofer and the
is very efficient, owing to the excellent matching of       high-pass filter to keep bass out of the tweeter each
the air column's acoustic impedance to the cone of          cut off at an ultimate attenuation rate of only 6 dB/
the speaker: this in turn results in a relatively small     octave. This is barely adequate in practice, where
peak-to-peak excursion of the cone even at low              12 or sometimes 18 dB/octave would be used, but
frequencies, minimizing distortion (such as caused          it shows the principle. Note that the arrangement
by the coil leaving the linear area of gap flux) and        is a constant resistance network, so that the
intermodulation. However, these three types of              amplifier sees a purely resistive load of its rated
speaker enclosure are all large if dimensioned for          value, at all frequencies- or would if the loud-
response down to 50 Hz or lower. With so many               speaker impedance were constant.
people living in 'little boxes', the demand is for             In practice the impedance of a loudspeaker is far
ever smaller loudspeaker enclosures; but the laws           from constant, rising steadily with frequency once
of physics are inexorable and such enclosures are           the reactance of the voice coil becomes appreci-
100    Analog Electronics

                 o     .          .1.                       very low inertia, as in the Quad electrostatic
                                                            loudspeaker. This results in a very clean sounding
          Output           L        T                       output with very little distortion, intermodulation,
          from                                              coloration or 'hangover' after transients. Unfortu-
                                                            nately the large size and high price of electrostatic
                         Bass R ~       Mid/treble
                                        unit                speakers restrict them to a rather limited market.
                 o R ~ _ unit

                                                            Signal sources

                                                            Having dealt first- for no very good reason - with
                                                            the back end of the audio chain, it's time to turn
Figure 4.12 Constant resistance crossover circuit.          from power amplifiers and loudspeakers to signal
  If L/C = R 2, the amplifier 'sees' a load resistance of   sources such as microphones, pick-ups and tape
R at all frequencies. At the crossover frequency f~,        recorders.
where f~ = 1/[2~v/(LC)], each speaker receives half
                                                               With the exception of purely electronic music,
the total power.
                                                            programme m a t e r i a l - be it speech or m u s i c -
                                                            originates from performers and the sound waves
able. This results in its drawing less current and          are turned into electrical signals by one or more
hence receiving less drive power at higher frequen-         microphones. The latter are called transducers
cies. However, this is offset by a number of factors,       since, like pick-ups and loudspeakers, they convert
including the speaker's greater directionality at           mechanical vibrations into electrical signals or vice
higher frequencies, resulting in the energy being           versa. In fact, the moving coil microphone is very
beamed forward rather than widely dispersed.                like a tiny loudspeaker in construction and pro-
Thus the moving coil loudspeaker produces a                 duces an audio-frequency output voltage from its
generally acceptable performance as far as the              'voice coil' in response to the movements of the
listener is concerned, though a graphical plot of           cone occasioned by sound waves incident upon it.
its acoustic output as a function of frequency              Sometimes two such microphones are mounted in
invariably looks rather like a cross-section of the         a common mounting with their axes at right
Alps. Speakers also produce distortion and inter-           angles. They thus respond principally to sounds
modulation due to non-linear and Doppler effects,           coming from left front or right front as the case
but hi-fi enthusiasts readily tolerate high percen-         may be, producing separate left and right channel
tage levels of distortion in a loudspeaker which            components of a stereo signal. Moving coil micro-
they would not dream of countenancing in an                 phones typically have an output impedance of 300
amplifier. It may be that the distortion produced           to 600 ohms and a sensitivity, at 1 kHz, of about
by the acoustomechanical deficiencies of a loud-            - 7 3 d B relative to 1 V per microbar. They are
speaker is less objectionable than that produced by         often supplied fitted with an internal 10:1 ratio
the electronic deficiencies of an amplifier, as some        step-up transformer providing an output of about
people maintain. But a more likely explanation is           - 5 3 d B at an impedance of about 50kf~. Con-
simply that loudspeaker manufacturers do not                denser microphones are also widely used. These
quote distortion figures for their products whereas         work on the same principle as the electrostatic
amplifier manufacturers do.                                 loudspeaker and, like the latter, early models
   Whilst the majority of loudspeakers are of the           needed a separate DC polarizing voltage. Nowa-
moving coil variety (first introduced by Kellogg            days this is furnished by an electret, in effect a
and Rice in the 1930s)piezoelectric tweeters are            capacitor with a positive charge trapped on one
now quite common. Much less common is the                   side and a negative on the other; it is thus the
electrostatic loudspeaker, in which a large dia-            electrostatic equivalent of a permanent magnet. As
phragm, forming one plate of a capacitor, vibrates          with moving coil microphones, the frequency
as a whole, providing a large radiating surface with        response of an electret condenser microphone
                                                                                                                                   Audio-frequency signals and reproduction         101

                                                                                                                                onto the tape. The relation between the record
                                                                                                                                current and the remanent magnetism is highly
                                                                                                                                non-linear, so a high-frequency (50kHz to
                                                                                                                                100kHz) bias current is added to spread the
                                                                                                                                recording to the linear parts of the characteristic
                                                                                                                                (Figure 4.14). The same head can be used for
                                                                                                                                playback, though the best machines use separate
 --                                       ton      over=: ma ne :sa :on
                                                                                                                                heads for record and playback, as each can then be
                                                                                                                                optimized for the job it has to do. This also allows

~             .a,. p.


                                                  ..   4

                                                                          n,. p   ..

                                                                                       p   J,.~.            .,i

                                                                                                                  .,j   4   i   monitoring from the tape, the almost simultaneous
                                                                                                                                playback which assures that the recording is going
                                                                                                                                well. Another head uses the AC bias current at a
                                                                                                                                much higher power to erase previous recordings.
 magnet isat ion
                                                                                                                                   The earliest tape recorders recorded just one
                                                                                                                                track using the full width of a 88 wide tape
 Alternating signal current
 + bias A.C. from amplifier
                                                                                                                                moving at 30 in or 15 in per second. As tapes and
                                                                                                                                recorders improved, it became possible to use
                                                                                                                                speeds of 71 and even 31 in per second and to use
                                                        --- .-.                                    ~-, Yo":'2~;;2'              half-track heads. One set of heads was used,
                                                                                                   head                         recording along one half of the width of the
                        ".           ~
                                                                                                                                tape; the take-up and supply reel were then inter-
                                                                                                                                changed (turning the tape reel over), so the same
                                                                                                                                heads could then record or play a second track
                                                                                                                                along the other half of the tape. Quarter-track
                                                                                                                                heads are now commonly used, permitting two
Figure 4.13 Magnetic tape recording.                                                                                            tracks to be recorded in each direction. The two
  (a) Unrecorded tape.
  (b) Saturated tape.                                                                                                           tracks can be used simultaneously for the left and
  (c) Recorded tape. In (a) to (c) the size of the                                                                              right channels of a stereo signal, or, on many
domains is very much exaggerated; arrows represent                                                                              machines, either can be used alone to provide
the direction of the magnetization.                                                                                             four mono tracks per tape. The two tracks are
  (d) Recording.                                                                                                                then called A and B rather than left and right.
  (e) Playback.
                                                                                                                                Because of the interleaving of tracks in each
                                                                                                                                direction, a four-track recording cannot be played
can be very wide, typically 50 to 16 000 Hz for a                                                                               on an older two-track machine (unless two of the
good quality type, and an FET preamplifier is                                                                                   four tracks - say the B track in each d i r e c t i o n -
usually built in. This is operated by a single                                                                                  were left blank), although a two-track recording
miniature button cell and provides an output at                                                                                 can be played on a four-track A/B machine.
1 kHz of a b o u t - 6 0 d B relative to 1 V/gbar.                                                                                 Mono cassette recorders use half-track heads in
   The signal from a microphone may be broadcast                                                                                the same way as older half-track reel-to-reel
direct or recorded on magnetic tape for later use.                                                                              recorders. With stereo cassette machines the two
Magnetic recording tape consists of thin plastic,                                                                               tracks of the stereo signal are on the same half of
coated with finely divided iron oxide or other                                                                                  the tape, and both are erased at the same time by
suitable magnetic powder. Once pins have been                                                                                   the erase head. (As in reel-to-reel recorders, the
picked up with a magnet, they will tend to stick                                                                                erase head is energized during recording, and the
together even in the absence of the magnet. This                                                                                tape passes over it before reaching the record
effect is called remanent magnetism and is the basis                                                                            head.) Thus four-track mono use is not available,
of tape recording. Figure 4.13 shows diagramma-                                                                                 but mono cassettes can be played on stereo cas-
tically how a recording head records the signal                                                                                 sette decks and vice versa. Of course, the resultant
102      Analog Electronics

                                 Magnet isat ion I                                                                the pre-emphasized low-level high-frequency com-
                                           of tape
                                                                                       +I                         ponents to restore them to their proper level, the
                                                                                                                  tape hiss on playback is reduced by 10 dB relative
                                                                                                                  to a non-Dolby recorder. On the other hand, if the
                                                                                                                  high-frequency components of the signal are of
                                                  -s       "     !    "   "                                       large amplitude, no emphasis is applied during
                                                                                                                  recording, avoiding any possibility of distortion.
                                       /                   i
                                                             produce little
                                                                                                                  On playback the hiss is not attenuated, but is not
                               J                           I mag net'sati~
                                                             (exaggerated for clarity)                            noticed as it is masked by the high level of the
                           direction)                                                                             treble part of the signal. The Dolby system for
                                                                                                                  professional applications works by dividing the
             Magnet lsat
                of tape                                                                                           audio spectrum into a number of bands and
                                                                                                                  applying the above principle to each indepen-
                                                                                                                  dently. Thus an improvement in the effective
                                                       I                                                          dynamic range of an analog tape recorder is
                                                  i !                                             Maximum
                                                                                                                  obtained across the whole audio spectrum. The
                                                                                               p, range of
                                                                                                  magnet,sat,on   dynamic range is simply the volume range in
                     I !
                                            I I        ,                                                          decibels separating on the one hand the highest
  current causes
  magnet,sat ion
                                            It         i       H,gh frequency b,as
                                                               alone -- extends to centre of
                                                                                                                  signal that can be handled without exceeding the
                                                               linear region each s!de                            recorder's rated distortion level, and on the other
                                                                                                                  the no-signal noise level on playback.
                                                         Bias plus peak-amplitude recorded s,gnal.
                                                       I just stays within hnear range. The playback
                                                       I amplifier rejects the high frequency bias
                                                                                                                     The Dolby svstem is only one of a number of
                                                       , and only amplifies the signal
                                                                                                                  schemes designed to increase the dynamic range
                                                                                                                  (lower the effective noise floor) of an analog tape
Figure 4.14 Use of high-frequency bias.                                                                           recorder. For example, the Philips Signetics
  (a) No bias.                                                                                                    NE57IN is a 16-pin dual in-line (DIL) plastic IC
  (b) With bias.
                                                                                                                  dual compander in which each channel may be
                                                                                                                  used independently as either a dynamic range
sound is always mono, except when a tape re-                                                                      compressor or an expander. Each channel com-
corded in a stereo machine is played in a stereo                                                                  prises a full-wave rectifier to detect the average
machine.                                                                                                          value of the input signal, a linearized temperature
   With the low tape speed (17 in per second) and                                                                 compensated variable gain cell block, and an
the narrower tracks on ~ in tape, it is difficult to                                                              internally compensated opamp. Features include
make the background noise from a cassette recor-                                                                  a dynamic range in excess of 100 dB, provision for
der as low as on LPs, reel-to-reel recorders or FM                                                                harmonic distortion trimming, system levels ad-
radio. Various noise reduction techniques have                                                                    justment via external components, and operation
therefore been proposed, of which Dolby B is                                                                      from supply voltages down to 6 V DC. This versa-
widely used in domestic cassette recorders. This                                                                  tile component can be employed for dynamic noise
substantially improves the signal/noise ratio at                                                                  reduction purposes in analog tape recorders, vol-
high frequencies (where background hiss is most                                                                   tage-controlled amplifiers, filters and so on.
noticeable) during quiet passages of music. The                                                                      Increasingly nowadays, analog recording meth-
more sophisticated Dolby A system is used by                                                                      ods (LPs, reel-to-reel and cassette tape recorders)
broadcasting authorities and other professional                                                                   are being replaced by digital recordings such as
users. Dolby B works by sensing the level of the                                                                  compact disc (CD) and the RDAT or SDAT
upper frequencies in the audio signal and, if they                                                                digital tape formats. Fourteen-bit sampling at
are below a certain level, boosting them on record                                                                44 K samples/second can provide a 20 kHz band-
by up to a maximum of 10 dB. The reverse process                                                                  width with an 84dB dynamic range, more than
is carried out on playback. Thus on attenuating                                                                   adequate for the highest domestic hi-fi standards,
                                                                                                      Audio-frequency signals and reproduction                   103

   (a)                                          ~ h t

                                                    s   na
  signal                                                            Rsignal cut   L signalcut
                                                                    on thiswall   on thiswall

            C                                                                               Disc

              9   . . . .



                                   /                                               Cuts both
                  Cuts      Cuts                             Cuts   both              s~des
                  this      this
                  side      side                                sides
  L channel                            R channel             L+R equal              L + R equal
     only                                only                Jnantiphase             in phase
Figure 4.15 Recording a stereo disc.
  (a) Principle of recording two channels in one
groove.                                                                                                                           Stylus
  (b) Motion of the cutter.
(Reproduced from Electronics Questions and
Answers. I. Hickman. Newnes 1982.)                                                                        L channel o   n   l      y        ~   R channel only

while sampling devices with 18, 20 or even 22 bit
resolution, giving dynamic ranges in excess of
 100 dB, are available. On the pop scene, however,                                                                              Modulated
dynamic range is not an important consideration.                                                                                   wall

Consequently, on the cheaper sort of cassette tape
recorder, a record-level control can be dispensed                                                  Figure 4.16 Playing stereo discs.
with entirely. All programme material is automa-                                                     (a) Moving magnet cartridge.
tically recorded at maximum level, giving rise to                                                    (b) Motion of the stylus.
virtually zero variation in volume level on replay,                                                (Reproduced from Electronics Questions                        and
which fits in well with the output of pop records
                                                                                                   Answers. I. Hickman. Newnes 1982.)
and local radio stations. On better quality cassette
and reel-to-reel tape recorders, a record-level con-                                               the point rounded), mounted on the end of a
trol and a volume unit (VU) meter or peak                                                          short arm or lever. In one inexpensive type of
programme-level meter (PPM) are provided, per-                                                     pick-up, the lateral movements of the stylus lever,
mitting optimum use of the dynamic range of the                                                    as it traces the waggles in the grooves, bend a
recording m e d i u m - ferric, chromium dioxide or                                                piece of piezoelectric material (e.g. barium tita-
metal tape.                                                                                        nate), producing an output voltage which is
   The other common programme source for                                                           applied to an amplifier. Figure 4.16 shows a
sound broadcasting is the disc or gramophone                                                       magnetic pick-up for stereo records, and indicates
record. Historic recordings on 78 RPM records                                                      how it responds separately to the left and right
and early 'long-playing' 331RPM discs (LPs)                                                        channel signals, recorded one on each wall. When
were originally recorded monophonically; these                                                     the same sound is present in both channels (or on
are usually recorded or 'transcribed' onto tape                                                    a mono record), the stylus movement is purely
to avoid further wear on the irreplaceable origi-                                                  horizontal.
nals. Modern LP records are stereophonically                                                          A useful feature of the cheaper ceramic pick-up
recorded, as indicated diagrammatically in                                                         is that the output voltage is almost independent
Figure 4.15. For playback a lightweight pick-up                                                    of frequency, for a given amplitude of groove
is used, the 'needle' or stylus being a conical                                                    deviation. This is not so with magnetic pick-ups,
artificial sapphire or a diamond (with the tip of                                                  whether of the moving magnet, moving coil or
104   Analog Electronics

variable reluctance variety. In fact, when gramo-       above the frequency of maximum output, to
phone records are made, the frequencies above           salvage an octave or two before the roll-off
about 2000Hz are boosted and those below 500            becomes too severe to cope with.
Hz down to 50Hz are attenuated relative to                 Now not only does the disc replay characteristic
middle frequencies. The former improves the             of Figure 4.17b equalize the output of a magnetic
replay signal/noise ratio because the replay equal-     pick-up back to a level frequency response, but the
ization restores the high-frequency level of at-        phase characteristic shown also restores the origi-
tenuation, which at the same time reduces the           nal relative phases of all the frequency components
objectionable hiss of high-frequency noise. The         of the recorded signal. The same may be said for
latter ensures that the amplitude excursion of the      the 6 dB octave bass boost used for tape playback,
recording stylus is restricted to a value that          but compensation for the high-frequency loss is a
avoids the need for excessively wide groove             different case entirely. The fall in high-frequency
spacing when large amplitude low-frequency sig-         response on tape playback is due to factors such as
nals are being recorded. The low-frequency bal-         tape self-demagnetization and head gap losses, and
ance is likewise restored by the equalization           these are not 'minimum phase' processes, i.e. the
circuit on playback. The playback characteristic        attenuation is not accompanied pro rata by a
is shown in Figure 4.17b; the recording character-      phase shift. Consequently, the top lift on playback
istic is the inverse of this. As a result, the          -  together with pre-emphasis applied during re-
magnitude of the groove modulation correspond-          cording (Figure 4.18b) - introduces phase shifts
ing to a given magnitude of the signal to be            between middle- and higher-frequency compo-
recorded is more or less independent of fre-            nents which were not present in the original signal.
quency, apart from the slight step in the 500-          These are usually of no consequence for speech or
2000Hz r e g i o n - a convenient result for the        music, but often had dire consequences in the early
ceramic pick-up already described. With magnetic        days of home computers, when storing programs
pick-ups of all varieties, the output is propor-        on a cassette recorder, causing errors on replay
tional to the rate of flux cutting and hence, for a     during program loading.
given amplitude of groove deviation, to fre-               Whether an audio signal originates from a high-
quency. This is restored to a flat response by an       grade studio microphone, a gramophone pick-up
equalization circuit such as that shown in Figure       or a tape replay head, it will be of low amplitude
4.17a.                                                  and will require considerable amplification to
   In tape recording, the playback process is again     bring it up to a usable level. A low-noise pream-
one of flux cutting the turns of a coil; so, for        plifier is employed to raise the level of the signal,
maximum recorded level on the tape, the replay          which is then, if necessary, equalized by an appro-
head output voltage increases with frequency, at        priate disc or tape equalization circuit. Low-noise
least at lower frequencies. Above a certain fre-        audio preamplifier stages may use discrete com-
quency, however, determined mainly by tape speed        ponents, e.g. a low-noise bipolar transistor such as
but also by head design and the nature of the           a BC109C or a FET. However, frequently nowa-
magnetic coating on the tape, the response begins       days a low-noise opamp is employed, such as an
to fall again, giving a playback characteristic as a    NE5534 (3.5VnV/v/-H-z noise) or OP-27 (3.2nV/
function of frequency for maximum recorded level,       v/-H~) (both general purpose low-noise single
before equalization, as in Figure 4.18a. In part, the   opamps), an LM381 low-noise dual audio pream-
problem is that the higher the frequency the            plifier for stereo tape or magnetic pick-up car-
shorter the length of each recorded half-wave-          tridge, or an HA12017 low-noise low-distortion
length on the tape, so the magnetic areas tend to       single audio preamplifier.
demagnetize themselves. The required playback                Having now considered the entire audio chain
characteristic is simply the inverse of the Figure      from input transducer (microphone, disc or tape)
4.18a characteristic, i.e. a 6dB per octave bass        right through to the output transducer (loud-
boost and a rather more vigorous treble boost           speaker or h e a d p h o n e s ) - a word about those
                                                                                  Audio-frequency signals and reproduction   105

                            C4                  +12 V      C3
                            0.22 gF                                    C5
                                                                       0.22 gF

                                                                      C2                 10-
                                                                      0.0056 gF      m    5-
                                                                                     "~ - 5 -
                                            _    _

                R3 1
               50 kf~ I
                                           820 kf/
                                                                                        -20     --
                                                                                                     Frequency (Hz)
                                                                      15 gF
                                                                 _         o0V

Figure 4.17 Equalization.
   (a) The reactive elements in the feedback circuit cause the gain to fall (feedback increasing) with increasing
   (b) Approximate response.


                      ,.~ tq





Figure 4.18
106    Analog Electronics

                      Threshold of feeling        Loudnesslevel (phons)

                120   _••          / ......

                                  ~ ~ 1 2 0 _ . . ~
                                                            /   ..

                                                                                         Input audio signal
                100          '
                     ~--..._--~_- ~ .... ', --, _ 9 " ~ . ~ 0 0 ~      2           G"    ---~o              '      Total resistance R T,
               80 ~ k ~ ' ' ~ - - -         , =-80                   " 0.2                                         linear law
                     ~ ' ~ ~ ~'~""-'.--.----- -70 ---:-'~~_~'_--                   ~             40
               60 ~ N ~ N ~ , , , , - ~ _ _    _ -60 - ~ " J ~ ' -     0.02        ,,,                             To amplifier
           ,-,     - "~"< ~'--~'-                -50 ~ / - "       ~               ~       R =
               4 0 -- .N ~. ~ ' ~ ~ ~~ : ~ 0 ~ ~ ~
                             . ~                                       0.002       ~
                                                                                   ~       RT/IO Y               I and loudspeaker

                 20- "      ,,    /~~'~-----20~._0t"/~'/-j                0.0002
                  0 ~Threshold of .hearin.g. -. ~ _ 1 0 ~ ~
                      ,,    ,,,     . . . . .   .                                                                       o

                      20         100          500 1000      5000 10000                                 "/7"]77
                                       Frequency in hertz

Figure 4.19
  (a) Fletcher-Munson equal loudness contours.
  (b) Compensated volume control provides bass boost at low levels, at frequencies below about f = I/2~CRT.
Additional taps and CR shunt networks may be used to maintain compensation over a wider range of volumes.

important items, volume and tone controls. The                            falls off at both low and high frequencies, but
operation of a volume control has already been                            particularly at the former. Thus with a normal
covered in Chapter 1, so it will not be mentioned                         volume control, reproduction of music sounds
f u r t h e r - except to say that its exact location in the              distinctly thin and cold at lower listening levels.
audio chain is not a trivial matter. The later it is                      A compensated volume control reduces the level of
placed (i.e. the nearer the large-signal back end)                        the low frequencies less rapidly as it is turned
the less chance of it contributing noise to the                           down, approximating to the curves of Figure
output, but the greater the possibility of overload                       4.19a. Thus reproduction at lower levels of volume
in the last stage preceding the volume c o n t r o l -                    retains the warmth and richness of the higher
and vice versa if it is located nearer the front.                         volume levels.
Likewise, tone controls must be suitably located in                          Figure 4.19b shows one way of arranging the
the audio chain. Simple bass and treble boost and                         required characteristic. Instead of the usual loga-
cut circuits have been covered in Chapter 2. The                          rithmic track, the volume control is a linear one
more sophisticated type of control is discussed in a                      with a tap at 40%. This point is shunted to ground
later chapter.                                                            via a series combination of C and R. At middle
    In some amplifiers and music centres, and even                        and high frequencies, R is in parallel with the
in the better class of transistor portable radio, the                     lower section of the volume control, giving it an
volume and tone controls are designed to interact;                        approximately logarithmic characteristic. At lower
this is called a compensated volume control or                            frequencies where the reactance of C is high, the
loudness control. Figure 4.19a shows the reason                           law is still linear. Thus as the volume is turned
for employing this arrangement. At high levels of                         down from maximum, the bass is attenuated less
sound, the apparent l o u d n e s s - measured in                         than the middle and upper notes, giving the
p h o n s - is more or less directly related to the                       required characteristic.
level of acoustic power regardless of frequency,                             Conventional rotary or slider potentiometer
except for a region of greater sensitivity at around                      types of volume control can become noisy in
4000 Hz. At lower volume the sensitivity of the ear                       operation or even intermittent, especially in
                                                                                                                                               Audio-frequency signals and reproduction 107



                                                                                                          50~F                                 620 pF               i

                                                                                              ' ~-~                 - IJ'--
                                                                                                   ~o~                       j~ ,
                                                                                                                        "l         RC


                               %                ~'xl               -1"xl                          I           I
                                    \                '8.o~c , ' x
                                                          ..~ClC: ,, v~                           I           I
                                          \Vcc                                     \                                                         2O

                                                                                         \                                          ~;       40
                                                  '~.                   \~,                   ~
         z                                             "~,               \;~.                     '!,%.                             z= 60

                                                                \x',,.             \( \',,.               'x'X:

                                         2.5               3.5       4.5                          5.5             6.5                              4.0             6.0        8.0 10           15     20         30        40
                                                     V2, CONTROLVOLTAGE(VOLTS)                                                                                               Rc. CONTROL RESISTOR(k OHMS)

                               FIGURE 5 -              FREQUENCY RESPONSE                                                                                    FIGURE 6 - OUTPUT VOLTAGE SWING

                                I                                                                                                             10 -

                                        1 1 Iili
                                         I I llll I llIl I-
                                                               1 lltl      I
                                                                                              ~lll IIU
                   lO-                  I[ i ill I i]lI I                                        ][ 1 llli                                   s.o

                                                [fill          lliil[
                                           Input voltagl (ein) = I0 .mV
                                                                                                                                     o~ 6.0                                                     .1
             <     8.0                                                                                                               o
                                                                                                                                     <              7
             ~.- 6 . 0 - -
                                           Pin 6 uncompensated
                                                11      II l i l l l

                                                                               1               i Il~!                                s
                                                                                                                                        >    4.0
                                                11 ii I Illl ]
                                                11 l[]II[li                                    l[ll                                     o    2.0

                    0[I  100            1.0 k
                                                        10 k           100 k
                                                                                   i   1.0M
                                                                                               liil llIIll
                                                                                                        10M        100M
                                                                                                                                               0   8.0       9.0        10    11     12    13   14   15     16        17
                                                               FREQUENCY (Hz)                                                                                                      SUPPLYVOLTAGE (VOLTS)


Figure 4.20
  (a) Typical DC 'remote' electronic volume control (MC3340P).
  (b) Typical attenuation, frequency response and output voltage swing characteristics
(Vcc = 16 V DC, TA = +25~ unless otherwise noted).
(Reproduced by courtesy of Motorola Inc.)
108   Analog Electronics

lower priced equipment, while the wiring to vol-       Questions
ume controls can be subject to hum pick-up or
                                                        1. The gain of a small-signal transistor amplifier,
other forms of interference. In some audio equip-
                                                           driven from and feeding similar stages, can
ment any possibility of such problems is eliminated
                                                           often be approximated as hfe/2. Explain how.
by using an electronic volume control. This is an
                                                        2. Derive from first principles the maximum
integrated circuit comprising either a variable gain
                                                           theoretical efficiency of (i) a push-pull class
amplifier or a voltage-controlled attenuator. The
                                                           A output stage, (ii) a push-pull class B output
user's volume control then consists of a variable
resistor or potentiometer which varies the voltage
                                                        3. In a stereo reproduction system, why is the
or current at the control input of the electronic
                                                           tone control always designed to operate
volume control IC: this pin can be heavily de-
                                                           equally on both channels?
coupled since the control input is simply a DC
                                                        4. It is not possible to open the feedback loop of
level. The IC can be located on the main printed
                                                           a particular power amplifier, yet it is desired to
circuit board, avoiding any long audio signal
                                                           measure its open loop distortion. Is this poss-
leads, whilst the volume control can be located
                                                           ible, and, if so, how?
on a remote front panel at the end of wires as long
                                                        5. Explain why and how a transitional lag is
as need be. A typical example is the MC3340P (see
                                                           often used in the design of an amplifier with
Figure 4.20).
                                                           heavy overall negative feedback.
                                                        6. Name and describe two different types of
                                                           distortion to which moving coil loudspeakers
                                                           are subject.
                                                        7. In tape recording, is the high-frequency bias
                                                           added to the audio signal, or modulated by the
1. Ultra-Low Distortion Class-A Amplifier. L.              audio signal?
   Nelson-Jones, p. 98, Wireless World, March           8. Describe the operation of the Dolby|
   1970.                                                   system. What level of high-frequency noise
2. 15-20W Class AB Audio Amplifier. J. L.                  reduction is achieved on quiet passages?
   Linsley Hood, p. 321, Wireless World, July           9. Describe the considerations that influence how
   1970.                                                   early or late in the chain the volume control in
3. Bass reflex- acoustical phase inverter- vented          a high fidelity reproducing system should be
   baffle, p. 845, Radio Designers' Handbook, F.           placed.
   Longford Smith, lliffe and Sons Ltd. 4th edition    10. Explain the operation of a compensated
   1953.                                                   volume control.

    5 Passive signal processing
       and signal transmission

Leaving aside the physical transportation of written      communication, and many services originally pro-
messages (this includes everything from the runner        vided by the latter are now carried by the former.
with a message in a forked stick, through carrier         Intercontinental telephone circuits are carried by
pigeons, to the Post Office), there are three main        radio via satellites, and now by fibre optic submar-
ways of transmitting information between widely           ine cables, supplementing the various transoceanic
separated locations. These are communication over         submarine telephone cables. These cables them-
wire lines, that is telegraphy and telephony; wireless    selves were only installed in the years following the
communication, that is radiotelephony, radiotele-         Second World War; prior to that transoceanic
graphy, sound and vision broadcasting; and optical        cables carried only telegraphy, the few interconti-
communication. Of these, by far the oldest is optical     nental telephone circuits being carried by HF radio.
communication. It includes the semaphore tele-            Radio communication in all its forms is so impor-
graph, heliograph and smoke signals for daylight          tant a topic that it is dealt with at length separately
use, and beacon fires or Aldis lamps for use after        in a later chapter.
dark. However, these are all examples of'broadcast'
light signals even if, as with the heliograph and Aldis
                                                          Transmission line communication and attenuation
lamp, there is an important element of direction-
ality. But optical communication is also the newest       It was found in the earliest days of line commu-
form, thanks to high-grade optical fibres or 'light       nication that signals were not transmitted without
pipes' and electro-optic transducers such as light        impairment. This took the form of attenuation: the
emitting diodes (LEDs), lasers and photodetectors.        signals were much weaker at the receiving end of a
These potentially offer huge bandwidth systems            long line than at the sending end. Furthermore,
and, in conjunction with digital voice communica-         higher-frequency signals were found to be more
tions, are rapidly emerging as the new backbone of        heavily attenuated than low-frequency signals,
national trunk telephone networks. These will carry       restricting signalling rates on very long telegraph
both voice and data traffic in digital form over an       lines to an impractically slow speed. Indeed, on the
integrated services digital network (ISDN).               first transatlantic telegraph cable the effect was so
   Practical line communication dates from 1837,          severe as to render it useless: the impatient backers
the original telegraphic systems preceding tele-          insisted on higher and higher sending end voltages
phony by about 40 years. Line communications              in an attempt to get the signals through, with the
form a major topic of this chapter. Practical com-        result that the cable burnt out. The famous
munication by means of radio waves, on the other          Scottish physicist William Thomson was called in
hand, dates only from Marconi's experiments of            and his analysis showed that the original design of
1895, following Hertz's demonstration in 1888 of          cable was doomed to failure due to its enormous
the actual existence of electromagnetic waves (these      capacitance. He further showed that although the
had been predicted by Maxwell in 1864). They have         capacitance could not be substantially reduced, its
now assumed at least equal importance with line           effect could be compensated by the addition of an
110   Analog Electronics

appropriate amount of distributed series induc-           conductor to screen the other, providing good
tance. In 1866 a new transatlantic cable incorpor-        isolation from electrostatic pick-up but less protec-
ating his ideas was completed and proved him              tion from unwanted magnetic coupling. This is
right. The backers got a cable that worked, and           called a coax&l cable and is illustrated in Figure
William Thomson got a knighthood and later a              5.1c. Here one of the two conductors is tubular
barony as Lord Kelvin of Largs.                           and completely surrounds the other. The induc-
   Figure 5.1a shows a simple telegraph system            tance and resistance of the outer are smaller than
suitable for the transmission of information by           those of the inner, so the line is usually represented
Morse or one of the other telegraph codes. The            by the unbalanced equivalent circuits shown. One
detector could be a buzzer or a bulb (either              is a ~ section, being directly equivalent to the bal-
connected directly or via a sensitive relay and           anced ~ or 'box' section of Figure 5.1b; the other is
local battery), or an 'inker' to record the 'marks',      a T section, which is the unbalanced equivalent of
e.g. dots and dashes, on a moving paper tape. In          the H or balanced T section (not shown). Of
Figure 5.1b a very short section of the two-wire line     course, neither the T nor the ~ circuit is really
is shown, together with its equivalent circuit. Each      equivalent to the cable, where the impedances R,
of the two wires will have series resistance and          L, C and G are continuously distributed, not
inductance: furthermore, there will be capacitance        lumped as shown. But at frequencies where the
between the two lengths of wire and, at least in          phase shift through the section is very small
principle, there may be a resistive leakage path          compared with a radian, the equivalent circuit
between them also. This is indicated by two               behaves exactly like the real thing.
conductances G/2, where G is the reciprocal of               Now owing to the resistive components R and G
the leakage resistance between the wires. A two-          per unit length, the signal will be attenuated more
wire system as shown in Figure 5.1b is usually            and more the further it travels along the line,
balanced, that is to say that at any instant the          u n t i l - if the line is long enough - the energy fed
voltage on one line will be as many volts positive        in at the input is virtually all dissipated before
with respect to earth as the voltage on the other is      reaching the other end. The line is thus effectively
negative, and vice versa. The advantage of this           infinitely long, so that whatever happens at the
arrangement is that if interference is picked up on       receiving end can have no effect on the sending
the line, either by magnetic coupling or electro-         end; it will be impossible to tell from the sending
static (capacitive) coupling, this will appear as         end whether anything is connected to the receiving
equal in-phase voltages on the two wires, and it          end or whether it is open- or short-circuited. To
is called longitudinal noise, noise to ground or          get the maximum signal energy into the line, it is
common mode noise. The interference is a 'push-           necessary to drive it from a matched source, so one
push' signal, unlike the wanted balanced or 'push-        needs to know the infinite line's input impedance
pull' signal, which is called a transverse, metallic or   Z0. It turns out that
normal mode signal. The receiving end circuitry is
usually arranged to respond only to the latter,
whilst rejecting the former. This is easily achieved                      Zo-        e+j
by coupling the signal at the receiving end through
a 1:1 ratio transformer whose primary is                  which has the dimensions of ohms and is in general
'floating', i.e. its centre tap is not grounded.          a function of frequency. This is inconvenient, since
Alternatively, the receive end may use the                to match it one would need a generator whose
common mode rejection provided by an active               source impedance also varied with frequency. At
circuit incorporating operational amplifiers.             zero frequency, Z o = x / ( R / G ) . At very high
   One of the commonest forms of common mode              frequencies, where jcoL >> R and jcoC >> G,
noise is mains power frequency hum. Pick-up is            Zo = v/(L/C)       (see Figure 5.1d). It" now
usually localized and can be minimized by using a         L / R = C/G, Zo is independent of frequency: this
twisted pair. An alternative cable design uses one        is called the distortionless condition. In practice this
                                                                                      Passive signal processing and signal transmission                                 111

                                  ~ t t                                            overhead line                               ~ ; y

          Batte~     Detector~

                                                                       (or single wire plus earth return)

                                                                                      R/2            L/2
                                                                   C     A     A

                                  (b)                      =

                                                                                      R/2            L/2

                                        R                      L                                    R/2            L/2                       R/2          L/2

               ::    ~.                                                               cr2                                 a           "r'C
                                                 section                                t)'c"                                 T section

               Izd                                                             Izd
                                                                                                                              Z*o~ i


                                                                                                                              -20 ~
                                                                                                                              _30~    f
                                            '"    ""-                                       I   I    1     1   I    l"-               - 'I    I    i 'i    i"   i ~-y
                                        Frequency                                     0 10 20 30 40 50 60                             0 10 20 30 40 50 60
                                 (d)                                                     Frequency (kHz)                                Frequency (kHz)
Figure 5.1     Transmissionlines.
  (a)   Simple telegraph system.
  (b)   Two-wire line: balanced rc equivalent of short section.
  (c)   Coaxial line: unbalanced equivalents of short section.
  (d)   Variation of characteristic impedance Z0 of a line with frequency.
  (e)   Frequency and phase distortion for an air line.
112   Analog Electronics

condition is not met, since the conductance G             propagate along the line at a finite velocity: this
between the two conductors is usually insignif-           cannot exceed the speed of light but is not much
icant, so that all the loss is due to R. G could be       less, at least in the case of an air spaced line. Hence
artificially increased by bridging resistors across       the peak voltage at some point down the line will
the line at intervals, but that would double the          occur a little later in time than at the send end: the
a t t e n u a t i o n - which doesn't seem a good move.   greater the distance x, the greater the delay or
Fortunately, the dilemma only exists at low fre-          phase lag. This can be expressed by e -j~x where 13
quencies, as Figure 5.1e, relating to an air line,        is the phase constant per unit length of line. The
shows clearly. Nowadays audio-frequency air lines         wavelength ~ in metres along the line, i.e. the
are only used for short connections, such as the          distance between points having a phase difference
last few tens of metres of a domestic subscriber's        of 360 ~ is simply given by ~ = 2n/f3, assuming
telephone line, from an overhead distribution pole        that 13 is expressed as radians per metre. So the
to the house. In a multipair underground cable,           voltage vector at any point x along the line relative
such as from the distribution pole to the local           to the sending end voltage Es, and phase can be
exchange, the two wires of each pair are twisted          expressed in polar ( M / ~ ) form as
together; the result is that the capitance C between
them is very much larger, whilst the inductance is             Ex = Ese-~e -j~x = Ese -(~                = e-WEs
much lower, than for an air spaced line.
                                                          where the complex number 7 is the propagation
Consequently, at audio frequencies jcoC >> G but
                                                          constant. A little tedious algebra which can be
R >> jcoL, so that here Z0 varies with frequency
                                                          found in textbooks ancient I and possibly modern
since (approximately) Z0 = (R/jcoC). The fre-
                                                          (although transmission lines was always a scanda-
quency distortion arising from the variation of
                                                          lously neglected subject) enables one to express 7
Z0 can be largely removed by artificially increasing
                                                          in terms of the primary line constants R, L, C and
the inductance per unit length of the line. This is
                                                          G per unit length. It turns out that
achieved, on twisted pair lines which are long
enough to make such measures necessary, by                            y = v/[(R + jcoL)(G + jcoC)]                 (5.2)
inserting inductors in series with the conductors
at intervals: these are called loading coils. Clearly,    The term 13necessarily varies with frequency but, if
a loaded balanced line closely approximates to the        the magnitude of the resistive terms is much less
equivalent circuit of Figure The loading            than that of the reactive terms, then
results in Z0 being nearly constant with frequency.
                                                                      y = et + jr3 ..~ 0 + jcov/(L/C )             (5.3)
     However, in the absence of loading, variations
in Z0 are not the only cause of signal distortion.        i.e. the attenuation is negligible. The velocity of
To understand why, one must look at the other             propagation v is given by v =f~., i.e. frequency
secondary line constant, the propagation constant,        times wavelength, but k = 2n/13 so
denoted by 7. This, like Z0, is determined by the
primary line constants R, L, C and G per unit                               2nf     co          1
                                                                        v = - - ~ = -~ : x / ( L / C )             (5.4)
length. The propagation constant is a complex
number which describes the variation of amplitude
                                                          Now at last one can see why the first transatlantic
and phase of a sinusoidal signal along the length of
                                                          telephone cable was a failure. Telegraph signalling
the line. Owing to the losses, the amplitude of the
                                                          speeds are down in the frequency range where R is
signal will decrease exponentially along the line. If
                                                          significant, and, with the unloaded coaxial struc-
the sending end voltage is 0.707V RMS at some
                                                          ture, L was small, C was large and Z0 was conse-
frequency co rad/s, the peak sending end voltage E
                                                          quently very low compared with R. Further, the
will be just unity. The peak voltage at any distance
                                                          distortionless condition was not met as G was very
x metres along the line can then be simply ex-
                                                          small. So there were three separate problems:
pressed as e -ax where 0~is the attenuation constant
per unit length. Furthermore, the signal energy will      1. The characteristic impedance Z0 varied with
                                                          Passive signal processing and signal transmission           113

   frequency, so that the source could not be                large. If the length l is only a centimetre or so, L
   matched at all frequencies.                               and C will be very small; so Z0 and a will be
2. The attenuation constant a increased with fre-            constant and determined by R and G, while 13will
   quency attenuating dots more than dashes.                 be negligible, up to a few hundred megahertz.
3. The phase constant 13 was not proportional to             What you have in fact is an attenuator or pad.
   03, so that different frequency components                Its Z o = v / ( R / G )        and (13 being negligible)
   travelled at different speeds; dots and dashes            7 "~ ot = v/(RG). Discrete components can be
   arrived in a jumble, on top of each other.                used as in Figure 5.2a and, provided the attenua-
                                                             tion a is small, the total series resistance and shunt
                                                             resistance can be very simply related to R and G as
Resistive line sections: pads
                                                             shown. For this to apply, the attenuation must not
At radio frequencies the reactive primary line               exceed a decibel or so. Now 0t defines the attenua-
constants C and L predominate, so that Z0, 0t                tion of a length of line in exponential terms. The
and 13 ought to be constant; but of course things            loss in a line of length x where the attenuation
are never that simple. In particular, just because           constant is 0t per unit length is ax, i.e.
jcoL >> R and jcoC >> G, that doesn't mean that 0t           Ex/Es = e -~x. If ax is unity, the output is just
should be negligible, only that, like Z0, and co/13, it      e -1 times the input voltage or 0.368 per unit. This
should be constant with frequency. Unfortunately             is an attenuation of 1 neper, whereas we normally
even that is wishful thinking, for the resistance of a       work in decibels. An attenuation of 1 decibel
piece of wire is not constant but rises with                 implies an output voltage of 0.891 times the
frequency due to skin effect. This effect is due to          i n p u t - always assuming that, as here, one is
inductance and results in the current being                  working with a constant value of Z0 throughout.
restricted to the outer part of the wire's cross-            Eight 1 dB pads in series would provide just less
section as the frequency rises, eventually flowing           than 1 neper of attenuation, nine pads a little
entirely in a very shallow layer or 'skin'. For this         more; in fact 1 n e p e r = 8.686 decibels, since
reason the wire of high-frequency coils and RF               (0.891)8686= 0.368. Clearly you could obtain a
transformers is often plated with silver, or even            large a t t e n u a t i o n - say 20dB or 2.3 n e p e r s - by
gold. (Gold is a poorer conductor than silver, but           connecting the required number of 1 dB pads in
the surface of the latter can tarnish badly. The             series, simulating a lossy line, but it is obviously
high-frequency losses due to skin effect in silver           much more economical to use just three resistors in
can then exceed those in gold.) Furthermore, in a            a single T or n pad. However, the R and G
coaxial cable with solid dielectric, even if the             elements can then no longer, by any stretch of
dielectric constant is reasonably constant with              the imagination, be regarded as continuously dis-
frequency, there will be an associated series loss           tributed, so the design formulae change. To stick
resistance which increases with frequency; hence             with a in nepers, use the design formulae of Figure
foamed dielectric or partially air spaced feeders are        5.2b. To work in decibels, start by calculating the
used at UHF. The theory to date has only allowed             input/output voltage ratio N from the required
for a shunt loss G, such as might be experienced at          attenuation of D dB. Thus N = 10 D/2~ and sub-
the supports of an open wire line on a wet day, not          stituting N in the formulae in Figure 5.2c gives the
for a loss component in series with the capacitance          required resistor values. Note that if the voltage
C. In short, the secondary line 'constants' Z0 and           (or current) ratio is very large, then (1) the
can vary with frequency, even if the primary line            coupling between input and output circuit must
'constants' really are constant, which in any case           be very small, and (2) looking into the pad from
they aren't. Nevertheless, expressions (5.1) and             either side one must see a resistance very close to
(5.2) for Z0 and 7 are exceedingly useful and can            R0 even if the other side of the pad is untermi-
take us a long way.                                          nated. For if very little power crawls out of the far
   Consider for example a very short length of line          side of the pad, it must mostly be dissipated on this
in which R and G have deliberately been made                 side. Thus, when N is very large, (1) Rp in a T
114     Analog Electronics

                 R/2            R/2                          R                                            Rs            Rs

                            G                   G/2
                                                                                     Zo=Ro                                              =Ro
                                                                                     each way o                   -               ~.~
                        T                  (a)                                                                   (b)

                                          N-1                                                                  ,.. ( N 2 _ l "]
                                                                                                         Rs= KoCS~)

                                                                                                            N+I'~ =
                    "~'R0-~                                        "q"R0"~                          Rp= R 0 ~ j     Rp                   "4-'R0-'~
                        0               -              , 0                                 0   ..                                  ., -0
                                      T pad                                  (c)                               x pad

                                  Sla I           !      ]Slb                                        [                  ]
                  Coaxial                                                          .....

                                            Screen                                                        Screen
                                                1 dB             2dB   2dB 5dB 10dB 20dB 20dB
Figure 5.2 Attenuators.
  (a)   Lossy line section used as an attenuator. Zov/(R/G) and 0t = v/(RG), only true for small a (in nepers: see text).
  (b)   Attenuator design in exponential form: Rs = R0 tanh 0c/2, Rp = R0/sinh 0t, true for all a (in nepers).
  (c)   Attenuator design in terms of input/output voltage ratio N: attenuation D = 20 log10 N dB.
  (d)   0 - 6 0 dB attenuator with I dB steps.

circuit must be almost zero, and Rs in a rc circuit                                useful purpose other than wasting power? There
almost infinity, and (2) Rs in a T circuit will be                                 are many reasons, such as adjusting the level of the
fractionally less than R0, and Rp in a n circuit                                   signal in, for example, a TV camera video signal
fractionally higher than R0. In fact, as can be seen                               path: if the level is too high and the gain of the
from Figure 5.2c, the Rs in a T circuit is the                                     preceding amplifier is not adjustable, a suitable
reciprocal of the Rp in a rc circuit (in the sense                                 attenuating pad can be employed. Or, when the
that RsRp = R~) and vice versa, for all values of N.                               level of a signal is being measured, it can be
    So that's how to design a pad of any given                                     attenuated until it is equal to some standard test
characteristic impedance, for any given attenua-                                   level such as 0 dBm, i.e. one milliwatt. The level of
tion, though in practice it will only be necessary to                              the original signal is then + D dBm, where D dB is
do so if a non-standard value of attenuation is                                    the attenuation of the pad. Similarly, the gain of
required. Normalized resistor values for all                                       an amplifier can be measured by attenuating its
c o m m o n pad values are tabulated in Appendix 3.                                output until the level change through the two
But what is the p o i n t of networks that serve no                                together is 0 dB - no net gain or loss. The gain
                                                        Passive signal processing and signal transmission 115

in decibels then simply equals the attenuation of          loss of the pads would limit the measuring cap-
the pad. However, measurements like this presup-           ability undesirably. In this case it would be better
pose that the attenuation of the pad can be set to         to use minimum loss pads. Figure 5.3b shows that
any known desired value at will whilst keeping Z0          for a 1.5:1 impedance ratio such as 50 and 75 f~,
constant. Whether using a T or art pad, to achieve         the minimum loss is about 6dB. Whatever the
this, it would be necessary to adjust three resistors      impedance ratio, the minimum loss pad will turn
simultaneously, following a different non-linear           out to be a two-resistor L type network. That is, in
law for series and shunt resistors. Such attenuators       Figure 5.3a, if R2 > R1 then RA in the T pad will
have been produced, but a much more common                 be zero or Rc in the ~t pad will be infinity, whilst
solution is to use a series of fixed pads and connect      the remaining series and shunt resistors of course
them in circuit or bypass them as necessary as             work out the same using either the T or the rc
shown in Figure 5.2d. This type of attenuator,             formula. There is no reason why you shouldn't
properly constructed with miniature switches and           work out the resistor values for a mismatch pad
resistors, would be usable up to VHF. Screens              with less than the minimum loss: 0 dB, say, would
would not be necessary for the lower attenuation           be a very convenient value. The difficulties only
steps, but they would be essential for the 10 and          arise at the practical stage, for one of the resistors
20 dB steps to prevent stray capacitance shunting          will turn out to be negative.
Rs and thus reducing the pad's attenuation at very
high frequencies. The switches might be manually
                                                           Reactive line sections: delay lines
operated toggle switches, or might be operated in
sequence as required by cams on a shaft to provide         The basic transmission line equations (5.1) and
rotary knob operation. The electrical design would         (5.2) were used to see under what conditions the
be such as to ensure that with 0dB selected, the           line is distortion free and under what conditions it
internal wiring exhibited throughout a character-          can be represented by a string of sections with the
istic impedance Z0 equal to R0, usually 50 f~.             series and shunt impedances lumped: this turned
   Another use for pads is to enable accurate              out to be that 0t and 13 per section must both be
measurements to be made where two different                small. Then, looking at purely resistive sections
impedance levels are concerned, without errors             and extending this to the case where a is large, led
(or the necessity for corrections) for mismatch            to different design equations. Now let's look at
losses. For example, one might want to measure             purely reactive sections (~ = 0), both distributed
the gain at various frequencies of an amplifier            and short lumped ([3 small), and then extend this to
with 75 f~ input and output impedances, using              the case where 13 is large.
50 f~ test equipment. A 50 f~ to 75 f~ mismatch               Starting with the case where 13 is small, (5.3)
p a d - it should really be called a matching pad,         showed that 13= cox/(L/C) and (5.4) that the
as it is an anti-mismatch p a d - would be used at         velocity v = 1/v/(L/C) was independent of fre-
the amplifier's input and another pad (the other           quency. Thus different frequency components of
way round) at its output. The design equations             a non-sinusoidal waveform will all arrive unatten-
for a mismatch pad are given in Figure 5.3a.               uated at the end of a line as in Figure 5.4a, and in
Note that in this case N is not the input/output           the same relative phases as at the sending end. For
voltage ratio but rather the square root of the            any component of frequency o3, the phase delay
input/output power ratio, since R 0 i n does not           will be 13= o3x/'(LC) radians per unit length, i.e.
equal R0 out. D can be chosen to be a convenient           linearly proportional to the frequency. So at 0 Hz,
value such as 10 dB, making N = 10 10/20 = 3.162.          the phase shift per unit length of line or per section
Allowing for both pads, the gain of the 75 f~              (T or rc) will be 0 ~ You might think prima facie
amplifier will then actually be 20dB more than             that if the output is in phase with the input, the
the measured value.                                        delay through the section must be zero - but not
   If using the above set-up to measure the stop           so. From (5.4) v = 1/x/(LC), and this is indepen-
band attenuation of a 75 f~ filter, the extra 20 dB        dent of frequency co. So the time taken to travel
116   Analog Electronics

                                                T pad                                                                it pad
                                           RA               RC                                                        RB

                                                        B        "4"R2"~                                .~-. RI-.~              . ~ R2-.~


                                 30-                                                              -30
                     ~' 2 5 -                                                                     -20
                        20                                                                         l0
                         o                                                                                N
                     .~ 10
                                  5                                                                2
                                  0             I   I   ! I IIII!             I   !   I   1 ! I
                                       1        2 3 4 56 810         20 30 40 60 80
                                                Impedance ration R l/R 2 or R21R l
  (a) Mismatch pads.
RB = 2RON2 _ 1

        N2 .-b 1                         N                            T pad
RA = R1 N2---- -- 2Ro                  N 2- 1                         Ro = v/(R1R2)
           N2+l                          N
Rc = R2 N2-----      - 2Ro             N 2- 1
      RoN 2 - 1
RB=    2     N
              N 2 -- 1                                                    Ro = v/(RxR2)
RA = R1
           N 2 - 2NS + 1                                                  S = v/(R1/R2)
                  N 2-       1
R c = R2
           N 2 - 2(N/S)+               1

  (b) Minloss pads.
Figure 5.3

unit distance is 1 / v = v / ( L C ) seconds, however                                       sometimes referred to as the D C d e l a y , being the
low the frequency. It's true that as co becomes                                             limit of 13/r as co tends to zero. This is illustrated
infinitesimal, the phase -13 at the output of the                                           in Figure 5.4b. The limit dl3/dr is usually called
unit length or section relative to the input becomes                                        the g r o u p d e l a y and is the reciprocal of the velocity
infinitesimally small also. But co is the signal's rate                                     v. For a uniform lossless transmission medium, v is
of change of phase ~ with t, so ~ at the output                                             independent of frequency, i.e. the group delay is
only advances through 13 to 0 ~ at an infinitesimal                                         constant. The group delay of free space is
speed. The time delay x / ( L C ) seconds is in fact                                        3.33 x 10 -8 seconds per metre, i.e. the reciprocal
                                                         Passive signal processing and signal transmission     117

                           L       L        L             L          L

                       ~                                00000                         Zo = .~/~/-Ic )

                       Tc TcTcT                                                      ~= Co~(LC )





                ~l,,.,~                               Slope : ~   = x/,(LC)
               ~ "~                                                       n or unit length

                       ~                     (b)

Figure 5.4
  (a) Delay line.
  (b) DC delay and group delay.

of the free space velocity of radio or light waves.           may be required and the length of cable needed
The wave velocity in a balanced open wire trans-              can then become an embarrassment. So special
mission line, such as the feeder used to connect a            delay cable may be employed; this is constructed
high-power HF transmitter to a balanced antenna,              so as to have an unusually high value of [3. The
is about 2.94 • 108 m/s or 98% of the speed of                time delay per unit length can be increased by
light. In a coaxial cable with solid dielectric the           increasing L and C per unit length, e.g. by using a
wave velocity is in the range 60 to 70% of                    spirally wound inner conductor in place of a
3.0 • 10Sm/s, whilst for an underground tele-                 straight wire. A typical application is to delay the
phone cable with lumped loading it may be                     output of an oscilloscope's Y amplifier on its way
0.15 x 108 m/s or thereabouts, only one-twentieth             to the deflection plates. This enables one to see on
of the speed of light.                                        the screen the very same rising edge which trig-
   In all these examples the delay experienced by             gered the sweep. In early oscilloscopes, when a
the signal is incidental and of no importance, but            delay line was provided at all, it was composed of
in many applications delay is introduced into a               discrete LC sections and was bulky and expensive
transmission path deliberately. For example, by               since a large number of sections was required to
splitting the output of a transmitter and feeding it          provide an adequate constant signal delay, even on
to two spaced antennas via different length feeders,          an oscilloscope with a bandwidth of only 50 MHz,
so that the radiation from one antenna is out of              then the state of the art.
phase with the other, the radiated power can be                  Discrete section delay lines are still used in many
concentrated in a desired direction. For this pur-            applications, a typical example being the tapped
pose the delay can be provided by ordinary coaxial            delay lines employed to permit time alignment of
cable. For other purposes a rather longer delay               the various component signals of a colour TV
118     Analog Electronics


                                                                                                Z0             I
            <                                                        I
             0                                                       I                           0
                 0              c           oo             0                                         0        fc
                           Frequency                              Frequency                                   Frequency
                              (a)                                    (b)                                            (e)

                                                                           tan 1 / 2 ~ ~ t a n - I                                1110
                                                                               /       ~      ~
                                                                         ~~~~,.,..----,,---"- _               -'L     ~ ic2 = 1/20

                 ii             iL    XL =j 0.1       io = 1 A peak           tan-] 1/20                            io = 1
                 =- -            ~- ~             -     ~                                   Current vectors
                    I                                            eo =                                         ei a,,L/       eL~ 0.1
                          iCl                             A

                                            C/2 ,,~             RL=' ~                                                ~ tan_l 1/20

        o            ..
                                                                                 tan-1 1/10                         eo = 1
                                                                                           Voltage vectors

Figure 5.5 Delay line or simple low-pass filter: variation with frequency of
  (a)   attenuation
  (b)   phase shift
  (c)   characteristic impedance
  (d)   Operation at f~/10.

waveform. As in the oscilloscope application, for                          5.5b, indicating an increasing group delay. Beyond
this purpose also it is essential that all frequency                       2/v/(LC) the phase shift remains constant at
components are delayed in time by the same                                 radians per section, whilst the attenuation in-
amount, i.e. that the phase delay per section is                           creases. Thus a delay line composed of discrete
strictly proportional to frequency corresponding                           lumped LC sections behaves as a low-pass filter, in
to a constant group delay. With a delay line                               contrast to a line (such as coaxial cable) where L
consisting of a series of n sections as in Figure                          and C are continuously distributed. A lumped
5.4a this is approximately so only when 03 is much                         delay line will thus distort a complex signal
smaller than 2/v/(L/C). As o~ approaches                                   unless its cut-off frequency is much higher than
2/v/(L/C), the phase shift per section increases                           the highest frequency components of appreciable
more rapidly with frequency as shown in Figure                             amplitude in the signal waveform.
                                                           Passive signal processin9 and signal transmission    119

                                                              worth looking at this in a little more detail.
                                                              Figure 5.5d shows the situation in an LC low-
 An important aspect of passive signal processing is          pass n section at a frequency well below cut-off,
 filtering. Filters are used to limit the bandwidth of        where ]XL] = 1/10 of the load resistance and
 a signal, e.g. before applying it to a transmission          [ X c l - 10 times the load resistance. Everything
 system. For example, the bandwidth of voice                  else has been normalized to unity for simplicity,
 signals in a telephone system must be limited to             working back from the output. From the current
 fit within the allocated 4 kHz channel of a fre-             vector diagram, iL = 1 + j0.05. This enables
 quency division multiplex (FDM) group. (A group              ei = eo +eL to be marked in on the voltage
 is a block of twelve 4 kHz telephone channels                vector diagram, and hence ii = iL + iCl can be
 multiplexed to occupy the frequency range 60-                marked in on the current vector diagram. As
 108kHz.) The same requirement to eliminate                   you can see, ei--eo and ii = io to a very close
 energy at 4kHz and above applies equally to                  approximation, so the attenuation is zero and
 baseband voice signals which are applied to a                the phase delay is tan -1 1 / 1 0 = 5.7 ~ (one-tenth
 digital modulator in a pulse code modulation                 of a radian). Note, however that iL, the current
 (PCM) multiplexer, for application to an all-digital         in the inductor, is slightly greater than ii and io.
 telephone trunk circuit. In these cases, a low-pass          This is because of a small additional component
 filter is required. Other applications require a high-       of current circulating round the LC circuit; this
 pass filter, which passes only components above its          represents stored energy and is the Achilles heel
 cut-off frequency, or a band-pass filter, which              of LC filters. It is responsible for the group
 passes only frequencies falling between its lower            delay increasing as the cut-off frequency is
 and upper cut-off points. Also met with are the              approached. Meanwhile you can see that the
 band-stop f i l t e r - which passes all frequencies         input impedance is resistive when RL (the load
 except those falling between its lower and upper             resistance of the section) is the geometric mean
 cut-off frequencies- and the all-pass filter. The            of the impedance          of L       and    C, i.e.
 latter has no stop band at all, but is employed on           v/(XLXc) -- RL = characteristic impedance Z0.
 account of its non-constant group delay, to com-             Hence
 pensate delay distortion introduced by some other
 component of a system: it is often called a phase
                                                                 Zo =             ~
                                                                           jcoL-j-O-   =             = 4(L/C)
    The frequency response of the ideal low-pass
filter would be simply rectangular, i.e. it would
 pass all frequencies up to its cut-off frequency             If both L and C have some associated loss
 with no attenuation, whilst frequencies beyond               resistance (series resistance R in the case of L
 cut-off would be infinitely attenuated. In a prac-           and shunt conductance G in the case of C), the
 tical filter, the stop band attenuation only rises           section's input resistance will still look purely
 gradually       beyond     the   cut-off     frequency.      resistive if both L and C have the same loss
 Furthermore, in practice, the pass band does                 angle, i.e. tan -1(R/coL) = tan -I(G/coC). If you
 not display a sharp corner at the cut-off fre-               redraw the vector diagrams to a larger scale with
 quency as shown in Figure 5.5a, since for                    XL replaced by R + jcoL and Xc replaced by
 convenience one uses a fixed value of R0 for                 conductance G in parallel with the reactance of
 the terminations, where R0 = v/(L/C). However,               C, you will find that ii is in phase with ei and
just as the group delay is constant only for                  Z0 becomes v/[(R + jcoL)/(G + jo~C)], the distor-
 values of co<< 2/v/(LC), the input impedance                 tionless condition for a cable which was quoted
 of an L C n section is only equal to v/(LC)                  earlier. However, now ei is greater than eo, even
 with the same limitation, rising to c~ at the                at very low frequencies, showing that the filter
 'cut-off frequency COc, i.e. at 2 / x / ( L C ) Hz, and      section has a finite pass band loss. This is
 becoming reactive above this frequency. It is                usually assumed to be negligible in filter design,
120   AnalogElectronics

        c~o0000~O0000~                       ~      0t                             d

                                             <                                               log ca



               ~ 5 t h          order                                                        log to


                                                          dB~      1 dB tipple
                                                           0 ~L    V
                                                          -3 i    -T

                                                                                       dB octave

                                                                         ,                ~ log co


                                                                                          ..~ log to
                          5th order         (b)


            oT T To                                                                                    y

                             5th order                                                                 log
                                                          Passive signal processing and signal transmission         121

                          Jl                   J,                                       jco
                      i " l                 1 " 1                                 co5

                0T                 T                     To

                  i            ,, \
                               %      co4           m5
Figure 5.6 Passive low-pass filters.
  (a) Butterworth.
  (b) Chebyshev.
  (c) Bessel.
  (d) Elliptic.

but becomes important in high-order filters with              its pass band, since the response of such a filter will
sharp cut-off, e.g. Chebyshev and elliptic types.             be less affected by deviations from R0 in its driving
   The variation of Z0 over the pass band was                 source and load impedances. Nevertheless, m
studied by Zobel, 1 who derived a filter design               derived filters are now of purely historic interest,
procedure which minimized the problem and at                  modern filter design taking account only of a
the same time permitted a much more rapid                     filter's insertion loss versus frequency. The inser-
increase in attenuation at frequencies just above             tion loss (or gain) of any two-port network, passive
coc. This is achieved by introducing finite zeros,            or active, is defined as the resultant change of level
that is to say frequencies above C0c where the                at the receiving end of a measuring system when
attenuation becomes infinite, in addition to the              the network is introduced into the system's trans-
infinite attenuation occurring at infinite frequency.         mission path such that the network is working
Zobel filters use one or more basic constant K L C            between resistive source and load terminations,
sections (like a lumped delay line) together with             usually 50 f~. Often the network is inserted be-
specially modified m derived sections, m being a              tween two 50 f~ 10dB pads to ensure that the
parameter between 0 and 1, terminated with rn                 source and load are very close to the ideal, whilst
derived half-sections. Terminating half-sections              if the network has a nominal impedance other than
with m = 0.6 result in the value of Z0 remaining              50 f~ then mismatch pads may be used as described
within +4% and - 9 % of v / ( L / C ) , the low-              earlier. Modern filter design reflects various char-
frequency value, right up to 0.9c%. However,                  acteristic pole-zero plots which very conveniently
whilst the resultant filter corresponds more closely          describe the performance of both passive and
to the ideal 'brick wall' filter shape, the group             active filters. Many different basic types are used,
delay variations in the pass band of an m derived             of which the following are some of the most
filter are substantially worse than in the prototype          important (see Figure 5.6).
constant K filter.                                                The Butterworth or maximally flat amplitude
   There is some real virtue in a filter which                filter is a good general purpose design (Figure
presents a good match over the greater part of                5.6a). It has a flat pass band; a reasonably fast
122   Analog Electronics

cut-off, especially if a high-order design (many             An important point to bear in mind when
sections) is used; and a group delay characteristic       comparing a Butterworth design with Chebyshev
which may be acceptable (in those cases where it is       or elliptic filters is the different definition of cut-off
important) if a fairly low-order design is adequate.      frequency. For low-pass filters with a fiat pass
   The Chebyshev filter provides a sharper corner         band amplitude response, such as Butterworth
at the cut-off frequency; the price paid is small         and Bessel, the upper limit of the pass band is
ripples of attenuation in the pass band and a             taken as that frequency at which the attenuation
severely degraded group delay characteristic. As          has risen to 3dB. For Chebyshev and elliptic
can be seen from the pole-zero plot in Figure 5.6b,       filters, the cut-off frequency is the highest fre-
the poles of a Chebyshev filter are on an ellipse         quency at which the attenuation equals the at-
displaced to the right relative to those of a             tenuation at the troughs of the ripples: this is
Butterworth, which are equally spaced around a            clearly shown in Figure 5.6.
semicircle in the -ty half of the diagram. Thus              The elliptic (or Caur) filter (Figure 5.6d) has
attenuation due to the pole on the -c~ axis sets in       both ripples in the pass band like a Chebyshev
earlier, the response being held up at higher             filter, and finite zeros in the stop band like an m
frequencies by one or more pole pairs which               derived filter. Like the Chebyshev filter, it is not
have a higher Q (nearer the jc0 axis) than the            suitable for use with complex signals requiring a
corresponding pairs in a Butterworth. The resul-          constant time delay for all frequency components,
tant pass band ripples are due to mismatch loss as        i.e. where waveform preservation is important.
the filter's impedance varies with frequency. The         The elliptic filter design, with the aid of finite
mismatch loss depends upon the reflection coeffi-         zeros in the stop band, offers the fastest cut-off
cient p (lower-case Greek letter rho); so, for ex-        of any LC filter. However, each of these zeros is
ample, a ripple depth of 0.1 dB corresponds to            bought at the price of 12dB/octave reduction in
15% reflection, whilst p = 2 5 % corresponds to           the final rate of cut-off. It is thus convenient to use
0.28 dB ripple depth. Note that for an even-order         a design with an odd number of poles, not only on
filter the 0 Hz response occurs at the trough of the      account of its equal source and load impedances,
ripple, so the filter must be mismatched. This is         but also to ensure that eventually the attenuation
achieved by working the filter between unequal            increases to infinity rather than remaining indefi-
source and load impedances, resulting in mismatch         nitely at As. The inverse hyperbolic filter has a stop
loss at 0 Hz, whilst at the peaks of the ripple it acts   band like the elliptic filter but a fiat pass band, i.e.
as an impedance transforming filter, matching the         the ripple depth is 0 dB.
different source and load impedances to each                  If it is desired to limit the bandwidth of a square
other. At frequencies well into the stop band, the        wave or pulse train whilst introducing the mini-
Chebyshev filter has the same rate of cut-off as a        mum of waveform distortion, then a Bessel filter,
Butterworth, i.e. 30 dB per octave for the five-pole      with its maximally linear phase and fiat group
filters shown in Figure 5.6. The 'faster' cut-off of      delay, is a better choice, even though the rate of
the Chebyshev is in reality just an earlier cut-off,      attenuation increase in the stop band is much
its stop band characteristic approaching the 30 dB/       lower than for other filter types (Figure 5.6c).
octave asymptote from above rather than, like the             IfL and C are interchanged as in Figure 5.7a, then
Butterworth, from below. Projected back, the              a high-pass filter results. It shows infinite attenua-
asymptote cuts the 0dB level well down the pass           tion at 0 Hz; the attenuation then falls until, above
band, rather than at the corner (compare Figure           r the pass band has zero attenuation. If the series
5.6b and a). Like the Butterworth and Bessel              elements are series tuned circuits whilst the shunt
filters, the low-pass Chebyshev design is an all-         elements are parallel tuned circuits, a bandpass filter
pole filter, the number of poles equalling the            response is produced (Figure 5.7b). Interchanging
number of reactive components. Indeed, these              the position of the series and parallel tuned circuits
three filters all have the same circuit diagram, the      gives a band-stop response. Both band-pass and
only difference being in the component values.            band-stop filters of this type are narrow band de-
                                                                 Passive signal processing and signal transmission            123

                   C                           dB

                                      o tyi
        ~    2L          2L
                                      18dB _ _ ~ - ~ -

                                                                                  log o
                                                                                                  -    1    ~            "-

                                                                                                       Triple zero at S = 0
                                                                                                       (~ = o, jco = 0)



         o     I

                                                                                                riple zero, hence
                          ~, [ ~ 3dB bandwidthh                                                  ore zeros at infinity
                              !                     ,.- log co


              /.4~--------18 dB/octave ------~ X
Figure 5.7
  (a) Three-pole high-pass filter.
  (b) Six-pole band-pass filter (three-pole low-pass equivalent).

signs, suitable for a small percentage bandwidth                          As with attenuators, the modern electronic
only, say half an octave at the very most. For very                    engineer seldom needs to design a filter, be it
small percentage bandwidths (less than 1%), crystal                    low, high or band pass, from first principles.
resonators are used in place of resonant LC circuits.                  Tabulated filter designs are to be found in many
For pass bands of +20% or more of the centre                           books, of which References 3, 4 and 5 are ex-
frequency, series connected high- and low-pass                         amples. The designs are usually normalized to
filters with overlapping pass bands offer a more                       l o h m source and load and to 1 radian per
satisfactory alternative. Similarly, a wide band-                      second cut-off frequency. To convert the tabulated
stop response can be produced with parallel con-                       values to those for a 50ohm filter, divide all C
nected high- and low-pass filters with non-over-                       values by 50 and multiply all L values by 50; to
lapping pass bands. When connecting different                          convert to a cut-off frequency of e.g. 1 MHz,
filters in series and more particularly in parallel,                   divide all the L and C values by 2rt x 10 6.
care must be taken to ensure that unwanted inter-                          An all-pass filter or phase equalizer has no stop
actions between the filters' terminal impedances do                    band. It is employed where a specific non-constant
not cause problems.                                                    group delay response is required to counteract
124    Analog Electronics

undesired group delay variations in a filter or other            constants. Hence derive the values of R and
network. Most textbooks do not even mention the                  G/2 for the resistors of a 50 f t n pad where the
all-stop filter 6, which is in any case little used, for         amplitude of the voltage at the output is 90%
obvious reasons.                                                 of that at the input.
                                                            3.   You are required to measure the response of a
                                                                 low-pass filter with a characteristic impedance
                                                                 of 75 9t, using a 50 f~ measuring set. Design
1. Handbook of Line Communication, Volume 1,                     suitable minimum loss pads for use at the
   HMSO London 1947.                                             input and output. What loss must be sub-
2. Theory and Design of Uniform and Composite                    tracted from the measured loss, at any fre-
   Electronic Wave Filters. A. J. Zobel, Bell                    quency, to give the loss due to the filter itself?.
   System Technical Journal, Vol. 2, no. 1, 1923.          4.    What would be the (notional!) values of the
   See also Transmission Characteristics of                      two resistors of a 50 f~ to 75 f~ minimum loss
   Electric Wave Filters; Ibid., Vol. 3, no. 4, 1924.            mismatch pad if it were to have zero loss?
3. Simplified Modern Filter Design, P. R. Geffe,            5.   Calculate the phase delay of the output of 1
   Iliffe 1964.                                                  metre of the cable in question 1, relative to the
4. Handbook of Filter Synthesis. A. I. Zverev, 1967              input, at 100 MHz.
   John Wiley & Sons Inc.                                   6.   Define group delay. What is the group delay of
5. Reference Data for Radio Engineers (chapter 8).                (i) 1 metre of the cable in question 17
   Howard W. Sams & Co Inc, 6th edition, 1975.                   (ii) free space?
6. Simplified Filter Design Routine, B. Sullivan,           7.   'In a practical filter, the stopband attenuation
   Practical Wireless, Vol. 65, no. 4, p. 28, April              only rises gradually beyond the cut-off fre-
   1989.                                                         quency' (see earlier in the chapter.) Why
                                                                 does the attenuation not rise sharply at the
                                                                 cut-off frequency, as in Figure 5.5a?
                                                            8.   What is a finite zero? How many finite zeros
 1. State the distortionless condition for a trans-              are there in a 5 pole Caur (elliptic) filter?
    mission line, in terms of the fundamental line          9.   Describe and compare the relative merits and
    parameters of L, R, C and G per unit length.                 limitations of Bufferworth, Chebychev and
    Coaxial cable type RG62A/U exhibits a capa-                  Bessel filters. Why must an even-order
    citance of 48 pF per metre. Its characteristic               Chebychev (or elliptic) filter of standard
    impedance Z0 = 93 9t; what is its inductance                 design work between unequal source and
    per metre?                                                   load impedances?
 2. Considering an attenuator as a lossy transmis-         10.   Design a 50 9t 5 pole 0.1 dB ripple elliptic low-
    sion line where the phase constant [3 = 0,                   pass filter with a cut-off frequency of
    express the characteristic impedance and at-                 16.9 MHz, by denormalizing the appropriate
    tenuation in terms of the fundamental line                   values given in Appendix 8.

    6            Active signal processing
                 in the frequency domain

Chapter 2 looked at passive circuits whose re-            infinite load impedance. Opamps can be used to
sponse was frequency selective, and Chapter 3             provide the required source and load impedances,
discussed operational amplifiers. This chapter            avoiding the otherwise necessary step of designing
brings these two items together and looks at              finite source and load impedances into the network
active filters and related topics. Active filters are     itself- which in any case may not be practicable if
preferred at audio frequencies since they enable all      they are not themselves resistive, and hence not
types of filters and phase equalizers to be realized      frequency independent. Figure 6.1 shows two
with suitable combinations of resistors, capacitors       opamps used as buffer amplifiers, providing the
and opamps. Thus the inductors that would be              CR section with a low source impedance and a
necessary for a passive filter can be dispensed with      high load impedance. The output amplifier is used
entirely; this is a real blessing since good quality      in the non-inverting mode to provide a high input
high-value inductors, such as would be required           impedance and is shown connected to provide
for audio-frequency filtering purposes, are both          unity gain, though it could perfectly well provide
bulky and expensive.                                      a gain of greater than unity if required. The input
                                                          amplifier is also shown as non-inverting, providing
                                                          a very high input impedance and keeping the
First-order active circuits
                                                          section as a whole non-inverting.
The least sophisticated filter, be it low pass or high       Now a passive CR section buffered by a pair of
pass, is the first-order or single-pole section. As       opamps hardly constitutes an active filter: this
noted in Chapter 2, this has a fixed shape provid-        description is generally reserved for a circuit
ing a gradual transition from the pass band to a          where the frequency determining passive compo-
6dB/octave roll-off in the stop band. It was also         nents are built into an opamp's gain determining
noted that, to obtain exactly the calculated per-         network. So can one design, for example, a single-
formance, a passive CR section should be driven           pole low-pass active filter with very high input
from a zero source impedance and work into an             impedance and low output impedance, all with a


                                                      R                   i    o
                              Input              i,

                                                          -                     O
Figure 6. I   F~rst-order low-pass circuit.
126     Analog Electronics

                                                                                                                                                                       !       1"I   CRI

                                                                                                             Gain (dB)                                           1                    1
                                                                                                                                                       0,) = T2            CR 1R2/( R 1 + R 2)
                                                - O                                 201og,~x~ Ri + R2
                                                                                                                                      ~                dB/octave

                Input                                 Output
                                                                                                                                                       I                   ~
                                                                                                                                     tol           ~            Log frequency
                                                                                                                0           II

                    t~ -                             -o

                                                                                              (a)           -90 ~                                               Vo         R 1+ R 2 l+jtoT 2

                                                     I!                                                                                Vo _ -JtoCR2
                                       [             II c                           I                                                  v.l - 1 + jtoCR 1                        R2
                                                                                            --~       R 1 l + jtoCR 2
                                                                                                                                           |1      r
                                                                                                                                 e         liC

                   Input               ~                                                    Output                               Input                                                              ut

                     O                 6                                                         O                               O

                                               (b)                                                                                                              (c)

                                                            .   .   .   .   .   .

                                      q- X A ~ p
                     Input                                                                       ut

                        O.,-               -                                                 O

                               k Gain (dB)
             20 log A dB                                                                                                                   0 Vi = 1               //                      Avi

                   0dB                                                                                                                                                      to increasin
                                                                                I"-,.   1
                                                                                                         9 logf
                                 ca                             ~
                                                      co 0 = _C___                                            Vo                  A                        -I
                                                                                                                    m                                            for to > > toO/ A,to 0 = 1/CR
                                                                                                                        1 -( A -1)jtoCR - jtoCR
Figure 6.2      Active first-order circuits.
  (a)   Non-inverting transitional lag.
  (b)   Inverting low pass: ]3da = I/2rcCR2, low-frequency gain = RE/RI.
  (c)   Inverting high-pass: f3dB = I/2rtCRl, high-frequency gain = RE/RI.
  (d)   Almost ideal integrator (zero bias current and offset voltage).
                                                         Active signal processing in the frequency domain       127

 single opamp? Figure 6.2a is probably the nearest              Things really get interesting when we start to
 you can get; it provides a low-frequency gain of            combine a first-order c i r c u i t - be it low pass or
 (R1 + Rz)/R2, and this starts to roll off at a fre-         high p a s s - with one or more second-order circuits.
 quency of 1/2TcCR1. However, as noted earlier,              However, before moving on to that stage, let's
with the non-inverting circuit the gain can never be         look at a first-order circuit with a finite zero, but
 less than unity, so the circuit of Figure 6.2a              not at the origin as in the case of a high-pass
provides a transitional lag response. If the gain is         section. Figure 6.3a shows a passive low-pass CR
 to roll off indefinitely beyond the corner frequency,       circuit connected to the non-inverting input of an
 one must use the inverting connection as in Figure          opamp. You will find the corresponding circle
 6.2b. Here there is an independent choice of low-           diagram and transfer function described in Chap-
 frequency gain (set by Rz/R1) and corner fre-               ter 2 (Figures 2.1, 2.5). The opamp's inverting
quency (set by CR2), but of course the input                input is connected to the junction of two equal
impedance is now just R1 and the signal is                   resistors R between the circuit's input and output
inverted. Figure 6.2c shows the corresponding               terminals. The heavy negative feedback will force
high-pass circuit.                                          the output to take up whatever potential is neces-
    For a response falling at 6dB/octave over                sary to make the voltages at b and c, the opamp's
several octaves - say all the way from 20 Hz to             input terminals equal. Further, since the input
20 k H z - one could always use a passive lag. But          terminals draw negligible current, the same current
with its unity response at frequencies below the            must flow through both the resistors R, and the
corner frequency, the response at 20 kHz would be           voltages across them must be equal and in anti-
very small indeed. The problem does not arise with          phase with each other. This is shown in Figure
the active lag circuit of Figure 6.2b, as you can           6.3b for a typical frequency somewhat below the
adjust the gain independently of the corner fre-            corner frequency. Since the vectors db and ba are
quency. As R2 becomes higher and higher, the                equal whilst ec and ca (the voltages across C and
corner frequency becomes lower and lower, and               R1) are in quadrature, qb1 equals qb2; and as the
the pole moves closer and closer to the origin. But         frequency rises from zero to infinity, qb1 and ~2 will
the unity-gain frequency is unaffected; it occurs at        both increase from zero to 90 ~. In fact the two
the frequency where the reactance of C equals R1.           resistors R act as a pantograph, drawing out a
When R2 becomes infinite, the pole is (ideally) at          copy of the circle diagram at a scale of two to one
the origin and the circuit is called an integrator: as      as shown. Thus for unity input voltage vi, repre-
the input frequency falls, the response rises at            sented by the vector ea in Figure 6.3b, Vo is also of
6dB/octave for ever more. In practice it is not             unity amplitude but its phase, relative to the input,
possible to make a true integrator in this way, for         decreases from zero to - 1 8 0 ~ as the frequency of
the opamp's input current and offset voltage                the input rises from zero to infinite frequency. The
would cause its output to saturate at maximum               frequency at which the phase shift is 90 ~ is given
or minimum voltage. Even without this practical             by co = 1/CR1. The circuit is called an all-pass
difficulty, the gain can never exceed the opamp's          filter or phase equalizer. The transfer function
open loop gain A, however low the frequency. So             for a passive lag is, as found earlier, F(s)=
whereas for an ideal (inverting) integrator                 1/(1 +sT), where T equals (in this case) CR1.
F(s) =-1/jcoCR1, for the circuit of Figure 6.2d             As this circuit contains but the one time constant,
F(s) = A/[1 - ( A - 1)jc0CR]. Likewise for the              it is convenient to normalize the frequency to coc
practical differentiator obtained by setting R1 in          which equals l/T, so that F(s)= 1/(s + 1) de-
Figure 6.2c equal to zero, the gain can only                scribes the voltage at the opamp's non-inverting
continue rising at 6dB/octave until the opamp's             input. Denoting the transfer function of the com-
open loop gain is reached. Note that in any case           plete all-pass filter (APF) by F(S)APF , yOU can
the open loop gain A of an opamp falls with                 describe it in terms of F(s) by noting that it is
frequency, so practical opamp differentiators are           twice as large and shifted to the left (see Figure
even less ideal than integrators.                           6.3b) by unity, since the input voltage is normal-
128   Analog Electronics

                                          R             b             R                     -vi                   e                  a = vi


                   o                                                             do
              Input                                ''                          nutput"v
              vi                                                                vo
                                                   c " r"                             o
                    0                                       e

                        Vo       1-sT               T =          1
                        Ili - 1 + sT '                          ~'~r = CRI
                             l1+ s    i f to normalised                   to
                                                        jo~ i


                                 •                                                   x0
                                 -1                                                  +1


Figure 6.3         First-order all-pass circuit (phase equalizer).

ized like everything else. So                                                                     zero diagram predicts exactly the same behaviour
                                                                                                  as the vector diagram. Figure 6.3d shows a three-
   F(S)ApF = 2 F ( s ) - 1                                                                        dimensional view of the pole-zero diagram but
                          2                           2               s+l             1-s         with F(s) plotted on a logarithmic rather than a
                                              ___                              ___
                                                                                                  linear scale. The zero becomes an infinitely deep
                        s+l                         s+l               s+l             l+s
                                                                                                  well, exactly like an upside-down pole, and sym-
This expression has a pole at o = - 1 due to the s                                                metry shows the response on the j0~ axis to be 0 dB
in the denominator, but the numerator goes to                                                     (unity) at all frequencies.
zero at c~ = +1; so the pole-zero diagram looks                                                      If the right-hand R in Figure 6.3a is made
like Figure 6.3c, with a zero at + 1. Moving up the                                               progressively smaller, the vector db in Figure
jc0 axis from zero frequency, the distances ll and 12                                             6.3b shrinks and the zero in Figure 6.3c migrates
to the pole and the zero increase but are always                                                  further to the right. So at high frequencies the
equal to each other. Since the magnitude of the                                                   output is smaller than at zero frequency; the
response is inversely proportional to the distances                                               magnitude response looks like that of a transi-
to poles but directly proportional to the distances                                               tional lag, but the phase finishes up at - 1 8 0 ~
to zeros, the product remains unity. Also, as the                                                 instead of returning to 0 ~ As R tends to zero,
output lag increases as we move anticlockwise                                                     the zero in Figure 6.3c and d migrates along the
relative to poles and clockwise relative to zeros,                                                +c~ axis to infinity and the circuit becomes a
the phase increases twice as fast as does that at the                                             simple first-order low-pass section. Conversely, if
non-inverting terminal of the opamp. So the pole-                                                 the right-hand resistor R is made progressively
                                                         Active signal processing in the frequency domain                    129

larger than its mate, the zero in Figure 6.3c               cies as in Figure 6.4c, the low-pass and high-pass
migrates towards the origin and the output at               asymptotes are mirror images of each other while
high frequencies rises, since the 'pantograph'              the phase responses are of exactly the same shape,
draws a larger and larger version of F(s).                  passing through +45 ~ at co01 in the case of the
   The all-pass filter is useful as a phase equalizer       lead and - 4 5 ~ at 0)o2 in the case of the lag. The
to correct phase distortion. For example, the high          frequency of minimum attenuation and zero phase
frequency attenuation associated with tape record-          shift occurs at the geometric mean of co01 and co02
ing is not a minimum phase shift process, whereas           call this frequency coo as in Figure 6.4c. If you now
the compensation applied in record and playback             connect the output of the circuit in Figure 6.4b
amplifiers is. The resultant phase distortion in the        back to its input, you have a circuit which can
reproduced signal is of no consequence for audio            almost but not quite supply its own input at coo; it
signals but could cause errors when reading digital         is no surprise therefore that, injecting a small
data from a diskette or other magnetic media.               signal at this frequency in series with R at point
                                                            Y, it will be amplified considerably. Just how
                                                            much, can be determined by analysing the circuit
Higher-order active circuits
                                                            a little more formally. Starting at the output of the
Now on to second-order active circuits. Figure              circuit in Figure 6.4a, one can write Vo - v3 and
6.4a shows an active second-order low-pass circuit
called a Kundert filter. It is not the commonest                                      1/jcoC2                       1
                                                                    1~3 n    1~2                       __ 1~2
active low-pass filter (LPF), but it behaves just like                             R + (1 / jcoC2)              1 + jco T2
the more common Salen and Key filter (described
                                                            where T2 - RC2. Next, v2 - vl and
later) and is particularly convenient for the pur-
poses of analysis. It is clearly an LPF, for at zero                               [                   1 ] + Vo
                                                                                                   1 + jcorl
frequency the capacitors can be considered open-                        Vl - -      (Vi --   Vo)

circuit, so it is just a cascade of two unity-gain
buffer amplifiers; whilst at infinite frequency the         where T1 - RC1. The square bracket gives the part
capacitors can be considered as short-circuits,             of the total voltage ( V l - Vo) across R and C1,
preventing the passage of signals. What exactly it          which appears across C1; this plus Vo forms the
does in between depends on the values of the                input Vl to the first opamp. Writing Vo in terms of
resistors and capacitors. To investigate this,              the expression for v3, and this in turn in terms of
Figure 6.4b shows the circuit 'unwrapped' by                the expression for v2 etc. gives
breaking it at point X in Figure 6.4a; the input
voltage Vl, (from its ideal zero-impedance source)                                               1]
                                                                             1 {i( v i - Vo) 1 + ]mT1 } + Vo
                                                                Vo = 1 + jo~T2
is set to zero, indicated by the ground point at Y in
Figure 6.4b. We are left with a passive lead RC1,           Collecting all the Vo terms on one side and vi terms
and a passive lag RC2, with buffer amplifiers.              on the other, then taking the ratio, gives
Clearly this circuit will not pass either very low
or very high frequencies, whilst if the corner                              vo                    1
frequency of the lag or top cut is much lower                               Vi         (jco)2T1T2 + jcoT2 + 1
than that of the lead or bass cut, there will
always be a large attenuation through the circuit.          Now (1/RC1)(1/RC2)= 1/(T1/T2) - co01co02 - co~
                                                            say. Also, with a little shuffling,          Tz-
If the tWO corner frequencies coincide, then at that
frequency there will be a total of 6 dB attenuation         (1/mo)v/(rz/rl). So Vo/Vi can be rewritten as
through the circuit. If the corner frequency co02 of               Vo                         1
the lag is higher than that of the lead co0~ as in                 vi       (jco/coo) 2 + (jco/coo)v/(TzT1)+ 1
Figure 6.4c, there will be a region between these
two frequencies where the gain almost rises to                                             1
unity. On the usual logarithmic scale of frequen-                           (jcon) 2 + jconv/(T2/T1)+ 1
130    AnalogElectronics

                                 0---4 R ~                              v2 ,


                                 tZ /7~77                                (b)

                             oI       I              /f-           . . . . r---
                                                   ,&--~                   z
                       a (dB)                      z       I            tOO

                                                           I                          I
                                                           I                          I
                       +90~ -


                       -90 ~. . . . . . . . . .

                                             R                 R
                                  o--i                             "-                     |   o   Q_

                                                                                                       2 4 ( T 2 / T 1)

                                      vi          C1                              ]
                                                                                              Vo T 1 = RC v T 2 = RC 2

                                                                                                  f~ = 2~ R%)(CIC2)

Figure 6.4    Some second-order low-pass sections.
  (a) The Kundert second-order low-pass filter section.
  (b) As (a), but with the loop 'opened out'.
  (c) Frequency response of (b)(see text).
  The Salen and Key second-order section.
                                                         Active signal processing in the frequency domain           131

where (% represents co normalized to COo. More              case you never came across the binomial theorem
generally,                                                  or have forgotten it, it goes something like this.
                                                            First, ( l + x ) 2 = l + 2 x + x  2, and this applies
            F(s) = s2 + sx/(Tz/T~) + 1                      whatever the value of x. However, if x is very
                                                            small compared with 1, then x 2 is very small
This expression for F(s) is identical to that we            indeed. For example, using 8 (lower-case Greek
found in Chapter 2 for a series tuned circuit or            letter delta) to indicate a number much less than
two-pole passive low-pass filter (equation (2.6)).          one, if 8 = 0.001 then 1.0012 = 1 + 2 x 0.001+
So it must behave in exactly the same way, and              0.000001 ~ 1.002. Conversely, the square root of
v/(Tz/T1) = D = 1/Q.                                        (1+8)~1+8/2.             Also, since ( 1 + 8 ) ( 1 - 8 ) =
    Figure 6.4d shows the better known Salen                1 + 82 ~ 1, then (1 + 8) ~ 1/(1 - 8 ) .
 and Key active LPF. Here the lag circuit is not               Armed with these results, let's see what happens
 buffered from the lead, and in consequence                 to the attenuation of a third-order LPF at very low
 Q = 1/2v/(T2/T1) as against 1/v/(T2/T1) for the            frequencies. The normalized response of the first-
 Kundert circuit. As a result of the square root, in        order section is 1/(jCOn + 1), and the magnitude of
 the Salen and Key circuit the ratio of C1 to C2 for        this is 1/v/(CO2n+ 12) by Pythagoras's theorem,
 a given Q must be four times greater than in the           since the j indicates that the COn term is at right
 Kundert circuit. Furthermore, in the latter there is       angles to the 1. Writing 8 for COn t o indicate a very
 a free choice of R and C for each time constant; it        low frequency compared with the corner frequency
 is not necessary to use the same value of R for both       of unity, then M1 (the magnitude of the response
 as shown in Figure 6.4a. By contrast, in the Salen         of the first-order section) is 1/v/(82+ 12) and,
 and Key circuit, equal value resistors give the            using the results we derived above, v/(1 + 82)
 optimum arrangement.                                       1 + 82/2 and M1 ~ 1 - 82/2. Thus at a frequency
    The single passive lag circuit gives an ultimate        8 the response of a first-order section has fallen by
 rate of attenuation, well past the corner frequency,       82/2. By arranging that the second-order circuit
 of 6dB/octave. This is not a very sharp cut-off,           has a response at 8 of 1 + 82/2, the overall third-
 although it could always be increased by cascading         order response will be independent of frequency, at
 several sections: a cascade of three such sections         least at frequencies much smaller than the corner
 would give 18 dB/octave, for example. The snag is          frequency.
 that each would contribute its own 3dB of                     The response of a second-order section is
 attenuation at the corner frequency, resulting in          1/[(jC0n)Z+j0)nD nt- 1], where D = 1/Q =x/(Tz/T1)
 considerable attenuation near the upper edge of            for the Kundert circuit or 2v/(T2/T1) for the Salen
 the pass band and a slow initial rate of increase in       and Key circuit. Setting D = 1, this becomes
 attenuation in the stop band. A much better third-         1/(--C02 -k-jC0n + 1). At some low frequency 8
 order filter results from combining a first-order          the denominator becomes ( - 8 2 + j 8 + 1), the
 circuit with a second-order circuit, since one can         magnitude of which is v/[82+(1-82) 2]
 arrange the latter to have a peak in its response          v/[82+(1-282)] = v / ( 1 - 82) ~ 1 - 8 2 / 2 using the
just below the corner frequency, to compensate for          results of the binomial theorem. Then M2 (the
 the roll-off of the first-order circuit. Choosing the      magnitude of the response of the second-order
 right degree of peaking in the second-order circuit,       section) is M 2 = 1 / ( 1 - 8 2 / 2 ) ~ 1 + 8 2 / 2 ,      so
 obtains the sharpest possible corner frequency             M1M2 ~ 1, i.e. the frequency response is level.
 response without actually getting a peak of any            Although the binomial approximations we have
 sort in the pass band. This is called a maximally          used themselves only hold for 8 << 1, the magnitude
flat or Butterworth response, and you can derive            of the complete third-order response stays remark-
 the required value of D in the denominator of the          ably level, being only 0.5 dB down at 8 = 0.707 or
 second-order circuit without resort to the higher          half an octave below the corner frequency, as you can
 mathematics. All you need is algebra, with the aid         easily check with a pocket calculator. At COn= 1 the
 of a useful result from the binomial theorem. In           response of the second-order section is unity since
132    Analog Electronics

D = Q = 1, i.e. for this value of damping the peak is        The pole-zero diagram looks the same except that
entirely to the left of the corner frequency, as can be      the two zeros that were at infinity are now at the
seen in Figure 2.8b for the curve R = 1, correspond-         origin. Turning now to the general purpose filter
ing to Q = 1 for the active filter. Thus the complete        section par excellence, this is variously called the
third-order filter's response at the corner frequency is     state variable or the biquadratic filter, depending
- 3 d B , the same as the first-order section alone.         on how the damping term D is organized. For sim-
However high the order of a Butterworth filter, the          plicity, I may from time to time refer to all the
bandwidth is still defined as that frequency at which        versions as state variable filters (or SVFs for
the response is - 3 dB relative to the low- frequency        short). The SVF is perhaps a little prodigal in its
r e s p o n s e - or to the high-frequency response in the   use of opamps; it requires at least three where most
case of a high-pass filter.                                  second-order sections and some higher-order sec-
    Figure 6.5 shows the pole positions for the              tions (e.g. Figure 6.5b) require only one. However,
three-pole Butterworth filter. Note that the three           it does provide a choice of low-pass (LP), band-
poles are spaced around the semicircle at intervals          pass (BP) and high-pass (HP) outputs and, with
of 180~           the outermost poles being at half this     the addition of just two or three resistors, all-pass
angle from the jco axis. This is the general rule for        and notch responses as well!
a Butterworth filter; for example a two-pole maxi-               Figure 6.6a shows the circuit diagram and the
mally flat filter would have poles at +45 ~. The             relationships between the voltages at various
required value of D is 2cos45 ~ or 1.414; the                points in the circuit, for generality using s in
response has no peak and is - 3 d B at co= 1.                place of jc0 from the outset. To understand how
Given the rule for the pole positions of a Butter-           it works, imagine the loop broken at the two
worth filter, you can design any order filter with           points X and Y and RQ disconnected. At 0 H z
ease. For example, Figure 6.5c shows the pole                the gain will be an enormous A 2 as shown in
positions for a fifth-order filter. In addition to a         Figure 6.6b. However, at frequencies where the
first-order section one needs two second-order               reactance of the capacitors C is small compared
s e c t i o n s - Salen and Key, Kundert, or any of the      with A times the value of the resistors R, each of
other second-order circuits you fancy - with D               the two integrator stages will produce a 90 ~ phase
values of 2 cos 36 ~ and 2 cos 72 ~ or D = 1.618 and          shift. So the locus of the output at the terminal LP
0.618. At c o - 1, ]F(jc0)[ collapses to l/D, so the          as the frequency rises will be as shown, heading for
two second-order sections have a response at                  a gain of unity with no phase shift when the
the corner frequency of 20 lOgl0 1.618 and                    reactance of C equals the value of R, where
20 logl0 0.618 or +4.2 dB and - 4 . 2 dB. Of course           VLp = vi. If the loop were closed at X - Y the gain
you seldom need to design a filter, especially a             would be infinite and the circuit unstable. So one
commonplace type like the Butterworth, since the              must apply some damping, which is shown in
necessary C and R values are tabulated in numer-              Figure 6.6a as coming from yet another opamp,
ous publications. 1 But getting inside a filter and           but this is just to simplify the analysis at this stage.
taking it apart as I have just done will give you a           In Figure 6.6c is shown the closed loop vector
clear idea of how it is supposed to work, and this           ~diagram for the frequency co = 1/CR, i.e. at the
puts you in a strong position when faced with one             normalized frequency of unity since co0 = 1/CR.
that doesn't.                                                 Each integrator provides a lag of 90 ~ but it looks
                                                              like a lead since they are inverting integrators,
                                                              owing to the opamps being used in the inverting
                                                              connection. The additional input VD applied via
                                                              RQ means that the circuit can no longer supply its
State variable filters
                                                              own input; an external input vi equal t o - V D
Simply interchanging the C and R in each section              needed to close the vector diagram. If you sort
of the low-pass filter of Figure 6.4a turns it into a         out the various terms in Figure 6.6a and work out
high-pass filter with the same corner frequency.              F(S)LP, i.e. the ratio VLp/Vi, you will f i n d - surely
                                                                                    Active signal processing in the frequency domain               133

                        R                                         R                 R                  A2
             o.--t           )-

        (from zero-           !                       i
        soo~:,                                                                               [,                 ,       o


                        R,            R               R     ,_                A 1
             O---I       '

        (from zero-                                  .224
                              .[~1 .T~2 T                                                      o

                              [1.39F         ~3.55F         1


             O---i      R             R               R     I-         ,.     A1                   R        R               A2

             o. . . .                            ~                                      ~              I            ~            ~o
                                       (c)                                  18~ jo~

                                                            36~ "-. ~

                                                                 36* " ~ -

Figure 6.5 Butterworth low-pass filters.
  (a) Third-order low-pass filter. Normalized to 1 f~, I rad/s.
  (b) Single-opamp version of (a).
  (c) Fifth-order Butterworth low-pass filter.

no great surprise by n o w - that                                                           CR--T-1,             F(S)L P = - l / ( s 2 + s D + I ) . The
                                            -1                                              same sort of exercise will provide you with the
             F(s)Lp      -    s 2 T 2 -+- s T ( I / Q ) + 1                                 results   that         F(S)B P = s/(s 2 + sD + 1)        and
                                                                                            F(S)H P - - $ 2 / ( s 2 -I- sO -Jr- 1). Note that the three
where Q -      1/D. In the normalized case where                                            outputs at HP, BP and LP are always in
134   Analog Electronics

                                                                                                    R1                                  X    Y
                                                                                                l          !                            ~-   ~-   --

                                                                R1 -                                  II




                                                                                                                   VLp !k

                        co= 1/CR~
                                            0V                                                                 VD---             .- ~- vBp = 3 vi
                                                                                ~co = 0 Hz

           Increasing-1 V/I
                           co = infinity]                        co increasing I
                                                                                                                    VHp           (c)

                                                 0 dB                                                              +34 dB
         +1.17 d      ~                                                                              ~
         COLP / u.o,~                                   \
             /-                                             \     o.o=0.62                                                          +0.99

           '                                     /~._~             = 1.b2 ~ c r e a s m g             ~ P + 3 dB
                                                                                                           \co   = 0.5
               co = 0 (LP) ~ (HP)                ~          4~"            . . . ~J 0   dB

                                                                  co = 1.62

                                                                                                    High-Q case D - 0.02, Q = 50
          Low-Q case Q = 1, D = 1, slight peak

                                                              A c t i v e signal p r o c e s s i n g in the f r e q u e n c y d o m a i n   135

quadrature, but are only of equal amplitude as                     from low frequency to the resonant frequency, and
shown in Figure 6.6c at the corner or resonant                     by another 180 ~ from there to a much higher
frequency. How they vary with frequency is shown                   frequency, while the amplitude will remain con-
in Figure 6.6d and Figure 6.7. Notice that Q gives                 stant; this is an all-pass response. The higher the
the magnitude of the output of a second-order                      Q, the more rapid the phase change at the resonant
section (for unity input) at the corner or resonant                frequency; in fact the picture looks like Figure 2.8b
frequency, not the maximum output. This occurs,                    except that the limits are + 180 ~ and - 180 ~
in the case of the low-pass output, at zero fre-                      Figure 6.9a shows a commonly encountered
quency for Q values of less than 1/2, and at a finite              version of the SVF. This has the advantage of
frequency below the corner frequency for Q > 1/2.                  economy, using only three opamps. Instead of
For higher and higher values of Q, the frequency                   inverting vBp in A4 (Figure 6.6a) and feeding a
of maximum response moves closer and closer to                     fraction 1/Q of it back to the inverting input of
the corner frequency.                                              opamp A1, a fraction of vBp is fed to the non-
   Since the HP and LP outputs are always in                       inverting input of A1 instead. If the fraction is
antiphase, combining them as in Figure 6.9a,                       one-third then Q is only unity, not 3: this is
there will be zero output at the resonant-fre-                     because the feedback from vLe and Vile, each via
quency: a notch circuit. Adding in a suitably scaled               R1, is effectively attenuated to one-third by the
contribution from the band-pass stage as in Figure                 other resistors R1 at the inverting input of A1,
6.9b, the output phase will increase by 180 ~ going                whereas in Figure 6.6a the inverting input of A1

Figure 6.6
  (a) State variable filter.
     I                            -I
VLp=~VBp                       or ~ v B p ' T = C R ,   s=r

1,'Bp = ~       1;HP

                  ( vHP vBP vi )
VLp                       -1
 v - T - s2T 2 + sT(1Q) + 1' Q = RQ/R1

In th normalized case where T = l,
VLp              --   1
V--7 = S 2      + s D + 1 D = 1/Q
VBp               S
 vi        s2 + sD + 1
12Hp             --S 2

 vi        s2 + sD + 1
   (b) Open loop vector diagram at low-pass output: vi short-circuited, link X-Y open, input at X, R Q = oc.
   (c) Closed loop vector diagram at o3 - C o- 1I T , RQ/R1 - Q = 3. Since all four resistors at the inverting input of
A1 are equal, the currents ii,Iup, iLe and ix) are all directly proportional to vi, vi, vile, vLp and vD respectively. At
o3 = 1/CR, vI-iP - --vLp, SO iHp and iLp cancel out. Thus vi = -vx) = vBp/Q.
   (d) Frequency responses, low and high Q.
136    Analog   Electronics


                              LP                            coc                                  HP
                                                  -3 dB                                   log frequency

                                                                                                coc = I/CR

                                   uP / . ~ o -              (a)


                                                                                                    log frequency

                                                                                                  coc = 1/CR

                         BP .~,~,,~r                 /                      ~           6 ~        BP

                                    HP                                                     LP

Figure 6.7  High-pass, band-pass and low-pass responses of the SVF.
  (a) Low-Q response (asymptotes shown faint). Maximally fiat case, Q = 0.707, D = v/2, no peaking.
  (b) High-Q response (asymptotes shown faint). As in (a), at all points the BP response lies between the other two.

is a virtual earth. Figure 6.9b shows another
                                                                   Further high-order filter design
three-opamp second-order filter section, provid-
ing BP and LP outputs. This version is some-                       The SVF provides an inherently high Q; indeed, it
times called the b i q u a d r a t i c filter, presumably          is infinite unless some deliberate damping is ap-
b e c a u s e - like the A P F of Figure 6.8b - it can             plied. With the Salen and Key and many other
provide a transfer function where both the                         second-order circuits, on the other hand, high
numerator and the denominator are quadratic                        values of Q are only achieved with extreme
equations in s.                                                    component ratios (though there are ways round
                                                            Active signal processing in the frequency domain          137

this). High values of Q are required in filters with a         passive filter composed (ideally) of purely reactive
very sharp c u t - o f f - the proverbial 'brick wall'         loss-free components, it is impossible to produce a
f i l t e r - for the second-order section with its poles      finite insertion loss at 0 H z in the filter itself.
nearest the jco axis. One can obtain such a sharp              Therefore an even-order passive Chebyshev filter
cut-off either with a Butterworth filter of very high          works between different design source and load
order or, if willing to tolerate some ripple in the            impedances, the loss at 0 H z being due to the
frequency response in the pass band, with rather               mismatch loss. At the peaks of the ripple the filter
fewer sections using a filter with a Chebyshev                 acts as an impedance transforming device, match-
response.                                                      ing the load to the source. With active filters this
     Imagine a three-pole maximally flat filter such as        consideration does not arise, as the signal energy
in Figure 6.10a, where the complex pole pair has               delivered to the load comes not from the input but
been deliberately slid nearer to the jco axis; the             from the power supply as dictated by the last
resultant frequency response will have a peak as in            opamp in the filter, which in turn is controlled
Figure 6.10b. If you now make the single-pole                  by the earlier ones and ultimately by the input
stage cut off at a lower frequency by sliding the              signal. This highlights the other major difference
pole on the cy axis to the right, it will depress the          between passive and active filters: a passive filter
peak back down to unity gain. The result is a dip              passes signals equally well in either direction,
in the response, as shown in Figure 6.10c. A fifth-            whereas an ideal active filter provides infinite
order filter would have two dips, and if the poles             reverse isolation.
are located on an ellipse, on the level at which they             As with passive filters, so with active filters we can
would have been in a Butterworth design, the                   obtain a closer approach to the brick wall filter
peaks will all line up at unity gain and the troughs           response by designing in finite zeros- frequencies
will all be the same depth. The more flattened the             in the stop band where the attenuation is infinite. A
ellipse, the greater the ripple depth and the faster           'finite z e r o ' - a finite frequency where the response is
the rate of roll-off just beyond cut-off. However,             zero - has already appeared, see Figure 6.8a. If RHp
the ultimate rate of roll-off is the same as a                 is made twice as large as RLp, the frequency of the
Butterworth filter, namely just 6NdB/octave,                   notch or zero would be higher than the resonant
where N is the number of poles. The standard                   frequency co0. If you then adjust the circuit's Q to
mathematical treatment of this Chebyshev filter is             give a maximum response in the pass band of
rather heavy going, and I haven't yet worked out a             +0.5 dB relative to the 0 Hz response, this gives a
way to explain it with nothing more than algebra               0.5dB ripple second-order elliptical filter. Some
plus the odd complex variable, so you will have to             writers call these Caur filters, whilst others simply
take it on trust at the moment. As with the                    regard them as a variant of the Chebyshev. Either
Butterworth design, the normalized component                   way, you can see that sliding the notch down in
values for different orders of filter are tabulated            frequency towards COo whilst winding up the Q to
in the literature, only there are many more tables             retain the 0.5 dB ripple Chebyshev pass band shape,
for the Chebyshev design as there is a choice of               you can achieve as sharp a cut-off as you wish. There
ripple depth. Designs are commonly tabulated for               is just one snag: the closer the zero to COo,the less
ripple depths of 0.18, 0.28, 0.5, 1, 2 and 3 dB. If the        attenuation is left at frequencies beyond the n o t c h -
first two ripple depths look odd, it is because they           which only goes to prove that you don't get owt for
derive from designs for passive filters. 2 A ripple            nowt, as they say in Yorkshire.
depth of 0.18 dB corresponds to a 20% reflection                  Fortunately the maths required to work out the
coefficient in the pass band at the troughs of the             pole and zero locations for different orders of
ripple, and 0.28 dB to 25%. With an odd number                 elliptical filters has all been done and the corre-
of poles, zero frequency corresponds to a peak of              sponding normalized component values for circuits
the ripple, i.e. to zero insertion loss in the case of a       have been tabulated; it is a little too heavy to wade
passive filter. With an even number of poles, zero             through here, and you will find it in the many
hertz occurs at a trough of the ripple. Now in a               textbooks devoted entirely to active filters. Don't
138   Analog Electronics



                                                            VHp                                                                  VBp                                                                                          VLp

             i                                                                               gl                          "-"

                           :-   ,i
                                      ii :-3 . . . .
                                           .                ....... !iii                                                                                                                                                      I



                                                        -    -                 -        -    R '    .        .   .   .   .   .   .   .   .        .    .    .      .   .   .   .   .   .    .        .     .      .   .



                                                   --            ii   i   ji       j
                                                                                                    Oc                                       i             i,,                         |,

                                                                                                                                     % I/CR           =


                                      +900,                                            ~X~

                                *       O~ . . . . . . . . . . . . . . . .                                                                                                 lo'gfr'e;u~n
                                      -90~                                                               L                                                                             cy
                                     -180~                                                                   I                                   , .............               L                ,,                        ,

                                                                  Active signal processing in the frequency domain                139


                                I        9 R'

                                                    v    R             %            "~"   9       -- vBp   +

                                          RQ                                                         RQ'       I IR'


                                                                                          v   o


                                                Medium       High !
                              + 180~                              IQ
                               +90 ~
                              -90 ~
                              -180 ~

Figure 6.8    Notch and all-pass SVF filters.
   (a) SVF notch filter, vnp and Vcp are always 180 ~ out of phase. At e0c, vnp = vHp, therefore vHv + vcv = 0.
   (b) SVF second-order all-pass filter (phase equalizer). If R~ --- RQ, amplitude response is unity at all frequencies.
If not, then ]Vap/Vi]o,c = RQ/R'Q.

use the first elliptic filter design y o u c o m e across,                 time constant per pole. So a canonic s e c o n d - o r d e r
though; it m i g h t n o t be the m o s t economical on                    (two-pole) section need only use two capacitors,
c o m p o n e n t s . The m o s t economical sections are                  each with an associated resistor, even if it also
canonic, that is to say that they use only one CR or                       provides a pair of zeros. F o r zeros come 'for free',
140    Analog Electronics

                                                                             !              i1


                  Vin                                   vLp                                           VBp

                 2                                            RQ

                                                                                                                    RQ + R'
                                                                   (a)                                               3R'


                                                                                                                      RQ [iQ

                                          VBp                                      VLp                ii
                                                                                   vi                                       t o c - RC
                                                                                                                              Q= R

                                                                                                                               - ~ iQ
                  vBp Qv i                                                 v~'p

                                                                                                           Current vector
                        Voltage rector                                                                     diagram for
                        diagram                                                                            point X

Figure 6.9 More second-order filter sections.
   (a) Three opamp form of SVF.
   (b) The biquadratic second-order section. If the biquadratic filter is tuned by adjusting the two resistors R, then Q
is proportional to 0~c. This provides a constant bandwidth response. Owing to the input point chosen, the
biquadratic filter has no high-pass output, but on-tune band-pass gain is independent of frequency.
   (c) Vector diagrams at co = 0~c for the biquadratic section.
                                                                     Active signal processing in the frequency domain                       141

                          /         c
                                  co first-order section       DB
                               -" eoc second-order section
                                                                                  ~         ,       ,                     ,ou
                      6ooA                                                                                            \~Xx~ log frequency

                                                                                                    -18 dB/octave ~ ' ~ \ x
                                                                                                                      Original .iX\
                                                                                      (b)                             Butterworth

                               _ coc of first-order section

                                                                                                                    ~c of
                                                                                                                ~,, Butterworth
                                                               0              ~   _         -   x       ,
            I           ipse                            Ripple
                                                        depth -3 dB (say)                                            ~~\~\\x

                                                               (c)                                      -18 dB/oc                     \\

Figure 6.10 The Chebyshev filter (third-order example).
  (a) Complex poles of Butterworth filter displaced to right.
  (b) Resultant response.
  (c) The Chebyshev response has the same ultimate rate of cut-off as Butterworth, but the peaking holds up the
response at the corner, followed by a faster initial descent into the stop band.

even though they cost 12 dB/octave offthe final filter                  As is required. For a Chebyshev filter, the given fs
roll-off rate. The tabulations for elliptic filters are                 and As can be achieved with fewer sections at the
even more extensive than for Chebyshev filters. For                     expense of a larger pass band ripple, or with lower
a Butterworth filter the only choice to be made is the                  ripple but more sections. With the elliptic filter there
number of poles, which is set by the normalized                         are returns in the stop band as well as ripple in the
frequencyfs at which a given stop band attenuation                      pass band, so for a given order of filter and pass
142       Analog Electronics

                                         II       ~ X                                            R5
               'Oi          I


Figure 6.11 The SAB section and elliptic filters.
  (a) SAB (single active biquad) band-pass circuit, also known as the multiple feedback band-pass circuit.
  (b) SAB low-pass circuit with notch (finite zero-frequency response).
Design equations for first-order and SAB second-order sections in elliptic low-pass filters, normalized to e0c = I
radian/second and C = I F are as follows. For pole-zero pair f~ICyland f~2 (or ft3cr3 and fh etc., see Figure 6.12a)
put ~ p2 ~--- ~ 2 -Jr- 0"12, ~z --" ~2 and 0 Hz stage gain G = 1 Then
R5 = 1/cyl
R4   =                 2
         R5/ ( a z2/ a p - 1)

R6 = (R4 -+- R5)/ [R5f]p - 2(1 + R4 /R5) ]
                    2 2

Rl -- 1/(R5Qp - l/R6)
R2 = R5
R3 = R4 (or c h o o s e R3, then R2 -- R3R5/R4)
If the circuit has an odd number of poles, then R' in first-order section is given by R' = I/or0. Scale all results to re-
quired frequency and to convenient component values.

 band ripple depth there is a further trade-off be-              3 dB then the filters are directly comparable, but
 tween fs and As. For an even-order filter, the                  the response of say a 0.28dB ripple Chebyshev
 attenuation beyond the highest-frequency zero or                low-pass filter with a 1 kHz cut-off frequency will
f ~ remains at As indefinitely (or at least until the            not be 3 dB down until a frequency somewhat
 opamps run out of steam), or rolls off at 12 dB per             above 1 kHz.
 octave if you accept one less f ~ . In an odd-order                To make an active low-pass elliptic filter one
 filter, the odd pole on the axis ensures that the               could use SVF sections for the second-order low-
 response at infinite frequency dies away to nothing,            pass notch circuits, but a more economical filter
 at a leisurely 6 dB/octave.                                     results from a variant of the single-amplifier
     An important point to bear in mind, when                    biquadratic (SAB) circuit. 3 Let's creep up on this
 comparing the responses of non-ripple filters                   ingenious circuit by stages. Figure 6.1 l a shows the
 such as the maximally flat amplitude Butterworth,               SAB band-pass circuit; you can see it has no gain
 maximally flat delay Bessel etc. with the responses             at 0 Hz if you imagine the capacitors open-circuit,
 of filters such as the Chebyshev and the elliptic,              and equally it has none at infinite frequency since
 concerns the definition of cut-off frequency. For               the opamp's output is then effectively short-
 the former, the cut-off frequency is defined as the             circuited to its inverting terminal. If you imagine
 frequency at which the response is - 3 dB, whereas              the input grounded, the loop broken at point X
 for the latter it is defined as the frequency at which          and a signal inserted via the upper capacitor, you
 the attenuation last passes through the design                  have a passive lead feeding an active (inverting)
 pass-band ripple depth. If the ripple depth is                  differentiator. At frequencies below its corner
                                                           Active signal processing in the frequency domain     143

frequency the passive lead will produce nearly 90 ~           same amount. It is wise therefore when working
phase advance, whilst the active differentiator will          with small signals to have some gain ahead of the
produce that much at all frequencies. So we have              filter, to keep the signal at the non-inverting
almost 180 ~ of advance plus the inversion pro-               terminal well clear of the noise level.
vided by the opamp, giving an open loop phase                    The higher-order filters discussed are used for
shift of nearly zero. If the open loop gain is unity          critical filtering applications, where it is desired to
when the phase shift is nearly zero, considerable             maintain the frequency response of a system up to
peaking will result, i.e. the circuit will behave like a      the highest possible value consistent with negli-
tuned circuit. Note that the resistor from the                gible response at some slightly higher frequency. A
inverting input of the opamp to ground will have              typical application would be the low-pass filtering
no effect as it is at a virtual earth.                        of an audio signal before it goes into a sampling
   Now look at the modified SAB circuit of                    circuit of some sort, say the analog-to-digital
Figure 6.1lb. An attenuated version of the                    converter in a compact disc recording system. In
input signal is applied to the non-inverting                  this instance, as is often the case with high-order
input of the opamp; if the ratio of R2 to R3 is               filters, a fixed cut-off frequency is used. Where a
made the same as the ratio of R5 to R4, then the              variable corner frequency is required a very low-
gain at 0 Hz will be unity, whilst the peak will be           order filter is, fortunately, often perfectly ade-
there as before. However, if R6 and the other                 quate: a good example is the tone controls fitted
components are chosen correctly, there will be a              in the preamplifier 4 section of a hi-fi amplifier (see
frequency above the resonant frequency where                  Figure 6.13). These are usually first-order filters
the signal at the inverting input is in phase with            except for a high-frequency steep-cut filter, if
the input and attenuated in the same ratio as the             fitted, which usually has one or two switched
signal at the non-inverting input. Thus there is              cut-off frequencies as well as an out-of-circuit
only a common mode input to the opamp but no                  position. The reason for this is that, to vary the
differential input and hence no o u t p u t - a notch,        cut-off frequency of a filter whilst preserving the
in fact. Figure 6.11 gives the design formulae for            shape of its response, it is necessary to vary the
the second-order sections of an elliptic low-pass             time constants associated with each and every pole
SAB filter, and for the first-order section if the            in sympathy; thus, for example, a three-gang
design chosen has an odd number of poles. The                 variable resistor is required for a three-pole fil-
design requires for each second-order section the             ter.There are two-pole filter circuits which can be
   and f~ (i.e. co) values of the pole pair and the f~        tuned by a single variable resistor, but remember
value of the zero; the cy value of the zero is of             that the transfer function of such a filter will
course 0, as it is on the jo~ axis. For an odd-order          contain a term in T1T2, i.e. the product of two
filter, the cy value for the single pole is also              time constants. If the resistance associated with
required. These values are all listed for various             only one of these is varied, then a 10:1 ratio in
orders of filter with 0.1, 0.18 and 0.28dB pass               resistance will provide only a v/(10: 1) tuning
band ripple for various fs(2nfts) and As(Amin)                ratio. Electronic tuning of high-order filters, e.g.
combinations in Reference 2.                                  those realized with the SVF circuit, is possible by
    Figure 6.12 shows a five-pole elliptic low-pass           using transconductance amplifiers or multiplying
filter, calculated using the design equations in              digital-to-analog converters in place of the resis-
Figure 6.11. The published tables of poles and                tors in the integrator sections.
zeros associate the lower-frequency zero with the                 For a number of years now another type of
lower-Q pole pair, as this results in less sensitivity        filter, the switched capacitor filter (not to be
of the response to component value tolerances.                confused with the earlier and now little used N-
Note that although the gain in the pass band is               path filter) has been available. The earlier types of
between unity and -0.18dB, the signal is at-                  switched capacitor filter, such as the MF10 intro-
tenuated before being applied to the non-inverting            duced by Motorola and now second sourced by a
terminal of the opamp and then amplified by the               number of manufacturers, are based on the SVF
144        A n a l o g Electronics

                                           toe= 2nfc
                                           O~s = 2n f s
                                           t'14 = 2~f 4
                         joa 1 ~2                                              0
                                           f~2= 2~f2
                                    L'~                                Amax I                              '
             Ellipse , ~
                                    f~3                                                                    !
                  Onll                                                                                     I
                     "-!'_L i                                                                              I
                                -f~3                                                                       I
                                -f~l                                     Amin                              I
                                                                                                          ,I-. . . . . . .
                                                                                                                                          - 6 dl
                                -~4                                                                        I
                                                                   (a)                                  fc =    1            fs f4             f2             logf



                                                                                           (1F)                                                    ,.    c
                                              33k                  I       c

              120k              _             (0R546)                    ..(1F)            ., 56Ok^,   56k

      /~                                      560k
                                                          !                             100k           18Ok                                                          Vo
                                              (9R82)                                    (1R79)            75k                                      47k
                           C                 6k8
                                             (OR 115)/77      r]                   r/
                                                                                   ,                           /'i77 7       I I I I I I l L r'/
                                                                  100k                                                       47k

Figure 6.12     Fwe-pole active elliptic LPF.
   (a) Fwe-pole elliptic filter. Unlike Butterworth type filters, the pole pair nearest the jco axis actually lies outside
the unit circle, to compensate for the effect of the nearest on-axis zero. Values for cr0,f21 etc. are read from
Reference 2.
   (b) First-order section, required when the number of poles N is odd. It is useful to include the odd pole on the -cr
axis, to ensure an ultimate roll-off at 6 dB/octave, as in (a). R' in ohms as in FNure 6.1 1, for C - I F, COo- I racl/s.
   (c) Fwe-pole elliptic filter, Amax (ripple)= 0.28 dB, Amin - 53.78 dB, f~s =fs/fc = 1.6616. Scaled tofc = 2.7 kHz,
giving a 3 dB bandwidth of about 3 kHz, designed as in FNure 6.11, using cr and f~ values from Reference 2.
Reference 2 uses f~ to stand for normalized frequency in radians/second. In circuit, C = 1 nF. Resistors adjusted
to nearest preferred value. Normalized values shown 0 for the highest Q section.
                                         Active signal processing in the frequency domain                         145
     "'     ~                    I-                      I
                                                                 °        ,
                                                                                     i       ~   ~
                                                                                             .L.L.L   ~o--~
                        i         -
                    o       ~-                                                           .   T T T
          ~t'---t       ~                                                                             c.~ 0   ~
            ~   I ~ . . . . ~[~----"
                                                                                                      8 ~~
                                          --                     0
                                               t'~   ~               '~           I" ~
                                                                                                      ~~ ~
                                 :~                          <                ~                       ~   ~°~ ~
146    Analog Electronics

configuration and can provide LP, HP, BP, notch                (high) input impedance and zero (low) output
and all-pass responses. These are analog sampled               impedance.
data filters, so the output is a stepwise approxima-      2.   Design a single pole low-pass filter using a
tion to the desired low-pass filtered version of the           single opamp, having an input resistance of
input signal, the steps being at 50 or 100 times the           100 kf~, a passband gain of 35 dB and a cut-off
cut-off frequency. The steps can be effectively                frequency of 4 kHz.
suppressed or rounded off by passing the output           3.   In question 2, which would be the more
through a further fixed low-pass filter, which can             appropriate, and why; an all bipolar opamp,
often be a very simple low-order filter. The ad-               or one with a FET or CMOS input?
vantage of this type of filter is that it can be tuned    4.   An inverting opamp integrator circuit has a
by simply altering the necessary clock frequency,              gain of unity at 1 kHz. What is the gain at
which is supplied to the filter at 50 or 100 times the         (i) 10 kHz, (ii) 10 Hz? Ideally, as the frequency
required cut-off frequency. More recently, special-            falls indefinitely, the gain rises indefinitely.
ist analog IC manufacturers have introduced more               What sets the frequency at which the gain
complex filter ICs of higher order. For example,               ceases to rise?
the XR- 1015/1016 from EXAR Corporation is a              5.   Design a first-order all-pass filter as per Figure
seventh-order elliptic low-pass filter with the usual          6.3, to have 90 ~ phase shift at 330 Hz. At what
choice of x 50 or x 100 clock frequency. This filter           frequency will the phase shift be 135~
may be used with clock frequencies between 1 kHz          6.   Design a first-order all-pass filter to have a
and 2.5MHz, giving cut-off frequencies in the                  phase shift of 45 ~ at the frequency where that
range 10 Hz to beyond 20 kHz.                                  in question 5 has a phase shift of 135 c. If the
                                                               two filters are cascaded, at what frequency will
                                                               the phase shift be 315~ At this frequency,
                                                               what is the time delay suffered by the signal?
1. Typical examples are: Active Filter Design.            7.   What advantages does the Kundert second-
   A. B. Williams, Artech House Inc, 1975. Refer-              order low-pass filter have over the Salen and
   ence Data for Radio Engineers, 6th Edition,                 Key circuit? Design a Kundert second-order
   Howard W. Sams and Co Inc. 1967.                            high-pass filter with maximally flat response
2. Handbook of Filter Synthesis, A. I. Zverev,                 and a corner frequency of 100 Hz.
   John Wiley and Sons Inc. 1967.                         8.   What is the disadvantage of achieving a third-
3. Sensitivity Minimisation in a Single Amplifier              order low-pass filter by cascading three
   Biquad Circuit, P. E. Fleischer, lEE Trans.                 first-order passive lags? Design a third-order
   Circuits Syst. Vol. CAS-23, pp. 45-55, Jan                  Butterworth low-pass filter with a cut-off
   1976 (with 14 further references).                          frequency of 3.3 kHz.
4. Transistor High-Fidelity Pre-Amplifier, R.             9.   A state variable filter provides high-pass,
   Tobey, J. Dinsdale, p. 621, Wireless World,                 band-pass and low-pass outputs. Describe
   Dec. 1961.                                                  how to combine these to implement (i) a
                                                               notch filter, and (ii) an all-pass filter.
                                                         10.   What is the attenuation at the design cut-off
                                                               frequency of (i) a Butterworth filter, (ii) a
 1. Draw and explain the operation of a single                 0.5dB ripple Chebychev filter, and (iii) a
    opamp first-order low-pass filter with infinite            0.18 dB ripple elliptic filter?

    7           Active signal processing
                  in the time domain

This chapter looks at the processing of signals         an electrically noisy environment and must be
where the main emphasis is on waveform shape,           amplified to a usable level without contamination
and how it can be preserved or manipulated,             from undesired voltages, e.g. mains hum and
rather than on the frequency response of the            harmonics of the mains frequency, often originat-
circuit and how it affects the signal. Some of the      ing from phase-controlled rectifier motor drive or
circuits described in the following pages are linear,   heater circuits. The unwanted electrical noise often
whilst others are deliberately non-linear. The non-     takes the form of a common mode voltage, that is
linearity may be provided by diodes, or by allow-       to say a voltage of equal magnitude with respect to
ing an opamp to run out of its linear range, or by      earth or ground appearing on each of the two
means of a comparator. The linear circuits may          leads from the transducer- the device producing
have outputs very different from their inputs. For      an electrical output corresponding to some physi-
example, in a constant current generator the sign       cal parameter such as temperature, pressure or
and magnitude of the output current forced              whatever. The problem is particularly acute
through the load is set by that of the input voltage.   where the transducer presents a high source im-
Again, in the frequency-to-voltage (F/V) converter      pedance, e.g. a piezoelectric transducer, or micro-
the output voltage is proportional to the frequency     pipette electrodes used for physiological measure-
of the circuit's input signal, regardless of ampli-     ments in medical diagnostics. The instrumentation
tude or waveshape- and vice versa for its dual, the     amplifier is a useful tool in such situations.
V/F converter.                                             An operational amplifier exhibits a high degree
                                                        of rejection of common mode signals at its two
                                                        inputs, but when fitted with gain defining resistors
Amplifier and multipliers
                                                        the input impedance is high at the non-inverting
First, a look at the front end of the signal chain      input, i.e. in the non-inverting mode, but simply
and at how a signal can be raised from (in many         equal to the input resistor in the inverting mode.
cases) a very low level to a level at which it can      Unequal input resistance at the two input term-
conveniently be processed, either on the spot or        inals is undesirable as it may cause conversion of
after being transmitted to a central location in,       any unwanted common mode signal to transverse
say, a manufacturing plant. Many signals start life     or normal mode, which will then be indistinguish-
as a very small voltage, only a few millivolts or       able from the wanted signal. The instrumentation
even microvolts. Typical sources of such very small     amplifier is a combination of several opamps that
signals are thermocouples or platinum resistance        provides a very high input impedance at both of its
thermometers for the measurement of the tempera-        input terminals, which are 'floating' with respect to
ture of a furnace or processing oven, or resistive      earth and respond only to normal mode signals.
and piezoelectric strain gauges for the measure-        Figure 7.1a shows the usual arrangement, although
ment of forces, accelerations or pressures. In a        several other circuits provide equivalent perform-
factory environment such signals often originate in     ance. Each of the two amplifiers in the input stage
148    Analog Electronics

                                                       R4        R3

                                              IE                        l
                                              "i                        -    0


                                              IL I~


                                                            RG                   Vo         20K
                                                       ,                         vi - 5 +

                                      10K ~ 2 K ~                           , 10K~          RG
                                                      ": ]

Figure 7. I  Instrumentation amplifiers.
  (a) Instrumentation amplifier with three opamps. Note that the resistors R3 and R4 form a bridge which is
balanced to any common mode output from Al and A2. In normal mode (push-pull, transverse), the input gain is
RI/R2 and the output gain is R3/R4. In common mode (push-push, longitudinal), input gain is unity (0dB) and
output gain is zero (-oo dB), provided resistors are perfectly matched and A3 is ideal.
  (b) Two-opamp instrumentation amplifier.

is used in the non-inverting connection, the gain            balanced to common mode input signals to the
being defined by the feedback from the output to             output stage if the output voltage remains at
the inverting inputs in the normal way. Thus the             ground potential. There is then no push-pull signal
input stage provides gain and impedance conver-              at the input terminals of the output opamp due to
sion but does not of itself reject common mode               a common mode output from the first stage,
signals entirely; they are simply transmitted to the         provided the bridge is balanced. The output
input of the output stage unchanged, i.e. at unity           stage may also be used to provide gain and,
gain. If the input stage provides considerable gain          provided that the ratio of the two resistors at the
to normal mode signals, for example 40dB, then               non-inverting input is the same as the ratio of the
the input stage makes a useful contribution to the           input and feedback resistors at the inverting input,
overall common mode rejection, in this case 40 dB.           the common mode rejection is maintained. Instru-
The rest of the rejection is provided by the output          mentation amplifiers are available from many
stage, by virtue of the way the signal is applied to         manufacturers and combine three opamps, to-
it. The four resistors form a bridge which is                gether with the gain defining resistors, all in one
                                                         Active signal processing in the time domain    149

IC package. The early models were often of hybrid       used in the inverted mode, connected so as to
construction but today's designs are usually mono-      modify the gain determining feedback at the invert
lithic. (As indicated in Chapter 2, a hybrid is a       input of an opamp. 1 Another scheme, providing
module containing active devices such as opamps         binary coded decimal (BCD)control of gain, uses
together with passive components, mounted on a          CMOS analog switches to select the appropriate
substrate such as ceramic and encapsulated.) The        gain determining resistors. 2 Yet another scheme
AD524AD monolithic instrumentation amplifier            uses an AD7545 analog-to-digital converter's
from Analog Devices can be set for a gain of 1,         R/2R ladder as the gain defining network for an
10, 100 or 1000 without any external components,        opamp, using AC coupling to permit high gain
or for intermediate gain levels with only a single      without excessive offset voltage magnification. 3
external resistor. At unity gain it features 70dB          Nowadays, where a digitally programmable
common mode rejection ratio (CMRR) and a                gain is required, a programmable amplifier (called
1 MHz bandwidth. Its input offset voltage of            a PRAM or PGA) is often used. These are
250~tV maximum may be nulled with a preset              available from many manufacturers. The 3606
potentiometer. The INA101HP monolithic instru-          series from Burr Brown are programmable gain
mentation amplifier from Burr Brown offers an           instrumentation amplifiers whose gain is set with a
impedance of 10 l~ 9t in parallel with 3 pF at each     four-bit binary word: the available gain values are
input, 85 dB C M R R and a large output swing of         xl, x2, •      x8, ..., x1024 at a bandwidth of
+12.5V. The bias current at each input is only          100 kHz at unity gain and 10 kHz at the highest
+15 nA and the input offset voltage is trimmable        gain settings. By contrast, for use with non-float-
to zero.                                                ing signal sources, the SCl1310 programmable
   Isolation amplifiers are akin to instrumentation     gain/loss circuit (PGLC) from Sierra Semiconduc-
amplifiers with the additional feature of providing     tor has a single-ended ground referenced input and
complete electrical isolation between the input and     a - 3 dB small-signal bandwidth typically in excess
output circuits. The AD204JY from Analog De-            of 1.5 MHz when driving a 600 f~ load. It has nine
vices amplifies the input and uses it to drive a        gain control inputs, one selecting gain or loss and
modulator. The resulting signal is transformer          the other eight providing 256 gain/loss steps of
coupled to the output stage which converts it           0.1 dB, i.e. a +25.5dB gain range. Given that
back to a replica of the input signal. Isolated         distortion is typically 60 dB down on full output
power supplies for the input stage are provided         and that noise is very low indeed, the device
by rectifying and filtering a 25 kHz transformer        provides a usable dynamic range of well over
coupled supply. The device has a very high input        100dB.
impedance, a bias current of + 3 0 p A and a               Next, a look at some circuits which have already
bandwidth of 5kHz, and provides a continuous            appeared in these pages. The opamp integrator in
input/output voltage rating of 750V RMS at              Figure 7.2a is identical to the inverting integrator
60 Hz or + 1000 V DC.                                   described in Chapter 6, where, however, only its
   Instrumentation amplifiers are usually used at a     frequency response was considered. This, it turned
fixed gain, set by internal or external resistors       out, increased indefinitely as the input frequency
when the system of which they form part is              was made lower and lower, becoming (ideally)
manufactured. A requirement not infrequently            infinite at zero frequency. In Figure 7.1a, imagine
arises for an amplifier whose gain is settable to       that an input of + 1 V is applied at the input at an
any desired value by a control signal, either digital   instant when the output voltage is zero. The input
or analog. In the latter case, an operational           current of 1 ~tA must be balanced by 1 lxA current
transconductance amplifier can be used. Where           in the capacitor, assuming that the input impe-
digital control is needed, a number of different        dance of the opamp is virtually infinite. The charge
circuit arrangements are suitable. An early ex-         on the capacitor must therefore increase by 1 ~tC
ample used binary related resistors (R, 2R, 4R          (one microcoulomb) per second; since the charge
etc.) switched by means of 2N3642 transistors           Q = CV, if C = 1 ~tF then the opamp's output
150   Analog Electronics

                                             II C
                         I~A    R            l~,,: [ ,.. h
                       vi                   f"          vo                             0.5              1
                                                                                              Time (seconds)


                                                         (a)                                                Vo

                                                                             +     ! Mains     1 of 2

                                                                ,=~I             Mo     1      valve

                                                                           . . . . .    15----..

Figure 7.2
  (a) Opamp integrator.
  (b) Integrator as part of a process control system.

voltage must become 1 V more negative every                    which is 'noisy' owing to the action of the stirrer,
second. This is true in the long term even if the              and to drive its output towards one or other
input voltage is varying, provided that its average            switching point of the comparator circuit accord-
value is 1 V, for the charge on the capacitor will             ing to whether the mean liquid level is higher or
equal the average input current times the time for             lower than the desired level. A fraction of the
which it is flowing, i.e. the integral of the input            comparator's output is fed back to its non-invert-
current.                                                       ing terminal. This positive feedback determines the
   A comparator can be used to monitor the out-                hysteresis or 'dead space' between the voltage
put voltage and, when it reaches a certain value, to           levels at which the comparator switches, provid-
control, say, the inlet valves to a liquid blending            ing, in effect, a further filtering action to prevent
tank. Thus we have the basis of a simple bang-bang             the valves chattering on and off.
servo system. Figure 7.2b shows the arrangement,                  The integrator circuit also provides a means of
with the integrator and comparator controlling the             modifying the shape of a waveform. Figure 7.3a
inlet valves, to make up as and when necessary as              shows a square wave centred about ground applied
the product is drawn off. The integrator acts as a             to an integrator. Since a constant input voltage
filter to average the output of the level sensor,              results in a linear ramp output, the output wave-
                                                                                Active signal processing in the time domain     151

                                                                        | cl--c-.-

                             Input Vp:[ ]--- ~'--f----t'-- ["
                             Output       ..........
                                                 _    I     I~
                                          -VPo ~             i   m        e                t
                                                          T/2    3T/2

                                                                        !, , L -u-L

Figure 7.3 AC response of opamp integrator and differentiator.
  (a) Opamp integrator with square wave input. Note that whilst the input is positive, the output becomes steadily
more negative, as this circuit arrangement is an inverting integrator.
      ip(T/4)       (Vpi/R)(T/4)      Tvpi
Vpo = ~         =        C         = 4C------R
  (b) The opamp differentiator.

form is a triangular wave as shown. However,                                  could be partially compensated by adding R in
don't forget that an integrator's gain at zero                                series with the non-invert input as shown.) Even if
hertz is indefinitely large; so if there is a DC                              these causes are absent, it might be that the square
component at its input, be it ever so small, the                              wave is not exactly centred about ground, or, if it
mean level of the output waveform will wander up                              is, that its mark/space ratio is not exactly 50/50. If
or down, according as the component is negative                               the output of the integrator is connected to a
or positive respectively, until the opamp's output is                         comparator with hysteresis, the comparator will
stuck at a supply rail voltage. The DC component                              convert the triangular wave output of the integra-
could arise from a number of causes, for example                              tor back into a square wave, which can be used to
the input offset voltage of the opamp, or, if it is a                         provide the input to the integrator in the first
bipolar input type, the drop caused by the input                              place. The circuit is a bang-bang servo which
bias current flowing through the resistor R. (This                            maintains the DC component at the input to the
152   Analog Electronics

                        vi,                          OV                    ~               Vo

                                           R                   R[2

                                           R         I         D~              I

                   +Up   __~                                                                    vo-~_~p
                   OV           --

                                     R = IOK (say)

                                                                           ]   i

                                                                l"~e               w



                                          R                     R                      R

                                                                                    Active signal processing in the time domain   153

                                             A   9
                                         C                                                           I
                                    vi               :i                :
                                                                      D2          I'
                                                                      | l
                                                           ....       | II-----
                                                                      | l

                                                                                                                      - Ovo

                            (i)                      2                 R1 (10K say)
                                                                  R                            R
                                                     ,|               !-


                            (ii)         ~                                [~/~
                                                                      _ D11                                     V~7
                                                                                                               O o7
                                        vi                                    II,                  K say


                                                                       R                      vo

                                                                                                    ~      L


Figure 7.4 Full-wave rectification.
   (a) Full-wave rectifier using diodes.
   (b) Ideal rectifier circuit (accuracy depends on absolute values of all resistors). D2 improves high-frequency
performance by closing the NFB loop around A I on negative-going input half-cycles, thus preventing A I from
being overdriven and saturating.
   (c) Poor man's ideal rectifier. Only the two resistors R need be accurately matched; actual value is unimportant.
Poor at high frequencies, since the opamp is saturated during negative half-cycles of the input and hence requires
a recovery time. Make RL >> R.
   (d) As (b) but less susceptable to errors due to opamp input voltage at low input levels.
   (e) Two circuits requiring only one matched pair of resistors. In each case an extra stabilizing capacitor C may be
needed as shown, as the loop includes both AI and A2 (on input negative half-cycles, upper circuit: on positive
half-cycles, lower circuit). (i) For unity gain, set Rg to zero: Vpo = Vpi. Input resistance differs for +ve and -ve
input half-cycles. (ii) Unity gain, very high input resistance.
   (f) Another poor mans ideal rectifier. Requires Rs << R, R << RL.
154    Analog Electronics

integrator at zero; it is also an example of a class of   sensitivity would typically be scaled to make the
oscillator known as relaxation oscillators, which         instrument read correctly on a sine wave. Figure
are described in Chapter 9.                               7.4c shows another absolute value circuit which is
   Figure 7.3b shows another way of turning a             more remarkable for its ingenuity than anything
triangular wave back into a square wave. A                else. 4 On positive-going inputs the opamp's feed-
ramp of voltage applied to the capacitor will             back loop is closed via the diode and the lower
cause a current I = C dv/dt, where dv/dt is the           10 K resistor, whilst for negative-going inputs the
rate of change of the ramp voltage in volts per           opamp is open loop. Its output therefore flies up to
second. The opamp's output voltage will cause an          the positive rail, saturating the NPN transistor.
equal current I to flow through R and hence will          This connects the negative input directly to the
remain at a constant positive or negative voltage         output with no more than a few millivolts drop
on alternate half-cycles of the input triangular          across the transistor, since it is being used in the
wave. The amplitude of the output square wave             inverted mode. Used in this way, a transistor's
is proportional to the amplitude and frequency of         gain is low - of no consequence in this applica-
the input triangular wave and to the differentiator's     tion - but the offset voltage between collector and
CR time constant.                                         base is much lower than the Vccsat of the device
   If the input to a differentiator is a square wave,     used in the normal mode. The circuit works well
overload will usually result. This is because the         enough at low frequencies, but performance is
slope dv/dt of the positive- and negative-going           poor at higher frequencies owing to the time
edges of a square wave is very large (ideally             required for the opamp to recover from overload
infinite), so that the current through the capacitor      each time the input goes positive. Figure 7.4d, e
will exceed the maximum current that the opamp            and f show other ideal rectifier circuits.
can force through R.                                         An ideal rectifier can be used to rectify a ground
   The absolute value circuit, also called an ideal       centred triangular voltage waveform, the result
rectifier, is an important and commonly met tool          being a positive-going triangular wave of twice
in signal processing. Figure 7.4a shows a full-wave       the frequency and half the amplitude. If this is
rectifier of the biphase variety, such as is often used   translated back to a 0 V centred level and amplified
in power supply circuits. It can be used to provide       by a factor of two, the same process can be
the absolute value (i.e. the modulus) of a wave-          repeated many times, giving a series of octave
form, but suffers from two disadvantages: first, it       related triangular waves.
requires a centre tapped transformer; and second,            Another widely used function in analog signal
the smaller the peak-to-peak input voltage the            processing is the constant current generator.
poorer the performance, owing to the forward              Chapter 3 on active devices described how the
drop in the diodes. By enclosing a diode D1 within        collector slope resistance of a transistor is very
the negative feedback loop of an opamp, we can            high, so that to a first approximation the collec-
effectively reduce its forward voltage to negligible      tor current is independent of collector voltage.
proportions and thus simulate a perfect rectifier.        The slope resistance is even higher when the
Figure 7.4b shows one such circuit. A typical             transistor is used in the common base connection
application might be as an AC/DC converter in             rather than the common emitter connection, and
an audio-frequency millivoltmeter, where the cir-         the same comment applies to FETs. In Figure
cuit's ability to provide a linear relation between       7.5a a variety of constant current sources is
the peak amplitude of the AC input and the DC             shown; the output current may be fixed or
output right down to small-signal levels is a big         controlled by a signal voltage. In the latter case,
advantage over the passive circuit of Figure 7.4a.        the proportionality is upset by the Vbc of the
In this application, the DC output could drive a          transistor or the gate voltage in the case of an
meter whose scale is calibrated in millivolts RMS.        FET; this is a further imperfection on top of the
The circuit would actually respond to the average         output resistance which, though high, is not
value of the modulus of the input, but the meter          infinite.
                                                                             Active signal processing in the time domain                               155

                         R ~                                                          0V               I                           =      R

               0--I                        ~                                Vi    /77~                                                 NPN
               Vi   0V                                                                                                                 types
                  c                -                                        o-q

                         0V                           R

              V                                Vi 0 V                                                                          0--7             OF T
                                                                                                                                               M SE
                                               ~                                                                                               types
                                       I                              I
                                                                                          Vi 0
                                                                                           o ~, - -                            Pchannel
                                                                      (a)                      n-~
                                                                                          N channel

                  o ~ I                                                                                                        I         Vi
                                                     I-     hFE-1 .Vi                                                              I-
                                                             hFE   R

                  0V                                                                      R
                  o/.~    -                               (b)
                                                                            0--       -

                              +V                                                           +V

       iR -    R__._.~ _                                   = iR -2 ib                              2i
                                                                                           =='~- _~_~ (approx)

                    ib ~-          ib          .--
                                                                                  I                                        I
                                                                                           .   .   .   .   .   .   .   .   J


Figure 7.5      Constant currnt generators.
   (a) Unipolar (sink only or source only) constant currrent generators.
   (b) O p a m p aided (improved) current sinks.
   (c) Current mirrors. By connecting two transisors as shown, the collector current of the right-hand device will
'mirror' that of the other, apart from the reduction due to base current. The circuit will only work if the
transistors are perfectly matched, which means in practice that they must be a monolithic pair. Such devices are
available, and, by using multiple emitter transistors, ratios other than 1 : 1 can be obtained, e.g. 1 : 2 (see right
hand circuit) and 2 : 1. Current mirrors are widely used in monolithic opamps.
156    Analog Electronics

   A better constant current source can be ar-             or reactive, as under these conditions the negative
ranged by enclosing the active device, be it FET           feedback is stronger than the positive. With ZL
or bipolar transistor, in the feedback loop of an          open-circuit the net feedback is ideally perfectly
opamp as in Figure 7.5b. The opamp output will             balanced, but in practice, owing to resistor toler-
take up whatever voltage is necessary to pass a            ances, one or other will predominate. So the
current through the resistor R such that the               opamp's output will either fly off to one supply
voltage at the inverting input equals the input            rail or the other, or hover in the region of 0 V at a
voltage. In the case of the bipolar type, if a change      magnified version of the opamp's input offset
in collector voltage, due to some change in the            voltage. The output is truly bipolar, that is to
load receiving the constant current, causes a              say it can source or sink current at any voltage
change in the required Vuc, this will automatically        within the opamp's output voltage range. In this
be implemented by the opamp so as to keep the              respect it differs from the current mirror of Figure
voltage across R unchanged. However, variations            7.5c, which shares the limitations of the other
in the current gain of the transistor are not              constant current generators of Figure 7.5. If the
compensated for, as it is actually the emitter             load connected to a Howland current pump is a
current of the transistor which is the controlled          capacitor, then a ramp of output voltage is
current. Thus the version using an FET as the              produced; the rate of change of voltage depends
controlled source provides improved performance.           on the controlling voltage at the input and the size
   The constant or controlled current sources in           of the capacitor (Figure 7.6d). This application
Figure 7.5a, b and c are strictly unipolar, that is to     dates back many years; 6 the arrangement is a non-
say that the output current can be set anywhere from       inverting integrator, with the further advantage
zero to some large positive value when using the           that one end of the capacitor is grounded.
PNP version, or from zero to a negative value (a              The base/emitter voltage of a bipolar transistor
current sink rather than a source) with the NPN.           is logarithmically related to the collector current,
Likewise, the voltage range that the output can take       and this is the basis of a class of circuits which can
u p - the voltage compliance- is limited to values         be used as elements of an analog computer.
negative with respect to the PNP's base voltage and        Analog computers were widely used at one time
conversely for the N P N type. Another type of             and are still employed in certain areas where
constant current generator, called the Howland             limited accuracy is adequate but high processing
current pump, s circumvents both of these limita-          speed and lower power consumption are essential,
tions. As you can see, the circuit has feed-back from      e.g. certain military applications. Figure 7.7a
the output to both the inverting and the non-              shows a logarithmic amplifier, that is to say one
inverting terminals in equal measure (Figure 7.6a).        where the output voltage is proportional to the
The feedback dividers therefore form a balanced            logarithm of the input voltage. 7 The logarithmic
bridge, with the result that variations in the output      voltage/current relation of the transistor only
voltage produce only a common mode input to the            holds over a wide range if the collector/base
opamp, and this has no effect. Thus with Vii, and Vi2      voltage is zero or very nearly so; this requirement
both at the same voltage, zero or otherwise, the           is fulfilled by using the device in the feedback loop
output voltage is indeterminate. Now look what             of an opamp as shown. Under these circum-
happens when an offset is introduced between Vii           stances, the logarithmic relation can hold over up
and vi2 (Figure 7.6b). The only way the opamp's            to seven or even eight decades of current, for some
input terminals can be at the same voltage is if a         transistor types.
current IL -- (V2 - V1)/R2 flows through the load,            Figure 7.7b shows how two logamps, an opamp
as you can prove by substituting different values of       used as an adder, and an antilogamp can be used as a
V1 and/or V2 or, more satisfactorily, by analysing         multiplier to obtain the product of two quantities,
the circuit formally.                                      represented by the two input voltages. The arrange-
   The arrangement is short-circuit stable and is          ment shown is unipolar, so that both inputs must be
also stable for all other finite values of ZL, resistive   constrained to be always positive, for which purpose
                                                                                  Active signal processing in the time domain            157

                    R                  R                                         R1              nR 1
        Vii             .....               b--                         V1 o-4        :-     !          "I:- ]

        Vi2                                                             v2o-4

                                 io                                                    zL        IL = ( v 2 - v~)/r2
                           ~~7             (a)                                                           (b)

                                R1                          R2
                                                                                 ov_JL     Reset
                                                   R4             R5
                                                             I~ :---o-15 V                                                R1
                                           15 V 0---~--,
                                                    !         L

                                                              VL                                                   - 89
                   R 2 0/2 -V 1)
              IL= -~1 x   R5                                                                                       (d)
              but set   R3 =           R2
              and trim Ra for best Zo

Figure 7.6 Bipolar current generator with wide bipolar voltage compliance.
   (a) The basic Howland bipolar current generator. In this useful 'current pump' circuit, the output is taken not
from the opamp's output terminal but from its non-inverting input terminal. It is a 'bipolar' voltage-controlled
constant current generator, i.e. it can either source or sink current.
   (b) Supplying load current. When supplying large load currents to ZL (RE is low), opamp loading is minimized by
making Rl >> R2.
   (c) Further modification for even higher load currents. The circuit has bipolar voltage compliance: it can source
or sink current at both positive and negative voltages. Making n < I increases the voltage compliance range.
(Reproduced by courtesy of EDN magazine.)
   (d) Sawtooth generator using a non-inverting integrator with ground referenced capacitor. A circuit similar to
this appeared as long ago as 1970:6 did the author recognize it as an application of Howland's current pump?

an ideal rectifier circuit can be used. The correct                              as in Reference 8) is switched on, grounding the
polarity can be restored to the output by using                                  non-inverting input of the opamp, the circuit's gain
comparators to determine the polarity of the input                               is unity and inverting; whereas if the F E T is off, the
voltages and using these to control the polarity of a                            voltage at the non-inverting terminal of the opamp
sign switched amplifier. One of these is shown in                                is the same as the input voltage, so the gain is unity
Figure 7.7c. If the F E T (or bipolar transistor switch,                         and non-inverting.
158   AnalogElectronics

                                                         ~ llxA

       vi                                                                                   Vo = -k log Vi




                                                                                                        ~    +Vs


        89                                               ~'---~~ 1T)77""                       V~ VilVi2~
                                                         k' log Vii + k' log Vi2 = k' log Vii Vi2
                                                                           (b)                         Antilog circuit

                                               IOK                    IOK
                            ViO       -~   ~         1     -     L

                +~v         ,~             ~                                         ~
        0   V~ "-'~~c~176                                                        /~
                      IIInverting                                    (c)
        Non-          I
                                                                           Active signal processing in the time domain   159

                         I '1'
                                            220 pF                    Vo             100R!~
                                    ,-      ~
                                                     -[        lk ,
                                                                     T , 15k     !   -        A

                                                     R6                                           1
                                                "          '                                          !

                             10 k                    L__. 220 pF
                                    1                                            2k
                                                     T                       !

Figure 7.7 Log and antilog circuits.
  (a) A logarithmic amplifier, k depends upon RI, R2, R3 and Trl. Trl is a monolithic dual transistor, e.g. 2N3860.
  (b) Multiplier using Iogamps. Note that the usable range of Vo is no greater than in (a). Therefore the circuit
cannot handle the whole combined range of VI V2.
  (c) Selectable sign of gain circuit: gives • or x(-1).
  (d) Improved log ratio amplifier (reproduced by courtesy of Electronic Product Design).

   If the same voltage is applied to both inputs of a                     Analog operations with a mixture of discrete
multiplier circuit such as in Figure 7.7b, the output                  devices and opamps are subject to errors due to
is the square of the input voltage. If this is                         imperfect matching of the logarithmic character-
smoothed (averaged) and then put through the                           istics of the transistors and due to changes of
inverse process, i.e. the square root taken, the                       temperature. These errors, together with a host
result is the RMS value of the input voltage. The                      of preset adjustment potentiometers for setting up,
square root can be extracted by enclosing a                            can be very largely avoided by using one of the
squarer in the feedback loop of an opamp. Figure                       integrated multiplier/divider or RMS linear inte-
7.7d shows a circuit which returns the logarithm of                    grated circuits which are available from a number
the ratio of two input voltages. 9                                     of manufacturers. A typical example of a multi-
   Another type of low-speed analog multiplier,                        plier is the AD534 from Analog Devices, which is
popular in the days before multiplier ICs were                         laser trimmed to an accuracy of 1% or better and
available, used the fact that the average value of                     which is a full 'four-quadrant' multiplier, i.e. the
a unipolar (e.g. positive-going from zero) pulse                       inputs are not restricted to positive voltages. The
train of constant frequency is proportional to the                     AD536 is a typical RMS-to-DC converter from
product of the pulse width and the pulse height.                       the same manufacturer.
A not dissimilar scheme employed pulses of
constant width, but used one of the inputs to
control the frequency of the pulse train, i.e. as a
voltage-to-frequency converter, l~ This ingenious                       Frequency-to-voltage (F/V)     and    voltage-to-
circuit produced an output Eo = E1Ez/E3, i.e.                          frequency (V/F) converters are important items in
the product of two inputs divided by the third                         the analog engineer's toolbox of useful circuits, as
input. Even in the days of valves, applications                        they can be used for so many applications. Figure
such as analog multiplication and squaring were                        7.8a shows a simple discrete V/F circuit 12 which
carried out, using multielectrode valves such as                       produces an output in the range 0 to 10kHz
the nonode, l l                                                        linearly related to a positive input control voltage,
160   Analog Electronics

                                                            680 pF         A
          DC input                                       _..___J ~         1.
          (+ve)                                                                       -12 V
                           l                                                      a0K_ 41-~ 12V
               R3 11                                                                       2              7
                      LJ                                                                                              R1
                                                                                   A                                       7
               R4                         s1-~p               ~                        -
                                                                        1Slll                                         R2
               5K6                             FET                 1M                                                 4K7
                                          -_                  ..                                                  -    0V

                                                     vo DC               (a)
                                    v,          -
                                                     ripple                       vo
                                                                                 Vipp--t' --,~
         sine or                   D2          C2        RL                         [     /~~'-:=.~                            ~ Timeconstant
         square,~                                                                      I       ,~[                             -" C2 RL

                                   For low ripple, make C 2 >> C 1

                                                                                                              1                Time
                                                                           (b)                 C2/C 1 •
                     C]             Dl
               0     II,       -    ,-,
                     II             "                                                                                                            .,
              vi                                                                                                                       . . . .     ....
                                                                                                                                                 ~ 9

                                                                                   o                                                             "" RL

                                                                                                     vi                          C2                       - o

Figure 7.8 Voltage-to-frequency-to-voltage converter circuits.
  (a) V/F converter (voltage-controlled oscillator)(reproduced by courtesy of Electronics and Wireless World).
  (b) Simple pump and bucket F/V converter has exponential response when vi peak-to-peak volts at f are first
applied. Vo/Vi =fCIRL, fairly linear up to 10%.
  (c) Linear pump and bucket F/V converter. Because of the bootstrap connection to the anode of D2,CI is fully
charged on every negative-going edge, regardless of Vo.
  (d) Another linear F/V converter.

with a scaling of 1 kHz per volt. Figure 7.8b shows                                voltage drop), and on each positive-going edge a
a simple 'pump and bucket' frequency counter or                                    charge of C1 V coulombs is tipped into the bucket
F/V circuit. On each negative-going edge of the                                    capacitor (72. At least, that is what happens
input square wave the pump capacitor C1 is                                         initially. However, as the voltage across (72 rises
charged up to V volts (less one diode forward                                      to the point where charge is leaking away through
                                                                                 Active signal processing in the time domain      161

                                                                                      IC1 CD4046B
                                                                                      IC2 CD4069B

                         +12 V.
                             C1 D1_

                                             .   .   .   .   .             i          TR1ZTX500
                                                                                      TR2 XTX300
                                                                                      D1, D2 IN914

                     1/6 4069 330pF                              1     16                  4[ 0      69
                       2-2r~                                          19       IC1
                                                                                                          1/6 4069
                        1()0:: D2~        ~00nF~C3R                   1
                                                                                       100 pF
                                0 V IN914                            10K

Figure 7.9 Circuit to provide an output at f2 = nfl where n can be a non-integer, greater or less than unity,
f2 = Jl (CI / C2). For correct operation, the peak-to-peak voltage of inputs at fl and f2 to the two charge pumps must
be equal, ensured here as they are derived from inverting buffers in the same CD4069 hex inverter. (Reproduced by
 courtesy of Electronic Engineering.)

RE as fast as it is being added via C1, the latter is                          balanced by removal via a complementary polarity
no longer discharged completely on each cycle of                               pump. The voltage across the bucket is connected
the input, resulting in the non-linear (exponential)                           to the voltage-controlled oscillator forming part of
relation between input frequency and output vol-                               1C~ and the output frequency is connected to
tage shown. Figure 7.8c and d show two solutions                               one of the complementary V/F inputs. The
to this problem. In the first, the pump is charged                             output frequency will therefore settle at value
up on negative-going edges not with respect to                                 F2 = (C1/C2)F1, since any deviation from this
ground but with respect to the buffered output                                 ratio would result in inequality of the average
voltage. In the second, the charge flows into a                                current outputs of the two F/V converters in
short-circuit provided by the virtual earth at the                             such a sense as to change F2 so as to equalize
inverting input of an opamp. The bucket should be                              them again.
large compared with the pump in order to keep                                     As with multipliers and RMS circuits, V/F and
down the size of the ripple on the DC output of the                            F/V circuits are available in IC form from a number
circuit, but not so large that the response to a                               of manufacturers. The AD645JN from Analog
change of input frequency is unduly slow                                       Devices is an integrated V/F converter operating
C2 = 10C1 is generally a good compromise. In                                   with a full-scale frequency up to 500 kHz and with a
the case of Figure 7.8d, C2 can alternatively be                               linearity of 0.03% for a full-scale frequency of
connected round the opamp as shown dashed,                                     250 kHz. The AD651AQ from the same manufac-
turning it into a leaky integrator.                                            turer is a synchronous V/F converter, that is to say
   Yet another variant is to replace the second                                that its full-scale frequency is the same as that of an
diode by the base/emitter junction of a transistor,                            externally provided clock waveform. It uses the
with the bucket capacitor in its collector circuit.                            charge balancing technique mentioned earlier and
This again provides a linear output as the pump is                             accepts a clock frequency of up to 2MHz. At
always charged and discharged by the full input                                100 kHz full-scale clock frequency the linearity is
voltage s w i n g - except of course for the two diode                         typically 0.002%. The 9400CJ from Teledyne
drops. Figure 7.9 shows an ingenious application                               Semiconductors is an IC which may be used as
of this scheme. 13 Here, a single bucket receives an                           either a V/F or an F/V converter, with operation
average rate of charge input from one pump                                     up to 100 kHz and 0.01% linearity at 10 kHz.
162   Analog Electronics

   To an ever-increasing extent, analog signal             the analog input will produce a logic one at its
processing is being carried out in numerical               output whilst all the others will indicate logic zero.
form, by first converting an analog waveform to            On the rising edge of the clock input waveform the
a stream of numbers denoting the value of the              outputs of all the comparators are latched and
voltage at successive instants. The number stream          encoded to 7-bit binary form, and the result is
can be operated upon by a digital signal processor         transferred to a set of output latches.
(DSP) IC in any number of ways (e.g. filtering,               Although the device only provides 7-bit resolu-
raising to a power, integrating, taking the mod-           tion, the divider chain and comparators are accu-
ulus, autocorrelating, cross-correlating etc.) and         rate to a resolution of one part in 28. It is possible
then, if appropriate, reconverted to an analog             to take advantage of this by cascading an
voltage. From the point of view of the analog              MC10315L with an MC10317L to obtain 8-bit
engineer, the important stages are the conversion          accuracy. Because all the comparators are identical
to and from digital form, since any errors intro-          and their outputs are latched at the same instant,
duced, particularly in the analog-to-digital conver-       they form a 'snapshot' of the input voltage at a
sion, simply misrepresent the original signal and          time tad shortly after the positive clock edge, where
cannot afterwards be rectified.                            tad is called the aperture delay time. Its value is
   There are many different forms of analog-to-            generally not important except when using two
digital converter (ADC, or A/D converter). Con-            similar ADCs to compare the voltages at two
sider first the flash converter, in many ways con-         points in a circuit, as in a dual-channel digital
ceptually the simplest though not historically the         storage oscilloscope (DSO). However, even in a
earliest type of ADC. The mode of operation,               single-channel application, sample-to-sample var-
which should be clear from Figure 7.10, is as              iations in tad are clearly undesirable, especially at
follows. The reference voltage, or plus and minus          the 15 mega-samples per second maximum sam-
reference voltages if ground centred bipolar opera-        pling rate of the device when the input voltage is
tion is required, is connected to a string of equal        changing rapidly. The MC10315/7L devices can
value resistors forming a precision voltage divider.       cope with input slew rates of up to 35V/~ts and
The voltage difference between any two adjacent            exhibit an aperture uncertainty of only 80 ps. The
taps is equivalent to the least significant bit of the     successive approximation register is another type
digital representation of the input analog voltage,        of ADC, capable of providing much greater
which is thus quantized- in the case of the 7-bit          resolution than the flash converter. However, to
MC10315L from Motorola, to a resolution of one             understand how it works, look first at the converse
part in 27 or about 0.8% of full scale. Each               process of D/A conversion.
reference voltage tap is connected to one input of            Figure 7.1 l a shows a current output high-speed
a string of 128 comparators, the analog input              8-bit multiplying digital-to-analog converter (DAC,
voltage being connected to the other input of              or D/A converter). Imagine the reference ampli-
each and every comparator. Every comparator                fier's inverting input connected to ground and its
whose reference input is at a lower voltage than           non-inverting input to, say, a +2.56V reference

Figure 7.10 A high-speed analog-to-digital converter (ADC).
  (a) Block diagram of Motorola MC10315L/7L 7-bit flash ADC. Features: 7-bit resolution and 8-bit accuracy
plus overrange; direct interconnection for 8-bit conversion; 15 MHz sampling rate; input voltage +2.0 volts, input
capacitance < 70pF; power dissipation 1.2W; no sample-and-hold required for video bandwidth signals;
standard 24-pin package.
   (b) Equivalent circuit of reference resistor ladder network, showing the input Ain applied to 128 comparators.
R ~ 0.5 ~. Ceq is the lumped equivalent value of capacitance, representing the distributed capacitance for each
resistor R and the input capacitance for each comparator.
   (Reproduced by courtesy of Motorola Inc.)
                                                       Active signal processing in the time domain                 163

                       (11)              (9)                                (10,24)
                       Ain               CLK                                VCC

  (5) VRTe I                               .....      "1 r. . . . .
                                                       I 0          ~                           o OVR       (14)
                                                       I   I
(3) VRTaCe                                             I I            Ii
                                                       ) ,            '                         oOVR        (15)
                                               128     I I
 (6) VRM e                          e          to7     ~ Output!
                                               encode        latches ;                      i o D 6 (16)
                                                       I'l                                              I
(8) VRB acc                                            I ,            ,                     ,           i
                                                           i   I
                                                           )   )                    ,   ) )oD      O (22)
  (7)   VRBO     [       ,                 L ....      J       L . . . ,.       J
                                         Gnd2          Gndl                  VEE
                             (a)         (1, 13)       (23)                 (2; 12)



                                                - -r
                                               ..1.. Ceq 1.5 pF
                       RT R
          VRT acc
                       5 R~..                     ,

                        R/2~                    "                              To 127-to-7
            VRM C                                 9
                        R/2                       o
          VRB ac e

                             .lIl              5_.
                        R / 2 1 1 Analog---~Ceq 1.OpF
               VRB c
                         (b)              (Ain)
164   Analog Electronics

                                               (MSB)                      (LSB)
                                    V+      VLC B I B 2 B 3 B 4 B 5 B 6 B7 B8
                                     t 13                          [ l l
                                              t l t ' t6 t7 1 8 ~9~10~11~12-1
                                  -Bias I ~ Logicbuffers and level shifters I
                                    etwork.l I [_ t~ [~-]~ I~ [~ I i                        O   lout
                    Vref(+) o--
                    v,of(_) 5
                                  R e ~      ~   ~       2R~ 2 ~ 2 ~ 2 ~ ~          2~

                             Compensation V-


                                                         ~               0 t o + , f , RL
                                                         I   ~R     ,     255 ,
                                                     I       _~._   /fr = 2"~" lref

Figure 7.11                                           (b)
  (a) An 8-bit digital-to-analog converter (DAC).
  (b) An application producing a positive-going low-impedance output.
  (Reproduction by courtesy of Motorola Inc.)

voltage source via a 1 K resistor. Then 2.56 mA                     going low-impedance voltage output of 0 to
will flow via the left-hand transistor and resistor R               10.20V in 40mV steps, according to the 8-bit
to the negative supply V-. The transistor acts as a                 binary code applied at the logic inputs. The device
constant current generator in a negative feedback                   is called a multiplying DAC because its output is
loop; the feedback is taken to the opamp's non-                     the product of the binary input and the reference
inverting terminal as the transistor provides the                   current. For the device shown in Figure 7.1 l a the
necessary inversion. The transistor connected to                    settling time of the output current to within 0.5
the left-hand current switch will pass half as much                 least significant bit (LSB) following an input code
current, as its emitter resistance is 2R. You can                   change from 0 to 255 or vice versa is typically less
easily verify that the next transistor current switch               than 100 ~ts, whilst the reference current may be
will receive half as much current as that, and so on                slewed at up to 4mA/las at least and typically
for each successive stage to the right, except for the              more.
last. Theoretically the last isn't quite right as                      So to return to the successive approximation
shown: a remainder current equal to the least                       register (SAR) ADC, which works as shown in
significant bit is in fact shunted to ground, so                    Figure 7.12a. The control logic sets the DAC
that the maximum output current at lout with all                    output to half of full scale by setting the most
logic inputs B1 to B8 high (or at Iout with all low) is             significant bit (MSB) to 1 and all the other bits to
only (255/256)Iref, or 2.55 mA in this case. Figure                 0. If the comparator indicates that the DAC out-
7.11b shows the DAC arranged to give a positive-                    put is greater than the analog input voltage, the
                                                                      Active signal processing in the time domain      165

                                                         16-bit DAC


                                                           16-bit SAR



                                    High-speed diode
                                    sampling gate

                                                         t_]               Balanced narrow
               Input~          _
                            . __                        l~       /         samplingpulse         -"

                                                       cs 7                  t>-- / -r-
                      Input                       i"      1     I~""~}/               ~t
                      attenuator                        _t j    II  "~t

                                                         S ',
                                                        ]'lilf rever e bias                      Sorvofo~backof
                                                                                                 last sample voltage

Figure 7.12
  (a) A 16-bit successive approximation type of ADC (reproduced by courtesy of Electronic Product Design).
  (b) Simplified schematic circuit of a sampling gate as employed in a high-speed analog sampling oscilloscope.

MSB is reset to 0; otherwise it is left set, and the                 voltage. The resolution of the digital answer is one
control sets the next most significant bit (NMSB)                    part in 256 or 0.39% (for an 8-bit device or 1 in
to 1. The process is repeated so that after N                        65 536 for a 16-bit device as in Figure 7.12), but the
comparisons the register controlling the DAC                         accuracy is another matter entirely. It depends
holds an n-bit digital representation of the input                   upon the DAC's linearity and the accuracy of its
166   Analog Electronics

reference voltage. It is obvious also that, with the   rangement is a monolithic switch using two en-
sequential conversion algorithm used by the SAR        hancement mode FETs in parallel, one N channel
DAC, the final answer will apply to the instant        and one P channel. The gates are driven by
when the LSB comparison was made and can only          appropriate antiphase voltages derived from on-
be guaranteed correct if the input voltage changes     chip CMOS inverter stages; these require positive
negligibly during the course of a conversion.          and negative power supply rails, usually +15 V. By
   Owing to the sequential nature of the conver-       using two complementary FETs in parallel then,
sion, SAR ADCs are inherently slower than flash        when the switch is on, one or other FET is fully
types. They would effectively be very much slower      enhanced regardless of the input voltage. The
indeed, because of the requirement that the input      usable range of input voltage, with the switch off
voltage must not change significantly during the       or on, extends to over 80% of the supply rail
course of a conversion, if special steps were not      range, e.g. +14V or so with +15V supplies,
taken. The usual procedure is to precede the SAR       depending upon what maximum on resistance is
DAC with a sample-and-hold (S/H) circuit. The          acceptable, because this rises as the input voltage
LF398H from National Semiconductor is a mono-          approaches either rail. The DG series is a long-
lithic BIFET sample-and-hold IC which will ac-         standing industry standard range of CMOS analog
quire the current value of the input voltage within    switches available from many manufacturers. For
10 las of being switched from the hold to the          example, the DG300A from Maxim Integrated
sample mode, however much the input has chan-          Products provides two separate single-pole single-
ged in the meantime, and will hold the current         throw (SPST) switches, each with a maximum on
voltage with a low droop rate when switched back       resistance rdson of 50 f~. In digital control applica-
to the hold mode. Sample or hold mode is selected      tions, it is usually desirable to have the on/off
by means of a TTL and CMOS compatible control          control input for each switch latched. For ex-
input, and the signal input characteristics do not     ample, the AD759xDI series are TTL, CMOS
change during hold mode. The two-stage                 and microprocessor compatible di-electrically iso-
AD585AQ from Analog Devices features a fast            lated CMOS switches featuring overvoltage pro-
acquisition time of 3.0 ~ts to 0.01% of the steady     tection up to +25 V above the power supplies. The
state final voltage for a full-scale input change.     AD7590 has four latch inputs, activated by a logic
Both devices incorporate offset null adjustment        zero on the write (WR) input, each controlling a
facilities. The devices described were state-of-the-   SPST switch, whilst the AD7592 has two logic
art when introduced, but have long been out-           inputs each controlling a single-pole double-throw
classed by more recent introductions.                  (SPDT) switch. Single-pole multiway switches are
   A discrete sample-and-hold circuit could of         known as multiplexers, a typical example being the
course be arranged using an electronic switch of       DG508A. The Maxim version of this industry
some sort and a hold capacitor. The use of a           standard single-pole eight-way part has an rdson
bipolar transistor in the inverted mode as a switch    of 300 f~ maximum, and its switching time is 1 ~ts
has already been mentioned, but it is far from         maximum. Again, improved performance is of-
ideal. For one thing, it only works well for one       fered by more recently introduced devices.
polarity of the switched voltage. At one time             Whilst this is adequate for very many applica-
symmetrical transistors were available. These          tions, there are cases where a very much more
were fabricated with identical emitter and collector   rapid switching time is essential. A good example
junctions for use as cross-point switches in switch-   is the sampling gate used in a sampling oscillo-
ing matrices, and had the advantage of working         scope. The difference between a sampling oscillo-
equally well (perhaps equally badly would be a         scope and a digital storage oscilloscope (DSO) is
better description) in either direction. Nowadays      that, unlike the latter, the sampling oscilloscope
the FET is universally used as an electronic switch    does not apply the samples it takes to an ADC.
in applications up to a few megahertz. Whilst an       The sample is stored in a hold capacitor for one
individual FET can be employed, the usual ar-          sampling period only, after which it is replaced by
                                                          Active signal processing in the time domain        167

the next sample. The main requirement is for a          original signal as shown and will be heard by the
very fast analog gate driven by a very narrow           ear (so the theory runs) as at the original low pitch,
sampling pulse. In order to achieve the widest          even through a loudspeaker system incapable of
bandwidth and shortest rise time possible, the          reproducing such low notes. Figure 7.13b is an-
sampling pulse is made very narrow; in particular       other gem from the armoury of the with-it audio
the trailing edge, where it switches from on to off,    engineer. It provides even-order d i s t o r t i o n - or,
is particularly fast. In the Tektronix 7S11 plug-in     with the alternative feedback network shown, both
with sampling head type $4 in a 7000 series             odd and even order. The result is both harmonic
mainframe, the rise time was only 25 ps, corre-         and intermodulation distortion of precisely the
sponding to a bandwidth of 14 GHz. To achieve           sort that in earlier years engineers spent great
this, a sampling pulse 200ps wide but with a            efforts trying to eliminate. It produces a sort of
trailing edge aperture time of only 20ps was            back-ground furriness to the sound, which can be
used. Figure 7.12b shows a high-speed sampling          heard on the soundtrack of any prewar black-and-
gate using Schottky diodes as the switching ele-        white movie and which is now back in fashion
ment. A single diode acts as a switch, in the sense     under the label 'fuzz'. It's marvellous what you can
that it is on if the anode is positive to the cathode   sell to some people.
and off otherwise. By combining four diodes as             Here is another example of waveform proces-
shown in Figure 7.12b with a pulse transformer,         sing: in this instance, a scheme to multiply the
they can be pulsed into conduction to provide a         frequency of a waveform by three. The original
path between the input and the output, and              application 14 was to obtain a clock frequency of
switched off again to leave a sample of the input       around 100kHz, for a microcontroller system
voltage at the instant of switch-off stored in the      incorporating an MM58174 real-time clock chip,
hold capacitor. The absence of stored charge in         from the latter's 32 768 Hz crystal derived clock
Schottky diodes permits very fast switching times,      output. The circuit, shown in Figure 7.14a,
resulting in the performance described above.           accepts either a sine or a square wave input and
Whilst analog sampling oscilloscopes such as that       operates as follows. The first two inverters ICla
mentioned are now part of history, the type of fast     and IClb are biased as linear amplifiers by virtue
sampling gate described forms the critical signal       of being enclosed in a three-inverter loop with
capture stage in modern digital sampling oscillo-       overall NFB at DC and low frequencies, via R2.
scopes, the fastest of which offer bandwidths up to     Their high gain ensures a square waveform at A,
50 GHz.                                                 even if the input is a sine wave. The square wave
                                                        is integrated by IClc to give a triangular wave at
                                                        B, and again by ICld to give a parabolic wave-
Further applications of analog processing
                                                        form at C. This waveform is a good approxima-
A number of applications of analog signal proces-       tion to a sine wave, having only 3.5% total
sing have been mentioned, in the course of review-      harmonic distortion. The 'sine wave' is subtracted
ing the various circuits that are used for this         from the square wave to give the waveform
purpose, but here are a few more, starting with         shown at D in Figure 7.14b: this is a square
one that is frankly frivolous.                          wave less its fundamental component and, as
   Figure 7.13a includes an amplifier arranged to       you can see by counting the zero crossings, it
provide less gain to negative-going inputs than to      contains a substantial amount of third harmonic
positive. The result is to produce second-order         of the original frequency. C4 is included to allow
(and other even-order) harmonic distortion. If          for excess phase shift through 1Cla and IClb; it
this is applied only to the very lowest notes of an     would not be necessary if the circuit were operat-
audio signal, say those below 100 Hz (separated         ing at a lower frequency. Finally, ICle and IClf
out from the rest by a low-pass filter), correlated     slice waveform D through the zero crossing
energy at harmonics of those low notes will be          points to produce the wanted output at three
generated. This can then be added back into the         times the input frequency. The circuit makes
168    Analog Electronics


          Gain (dB)          /                                                            Set synthetic
                                                                                          bass level

                                       9 logf
                            100 Hz

              In       Buffer
                                                            I~1        A   I~i        A    I~1        _

                                                   I ....              T   v,
                                                                                                 .!                    'ypa si
                                                            "~'                            ,-,


                                                                  R3             R2              R1

                             Odd and
                             even                                                                                 dj t fuzz depth      ]

                                        --4    [ ~ tr " - - ,[                                              _

       0~/, Even

Figure 7.13 Audio processing circuits.
  (a) Harmonic bass generator.
  (b) Fuzz generator. RI to R4 are graduated to produce the sequential break points shown. In practice the diodes
exhibit a soft turn-on, providing a smoothly curved characteristic.

(approximate) use of the identity sin 30 -                                 the effect of reducing the modulation index,
3 s i n 0 - 4 s i n 3 0 , where 0 - c o t and co is the                    which in some cases may ease the measurement
radian frequency. This identity can be used in                             of the modulation index.
reverse 15 to extract a sine wave at one-third of                             An application frequently arises for a linear
the frequency of a given sine wave: in the case of                         circuit which will accept input signals over a
a frequency modulated (FM) wave this also has                              wide range of levels and produce an output of
                                                                                                       Active signal processing in the time domain    169

#                                                                      as                            varies sinusoidally between 0 V and 2 V peak-to-
                               !                                                                 /   peak at a rate of one cycle per second can be
                               I               cz                 c3 I i c 4       1             /
                               I        J I 31~,"            ,_Jil~         1Sp Io               /   analysed (see Chapter 8) into a 1 kHz tone or
       Cl                      I        I ll       I      I_ ll       m      l cs _ICle _IClf    m
in~,. m '                  A       R1     --   _       R3    --           R4   t   ~   l             'carrier', and two other tones or 'sidebands' at
                                                                                                     0.999 and 1.001 kHz. All three of these tones are
                                                                                       +SV   U
                                                                                                     within the amplifier's bandwidth and no other
                                                                                                     tones are present, yet the AGC action will result
                                                                                                     in an output of very nearly constant amplitude, i.e.
                   input                                                       "
                                                                                                     the sidebands are suppressed.
                                                       (a)                                              Figure 7.15a shows the block diagram of an
                                                                                                     AGC circuit. The peak detector produces an out-
                                                                                                     put proportional to the peak level of the signal,
                                                                                                     and this is used to control the gain of the amplifier.
    AJ                     I                             1                             1             The larger the input signal the larger is the output
                                                                                                     from the peak detector, and the more this reduces
    B                                                                                                the gain of the amplifier. Figure 7.15b shows a
                                                                                                     typical sort of circuit where the gain-controlled
                                                                                                     amplifier is implemented by changing the effective
    C                                                                                                input resistor R1 of an opamp. The more positive
                                                                                                     the gate voltage of the FET, the lower its drain/
                                                                                                     source resistance. Performing the star/delta trans-
    D                                                                                                formation (Figure 1.2b) on R1A, RIB and the FET
                                                                                                     resistance rds, this is equivalent to increasing RI:
                                                                                                     the other two resistances of the delta simply shunt
                     F-L.P                               I I F7 I LI                                 the source and the opamp's input, which is a
                                                                                                     virtual earth, and so are of no consequence.
                                                       (b)                                           Since the amplifier's gain is determined by the
                                                                                                     ratio R2/R1, increasing the effective value of R1
Figure 7.14
  (a) Clock frequency tripler circuit.                                                               reduces the gain. Alternatively, the FET and R1A
  (b) Waveforms at designated points in the circuit.                                                 may simply be regarded as a passive attenuator
  (Reproduced by courtesy of Electronic Engineering.)                                                ahead of a fixed gain amplifier, although this is
                                                                                                     somewhat of an oversimplification, as it ignores
more or less constant l e v e l - that is, for an                                                    the fact that the source resistance seen by R~B
automatic gain control (AGC) circuit. At audio                                                       changes.
frequencies such circuits are readily implemented                                                       At very small signal levels the positive-going
using opamps and FETs. One such published                                                            peaks of the output may not be large enough to
circuit 16 claims a 100dB dynamic range and dis-                                                     overcome the forward voltage of the diode, so up
tortion-free operation over the frequency range                                                      to that point the gain remains constant at the
60 Hz to 30 kHz. However, any AGC circuit is                                                         maximum level: this is described as an AGC
only a linear amplifier in a limited sense. It will                                                  delay. If an amplifier is fitted between the peak
reproduce an input sine wave within the stated                                                       detector and the gate of the FET, a smaller change
range without distortion and likewise will, in                                                       in output voltage will be sufficient to change the
general, reproduce a complex waveform without                                                        gain of the amplifier by a given amount, giving a
distortion. But if there are variations in the                                                       'tighter' AGC characteristic: this is described as
amplitude of the signal occurring at a subaudio                                                      amplified A GC. However, it will be clear on
rate, then they will be suppressed by the AGC                                                        reflection that with a feedback AGC circuit such
action, since it maintains a constant output level.                                                  as Figure 7.15a the output must always rise
For example, a 1 kHz sine wave whose amplitude                                                       slightly as the input is increased (Figure 7.15c),
170   AnalogElectronics
                                                       Amplifier with
                                                       voltage variable
                                                                                                            Compressed (almost
                        input signal                                LPF,                                    constant) level output
                        Variable level     ~ontrol t
                                   Oa~n                         I~I                  I
                                   ~n~ut                        I'~I                  I       Peak level
                                                  Comer frequency
                                                  = 1.5 Hz (typically)           (a)
      Input from low-                            ._    ~                         I
      impedance source             ,,        ["-"t .-. -        1
                    ~,,,,                                             ou~pu,,,                                                           R3
                                                                -             ,                                               Trl

                                                                                 ,,                                                          R4

                                                                                 ', o-c::=

                         N channel

                                Straight AGC

                                                                                 (b)                     Delayed amplified AGC
                                                                                                                                     -15 V


                9                                                                                   ~o
                                           Input level (dB)
                                                                                                   AGC        Nearly constant
                                                                                     Constant gain threshold output level region
                                        Straight AGC                                 region
                                                                           (c)                       Delayed amplified AGC

                                 Input o                                                                   o Output

                                                i       I
                                                                                                    ntrol voltage

                                                       Detector or
                                                       ideal rectifier                (d)
Figure 7.15   Automatic gain control schemes.
  (a) Block diagram of a feedback AGC system.
  (b) Audio-frequency AGC schemes. Trl acts as both AGC diode and amplifier. Delay set by Vbe, R3 and R4.
  (c) AGC characteristics.
  (d) Feedforward AGC. If gain of VCGA is inversely proportional to Vc, then above threshold the output level is
                                                           Active signal processing in the time domain   171

since it is the rise in output level that causes the     which occurs with amplified AGC. Indeed, de-
required reduction in the gain of the amplifier. The     pending on the circuit's characteristics, the output
scheme is a form of negative feedback loop, and          may even fall as the input increases. Where a very
the usual stability requirements must be fulfilled.      tight AGC loop with fast response t i m e s - re-
The higher the loop gain (the tighter the loop), the     quirements which conflict- is needed, the best
slower the loop must act or the higher the low-          approach is a looser feedback AGC loop followed
frequency limit of the audio band must be to             by a narrow range feedforward AGC circuit.
enable the loop gain to be rolled off. In principle,
loop stability can be predicted analytically as in
                                                         Pulse modulation
the case of an NFB loop round any other ampli-
fier, but often in the case of an AGC loop the law       Being concerned with analog circuitry, this book
of gain reduction versus control voltage is un-          does not deal with digital signal processing (DSP)
known or incompletely specified.                         as such, only with the interfaces between digital
    Another point to note in connection with an          and analog circuitry, such as A/D converters.
AGC circuit such as Figure 7.15b is that the AGC         However, there is a form of signal processing
attack t i m e - the rate at which the gain is turned    which is intermediate in nature between analog
down following a sudden rise in input l e v e l - is     and digital, namely time-discrete analog processing.
determined by the capacitor C and the charging           Here, the instantaneous voltage of a wave-form is
circuit resistance, i.e. the diode slope resistance      sampled at regular intervals and represented by the
and the source impedance of the circuit driving          value of some parameter of pulses occurring at the
it. The rate of gain recovery following a sudden fall    sampling instants. Typical examples are pulse
in input level, however, is determined by the time       amplitude modulation (PAM), pulse position
constant CR of the smoothing circuit following the       modulation (PPM) and pulse width modulation
detector. This makes the AGC loop more compli-           (PWM). These are illustrated in Figure 7.16. PWM
cated to analyse theoretically. However, if the rate     has already been discussed in Chapter 4 and so will
of gain recovery is not critical and can be made         not be further covered here. PPM is used as the
fairly slow, e.g. 20dB per second or less, it is a       modulation method in some forms of model radio-
powerful factor in ensuring loop stability. A typi-      control systems.
cal example of an application where a fairly fast           Pulse amplitude modulation (PAM) is important
attack of around 5 ms and a much slower decay            since it is widely used in one particular form of
are acceptable, is the automatic record level cir-       audio signal processing, namely the bucket brigade
cuitry in a cheap cassette recorder/dictation ma-        delay line (BBD), also known as the charge coupled
chine. Another example is the voice compression          device (CCD), which provides a means of delaying
circuit used in single-sideband (SSB) transmitters       an analog signal in time. The BBD is named from
to maintain the modulation at near the permitted         a not entirely fanciful similarity between its mode
maximum level, in order to increase the average          of action and a line of firefighters passing buckets
output power or 'talk power'. Sophisticated ICs          from one to the next. It uses a series of capacitors
are available for this purpose, a typical example        to store charges proportional to the amplitude of
being the Plessey 5L6270 gain-controlled ampli-          the input waveform at successive instants corre-
fier or voice operated gain adjusting device             sponding to the rising edge of regular clock pulses.
 (VOGAD) circuit. It is designed to accept signals       The capacitors are connected each to the next by
from a low-sensitivity microphone and to provide         an FET: the gates of alternate FETs are connected
 an essentially constant output signal for a 50 dB       to one phase of the two-phase clock and those of
 range of input. The dynamic range, attack and           the remaining FETs to the other. By this means
delay times are controlled by external components.       the samples are effectively passed stage by stage
    Figure 7.15d shows a feedforward AGC circuit.        along the line, arriving at the other end after the
With this arrangement it is possible to avoid even       period occupied by N/2 clock pulses for an N,
 the slight rise in output level with increasing input   stage device; a pair of buckets is required to
172   Analog Electronics

                           t I i
                           Sampling instants

                                     i                   I

                                         Non-return-to-zero          Return-to-zero
                                         (NRZ) pulses                (RZ) pulses

                           Pulse amplitudemodulation (PAM),50% modulation shown

                            "            nn__n                n n n
                           Pulse position modulation (PPM)

                            Pulse width modulation (PWM)
Figure 7.16 Pulse modulation.

provide one clock pulse period of delay. A                    1194, 1726 and 2790 as well as the final output at
straightforward delay is useful in sound reinforce-           stage 3328. The taps are chosen so that the ratios
ment systems in large buildings such as a cathe-              of the various delays are surds, i.e. irrational
dral, to delay the signal from the speaker's                  numbers. Thus when the outputs of the various
microphone so that it is emitted from a loud-                 taps, which may be deliberately attenuated by
speaker column half-way down the hall at the                  varying amounts, are combined and recirculated
same time as the direct sound waves arrive. This              to the input along with the original signal, an
avoids a confusing post-echo from the direct                  approximate simulation is produced of the com-
sound. In other applications, an echo is deliber-             plex pattern of reverberation in a concert hall or
ately produced to enrich the sound e.g. from an               cathedral.
electronic organ. A discrete echo with a noticeable              An audio delay may also be used without
delay sounds very artificial and trying, as anyone            recirculation to produce a comb filter, by simply
who went to the Proms concerts in London's                    combining the original signal with the delayed
Royal Albert Hall fifty years ago can testify, so a           version. For example, if a short delay is used, say
shorter delay can be used and the delayed signal              10 ms, then a 100 Hz sine wave will be delayed by
added back into the input of the BBD along with               exactly one cycle, whilst 50 Hz will be delayed by
the original signal. This produces a multiple echo            half a cycle or 180 ~ If the original and delayed
which, whilst an improvement, can also sound                  signals (both at the same level) are now combined,
artificial if overdone. The MN3011 from National              the net 50 Hz signal will be zero owing to cancel-
Panasonic is a 3328-stage BBD audio signal delay              lation, whilst the 100Hz signal will double in
device with intermediate taps at stages 396, 662,             amplitude. Furthermore, sine waves of 200, 300,
                                                          Active signal processing in the time domain     173

400 Hz etc. will be delayed by an even number of         of the latest segment of the input in the delay line,
half-cycles and 150, 250, 350 Hz etc. by an odd          with older samples constantly 'falling off the end'
number. So the response is the comb filter shown         and being lost as new samples are fed in at the
in Figure 7.17a. If the delayed signal is inverted,      front of the line. Following a selected triggering
equivalent at any frequency to a 180 ~ phase shift,      event, the high-speed clock can be stopped and the
then the peaks and troughs in Figure 7.17a change        samples in the pipeline clocked out at a much
places and there is zero response at 0 Hz. If the        slower rate to an ADC, which is thus relieved of
delayed signal is attenuated relative to the original,   the task of operating at up to 400 MHz. If the
or vice versa, before combining, then the peaks          clock is stopped immediately on the occurrence of
and troughs are less pronounced. If the clock            the trigger, the segment of signal passed to the
frequency is altered, the delay changes and so do        ADC for digitization and subsequent storage in
the positions and spacing of the peaks and               the random access memory (RAM) waveform
troughs. By this means, the comb filter can be           store consists entirely of pretrigger information.
made to sweep up and down the audio band at              If on the other hand the high-speed clock is
will, giving rise to some novel and striking effects.    allowed to continue for several or many pulses
   Recirculation of the delayed sound also results       after the trigger event, the stored signal segment
in a frequency response which is anything but level      will bracket the trigger event, with part of the
and, if the feedback level is too high, the output at    waveform referring to time before the trigger and
one or more frequencies may build up indefinitely        part to time after, in any desired ratio up to 100%
to give an unstable condition. The reason for this       post-trigger. Even greater post-trigger delay can be
is not difficult to see, for if the delayed feedback     employed to give a stored waveform segment
signal at a frequency corresponding to a peak in         referring to some later detail of interest in the
Figure 7.17a is as large as the original signal, the     waveform, similar to the A delayed by B function
latter may be removed and the output will never          of a conventional (real-time) oscilloscope. What
die away. If the feedback signal is a whisker larger     the conventional oscilloscope cannot do, of course,
than this, then it will build up indefinitely, as        is to provide the pretrigger viewing capa- bility of
illustrated in Figure 7.17b. The effect of a regen-      the DSO.
erative comb filter on speech is intriguing, not to         Pulse amplitude modulation of audio signals is
say weird. It sounds as though the speaker is in an      provided by the switched capacitor type of filter.
echoey room with a set of tubular bells. Whenever        These are now available in types implementing
the pitch of his voice coincides exactly with the        more and more complex filters; however, as they
frequency of one of these 'bells' it rings for a         were mentioned in Chapter 6, they are not covered
considerable period, dying away only slowly if           further here. N-path band-pass filters are also an
the regeneration is only just short of oscillation.      application of the PAM principle in a way. They
With a close comb spacing, several 'bells' may be        operate as in Figure 7.18, which illustrates the case
sounding simultaneously, rendering the speech            where N = 4, i.e. a four-path filter. First, imagine
incomprehensible but not, strangely, preventing          that one of the four switches in Figure 7.18a is
one from identifying the accent of the speaker!          permanently closed and the other three open. Then
   Another application of CCDs is in digital stor-       clearly there is a low-pass response with a corner
age oscilloscopes (DSOs). Usually the bandwidth          frequency off~ = 1/2nCR. Now consider the case
of these is limited by the speed at which a flash        when the four switches are operated in sequence as
converter type of ADC can operate. However,              indicated, so that each is on for exactly 25% of the
several manufacturers use CCDs to circumvent             time. Suppose the frequency of the four switch-
this problem. The input to the oscilloscope is fed       control waveforms is f~ (where fs =f~lo~k/4) and
into a very high-speed CCD delay line, operating         the input is a sine wave of frequency f = f~. Then
basically in the same way as the audio BBDs              each switch is on during exactly the same quarter
described above, but with a clock frequency of           of the input sine wave every cycle, and so will in
up to 400 MHz. Thus at all times there is a record       time charge up to the average value of the input
174   AnalogElectronics

        Linear gain
       2-]-.-,,,    ~             ~                        ~ 100%
        1             ~                                       10%

      0"51. . . . f/~-2-'i    '3~2'               5~2-Linear frequency                                     f=l/T          2f
                         Erect response                                                                     Inverted response

                                          R                      R
                                                                                   RI 1o                          R

                                          Direct pat                 1                     .....

                                   Delay path      _   .

                             Time delay r(s)                               \
                             T    N fcllock                 /
                                                            R~           delayedpath gain


                                              ,                                ,   .   .   .   .
                                                                  Active signal processing in the time domain       175

                             R           R
                  Input                                                P' Output


                                                          BBD delay

Figure 7.17 Comb filters.
  (a) Audio comb filter using BBD device.
  (b) Comb filter with regeneration.

 voltage during that quarter-cycle. Thus the output             section may be realized using a Salen and Key
will be a four-step approximation to a sine wave as             (or, much better in this application, a Kundert)
 shown. If the frequency of the input sine wave                 circuit as in Figure 7.18c. Another circuit arrange-
 differs only very slightly from fs, the phasing of the         ment has the advantage that both switched N-
 four steps relative to the sine wave will gradually            capacitor banks are connected to ground. 17 Each
 drift through the patterns shown in Figure 7.18b,              of the N capacitors must 'look' only at that (1/
 but otherwise the circuit operation is unchanged.              N)th of the input waveform occurring whilst its
 However, if the input sine wave differs from fs by a           associated switch is closed, if the filter is to operate
 rather greater margin, either higher or lower in               correctly. Any stray capacitance to ground at the
 frequency, each switch will be closed during a                 junction of the N capacitors and the resistor R will
 rather different part of the input sine wave on                'smear' some charge of magnitude and polarity
 successive closures, and so its associated capacitor           corresponding to the equilibrium voltage on one
 won't have a chance to charge to an appropriate                capacitor into the next capacitor, and from that
 constant voltage. In fact, the circuit possesses a             into the following one, and so on. There is
 band-pass response similar to the low-pass re-                 inevitably some stray capacitance associated with
 sponse of the basic CR circuit, but mirrored                   this node, for example the input capacitance of the
 symmetrically around the frequency fs. Since                   opamp. In a second-order section such as that of
each capacitor is only connected to the input via               Figure 7.18c, this results in one of the peaks being
 R for 25% of the time rather than continuously,                of greater amplitude than the other. This makes it
 the effective low-pass bandwidth is in fact                    difficult to build up a high-order filter with a
 (1/4)(1/2~CR), and hence the pass bandwidth at                 Chebyshev pass band, since the complete filter
fs is 1/4~CR or more generally 1/N~CR.                          exhibits a general attenuation slope across the
    At audio frequencies, N-path filters can provide            pass band, superimposed upon the usual Cheby-
very high values of Q; for a two-pole band-pass                 shev ripple.
 (single-pole low-pass equivalent) filter such as in               As an example of a frequency selective filter, it
 Figure 7.18a, the ratio of centre frequency to 3 dB            can well be argued that the N-path filter should
 bandwidth can be in the range 10 000 to 100 000.               have been covered in the preceding chapter. How-
 However, such performance is only achievable                   ever, it has been discussed here because of the
with care in both circuit design and layout, es-                time-discrete nature of its implementation, and it
pecially if using second-order sections as well to              completes this review of analog signal processing
build up a higher-order filter. A second-order                  in the time domain.
176   AnalogElectronics


                           .2~2~ _~>---~176                                                 Pole

                                                                                                         - A~k/4

                                                                               o =-I/gxcR                ,91

            fclock                                                                                   I

                          Response (dB)                                        -~ ~--BW = 2 ( I / 4 1/2~CR)

                                        I/2~CR                               fs =fclock/4     log frequency

                                                           ..__l----I                   V
                                                                                               m w

               ofF7           --L_               or

               o2_F-I            ~F7
               o3__.l-I             .... J       or
                                                                         1        L~

               ~          Ft__
                     FT: l     l-                          (b)
                                                                             Active signal processing in the time domain    177


       vi                                                            .   .     .    .     .                 i         lpo

                          --   -                                                                                  /



      dB                           dB/octave
                                                                                                        x   2=i

                                                                                                            Two zeros
                                       logf                  fs                               logf

                 Low Q
                      z                              (c)
                                                           High Q




Figure 7.18 N-path filters.
   (a) One-pole low-pass equivalent (LPE) N-path band-pass filter section. A single I circulating in a shift register is
only one of many ways of producing the four-phase drive waveform shown in (b).
   (b) Waveforms associated with (a). The exact shape of Vo, when f =feloek/4 exactly, will depend upon the
relative phasing of vi and the clock waveform. For very small differences between f and f~iock/4, the output
waveform will continuously cycle between the forms shown, and all intermediate shapes.
   (c) Second-order N-path filter, showing circuit, frequency response and pole-zero plot. Q = I/2~/(Ci/C2),
exactly as for the low-pass case.
   (d) Stray capacitance. Showing the stray capacitance to ground, consisting of opamp input capacitance Cs2 plus
circuit and component capacitance to ground with all switches open Csl.

                                                                            Offset. C. Paul, L. Burgner, p. 186, EDN,
                                                                            August 21, 1986, Vol. 31, no. 17. See also
 1. Simple Digitally-Controlled Variable-Gain                               An Unusual Circuit Configuration Improves
    Linear DC Amplifier. A. Sedra, K. C. Smith,                             CMOS-MDAC Performance. N. Sevastopou-
    p. 362, Electronic Engineering, March 1969,                             los, J. Cecil, T. Frederikson, EDN, March
    Vol. 41, no. 493.                                                       1979.
 2. Controlled Gain Amp. Q. Rice, p. 18, New                             4. Inexpensive Circuit Generates Unipolarity
    Electronics, 14 April, 1987.                                            Output, p. 69, Electronic Design, November
 3. High-Gain Amp Yields Low DC Output                                      22, 1969, Vol. 17, no. 24.
178         Analog Electronics

          Improve Circuit Performance With A 1-opamp       Questions
          Current Pump. P. A. Pease, p. 85, EDN, Jan-
                                                            1. Describe a three opamp instrumentation am-
          uary 20, 1983, Vol. 28, no. 2.
                                                               plifier and explain how it works. What im-
      .   Improved Saw Tooth Generator Has
                                                               provement in the ratio of normal mode to
          Grounded Reference Point, p. 80, Electronic
                                                               common mode signal is contributed by (i)
          Design, February 15, 1970, Vol. 4, no. 18.
                                                               the first stage, (ii) the output stage?
  .       See e.g. The BIFET Design Manual. P. F.
                                                            2. (i) Draw, and deduce the gain of; a two opamp
          Nicholson, Texas Instruments Ltd, August
                                                               instrumentation amplifier.
          1980, and A Circuit With Logarithmic Trans-
                                                               (ii) How does an isolation amplifier differ from
          fer Response Over 9 Decades. J. F. Gibbons,
                                                               an instrumentation amplifier?
          IEEE Trans on Circuit Theory, Vol. CT-I1,
                                                            3. Derive from first principles the time response
          September 1964, p. 378.
                                                               of an ideal inverting integrator with input and
  .       Simple Control For Sign Of Op-Amp Gain.
                                                               feedback components R and C respectively, to
          Electronic Design, November 8, 1970, Vol. 18,
                                                               a unit step input, i.e. one which changes
          no. 23.
                                                               instantly from 0 V to, and remains at, + 1 V.
  .       Improved Log Ratio Amplifier. D. J. Faul-
                                                            4. Describe three different full-wave signal recti-
          kner, p. 23, Electronic Product Design, March
                                                               fier circuits, and analyse their mode of opera-
10.       Integrated Analogue Divider/Multiplier. V. C.
                                                            5. An adjustable unipolar current sink of 0-
          Roberts, Electronic Engineering, April 1969,
                                                                100mA, controlled by an input voltage of
          Vol. 41, no. 494.
                                                               exactly 0-10mV is required. Design such a
11.       Multi-Electrode Valve Multipliers. V. A. Ste-
                                                               circuit using an opamp, an FET and such
          phen and W. W. Forest, p. 185, Electronic
                                                               other components as may be required.
          Engineering, March 1964.
                                                            6. Describe the operation of the Howland bipo-
12.       Voltage Controlled Triangle Generator. J. W.
                                                               lar current pump. Show how it may be used to
          Howden, p. 540, Wireless World, November
                                                               implement a non-inverting integrator, where
                                                               one end of the integration capacitor is earthed.
13.       Charge Balancing Frequency Multiplier. H. R.
                                                            7. The basic 'pump and bucket' F to V converter
          Goodwin, p. 25. Electronic Engineering, De-
                                                               is only approximately linear over the lower
          cember 1983.
                                                               portion of its output voltage range. Explain
14.       Clock Frequency Multiplier Using MM58174.
                                                               how an opamp may be added to the circuit to
          I. March, p. 27, Electronic Engineering, July
                                                               give a wide, linear output range.
                                                            8. Compare the operation of an SAR ADC with
15.       Index Reduction of FM Waves By Feedback
                                                               that of a Flash ADC. What are the main
          And Power-Law Non-Linearities. V. Bene~,
                                                               advantages and disadvantages of each?
          p. 589, BSTJ, April 1965, Vol. XLIV, no. 4.
                                                            9. Explain why a conventional (feedback) AGC
16.       A 100 dB AGC Circuit Using The CA3130. Y.
                                                               system cannot produce an output whose am-
          Gopola Rao, P. U. Mesh, p. 35, Electronic
                                                               plitude (above threshold) is exactly constant.
          Engineering, November 1977.
                                                               How can a feedforward AGC circuit circum-
17.       A New Type of N-Path N Filters with Two
                                                               vent this limitation?
          Pairs of Complex Poles. E. Langer, Session 11,
                                                           10. Describe how a BBD may be used in a circuit
          WAM 2.5, International Solid-State Circuits
                                                               to provide (i) an echo effect, (ii) a comb filter.
          Conference, 1968.

    8                 Radio-frequency circuits
Radio-frequency equipment is used for a vast                 carrier wave does not change, but its amplitude is
range of purposes, including heat treating special           modulated in sympathy with the programme
steels, medical diathermy treatment for cancer,              material, usually speech or music. This gives rise
heat sealing plastic bags, and experiments in                to sidebands, which are limited to +4.5 kHz about
atomic physics. Nevertheless, as the name implies,           the carrier frequency by limiting the bandwidth of
the original use was in connection with the trans-           the baseband modulating signal to 4.5 kHz maxi-
mission of information by radio waves. The                   mum. This helps to minimize interference between
earliest form of this was wireless telegraphy (WT)           adjacent stations on the crowded MW band, where
using Morse code. This was followed by wireless              frequency allocations are only 9 kHz apart (10 kHz
telephony and, much later, b r o a d c a s t i n g - radio   in the USA). With maximum modulation by a
and television. So, before diving into RF circuits in        single sinusoidal tone, the transmitted power is
detail, a word might be in order about the different         50% greater than with no modulation; this is the
forms of modulation employed to impress the                  100% modulation case. Note that the power in the
information to be transmitted onto the radio                 carrier is unchanged from the 0% or unmodulated
wave. It is only a brief word, though, as this is a          case. Thus at best only one-third of the transmitted
book particularly about analog electronic cir-               power actually conveys the programme informa-
cuitry, not a general light-current electrical engi-         tion, and during average programme material the
neering textbook.                                            proportion is much lower even than this. For this
                                                             reason, the single-sideband (SSB) mode of modula-
                                                             tion has become very popular for voice com-
Modulation of radio waves
                                                             munication at HF. With this type of modulation,
Figure 8.1a shows how information is transmitted             illustrated in Figure 8.1c, only one of the two
by means of an interrupted continuous wave, often            sidebands is transmitted, the other and the carrier
called simply continuous wave (CW). This type of             being suppressed. As there is no carrier, all of the
modulation is frequently employed in the high-               transmitted power represents wanted information,
frequency (HF) band, i.e. from 1.6 to 30 MHz. In a           and as all of this is concentrated in one sideband,
simple transmitter either the oscillator would be            'spectrum occupancy' is halved. At the receiver,
'keyed' on and off with a Morse key, or alterna-             the missing carrier must be supplied from a carrier
tively the drive signal or the power supply to the           reinsertion oscillator at exactly the appropriate
output stage would be likewise keyed. In the                 frequency in order to demodulate the signal and
simplest possible transmitter there would be no              recover the original. Although this is a trivial
separate output stage, only a keyed oscillator.              exercise with modern synthesized receivers, histori-
Using CW, amateur radio enthusiasts can contact              cally it was difficult. Amplitude modulation, with
others in any country in the world using only a few          its uncritical tuning requirements, continues to be
watts, but only as and when propagation con-                 used by broadcasters for both local audiences on
ditions are favourable.                                      MW and international broadcasting on SW. There
   Broadcasting on medium wave (MW) uses am-                 are a number of bands of frequencies allocated by
plitude modulation which is illustrated in Figure            international agreement to broadcasting in the Here, the frequency of the radio-frequency or          short-wave band between 1.6 and 30 MHz.
180   Analog Electronics

                                                                                                                                      ] I[II RF output either

             Amplitude                           C
                                                                                                                                         !!11 ~176176
                                                                                                                                                Instantaneous resultant
                                                                                                                                                amplitude of RF wave,
                                                  Carrier                                                                              ....     corresponds to A-A in
               RF voltage                                                                                                                       ume domain representation
                  r            i   ....            i_,ino                                 ,    ~

                                                                                                       .       .,|

                                                                                                    I, Carrier "- ~
                                                                                                                             ......      ~

                               [          fc       l         frequency          coc assumed zero                .                 i;' / -
                    ]i = fc-fm   fu=fc+fm                                       for purposes of      ~,:omponent - - ~ a _ _ ' ~ ~
                  Amplitude of upper and lower                                  vector representation x           -X~        /
                               I m%
                  sidebands ='2('i"~') each
                      Spectrum (frequency                                                                  Vector representation coc = 2refc
                      domain representation)                                                                                     corn= :'~fm

                                                A I RF waveform
                                                                           ~                 7
                                                                                  fEnq:le:cPey : odulating

                                                                          HttttHtttHHi                                                                     Time

                                                                                              omaia i : ; : : : : :
                                               Modulation m -- 100%              (b)

                                                                                               COm= r                      COC
                                                       USB modulation
                                                       with single tone

                             "- frequency
                                                       of frequencyfro                                       Reference phase
                                                                                                             of suppressed
                                                                                                             carrierfc                           I i111111111Ti-
                  fc fu = fc + fm
                           Two equal amplitude USB modulation with two                                                                                             fu2-ful
             RF            tones shown         tones fm1and fm 2                                                                              Dashed frequency =      2

                   '           Line~requency                                                                                                                       Time
                  fc ful fu2
                  fal -'-fc +fro 1
                                                                                   )  Time domain representation
                  fu2 =fr +fro2                                          (corn2 + C~
                                                                    Ref. phase of fc
                     Spectrum                                    co rn'=        2
                                            (c) Vectorrepresentation
Figure 8.1 Types of modulation of radio waves.
  (a) CW modulation. The letters CQ in Morse (seek you.:?) are used by amateurs to invite a response from any
other amateur on the band, to set up a QSO (Morse conversation).
  (b) AM: 100% modulation by a single sinusoidal tone shown.
  (c) SSB(USB) modulation. Note that with two-tone modulation, the signal is indistinguishable from a double-
sideband suppressed carrier signal with a suppressed carrier frequency of (ful +fu2)/2. This can be seen by
subtracting the carrier component from the 100% AM signal in (b). The upper and lower halves of the envelope
will then overlap as in (c), with the RF phase alternating between 0 ~ and 180 ~ in successive lobes.
                                                                                                                                    Radio-frequency circuits 181

                                                                                                                         iI         sidebands
                                                                                                                       I           _..,....,.... ~ . _ 20t)m
                                                                            eorresl~nds to A-A               //
                                                                            in time domain               /                    /
                                        ,     ,               ._                                                                        +2o)m
                                                                                                                                                       I I
           fc -       2fro I    fc fc   +fro fc + 2fro        F"requency                                                                           l'
                        ~o-~o   ~                                                               i ll                                               I         I

                                                                                                                                                   e         I
                           +, CarrierI
                      First-order sidebands

                  Second-order sidebands
                             Spectrum representation
                                                                                                       Vector representation

                                             Voltage                A
                                              *l                       '
                                                       I'              !       t
                                                                                           ! - Time

                                              -                        i Modulating waveformfm e.g. 1 kI-Iz
                                             Voltage                   I
                                              +t                       I                 RF
                                 Amplitude                   IA~~![~        /~/~~/~   ~"~Time
                                 (constant) 1
                                                             ~WI~V~/       U ~J~llVU~-
                                                    Frequency modulated RF carrier
                                             (Frequency variation grossly exaggerated for clarity.
                                             Actual RF carrier frequency would be much higher
                                             than shown, e.g. 100 MHz)
                                                        Time domain representation

                                                                                                Modulation index of 0.1, i.e. _+0.1
                                                                                                radian peak phase deviation
        +5"7~ I-~.7
         Reference phase
              [1',~I I[I III Ill I!1 I11 I11 [ I!1 m,L                                          m~176176

                                                                                                After divide-by-2 frequency division
              ill                ,I,                   ,ll         ,I,              Ill         circuit, modulation index reduced to 0.05

                                                                                                After further divide-by-2 stage, modulation
              rl',....                                 '1'                          /1,         index reduced to 0.025 or +1.4 ~


  (d) FM. For maximum resultant phase deviation qb up to about 60 ~ as shown, third- and higher-order sidebands
are insignificant.
  (e) Reduction of phase deviation when a phase modulated signal passes through a frequency divider chain,
showing- for example- how a divide-by-4 (two-stage binary divider) reduced modulation index by a factor of 4.
182   Analog Electronics

   Figure 8.1d illustrates frequency modulation. FM    sponding to a low amplitude of the modulating
was proposed as a modulation method even before        sine wave (frequency fm). Even so, it is clear that
the establishment of AM broadcasting, but any          if only the sidebands at the modulating frequency
enthusiasm for it waned as a result of an analysis     are considered, the amplitude of the signal would
which showed that it produced sidebands exceed-        be greatest at those instants when its phase
ing greatly the bandwidth of the baseband signal. 1    deviation from the unmodulated position is great-
With the limited bandwidth available in the LW         est. It is the presence of the second-order side-
and MW bands, this was obviously an undesirable        bands at 2fm which compensate for this,
characteristic. However, following the Second          maintaining the amplitude constant. At wider
World War the technology had advanced to the           deviations many more FM sidebands appear, all
point where it was possible to use the considerable    so related in amplitude and phase as to maintain
bandwidth available in the then largely unused         the amplitude constant. They arise automatically
very high-frequency (VHF) band. The lower part         as a result of frequency modulating an oscillator
of the 30-300 MHz VHF band had already been            whose output amplitude is constant; their exist-
used before the war for television, and now a high-    ence is predicted by the maths and confirmed by
quality sound broadcasting service was established     the spectrum analyser.
using FM in the band 88-108 MHz. The standard             Note that the maximum phase deviation of the
adopted was a maximum deviation from the centre        vector representing the FM signal will occur at the
or carrier frequency of 4-75 kHz, and a baseband       end of a half-cycle of the modulating frequency,
frequency response extending from 50Hz to              since during the whole of this half-cycle the
15 kHz. This represented real hi-fi compared with      frequency will have been above (or below) the
the 4.5kHz limitation on MW, and the much              centre frequency. Thus the phase deviation is 90 ~
lower level of interference from unwanted stations     out of phase with the frequency deviation. Note
was a real blessing. The modulation index for an       also that, for a given peak frequency deviation, the
FM signal is defined in terms of a single sinusoidal   peak phase deviation is inversely proportional to
modulating tone, as 'm', where m =fd/fm, the           the modulating frequency, as may be readily
peak frequency deviation of the carrier, divided       shown. Imagine the modulating signal is a
by the modulating frequency. It is shown below         100 Hz square wave and the deviation is 1 kHz.
that m is also equal to the peak phase deviation of    Then during the 10 ms occupied by a single cycle of
the carrier in radians. With the 75kHz peak            the modulation, the RF will be first 1000 Hz higher
deviation being five times the highest modulating      in frequency than the nominal carrier frequency
frequency, broadcast FM (also known as WBFM            and then, during the second 5 ms, 1000 Hz lower in
- wide band FM) is a type of spread spectrum           frequency. So the phase of the RF will first
signal. This confers a degree of immunity to           advance steadily by five complete cycles (or
adjacent- and co-channel interference due to the        10rt radians) and then crank back again by the
'capture effect'. This is particularly effective on    same amount, i.e. the phase deviation is +5re
mono reception, the advantage being much less          radians relative to the phase of the unmodulated
for stereo reception.                                  carrier. Now the average value of a half-cycle of a
   Figure 8.1 shows the characteristics of the         sine wave is 2/rt of that of a half-cycle of square
various modulation methods in three ways: in           wave of the same peak amplitude; so if the
the frequency domain, in the time domain, and          modulating signal had been a sine wave, the
as represented by vector diagrams. Each illus-         peak phase deviation would have been just
trates one aspect of the signal particularly well,     •      radians. Note that the peak phase deviation
and it is best to be familiar with all the repre-      in radians (for sine wave modulation) is just fd/fm,
sentations. Choosing one and sticking to it is         the peak frequency deviation divided by the mod-
likely to be misleading since they each tell only      ulating frequency: this is known as the modulation
a part of the story. Note that in Figure 8.1d a        index of an FM signal. If the modulating sine wave
very low level of modulation is shown, corre-          had been 200 Hz, the deviation being 1 kHz as
                                                                           Radio-frequency circuits    183

before, the shorter period of the modulating           how dividing the frequency of a phase or fre-
frequency would result in the peak-to-peak phase       quency modulated wave divides the modulation
change being halved to -1-5 radians; that is, for a    index in the same proportion. In the figure, a
given peak frequency deviation the peak phase          sinusoidal modulating waveform has been as-
deviation is inversely proportional to the modulat-    sumed; in this case the peak phase deviation in
ing frequency.                                         radians is numerically equal to the modulation
   For monophonic FM broadcasting the peak             index, i.e. to the peak frequency deviation divided
deviation at full modulation is 75kHz, so the          by the modulating frequency, as noted above.
peak phase deviation corresponding to full sine        However, whatever the modulating w a v e f o r m -
wave modulation would be -t-5 radians at 15 kHz        and even in the case of a non-repetitive signal such
and 4-1500 radians at 50 Hz modulating frequency.      as noise- passing the modulated carrier through a
If the modulation index of an FM signal is much        divide-by-N circuit will reduce the peak deviation
less than unity, the second-order and higher-order     by a factor of N, as should be apparent from
FM sidebands are insignificant. If, on the other       Figure 8.1e. For the time variations on the edges
hand, the modulation index is very large compared      of the divider output remain unaffected but they
with unity, there are a large number of significant    now represent a smaller proportion of a complete
sidebands and these occupy a bandwidth virtually       cycle. Conversely, if a phase or frequency modu-
identical to 2fd, i.e. the bandwidth over which the    lated signal is passed through a frequency multi-
signal sweeps. The usual approximation for the         plier (described later in this chapter), any phase
bandwidth of an FM signal is B W = 2(fd +fm)-          noise on the signal is multiplied pro rata.
   You can see in Figure 8.1b that the vectors
representing the two sidebands of an AM signal
                                                       Low-power RF amplifiers
are always symmetrically disposed about the
vector representing the carrier. As they rotate at     Having looked at some typical radio-frequency
the same rate but in opposite directions, their        signals- there are many other sorts, for example
resultant is always directly adding to or reducing     frequency shift keying (FSK), numerous varieties
the length of the carrier vector. The second and       of digital modulation, and of course television - it
higher even-order sidebands of an FM signal            is time to look at some of the wide range of RF
behave in the same way. But as Figure 8.1d             circuits, both passive and active, used to process
shows, the first-order sidebands (at the modulating    them. These include amplifiers of all sorts, but
frequency) are symmetrical about a line at right       only low-power RF amplifiers are discussed, for
angles to the carrier, and the same goes for higher    a very good reason. This is a very exciting time for
odd-order sidebands. Note that if one of the first-    the high-power RF engineer, with devices of ever
order FM sidebands was reversed, they would            higher power becoming available almost daily.
look exactly like a pair of AM sidebands: this is      There are regular improvements in high-power
why one of the first-order FM sideband signals in      bipolar RF transistors. RF MOSFETs are improv-
the frequency domain representation in Figure 8.1d     ing in terms of both power handling and reduced
has been shown as inverted. A spectrum analyser        capacitances, particularly the all-important drain/
will show the carrier and sidebands of either an       gate feedback capacitance Cdg; they are also avail-
AM or a low-deviation FM signal as identical, as       able now as matched pairs in a single package, for
the analyser responds only to the amplitudes of the    push-pull applications. Meanwhile other exciting
individual sidebands, not their phases. However, if    developments are on the horizon, including the
the first-order sidebands displayed on the analyser    static induction transistor (SIT). This device is
are unequal in amplitude, this indicates that there    half-way between a bipolar and an FET, and its
is both AM and FM present on the modulated             notable feature is an unusually high voltage cap-
wave.                                                  ability. This eases the difficulties associated with
   An important principle in connection with phase     the design of high-power RF circuits due to the
modulation is illustrated in Figure 8.1e. This shows   very low impedance levels at which lower-voltage
184   Analog Electronics

                                                                                                                                                                To next stage

                           1             !
                                         .   .   .    .   .   .   .   .   .   .   .   .   .   .   .   .       m   .   .       .   .   .   .   .


                                 5T                                                           C4
                        SK1                                                                   In F ~                                              PL1
                                                     Tr 1
                                  1 nF               2N3563                                   23/4T


                                 1KS   ]•        " F 1 nF
                                                                  C6                                                                      R3
                                                                                                                                              2K2   n"      ~ 9V

Figure 8.2 RF amplifier stages.
  (a) Common emitter RF amplifier stage with both input and output circuits tuned. Co are decoupling capacitors.
  (b) Common base RF amplifier with aperiodic (broad band) input and tuned output stages (reproduced from
'VHF preamplifier for band II', lan Hickman, Practical Wireless, June 1982, p. 68, by courtesy of Practical

devices necessarily work. Even more exciting is the                                                       the input and output circuits are tuned. This is by
prospect of high-power devices using not silicon or                                                       no means the invariable practice but, for the
gallium arsenide (GaAs), or even indium phos-                                                             input RF stage of a high-quality communications
phide (InP), but diamond. The technology is                                                               receiver, for example, it enables one to provide
currently being researched in the USA, Japan                                                              more selectivity than could be achieved with only
and the USSR, and already diodes (operating up                                                            one tuned circuit, whilst avoiding some of the
to 700~      have been produced. With a carrier                                                           complications of coupled tuned circuits. The
velocity three times that of silicon and a thermal                                                        latter can provide a better band-pass shape - in
conductivity twenty times that of silicon (four                                                           particular a flatter pass b a n d - but, for a com-
times that of copper, even) the possibilities are                                                         munications receiver covering say 2 to 30 MHz,
immense. So any detailed discussion of RF power                                                           two single tuned circuits such as in Figure 8.2a
devices is fated to be out of date by the time it                                                         provide an adequate pass band width in any case.
appears in print. So only low-power amplifiers are                                                        With the continuing heavy usage of the 2 to
discussed below.                                                                                          30 MHz HF band, which seems to become even
   Figure 8.2 shows two class A NPN bipolar                                                               more congested yearly rather than dying as the
transistor amplifier stages. In Figure 8.2a, both                                                         pundits were once predicting, RF stages are
                                                                                                  Radio-frequencycircuits        185

                      a=Q                Hz off resonance
                                Resonant frequency (Hz)                                           voltage
                1.0                                                                               generator
                0.9             e~

                0.7 -75 ~                                                                                   i_

                0.6   -50                                                                               T          o
            ~o 0.5 -25

                                                                                                        i Li ~

                0.3 --25                                                                               ii         TC        v~

                0.2                                                                                Constant
                0.1                                         For Q very large                       generator
                                                                   I    I
                                                                                          I   I
                            I        I       I    I    I      I
                  3.0 2.5 2.0 1.5 1.0 0.5            0.5 1.0 1.5 2.0 2.5 3.0
                  Frequency below resonance             Frequency above resonance
                                            Values of a
Figure 8.3 Universal resonance curve for series resonant circuit. For a Q of greater, the phase and amplitude
curves depart by only a very small amount from the above. Also applies to the response of a parallel tuned circuit,
for Q > 20. In both cases, curves give Vo/Vmaxin magnitude and phase.

coming back into favour again. However, an R F                         gain and the higher the Q of the tuned circuits,
amplifier with both input and output circuits                          the more likely is the feedback to be sufficient to
tuned needs very careful design to ensure stabil-                      cause oscillation, since when the phase shift in
ity, especially when using the common emitter                          each tuned circuit is 45 ~, its amplitude response is
configuration. The potential source of trouble is                      only 3 dB down (see Figure 8.3). Even if oscilla-
the collector/base capacitance, which provides a                       tion does not result, the stage is likely to show a
path by which energy from the output tuned                             much steeper rate of fall of gain with detuning on
circuit can be fed back to the base input circuit.                     one side of the tuned frequency than on the
The common emitter amplifier provides inverting                        o t h e r - a sure sign of significant internal feed-
gain, so that the output is effectively 180 ~ out of                   back. The grounded base stage of Figure 8.2b may
phase with the input. The current fed back                             prove a better choice, since some bipolar transis-
through the collector base capacitance will of                         tors exhibit a significantly smaller feedback capa-
course lead the collector voltage by 90 ~ At a                         citance in the grounded base connection, i.e. C~e
frequency somewhat below resonance (Figure 8.3)                        is smaller than their Ccb. The N channel junction
the collector voltage will lead the collector cur-                     depletion FET (JFET) is also a useful R F ampli-
rent, and the feedback current via the collector/                      fier, and can be used in either the grounded
base capacitance will produce a leading voltage                        source or grounded gate configuration, corre-
across the input tuned circuit. At the frequency                       sponding to the circuits of Figure 8.2. It is
where the lead in each tuned circuit is 45 ~, there                    particularly useful in the grounded gate circuit
is thus a total of 180 ~ of lead, cancelling out the                   as a V H F amplifier.
inherent phase reversal of the stage and the                               For ease of reference, Figure 8.3 is repeated as
feedback becomes positive. The higher the stage                        Appendix 2.
186    Analog Electronics

                                +Vs                                    +Vs
                                            (essential)                               ommended)


                Inpu                                      I~                          Output

                              (a)                                        (b)

Figure 8.4
  (a) Cascode amplifier.
  (b) Complementary cascode. The load may be a resistor, an RL combination (peaking circuit), a tuned circuit or
a wide band RF transformer. CD are decoupling capacitors.

Stability                                                      the output port and the 'output' taken from the
                                                               input port. This is easily done in the case of a
There are a number of circuit arrangements which
                                                               stand-alone amplifier module, but is not so easy
are used to ensure the stability of an RF amplifier
                                                               when the amplifier is embedded in a string of
stage. One of these, the cascode, is shown in Figure
                                                               circuitry in equipment. In the days of valves, one
8.4a. The cascode stage consists of two active
                                                               could easily derive a stage's reverse isolation
devices; bipolar transistors are shown in the
                                                               (knowing its forward gain beforehand) by simply
figure, but JFETs or RF MOSFETs are equally
                                                               disconnecting one of the heater leads and seeing
applicable. The input transistor is used in the
                                                               how much the gain fell! When a valve is cold it
grounded emitter configuration, which provides
                                                               provides no amplification, so signals can only pass
much more current gain than the grounded base
                                                               via the interelectrode capacitances, and these are
configuration. However, there is no significant
                                                               virtually the same whether the valve is hot or cold.
feedback from the collector circuit to the base
                                                               With no gain provided by the valve, the forward
tuned circuit since the collector load of the input
                                                               and reverse isolation are identical. Much the same
transistor consists of the very low emitter input
                                                               dodge can be used with transistors by open-circuit-
impedance of the second transistor. This is used in
                                                               ing the emitter to DC but leaving it connected as
the grounded base configuration, which again
                                                               before at AC. However, the results are not nearly
results in very low feedback from its output to
                                                               so reliable as in the valve case, as many of the
its input. With a suitable type of transistor the
                                                               transistor's parasitic reactances will change sub-
cascode circuit can provide well over 20 dB of gain
                                                               stantially when the collector current is reduced to
at 100 MHz together with a reverse isolation of
                                                               zero. For an RF amplifier stage to be stable,
70 dB. Reverse isolation is an important parameter
                                                               clearly its reverse isolation should exceed its for-
of any RF amplifier, and is simply determined by
                                                               ward gain by a reasonable margin, which need not
measuring the 'gain' of the circuit when connected
                                                               be anything like the 40 to 80dB obtainable with
back to front, i.e. with the signal input applied to
                                                                                Radio-frequency circuits       187

the cascode mentioned above. A difference of              to drive current through the feedback capacitance
20dB is fine and of 10 dB adequate, whilst some           (Cob in a bipolar transistor, Cdg or Crss in an FET)
commercially available broad band RF amplifier            is reduced pro rata. Likewise, if the source im-
modules exhibit a reverse isolation which falls to        pedance seen by the base (or gate) is reduced, the
as little as 3 dB in excess of the forward gain at the    current fed back will produce less voltage drop
top end of their frequency range.                         across the input circuit. Both measures reduce gain
   An interesting feature of the cascode stage of         and increase stability: it may well be cheaper to
Figure 8.4a arises from the grounded base connec-         recover the gain thus sacrificed by simply adding
tion of the output transistor. In this connection its     another amplifier stage than to add circuit com-
collector/base breakdown voltage is higher than in        plexity to obtain the extra gain from fewer stages
the common emitter connection, often by a con-            by unilateralization. This cumbersome term is used
siderable margin, as transistor data sheets will          to indicate any type of scheme to reduce the
show. This fact makes the cascode circuit a               effective internal feedback in an amplifier stage,
favourite choice for amplifiers which have to             i.e. to make the signal flow in only one direction-
handle a very wide range of frequencies whilst            forward. Data sheets for RF transistors often
producing a very large peak-to-peak output vol-           quote a figure for the maximum available gain
tage swing. Examples include the range from DC            (MAG) and a higher figure for maximum unilater-
to RF in the Y deflection amplifier of an oscillo-        alized gain (MUG).
scope, and that from 50 Hz to RF in the video                The traditional term for unilateralization is neu-
output amplifiers in a TV set. Figure 8.4b shows a        tralization, and I shall use this term hereafter as it is
complementary cascode stage. This has the advan-         just a little shorter, even though they are not quite
tage of not drawing any appreciable RF current            the same thing. Figure 8.6a shows one popular
from the positive supply rail, easing decoupling         neutralization scheme, sometimes known as bridge
requirements.                                            neutralization. The output tuned circuit is centre
   Figure 8.5a shows what is in effect a cascode         tapped so that the voltage at one end of the inductor
circuit, but in the dual-gate RF MOSFET the two          is equal in amplitude to, and in antiphase with, the
devices are integrated into one, the drain region of     collector voltage. The neutralizing capacitor Cn has
the input device acting as the source of the output      the same value as the typical value of the transistor's
section. Thus the dual-gate MOSFET is a 'semi-            Cob, or Cn can be a trimmer capacitance, set to the
conductor tetrode' and, as in the thermionic             same value as the Ccb of the particular transistor.
tetrode and pentode, the feedback capacitance            The criterion for setting the capacitor is that the
internal to the device is reduced to a very low          response of the stage should be symmetrical. This
level (for the Motorola MFE140 the drain/gatel           occurs when there is no net feedback, either positive
capacitance amounts to little more than 0.02 pF).        or negative. The series capacitance of Ccb and Cn
The dual-gate MOSFET exhibits a very high                appears across the output tuned circuit and is
output slope resistance, again like its thermionic       absorbed into its tuning capacitance, whilst the
counterpart, and also an AGC capability. The             parallel capacitance of Ccb and Cn appears across
circuit of Figure 8.5b provides up to 27 dB gain         the input tuned circuit and is absorbed into its
at 60 MHz when the AGC voltage Vgg is +8 V and           tuning capacitance. Neutralization can be very
up to 60 dB of gain reduction as Vgg is reduced to       effective for small-signal amplifiers, but is less so
below 0 V.                                               for stages handling a large voltage swing. This is
   A common technique to increase the stability          because the feedback capacitance C~b, owing to the
margin of transistor RF amplifiers is mismatching.       capacitance of the reverse biased collector/base
This simply means accepting a stage gain less than       junction, is not constant but varies (approximately)
the maximum that could be achieved in the                inversely as the square of the collector/base voltage.
absence of feedback. In particular, if the collector         Neutralization can be applied to a push-pull
(or drain) load impedance is reduced, the stage will     stage as in Figure 8.6b, but great care is necessary
have a lower voltage gain, so the voltage available      when so doing. The scheme works fine just so long
188   AnalogElectronics

                                                                                                                     t                          _~       O+15V
                                                                         120 k                                  47                                   0.001 /AF
                                                                                                                           0.001 FF             _

                                            82 k O.O0!pF kO0t                          Forrite Beads                 ,[          l~        _L
                                                        1                                                                  L2                         C4
                                                                                                                                                       II.qr     RF Output
                                                                    _--                                     ~        J"                              ,,~         Zou t - S0 ~1
              RF Input ~'~              -       J
              Zin ,, 50 1"1 /


                                                                ~   0.     1                                                         0.001

                  The following component values are for 9 |mr____n
                                                                  stability factor - 3.0.                       C2 Nominal 4.0 pF ARCO 402
                  L.1,L.2 13Q nN PAUL. SMITH CO. 11K-138-1                                                      C3 Nominal 13.73 pF ARCO 403
                            4.~ Turns (yellow)                                                                  C4 Nominal 4.36 pF JOHANSON JMC2961
                      C1 Nominal ?.0 pF Adjuslld for source Impedance of                                    All Decoupltn 9 Caplcitors are Cerami9 Discs.
                            approximately 1000 N, JONANSON JMC3961

                                                                                       ,'   IO~.'H,
                                                 - . t.O[
                                                            \                                                                    /
                                                 = 3.S
                                                 8 3.0
                                                                                       l"lmI ~ i~ l
                                                                                       b - v

                                                      2.0 I
                                                       300           400          600   800 1.0k                         2.0 k         3.0 k


                                                                                       ~ +15 V
                                     150 k                                 ( 82k            ~0.11.LF            1.8uH
                                                                           ~                --_                    '

                  Optional AGC
                                                                                                                                                           3.0-15 pF
                                                                                               ( , I'2                                                     C4
                                                                                                                                                                                 50 ohrr
                                                                                                                                                     C3        ~.[-
      50 ohm .. k , ~                          J,f                                             Gll r-ls                                              3.0-15 pF --'-
      ,o ut   -                                        ;
                      ..=.      C1                          C2
                                3.0-15 pF1                  3.0-15 pF                                 270                             0.1 ,uF
                                                                                                     Radio-frequency circuits      189

                                    0 --
                                    I0                                      .,j ~r"

                                w   40-           /,'
                                z         -       ii
                                        1i    -

                                    6o- I
                                     -2.0                O        *2.0       +i.O      +6.0   +8.0
                                                          VG2.GATE2 TO GROUNOVOLTAGE(VOLTS)


Figure 8.5   Dual-gate MOSFET RF amplifiers.
  (a) Low-noise dual-gate MOSFET VHF amplifier stage and noise figure curve. The Motorola MFE140 shown
incorporates gate protection Zener diodes, to guard against static electricity discharge damage.
  (b) Dual-gate MOSFET VHF amplifier with AGC, with gain reduction curve. Maximum gain 27 (20)db at 60
(200) MHz with no gain reduction (Vg2 at +7.5 V). The Motorola MPF131 provides an AGC range featuring up to
60 dB of gain reduction.
  (Reproduced by courtesy of Motorola Inc.)

as the voltage at the collectors can be guaranteed                            though the collector current is not. Thus a tran-
to be in antiphase. This will indeed be the case at                           sistor can amplify the signal even though it con-
the resonant frequency where the collector load is                            ducts for more than 180 ~ but less than 360 ~ - i . e .
a tuned circuit, or over the desired band of output                           operates in class AB. Likewise for class B (180 ~
frequencies where the collector load is a wide band                           conduction angle) and class C (conduction angle
RF transformer. However, at some other (usually                               less than 180~ These modes offer higher efficiency
higher) frequency this may no longer apply, owing                             than class A, but whether one or other of them is
to leakage inductance between the two halves of                               appropriate in any given situation depends upon
the collector circuit's inductor or transformer. The                          the particular application. Consider the earlier low
two collector voltages may then be able to vary in                            and intermediate power amplifier stages of an FM
phase with each other, and the circuit simply                                 transmitter, for example. Here, the amplitude of
becomes two identical amplifiers in parallel, each                            the signal to be transmitted is constant and it is the
with a total feedback capacitance equal to twice its                          only signal present; there are no unwanted signals
internal feedback capacitance. If the amplifier                               such as one inevitably finds in the earlier stages of
devices still have substantial power gain left at                             a receiver. Consequently a class B or C stage is
the frequency at which this condition exists, then                            entirely appropriate in this application. However,
the circuit can oscillate in a parallel single-ended                          an AM or SSB transmitter requires a linear
mode.                                                                         amplifier, i.e. one that faithfully reproduces the
                                                                              variations in signal amplitude which constitute the
                                                                              envelope of the signal.
                                                                                 In the receiver the requirement for linearity is
All of the amplifier circuits discussed so far have                           even more pressing, at least in the earlier stages
operated in class A, that is to say the peak current                          where many unwanted signals, some probably
swing is less than the standing current, so that at                           very much larger than the wanted signal, are
no time is the transistor cut off. Where the                                  present. Chapter 4 covered the mechanism by
collector circuit of an amplifier is a tuned circuit,                         which second-order n o n - l i n e a r i t y - second-har-
this will have a 'flywheel' effect so that the                                monic d i s t o r t i o n - results in sum and difference
collector voltage is approximately sinusoidal even                            products when more than one signal is present,
190    Analog Electronics

                                                                               m~ cD


                                                               Ccb 9 9
                                                                T.---I I
                                                                I     ii




                                   A                                                        A      A


                                                                                                RF choke
                                                 "l" /      ,,cD-
                                                                                           -l-c D

                                                                                                   o Output


Figure 8.6 Neutralization.
  (a) Bridge neutralization. The internal feedback path is not an ideal capacitor Cob as shown, but will have an in-
phase component also. If the phase angle of the neutralization via Cn is adjusted, e.g. by means of an appropriate
series resistance, the neutralization is more e x a c t - at that particular frequency. The stage is then described as
'unilateralized' at that frequency.
  (b) Cross-neutralization, push-pull amplifier.
                                                                             Radio-frequency circuits 191

and third-order non-linearity in products of the        will be 26.4MHz, and an unwanted signal at
form 2fl +f2. These latter intermodulation prod-        27.8 MHz will also produce an IF output from
ucts, resulting from two unwanted frequencies fl        the mixer at 1.4 MHz. This represents a fractional
and f2, are particularly embarrassing in radio          detuning of only 11.2%, and reference to the
reception. Imagine that fl is, say, 20 kHz higher       universal tuned circuit curves of Figure 8.3 will
than the wanted signal at f0, and that f2 is 20 kHz     verify that even with high-Q tuned signal fre-
higher still. Then 2fl - f 2 turns out to be exactly    quency circuits, it is difficult adequately to sup-
at f0. If the two unwanted frequencies were on          press the image response; hence the popularity of
the low-frequency side, f2 being 20kHz lower            the up-converting double superhet of Figure 8.7b.
than f0 and fl 20 kHz lower still, then it would        Here, signal frequency tuned circuits can be
be the intermodulation product 2 f 2 - f l that falls   replaced by suboctave filters (band-pass filters
on the wanted frequency. The intermediate-              each covering a frequency range of about 1.5 to
frequency (IF) amplifier section of a superheter-       1), or simply omitted e n t i r e l y - although this
odyne receiver- or superhet for short, shown in         sacrifices the protection against second-order
block diagram form in Figure 8 . 7 a - is preceded      intermodulation products afforded by suboctave
by a highly selective filter which, in a good           filters.
quality communications receiver, will attenuate            The linearity of amplifiers, both discrete com-
frequencies 20 kHz or more off tune by at least         ponents and multistage amplifier stages, and of
80 dB. However, it is not possible to provide that      mixers is often quoted in terms of intercept points.
sort of selectivity in a tunable filter; the compara-   You may not realize it, but the theory behind
tive ease of obtaining high selectivity at a fixed      these has already been covered in Chapter 4. You
frequency is the whole raison d'Ftre of the super-      may recall that if the input to an amplifier with
het. So the RF amplifier stage (if any) and the         some second-order curvature in its transfer char-
mixer must be exceedingly linear to avoid inter-        acteristic is increased by 1 dB, the second-harmo-
ference caused by third-order intermodulation           nic distortion rises by 2dB, and the sum and
products.                                               difference terms due to two different input fre-
    In a double-conversion superhet such as shown       quencies applied simultaneously behave likewise.
in Figure 8.7b, this requirement applies also to        Also, with third-order (S-shaped)curvature, both
the first IF amplifier and second mixer, although       third-harmonic and third-order intermodulation
the probability of interference from odd-order          products rise three times as fast as the input, at
intermodulation products is reduced by the roof-        least for small inputs.
ing filter preceding the first IF amplifier. This is       Of course, for very large inputs an amplifier will
always a crystal filter offering 30 or 40dB of          be driven into limiting and the output will even-
attenuation at frequencies 30kHz or more off            tually cease to rise: the output is said to be
tune. Indeed, recent developments in crystal            compressed. Figure 8.8a illustrates this: the point
filter design and manufacture permit the roofing        where the gain is 1 dB less than it would have been
crystal filter to be replaced by a crystal filter,      if overload did not occur is called the compression
operating at 70 MHz, with the same selectivity as       level. It is found that for levels up to about 10 dB
previously obtained in the second IF filter at          below compression, it is a good rule that Nth-
1.4 MHz, enabling the design of an 'up-convert-         order intermodulation products rise by N dB for
ing single superhet'. An up-converting superhet         every 1 dB by which the two inputs rise. Figure
removes the image problem encountered with a            8.8b shows the behaviour of an imaginary but not
down-converting single superhet such as in Figure       untypical amplifier. The level of second- and third-
8.7a. With a 1.4MHz IF and a local oscillator           order intermodulation products, as well as of the
tuning from 3 to 31.4MHz, it is difficult to            wanted output, have been plotted against input
provide enough selectivity at the top end of the        level. All three characteristics have then been
HF band. For example, when the receiver is              produced on past the region of linearity, and it
tuned to 25MHz the local oscillator frequency           can be seen that eventually they cross. The higher
192       Analog Electronics

                                                                                             IF strip
                                                                                         A      _
                                                                                                                      \           AF
                         RF stage                                                                                           Audio amplifier
                         .(if fitted)            _     Mixer ~ I F            amplifier                 IF amplifier

          RF'un ! Mixer
          circuit(s) tuned
                                                                   IF filter
                                                                                       ! Ban  pass
                                                                                         IF filter                    Detector
                                                                                                                                            AF output
          (if fitted)                   circuit(s)                                                                    and AGC
                                                                    Local                                             rectifier
                                        (if fitted)                 oscillator
                                                                    (may be


                                                                                                          Second IF                   AF
  I                 RF stage                        Roofing                                               amplifier
                    (if fitted)               First filter                    Second                      stages                Audio amplifier
                                              mixer                           mixer                                                                    LoudspeakeJ

      RF tuned                                        e.g. 70 MHz FirstIF                Band-pass                                           AF output
      circuit(s)                                                  amplifier              IF filter                                           stage
      (if fitted)
                                             ~        First                   i~         Second
                                                      local                              local
                                                      oscillator                         oscillator
                                                 !                                 t
                                                                                                                                         AGC voltage
                                                             [~         Crystal-controlled
                                                                        reference frequency


Figure 8. 7 Supersonic heterodyne (superhet) receivers.
  (a) Single-conversion superhet. Several filters may be used throughout the IF strip.
  (b) Double-conversion superhet, with synthesized first local oscillator and second local oscillator both crystal
reference controlled.

the level at which an amplifier's second- and third-                                          with the largest signals. A mixer (frequency chan-
order intercept points occur, the less problem there                                          ger) can be characterized in a similar way, except
will be with unwanted responses due to intermo-                                               of course that the intermodulation products (co-
dulation products, provided always that it also has                                           loquially called 'intermods') now appear translated
a high enough compression point to cope linearly                                              to the intermediate frequency.
                                                                                                         Radio-frequency circuits   193

                  Output (dBm)

                                          1 dB                                     /
                      Output _ _         =_~. . . . . .                            [

                       (x + G) dBm

                                          "        I                               i
                                                   i     ,,            ,           I                                      r

                                                 x dBm                      Input                               Input (dBm)

                      Output ( d B m )
                                                                                       -       I
                                                                                           I   I

                                          /                   "/12   ~/13


                                                                                           I   ,I    ,
                                                                                           II3 II2       Input (dBm)


Figure 8.8 Compression and intermodulation.
  (a) Compression point of an amplifier, mixer or other device with gain G dB. (Single tone input).
  (b) Second- and third-order input and output intercept points (II and O1); see text. (Two tones of equal amplitude)

                                                                             noise figure, as indeed must all the stages preced-
Noise   and dynamic      range
                                                                             ing the IF filter defining the final bandwidth. For it
For an amplifier forming part of a receiver, high                            makes sense to supply most of the gain after this
linearity is only one of several very desirable                              filter; this way, large unwanted signals are
qualities. The input stage must exhibit a low                                amplified as little as possible before being rejected
194   Analog Electronics

by the filter. Remember that unwanted signals may        v/(kTRB), where R is 50f~, k is Boltzmann's
be 60, 80 or even 100 dB larger than the wanted          constant = 1.3803 x 10 -23 joules per kelvin, T is
signal!                                                  the absolute temperature in kelvin (i.e. degrees
   The noise figure of an amplifier is related to the    centigrade plus 273) and B is the bandwidth of
amount of noise at its output, in the absence of         interest. At a temperature of 290K (17~ or
any intentional input, and its gain. Noise is an         roughly room temperature) this works out at
unavoidable nuisance, and not only in amplifiers.        24.6nV in 50f~ in a 3kHz bandwidth. If the
Chapter 1 showed how a current in a metallic             amplifier were perfectly noise free and had a gain
conductor consists of a flow of electrons jostling       of 20 dB (i.e. a voltage gain of • 10, assuming its
their way through a more or less orderly jungle of       output impedance is also 50 f~), we would expect
atoms, of copper maybe or some other metal; and          0.246 ~tV RMS noise at its output: if the output
Chapter 3 how current is produced by carriers-           noise voltage were twice this, 0.492 laV RMS, we
electrons or holes - flowing in a semiconductor.         would describe the amplifier as having a noise
Since at room temperature- indeed at any tem-            figure of 6dB. Thus the noise figure simply ex-
perature above absolute z e r o - the atoms are in a     presses the ratio of the actual noise output of an
state of thermal agitation, the flow of current will     amplifier to the noise output of an ideal noise-free
not be smooth and orderly but noisy, like the            amplifier of the same gain. The amplifier's equiva-
boisterous rushing of a mountain stream. Like            lent input noise is its actual output noise divided
the noise of a stream, no one frequency predomi-         by its gain.
nates. Electrical noise of this sort is called thermal      The dynamic range of an amplifier means the
agitation noise or just thermal noise, and its inten-    ratio between the smallest input signal which is
sity is independent of frequency (or 'white') for        larger than the equivalent input noise, and the
most practical purposes. The available noise power       largest input signal which produces an output
associated with a resistor is independent of its         below the compression level, expressed in decibels.
resistance and is equal to - 174 dBm/Hz e.g.
- 1 3 9 d B relative to a level of 1 milliwatt in a
                                                         Impedances and gain
3 kHz bandwidth.
   This means that the wider the bandwidth of a          The catalogue of desirable features of an amplifier
filter, the more noise it lets through. It would seem    is still not complete; in addition to low noise, high
that if we have no filter at all to limit the            linearity and wide dynamic range, the input and
bandwidth, there would be an infinite amount of          output impedances need to be well defined, and the
noise power available from a resistor- free heating      gain also. Further, steps to define these three
for evermore! This anomaly had theoretical physi-        parameters should not result in deterioration of
cists in the late nineteenth century worrying about      any of the others. Figure 8.10a shows a broadband
an ultraviolet catastrophy, but all is well; at room     RF amplifier with its gain, input impedance and
temperature thermal noise begins to tail off             output impedance determined by negative feed-
beyond 1000 GHz (10% down), the noise density            back. 2 The resistors used in the feedback network
falling to 50% at 7500 GHz. At very low tempera-         necessarily contribute some noise to the circuit.
tures such as are used with maser amplifiers, say        This can be avoided by the scheme known as
1 kelvin (-272~        the noise density is already      lossless feedback, 3 shown in Figure 8.10b. Here,
10% down by 5 GHz (see Figure 8.9b).                     the gain and the input and output impedances are
    Returning to RF amplifiers then, if one is driven    determined by the ampere-turn ratios of the wind-
from a 50 f~ source there will be noise power fed        ings of the transformer.
into its input therefrom (see Figure 8.9a). If the          Whilst in a high-quality receiver the stages pre-
amplifier is matched to the source, i.e. its input       ceding the final bandwidth crystal filter need to be
impedance is 50 f~ resistive, the RMS noise voltage      exceedingly linear, this requirement is relaxed in the
at the amplifier's input Vn is equal to half the         stages following the filter; a little distortion in these
source resistor's open-circuit noise voltage, i.e. to    will merely degrade the wanted signal marginally,
                                                                                            Radio-frequency circuits     195

                                      ~   R

                                                               Vn     0 RI

                            Noise source,           vn - R +         en
                            e.g. resistor R                     RI
                            en='x/4kTRB         I If R1 = Rthen vn =~-e n =
                                                                     '            ~        k
                                                                                         "t )
                                                                                       - -.(TB




        9 1.0           '
        "-" 0.5
        '~          ,                 I                !

        o.,(                       109T       1010T    tt       1011T Frequencyx temperature (log scale)
          o                                            I
        2:                                      2.6 x 1010 T

Figure 8.9    Thermalnoise.
   (a) A noisy source such as a resistor can be represented by a noise-free resistor R of the same resistance, in
series with a noise voltage generator of EMF en = v/(4kTRB) volts. Available noise power--
Vn2/R -- (en/2)2/R - - P n say. At room temperature (290 K) P n - - -204 dBW in a 1 Hz bandwidth = - 174dBm in a
1Hz bandwidth. If B = 3000Hz then P n - - - 1 3 9 d B m ,    and if R = R1 = 50f~ then Vn =0.246pV in 3kHz
   (b) Thermal noise is 'white' for all practical purposes. The available noise power density falls to 50% at a
frequency of 2.6 • 101~ i.e. at about 8000 GHz at room temperature, or 26 GHz at T = 1 K.

since by that stage in the circuit all the unwanted                   stage and progressing towards the earlier stages
signals have been rejected by the filter. It is usual to              the larger the gain reduction required. The final IF
apply automatic gain control so that the level of the                 stage may also be gain controlled, but this must be
wanted signal at the receiver's output does not vary                  done in such a way that it can still handle the largest
by more than a few decibels for an input level change                 received signals. Finally, in the presence of a very
of 1O0 dB or more. This is achieved by measuring the                  large wanted signal it may be necessary to reduce the
level of the signal, for example the level of the carrier             gain of the R F amplifier. The application of A G C is
in the case of an A M signal, and using this to control               usually 'scheduled' to reduce the gain of successive
the gain of the receiver. Since most of the gain is in                stages in the order described, as this ensures that the
the IF amplifier, this is where most of the gain                      overall noise figure of the receiver is not compro-
reduction occurs, starting with the penultimate                       mised.
196   Analog Electronics

                                                                   +V s
                               RF choke              ~=d,=: ~
                                                    T ~ - Decoupling
                                    CD              /7"nZ                                 Input      i

                      ,                  II
                      !             -

                                                    RE      Output~ Z     o
       Zi ~   Input           Rb2                                                                        (b)

              /~          .         .    .           .              o0V

Figure 8. I 0 Input and output impedance determining arrangements.
  (a) Gain, input and output impedances determined by resistive feedback. Rbl, Rb2 and Re determine the stage DC
conditions. Assuming the current gain of the transistor is I0 at the required operating frequency, then for input
and output impedances in the region of 5 0 ~ , R v = 5 0 2 / R E . For example, if RE = 10~,Rv=2509t, then
Zi ~ 3 5 ~ , Z o ~659t and stage gain..~ 10dB, while if RE =4.79t, Rv = 4709t, then Zi ~ 2 5 ~ , Z o ~959t and
gain ~ 15 dB. CD are blocking capacitors, e.g. 0.1 ~tF.
  (b) Gain, input and output impedances determined by lossless (transformer) feedback. The absence of resistive
feedback components results in a lower noise figure and higher compression and third-order intercept points.
Under certain simplifying assumptions, a two-way match to Zo results if N = M 2 - M - 1 . Then power
gain = m 2, impedance seen by emitter = 2Zo and by the collector = (N + M)Zo. This circuit arrangement is used
in various broadband RF amplifier modules produced by Anzac Electronics Division of Adams Russel and is
protected by US Patent 3 891 9 3 4 : 1 9 7 5 (DC biasing arrangements not shown). (Reprinted by permission of
Microwave Journal.)

   A number of different schemes are used to vary                             increased rather than reduced with large signals.
the gain of radio-frequency amplifier stages, one of                          The change of gain was brought about by a
which, the dual-gate FET, has already been men-                               spectacular fall in the f x of the transistor as the
tioned. The gain of a bipolar transistor can also be                          collector current increased. At the constant inter-
reduced, by reducing its collector current, but this                          mediate frequency at which the device was de-
also reduces its signal handling capability, so that                          signed to operate, this resulted in a fall in stage
only a few tens of millivolts of RF signal may be                             gain.
applied to the base. The available output is also                                Discrete transistor IF stages are giving way to
reduced when AGC is applied. At one time,                                     integrated circuits purpose designed to provide
bipolar transistors designed specifically for gain-                           stable gain and wide range AGC capability. A
controlled IF amplifier stages were available.                                typical example is the Plessey SL600/6000 series
These used forward rather than reverse control,                               of devices, the SL6IOC and 611C being RF
i.e. the collector current was increased to reduce                            amplifiers and the 612C an IF amplifier. The
gain. This had the advantage that the signal                                  devices provide 20 to 34dB gain according to
handling capability of the stage was actually                                 type, and a 50dB AGC range. The range also
                                                                               Radio-frequency circuits    197

contains the SL621 AGC generator. When receiv-            circuits will be mismatched when attenuation is
ing an AM signal, the automatic gain control              introduced. Two or more diodes can therefore be
voltage can be derived from the strength of the           used, and the current through each controlled in
carrier component at the detector. With an SSB            such a manner as to implement an L pad, 4 which is
signal there is no carrier; the signal effectively        matched in one direction (see Figure 8.11 a), or a T
disappears in pauses between words or sentences.          or n pad, which is matched from both sides. In
So audio derived AGC is used, with a fast attack          principle, an attenuator matched both ways can be
capable of reducing the gain to maintain constant         implemented with only two diodes if the bridged T
output in just a few milliseconds, and at a rate of       circuit is used (Figure 8.1 l b).
decay or recovery of gain of typically 20 dB per             It is only when receiving signals where the
second. The disadvantage of this scheme is that a         modulation results in variations of signal ampli-
stray plop of interference can wind the receiver's        tude, such as AM and SSB, that AGC is required.
gain right down, blanking the wanted signal for           With FM, PM and certain other signal types, no
several seconds. The SL621 avoids this problem. It        information is contained in the signal amplitude-
provides a 'hold' period to maintain the AGC level        other than an indication as to how strong the
during pauses in speech, but will nevertheless            signal is. Any variations in amplitude are therefore
smoothly follow the fading signals characteristic         entirely adventitious and are due to fading or noise
of HF communication. In addition, interaction             or interference. The effect of fading can be sup-
between two detector time constants, a level              pressed, and that of noise or interference reduced
detector and a charge/discharge pulse generator,          by using a limiting IF strip, i.e. one in which there
prevent stray plops and crashes from inappropri-          is sufficient gain to overload the last IF stage even
ately winding the receiver gain down.                     with the smallest usable signal. With larger signals,
   In critical applications such as the RF stage of a     more and more of the IF stages operate in over-
professional communications receiver, a different         load; all the stages are designed to overload
approach to gain variation is often employed. As          'cleanly', that is to accept an input as large as
noted above, with an increasing input signal level        their output. Thus stages in limiting provide a gain
the AGC scheduling would reduce the RF stage              of unity; in this way the effective gain of the IF
gain last. But if it is difficult to achieve sufficient   strip is always just sufficient to produce a limited
linearity in the RF stage in the first place, it is       output, however small or large the input, without
virtually impossible to maintain adequate linearity       the need for any form of AGC. Here again, ICs
if the gain is reduced. So instead the gain is left       have taken over from discrete devices in limiting
constant and an electronically controlled attenua-        IF strips, and other stages as well. For example,
tor is introduced ahead of the RF stage. The              the Plessey SL6652 is a complete single-chip mixer/
attenuator uses PIN diodes, whose mode of oper-           oscillator, IF amplifier and detector for FM cel-
ation was described in Chapter 3. PIN diodes can          lular radio, cordless telephones and low-power
only operate as current-controlled linear variable        radio applications. Its limiting IF strip has a
resistors at frequencies at which the minority            maximum gain to small signals, before limiting
carrier lifetime in the intrinsic region is long          sets in, of 90dB, whilst the whole chip typically
compared with the period of one cycle of the              draws a mere 1.5 mA from a supply in the range
RF. Even so, PIN diodes are available capable of          2.5 to 7.5 V.
operation down to 1 MHz or so, and can exhibit               In contrast to FM and PM signals, for some
an on resistance, when carrying a current of              signals the amplitude is the only useful informa-
several tens of millamperes, of an ohm or less.           tion. For example, in a low-cost radar receiver a
When off, the diode looks like a capacitance of           successive detection log IF strip is used to detect
1 pF or less, depending on type. Whilst a single          the returns from targets. As the strength of a
PIN diode can provide control of attenuation              return varies enormously depending upon the
when used as a current-controlled variable resistor       range and size of the target, an IF strip with a
in series with the signal path, the source and load       wide dynamic range is needed. The Plessey
198 Analog Electronics

                                                                        ~ ~-----.o Attenuated
                                                                -                  RF output
     RF  o       ~        ,,         D   ~t           D2
     input        Z0   Z0
                                                  ~I2 = I 1 -13                18K      H / f 18K

                                                                12 V o--i
                                                                               Set I'
                                                                                         " i
                                                                                                              10 I.tH

                                                                                                                     | 2 nF

                                                                        IN998       D3
                                                                                                              10~H                         1M,,

                                                                        IN998       D4
                                                                                                              0,        I00.F0 ]
                                                                                                                        ~          100K~ Attenuation
                                                                                 (ii)                                                     control
                                                                                                                   Min. atten.-2 V
                                                                                                                   Max. atten. +3 V

                                         1        ~                       Attenuation
                 Reference               9                                control
                               diodes                                     10
        D4                                   D2

                                         ~--                        ~                      (a)
                                                           R1                                      R 1 t #,

                                                                                                   i Z0             Loid Z0

                                                                        L pad attenuator

                                                           _z~            " ~R2"                 ~-z0
                                                           o-                  --r~       "        o
                               (b)                                  Bridged T attenuator
                                                                                  Radio-frequency circuits     199

SL1613C is an IC wideband log IF stage with 12              the USA on that frequency) the local oscillator
dB gain R F input to RF output and a rectified              frequency could be either 8.6 MHz or 11.4 MHz,
output providing 1 mA video current for a 500 mV            since in either case the difference frequency is equal
RMS signal input. The video output currents of              to 1.4 MHz, the intermediate frequency. The sum
successive stages are summed to provide an output           frequency will also appear at the ouput of the
whose amplitude is proportional to the logarithm            mixer, but the IF filter rejects not only the sum
of the signal amplitude, with a video rise time of          frequency but the original RF and local oscillator
only 70 ns. Six or more stages may be cascaded to           signals as well, accepting only the wanted 1.4 MHz
provide 60 MHz IF strips with up to 108 dB gain             IF. In many cases the local oscillator frequency
with better than 2 dB log linearity.                        will be higher than the signal frequency ('high side
                                                            injection', 'LO runs high'); for example, the first
                                                            LO in the 100kHz to 3 0 M H z double superhet
Mixers                                                      of Figure 8.7b would run from 70.1MHz to
Most modern receivers are of the superheterodyne            100 MHz.
type, with most of the amplification provided by               It has already been noted that any device with
the IF stages. This applies to broadcast receivers of       second-order curvature of its transfer character-
all sorts, both sound and television; to professional       istic will produce not only second-harmonic dis-
communications, both civil and military, whether            tortion but also second-order intermodulation
at H F (up to 30MHz), V H F or UHF; and to                  products, i.e. sum and difference tones. The
receivers of other sorts, such as radar and naviga-         mixer in an early valve superhet, also called the
tion beacons. A frequency changer, converter or             'first detector', worked in exactly this manner: a
m i x e r - all names for the same t h i n g - is used to   half-wave rectifier circuit would do just as well.
translate the incoming signal from whatever fre-            However, this type of mixer exhibits a large
quency it was transmitted at to a fixed frequency,          number of spurious responses. At its broadest, a
which is more convenient for providing high                 receiver spurious response is any frequency at
selectivity. In a single superhet such as Figure            which a receiver produces an output other than
8.7a the RF signal is applied, following amplifica-         the wanted frequency to which it is tuned. One
tion by one or more R F stages if fitted, to a mixer.       example, the image frequency (formerly called the
This stage has two input ports and one output               'second channel'), has already been noted: this is
port. To the second input port is applied an R F            really a special case. Given sufficient front end
signal generated locally in the receiver; this is           selectivity, there will be no image response since
called the local oscillator (LO). The mixer is a            no energy at that frequency can reach the mixer. In
non-linear device and thus produces sum and                 the up-converting superhet of Figure 8.7b, the
difference frequency components. For example, if            image frequency will always be higher than
the receiver of Figure 8.7a were tuned to receive a         70.2 MHz, so a low-pass filter at the front end
signal at 10 MHz (it might be the WWV standard              can suppress the image response entirely. This
time and frequency transmission, broadcast from             same filter will also prevent a response at the IF

Figure 8.11 Voltage-controlled RF attenuators using PIN diodes.
  (a) (i) Pair of PIN diodes in L pad configuration, used to attenuate RF signals controlled by DC. Both 11 and/2
must be varied appropriately to control attenuation and keep Zo constant. (ii) Working PIN diode attenuator must
provide separation of the DC control current and RF signal paths. (iii) Constant attenuator impedance and
temperature compensation are attained when the PIN diodes are matched against reference diodes in this
arrangement. Opamp ICI keeps the voltage drive to both sets of diodes equal, and IC2 acts as a current sink
control for the PIN diodes and as a temperature compensator. Control of attenuation is logarithmic (dB law).
  (b) L pad attenuators can provide a constant characteristic impedance Zo as the attenuation is varied, but only at
the input terminals. A bridged T configuration can keep Zo constant at both input and output terminals.
200   Analog Electronics

frequency by preventing any signals at 70 MHz          ideally only produce an output due to the RF
reaching the mixer. However, the image and IF          and LO signals themselves, not their harmonics.
rejection are usually quoted separately in a recei-       The spurious responses just described are
ver's specification, the term 'spurious response'      termed external spurious responses, in that they
being reserved for unwanted responses due to           appear in response to an externally applied signal
much subtler and more insidious causes.                which bears a particular relation to the LO
   A mixer necessarily works by being non-linear.      frequency, and thus to the wanted frequency.
It would be nice if the mixer produced only the        Internal spurious responses, on the other hand,
wanted IF output, usually the difference frequency     are totally self-generated in the receiver. Most
between the RF signal and local oscillator inputs.     professional communications receivers nowadays
In practice the mixer may also produce an output       contain a microprocessor to service the front
at the intermediate frequency due to signals not at    panel, to accept frequency setting data from a
the wanted RF at all. A mixer, being a non-linear      remote control input, to display the tuned fre-
device, will produce harmonics of the frequencies      quency, and so on. Harmonics of the micropro-
present at its inputs, and these harmonics them-       cessor's clock frequency can beat with either the
selves are in effect inputs to the mixer. So imagine   first or the second local oscillator, to produce the
the single superhet of Figure 8.7a tuned to receive    same effect as an externally applied CW interfering
a signal at 25 MHz. The LO will be at 26.4 MHz,        signal. Needless to say, in a well-designed receiver
and the second harmonic of this, at 52.8 MHz, will     such responses are usually at, or below, the
be lurking in the mixer just waiting to cause          receiver's noise level. However, there is also the
trouble. Imagine an unwanted signal at                 possibility of the odd spurious response due to
25.7 MHz, too close to the wanted frequency to         interaction of the first and the second LO, which
be much attenuated by the RF tuned circuits. The       makes the up-converting single superhet an attrac-
second harmonic of this, at 51.4 MHz, is exactly       tive proposition now that advances in crystal filter
1.4 MHz away from the second harmonic of the           technology make it possible. Most modern com-
local oscillator and will therefore be translated to   munications receivers have the odd internal 'spur'
IF. This is variously called the 2-2 response or the   in addition to the inevitable external spurious
'half IF away' response, being removed from the        responses or 'spurii'.
wanted frequency by half the IF frequency.                A dual-gate FET can be used as a multiplicative
Similarly, the 3-3 response will occur at a fre-       mixer by applying the RF and LO voltages to gate
quency removed from the wanted frequency by             1 and 2 respectively. If the RF and LO voltages are
 1.4/3 MHz. Clearly these responses will not be a      represented by pure sinusoidal waveforms sin r
problem in the up-converting superhet of Figure        and sin L, where sin r stands for sin(2nfRvt) and
8.7b, which is one reason for the popularity of this   sin (L) for sin (2nfLot), then, ignoring a few con-
design. It is not, however, entirely immune from       stants, the mutual conductance can be represented
spurious responses. Imagine that it is tuned to        by sin r sin L. So the drain output current can be
23 MHz, so that its first LO is at 93 MHz, and         represented by [cos(r - L) - cos(r + L)]/2, cour-
that there is a strong unwanted signal at              tesy of your friendly neighbourhood maths text-
23.2 MHz. The fifth harmonic of the latter, at         book, i.e. it contains the sum and difference
 116 MHz, is removed from the second harmonic          frequencies. The constants ignored in such a
(186 MHz) of the LO by 70 MHz. Admittedly this         cavalier fashion are responsible for the presence
is a seventh-order response, and fortunately the       in the drain current of components at the RF and
magnitude of spurious responses falls off fairly       LO frequencies, so the dual-gate FET mixer is
rapidly as the order increases. But it does indicate   described as unbalanced. However, if its operation
that the ideal mixer is a very peculiar device: it     were ideally multiplicative then these would be the
must be very linear to two or more unwanted            only unwanted outputs, i.e. it would be free of
signals applied at the RF port (to avoid unwanted      spurious responses.
responses due to intermodulation), and should              The presence in a mixer's output of components
                                                                               Radio-frequency circuits     201

at the RF and LO frequencies can be a serious             applied differently. For example, the LO can be
embarrassment. Consider the communications re-            applied to the DC coupled port and the IF output
ceiver of Figure 8.7b, for example. Such a receiver       taken from one of the transformer coupled ports.
is typically specified to operate right down to an        Whilst this has certain advantages in special cases,
input frequency of 10 kHz. At this tuned frequency        it is not usually used in a receiver, since LO
the LO will be running at 70.001 MHz, which is            radiation via the receiver's input port is then
uncomfortably close to the IF at 70 MHz, bearing          likely to be worse.
in mind that the LO signal is very large compared            Another well-known scheme (not illustrated
with a weak RF signal. So a balanced mixer is             here) uses MOSFETs instead of diodes as the
used. A single-balanced mixer is arranged so that         switches. 5 It is thus, like the Schottky diode ring
the signal at one of the input ports (usually the LO      DBM, a passive mixer, since the active devices are
port) does not appear at the output port; thus it         used solely as voltage-controlled switches and not
can effectively 'reject' the LO. In a double-balanced     as amplifiers. Reference 6 describes a single-
mixer (DBM) neither of the inputs appears at the          balanced active MOSFET mixer providing 16 dB
output, at least in the ideal c a s e - and in practice   conversion gain and an output third-order inter-
this condition is nearly met, with RF and LO              cept point of +45 dBm.
rejection figures typically greater than 20 dB.              Figure 8.12b shows a double-balanced active
   Figure 8.12 shows three DBMs. The first is the         mixer of the seven-transistor tree variety; the
basic diode ring mixer, so called because if you          interconnection arrangement of the upper four
follow round the four diodes you will find they are       transistors is often referred to as a Gilbert cell.
connected head to tail (anode to cathode) like four       The emitter-to-emitter resistance R sets the con-
dogs chasing each other in a circle. On positive-         version gain of the stage; the lower it is made the
going half-cycles of the LO drive two of the diodes       higher the gain but the worse the linearty, i.e. the
conduct, connecting one phase of the RF input tO          lower the third-order intercept point. This circuit is
the IF port. On the other half-cycle the other two        available in IC form (see Figure 8.12c) from a
diodes conduct, reversing the phase fed to the IF         number of manufacturers under type numbers
port. A very large LO drive is used, so that for          such as LM 1496/1596 (National Semiconductor),
virtually all the time either one pair of diodes or       MC1496/1596 (Motorola, Mullard/Signetics) and
the other is conducting heavily: the diodes (which        SG1496/1596 (Silicon General), whilst derivatives
are selected for close matching, or are monolithic)       with higher dynamic range are also available.
are in fact used simply as switches. The ring DBM            Finally, in this whistle-stop tour of mixers,
is double balanced, produces the sum and differ-          Figure 8.12d shows one of the simplest of the
ence frequencies, and exhibits about half as many         many ingenious ways in which the performance
spurious responses as an unbalanced mixer. The            of the basic Schottky diode ring DBM has been
conversion loss (ratio of IF output power to RF           i m p r o v e d - almost invariably, as here, at the
input power) is about 7 dB; this is attributable to       expense of a requirement for greater LO power
several different causes. Half of the input RF            (up to +27 dBm is not uncommon). The resistors
energy will contribute to the sum output and half         in series with the diodes waste LO power and in-
to the difference: as only one of these is required       crease the insertion loss, but they have beneficial
there is an inherent 3 dB conversion loss, the other      effects as well. They permit a larger LO drive to be
3 or 4dB being due to resistive losses in the on          applied, which reduces the fraction of the LO cycle
resistance of the Schottky diodes, and to transfor-       which is taken up by commutation, that is chang-
mer losses. The IF port is 'DC. coupled', and thus        ing from one pair of diodes conducting to the
operates down to 0Hz. This is the mode of                 other pair. They stabilize the effective on resistance
operation when the diode DBM is used as a                 of the diodes, which would otherwise vary
phase sensitive detector, the RF and LO frequen-          throughout each half-cycle owing to the sinusoidal
cies then being identical. Where an IF response           current waveform. Finally, they cause an
down to DC is not required, the inputs can be             additional voltage drop across the on diodes; this
202    AnalogElectronics

                    (        ,,

                    Ferrite tomidal core
                                                   ,~,      ~

                                                         X (IF) (Also known as
                                                                                       Equivalent circuit on
                                                                                       positive half-cycle of LO
                                                                the l port)

                                                           /~                Ca)

                             Balanced outputs

            . ij
      (switching)                          3L

      (linear) inputs
            0      '"

            0 -
         Set tail
         (bias)                            Rbias

                                      _     -V S

                                    (d)                                                                        (c)

Figure 8 . 1 2  Double-balanced mixers (DBMs).
  (a) The ring modulator. The frequency range at the R and L ports is limited by the transformers, as also is the
upper frequency at the X port. However, the low-frequency response of the X port extends down to 0 Hz (DC).
  (b) Basic seven-transistor tree active double-balanced mixer. Emitter-to-emitter resistance R, in conjunction with
the load impedances at the outputs, sets the conversion gain.
  (c) The transistor tree circuit can be used as a demodulator (see text). It can also, as here, be used as a modulator,
producing a double-sideband suppressed carrier output if the carrier is nulled, or AM if the null control is offset.
The M C 1 4 9 6 includes twin constant current tails for the linear stage, so that the gain setting resistor does not
need to be split as in (b). (Reproduced by courtesy of Motorola Inc.)
  (d) High dynamic range DBM (see text).
                                                                                                    Radio-frequency circuits   203

                                   Ring modulator

                                   [R +
                                         l Audio
                            Level 1r               ,,
                                                              = Frequency              Typically
                               0.3-3 ld-lz
                                                                                       -3 dB bandwidth:
                                                                                       1.4003- 1.4030 MHz
                                                                                       -20 dB at 1.4000 MHz
                                                                   Carrier at IF       -40 dB at 1.3997 MHz
                                                    1.4 MHz

          Double-sideband =- l                                y

                                                                  1.4 MHz upper i
          suppressed carrier                                      side-band
          output_._.               L                              filter

          To mixer (translate to
          transmit frequency)
          and power amplifier                           (a)

                                             Ring modulator
                                                                                       ~ ~ 1 r 1.400 MHz carrier
                                                                                               reinsertion oscillator
                       1.4 MHz USB signal
                       from IF strip
                                                                         Clarifier provides +750 Hz
                                                                         fine frequency adjustment

                                         Recovered audio output

Figure 8.13      DBM used as modulator and demodulator.
   (a) DBM used as a modulator in an HF SSB transmitter. The carrier rejection of the mixer plus the 20 dB
selectivity of the USB filter at 1.4 MHz ensure that the residual carrier level is more than 40 dB down on the peak
transmitter power.
   (b) DBM used as an SSB demodulator in an HF SSB receiver.

increases the reverse bias of the off diodes, thus                     transmit frequency by a mixer, for amplification in
reducing their reverse capacitance.                                    the power output stages. In an SSB transmitter, the
                                                                       voice signal to be transmitted can be applied to the
                                                                       DC coupled port (also known as the X or I port) of
                                                                       a double-balanced mixer, whilst the LO signal is
The DBM is also popular as both a modulator and                        applied to one of the transformer coupled ports as
a demodulator. A modern transmitter works rather                       in Figure 8.13. The output from the other trans-
like a superhet receiver in reverse, that is to say that               former coupled port is a double-sideband sup-
the signal to be transmitted is modulated onto a                       pressed carrier signal as shown, which can then be
carrier at a fixed IF and then translated to the final                 filtered to leave the SSB signal, either upper side-
204    Analog Electronics

                              ,,         +Vs
                       =L.       -                             DC blocking
                      ~    ~ ~                 ~-I -       _ I capacitor

              Doouplin                            2.          '                    To a u d i o

                  transformer                                      Volume
                  (e.g. 455 kHz)                  ~                control
                                        AGC voltage to
                                        IF stages (with
                                        a path to-V s)       (a)

                          Both tuned
                          to 10.7 MH
                               ; +V s


                                                                                                  Vectors relative to point A showing
              From last stage                                      ~                              variation of rectified voltages
              of limiting IF              A               i_.~.~pacltor "t~                       VRI, and VR2with frequency
              strip, 10.7 MHz                             1 ~ - - - - Audio

                                   C                      /77~ C3

Figure 8.14     AM and FM demodulators (detectors).
  (a) Diode AM detector. In the 'infinite impedance detector', e.g. Tr3 in Figure 8.21, a transistor base/emitter
junction is used in place of the diode. The emitter is bypassed to RF but not to audio, the audio signal being taken
from the emitter. Since only a small RF base current is drawn, the arrangement imposes much less damping on
the previous stage, e.g. the last IF transformer, whilst the transistor, acting as an emitter follower, provides a
low-impedance audio output.
   (b) Ratio detector for FM, with de-emphasis. C' =RF bypass capacitor, 330 pF.

band (USB) or lower sideband (LSB) as required.                              reinserted carrier is not at exactly the appropriate
(Amateur radio practice is to use LSB below                                  frequency. This results in reduced intelligibility and
10 MHz and USB above, but in commercial and                                  has been likened to the sound of Donald Duck
military applications USB is the norm regardless of                          talking through a drainpipe. A control called a
frequency.) In the receiver, the reverse process can                         clarifier is usually provided on an SSB receiver to
be applied to demodulate an SSB signal, i.e. the                             permit adjustment of the frequency of the rein-
output of the IF strip is applied to one of the                              serted carrier for maximum intelligibility. In prac-
transformer coupled ports of a diode ring mixer,                             tice an IC such as 1496 DBM is often used for the
and a carrier wave at the frequency of the missing                           demodulator: linearity is not of paramount impor-
suppressed carrier at the other. The beat frequency                          tance in this application, since any signal in the
between the two is simply the original modulating                            pass band of the IF is either the wanted signal or
voice signal, but offset by a few cycles if the                              unavoidable cochannel interference.
                                                                               Radio-frequency circuits     205

   Having touched on the subject of SSB demodu-          the recovered audio appears in antiphase across R1
lation, it is appropriate to cover here demodula-        and R2, whilst the voltage across CA is constant.
tors - often called detectors - for other types of       R3C3 provides de-emphasis to remove the treble
signals as well. Figure 8.14a shows a diode detec-       boost applied at the transmitter for the purpose of
tor as used in an AM broadcast receiver. It re-          improving the signal/noise ratio at high frequen-
covers the audio modulation riding on a DC level         cies: the time constant is 50 ~ts (75 ~ts is used in the
proportional to the strength of the carrier compo-       USA). This type of frequency discriminator,
nent the signal. This DC level is used as an AGC         known as the ratio detector, was popular in the
voltage, being fed back to control the gain of the       early days of FM broadcasting, since it provided a
IF stages, so as to produce an effectively constant      measure of AM rejection to back up the limiting
signal even though the actual level may change due       action of the IF strip. Any rapid increase or
to fading. The result is usually acceptable, but         decrease in the peak-to-peak IF voltage applied
AGC can give rise to unfortunate effects. For            to the diodes would result in an increase or
example, on medium wave after dark, signals              decrease of the damping on the centre tapped
from distant stations can be received but the            tuned circuit by the detectors, as C4 was charged
nature of the propagation (via reflections from          up or discharged again. This tended to stabilize the
the ionosphere) can give rise to frequency selective     detected output level, whilst slow variations in
fading, resulting in quite sharp notches in the          level, due to fading for example, were unaffected.
received RF spectrum. If one of these coincides          Modern FM receivers use IC IF strips with more
with the carrier component of an AM signal, the          than enough gain to provide hard limiting on the
AGC will increase the IF gain to compensate. At          smallest usable signal, so an on-chip discriminator
the same time, as the sidebands have not faded in        based upon quadrature detection by the Gilbert
sympathy, the result is that the signal is effectively   cell is normally used.
modulated by greater than 100%, resulting in
gross distortion in the detected audio. It is un-
fortunate that this coincides with the increased
output due to AGC action, resulting in a very            The next major category of circuit considered in
loud and unpleasant noise!                               this chapter is the RF oscillator. Every transmitter
   Figure 8.14b shows one of the types of demodu-        needs (at least) one, and receivers of the superhet
lator used for FM signals. It depends for its action     variety also need one in the shape of the local
upon the change of phase of the voltage across a         oscillator. The frequency of oscillation is deter-
parallel tuned circuit relative to the current as the    mined by a tuned circuit of some description. The
signal frequency deviates first higher then lower in     hallmarks of a good oscillator are stability (of both
frequency than the resonant frequency. The refer-        output frequency and output level), good wave-
ence voltage Vref in the small closely coupled wind-     form (low harmonic content) and low noise. An
ing at the earthy end of the collector tuned circuit     oscillator can be considered either as an amplifier
is in phase with the voltage across the latter. The      whose output is applied via a band-pass filter back
centre tapped tuned circuit is very lightly coupled      to its input, so as to provide positive feedback with
to the collector tuned circuit, so the reference         a loop gain of just unity at one frequency; or as a
voltage is in quadrature with the voltage across         circuit in which an active device is arranged to
the centre tapped tuned circuit. The resulting           reflect a negative resistance in parallel with a tuned
voltages applied to the detector diodes are as           circuit, of value just sufficient to cancel out the
indicated by the vector diagrams. Capacitors C'          losses and raise its Q to infinity. In practice, there
have a value of around 330 pF, so that they present      is seldom any real difference between these appar-
a very low impedance at the usual FM IF of               ently divergent views: Figure 8.15 illustrates the
 10.7MHz but a very high impedance at audio              two approaches.
frequencies. As the detected output voltage from            In Figure 8.15a a single tuned circuit with no
one diode rises, that from the other falls, so that      coupled windings is employed. For the circuit to
206   Analog Electronics

                                                _+          = .~         or _ ~   or.~~   or   "1   ~- etc.


                           I                                  i
                           Tuned circuit        Amplifier          (b)
                           (band-pass filter)

Figure 8.15 Oscillator types.
  (a) Negative resistance oscillator: see text.
  (b) Filter/amplifier oscillator.

oscillate, Z2 and Z 3 must be impedances of the                device depends upon the Q of the tuned circuit.
same sign (both positive, i.e. inductances, or both            True, the Q is infinity, in the sense that the
negative, i.e. capacitances) whilst Z1 must be of the          amplitude of the oscillation is not dying away,
opposite sign. The funny symbol is a shorthand                 but that is only because the active device is making
sign for any three-terminal active device, be it               up the losses as they occur. As far as rate of change
valve, bipolar transistor or FET. Figure 8.16                  of phase with frequency in the tuned circuit is
shows a number of Figure 8.15a type oscillators,               concerned, the Q is determined by the dynamic
with their usual names. Of these, the Clapp (or                resistance Ro of the tuned circuit itself, in parallel
Gouriet) is a circuit where the value of the two               with the loading reflected across it by the presence
capacitors of the corresponding Colpitts oscillator            of the active device. In the case of a valve or FET,
has been increased and the original operating                  the anode or drain slope resistance is often the
frequency restored by connecting another capaci-               main factor: in the case of a bipolar transistor, the
tor in series with them. To understand how this                low base input impedance is equally important.
improves the stability of the oscillator, remember                The additional capacitor C1 in the Clapp
that any excess phase shift through the active                 circuit effectively acts in the same way as a
maintaining device, resulting in its phase shift               step-down transformer, reducing the resistive
departing from exactly 180 ~ must be compensated               loading on the tuned circuit, so that its loaded
for by a shift of the frequency of oscillation away            Q approaches more nearly to its unloaded Q.
from the resonant frequency of the tuned circuit,              This improves the frequency stability by increas-
so that the voltage applied to the 'grid' lags or              ing the isolation of the tuned circuit from the
leads the 'anode' current by the opposite amount.              vagaries of the maintaining circuit, but of course
This restores zero net loop phase shift, one of the            does nothing to reduce frequency drift due to
necessary conditions for oscillation. By just how              variation of the value of the inductance and of
much the frequency of operation has to change to               the capacitors with time and temperature varia-
allow for any non-ideal phase shift in the active              tions. The improved isolation of the tuned circuit
                                                                                                Radio-frequency circuits       207

                                        Reversed feedback               Tickler feedback

              .TIi,                   T
               Hartley oscillator                   coupled Hartley oscillators
                                           Transformer                                                   Colpitts oscillator

                     .~L Ic1_                                                                   ""

                                                                                           --        w       -

                   Clapp (Gouriet) oscillator       Pierce oscillator                           TATG
                                                                                  C is internal to the active device.
                                                                                  No magnetic coupling between
                                                                                  LI and/,2

                                     Line stabilized TATG           T,-J      ~      v
                           Length 1= (2n + 1) L/4 at frequency
                           of oscillator, e.g. l = L/4.                 Dual-gate FET solid state version
                           Line has short-circuited ends.               of the electron coupled oscillator

Figure 8.16   Negative resistance oscillators (biasing arrangements not shown).

from the active device cuts both ways. There is                   oscillator is a deservedly popular configuration
less drive voltage available at the active device's               for a high-stability crystal oscillator.
input and, at the same time, the load resistance                     Figure 8.17 shows oscillator circuits of the
reflected into its output circuit is reduced: both                Figure 8.15b variety. The TATG circuit in Figure
of these factors reduce the stage gain. Thus the                  8.16 (named from its valve origins: tuned anode,
Clapp circuit needs a device, be it valve, transis-               tuned grid) is like the Meissner oscillator in Figure
tor or FET, with a high power gain. Clearly the                   8.17, except that the feedback occurs internally in
higher is the unloaded Q of the tuned circuit, the                the device. The line stabilized oscillator is an
lower are the losses to be made up and hence the                  interesting circuit, sometimes used at UHF where
less gain is demanded of the maintaining circuit.                 a coaxial line one wavelength long becomes a
Assuming a high output slope resistance in the                    manageable proposition. By increasing the rate
active device, the losses will nearly all be in the               of change of loop phase shift with frequency, the
inductor. If this is replaced by a crystal, which at              line increases the stability of the frequency of oscil-
a frequency slightly below its parallel resonant                  lation. A surface acoustic wave (SAW) device can
frequency will look inductive, a very high-Q                      provide at U H F a delay equal to many wave-
resonant circuit results, and indeed the Clapp                    lengths. If the SAW device provides N complete
208    AnalogElectronics

                                               I T1                       "i C4,iaxiallile        ..

                         Meissner oscillator                                  4 - l --~
                                                               Line stabilized          Length 1= n ~'2, n odd
                                                                                         or even depending on
                                                                                         phasing of feedback

                                 h !l-sAwi F-7 High-Q tuned circuit to
                                      delay I ~ --
                                                   select frequency from comb
                                    i line i               at which phase-shift
                           _                          !T   through SAW is n 360~

                       Surface acoustic wave delay,
                       line stabilized
Figure 8.17 Filter/amplifier oscillators.

cycles of delay, the rate of change of phase shift              use two active devices. In principle, two devices
with frequency will be N times as great as for a                can provide a higher gain in the maintaining
single-wavelength delay. The SAW stabilized                     amplifier and thus permit it to be more lightly
oscillator can thus oscillate at any one of a                   coupled to the tuned circuit, improving stability.
'comb' of closelyspaced frequencies, a conven-                  But on the other hand the tuned circuit has to cope
tional tuned circuit being used to force operation              with the vagaries of two active devices instead of
at the desired frequency.                                       just one. The maintaining amplifier need not use
   Figure 8.18 shows two oscillator circuits which              discrete devices at all. The maintaining device can

                      v~                              v~                v~                   v,

                            C              C


Figure 8.18 Two-device oscillators (see also Figure 9.14a).
   (a) Franklin oscillator. The two stages provide a very high non-inverting gain. Consequently the two capacitors
C can be very small and the tuned circuit operates at close to its unloaded value of Q.
   (b) Emitter coupled oscillator. This circuit is unusual in employing a series tuned resonant circuit. Alternatively it
is suitable for a crystal operating at or near series resonance, in which case R can be replaced by a tuned circuit
to ensure operation at the fundamental or desired harmonic, as appropriate.
                                                                                           Radio-frequency circuits         209

                 8 MHz                                              (TCXO) is used. In this, the ambient temperature
           ]                    ,,
                                               R = 470R for 7404    is sensed by one or more thermistors and a voltage
                                                 = 4K7 for 74LS04   with an appropriate law is derived for application
                                                                    to a voltage-controlled variable capacitor (vari-
                                                                    cap). Both OCXOs and TCXOs are provided
                                               Buffered             with adjustment m e a n s - a trimmer capacitor or
                                               output               varicap diode controlled by a potentiometer- with
                                                                    sufficient range to cover several years drift, allow-
                     (a)                                            ing periodic readjustment to the nominal fre-
                                                                       In any oscillator circuit, some mechanism is
                           10M- ~ ~ - ~           Buffered          needed to maintain the loop gain at unity at the
                                     2-~          output            desired amplitude of oscillation. Thus the gain
                                                                    must fall if the amplitude rises and vice versa. In
                                                                    principle, one could have a detector circuit which
         15p     _         ..         _ CC I   C2 =15 pF
                                                                    measures the amplitude of oscillation, compares it
                                                                    with a reference voltage and adjusts the amplifier's
                                                                    gain accordingly, just like an AGC loop. In this
                                                                    scheme, called an a u t o m a t i c level c o n t r o l (ALC)
                                                                    loop, the amplifier operates in a linear manner, for
             Crystal-controlled computer clock oscil-
Figure 8 . 1 9                                                      example in class A. However, it requires a detector
lators.                                                             circuit with a very rapid response, otherwise the
   (a) TTL type with crystal operating at series                    level will 'hunt' or, worse, the oscillator will
resonance.                                                          'squegg' (operate only in short bursts). Most
  (b) CMOS type wih crystal operating at parallel                   oscillator circuits therefore forsake class A and
                                                                    allow the collector current to be non-sinusoidal.
                                                                    This does not of itself ensure a stable amplitude of
be an integrated circuit amplifier, or even an                      oscillation, but the circuit is arranged so that as the
inverting logic gate used as an amplifier, as                       amplitude of oscillation increases, the device biases
shown in Figure 8.19.                                               itself further back into class C. Thus the energy
   Where high stability of frequency is required, a                 delivered to the tuned circuit at the fundamental
crystal oscillator is the usual choice. For the most                frequency decreases, or at least increases less
critical applications, an ovened crystal oscillator                 rapidly than the losses, leading to an equilibrium
can be used. Here, the crystal itself and the                       amplitude. In a transistor oscillator, stability is
maintaining amplifier are housed within a con-                      often brought about by the collector voltage
tainer, the interior of which is maintained at a                    bottoming, thus imposing heavy additional damp-
constant temperature such as +75~         Oven-con-                 ing upon the tuned circuit. This is most undesir-
trolled crystal oscillators (OCXOs) can provide a                   able from the point of view of frequency stability,
temperature coefficient of output frequency in the                  and the current switching circuit of Figure 8.20a is
range 10 -7 to 10 -9 per ~ but stabilities of much                  much to be preferred.
better than one part in 10 6 per annum are difficult                   Figure 8.20 also shows various ways in which
to achieve. The best stability is provided by the                   the net loop gain of an oscillator can vary with
glass encapsulated crystal, the worst by the solder                 amplitude. The characteristic of Figure 8.20b is
seal metal can crystal, with cold weld metal cans                   often met though not particularly desirable. That
providing intermediate performance. Where the                       of Figure 8.20c will not commence to oscillate
time taken for oven warm-up is unacceptable                         unless kicked into oscillation by a transient such
and the heater cannot be left permanently switched                  as at switch-on, a most undesirable characteristic.
on, a temperature-compensated crystal oscillator                    That of Figure 8.20d is representative of the
210    Analog Electronics

                                Decoupling I-                                       +Vs
                                capacitor /77~                             RF choke

                                                           II          -      II
                                                  [        "                  "
                                                           C                  C      1

                                                               R               R



                    xl                                                             xl I

                  x0.1                                                                                         y

                                   (b)        Input signal level                                  (c)

                                                       -                           xl I

                                     ,             y                                                           r

                                    (d)                                                           (e)
Figure 8.20 Oscillator feedback: degree of coupling.
  (a) Class D or current switching oscillator, also known as the Vakar oscillator. With R zero, the active devices act
as switches, passing push-pull square waves of current. Capacitors C may be replaced by a feedback winding. R
may be zero, or raised until circuit only just oscillates. 'Tail' resistor approximates a constant current sink.
  (b-e) Characteristics (see text).

current switching and Vakar oscillators and is very                        simple radio receiver designed to achieve most of
suitable for a high-stability oscillator. That of                          its sensitivity by means of reaction, also known as
Figure 8.20e results in an amplitude of oscillation                        regeneration, such as that shown in Figure 8.21. 7
which is very prone to amplitude variations due to                         With this circuit, as the reaction is turned up, the
outside influences. It is therefore excellent for a                        effective circuit Q rises towards infinity, providing
                                                                                             Radio-frequency circuits   211

    Ae ( ~


       ," x

                                                                    Tr3       Trl,
                                          Slc                       BCIO9B    BC21/,
                                                                         RS    ~

              /     o_~l"lOP I
                               I                                                                                        P3
                          3               Sld
              0,,     -        I
              SIo         ~.
                                                                                                LT 700

                                                               .a_ c7

Figure 8.21  A straight receiver with reaction (regeneration).
  (Reproduced by courtesy of Practical Wireless.)

a surprising degree of sensitivity. The greatest                 only class A operation provides distortion-free
sensitivity occurs when the RF amplifier is actually             amplification. The output tuned circuit selects the
oscillating very weakly; it is thus able to receive              desired harmonic. In principle, the.bias and drive
both CW and SSB signals. With AM signals its                     level can be adjusted to optimize the proportion of
frequency becomes locked to that of the incoming                 the desired harmonic in the collector current;
signal and its amplitude varies in sympathy;                     however, whilst this is worth doing in a one-off
anyone who has never played with a 'straight' set                circuit, it is difficult to achieve in production.
(i.e. not a superhet) with reaction has missed an                   It is important at any frequency, and particu-
experience.                                                      larly in RF circuits, to ensure that the signals to be
   It is not always convenient to generate an RF                 amplified, multiplied, converted to another fre-
signal using an oscillator running at that fre-                  quency or whatever, only proceed by the intended
quency: an example is when a crystal-controlled                  paths and do not sneak into places where they are
VHF or U H F frequency is required, as crystals are              not wanted, there to cause spurious responses,
only readily available for frequencies up to around              oscillations or worse. The main means of achieving
70 MHz. A common procedure in these cases is to                  this are decoupling, to prevent RF signals travel-
generate the signal at a frequency of a few tens of              ling along the DC supply rails, and screening, to
megahertz and then multiply it in a series of                    avoid unintended capacitive or inductive coupling
doubler and/or tripler stages. A multiplier stage                between circuits. At radio frequencies, screens of
is simply a class C amplifier with frequency f MHz               non-magnetic metal are equally effective at sup-
applied to its input and with a tuned circuit                    pressing unwanted magnetic coupling as well as
resonant at N f M H z as its collector load. The                 electrostatic coupling. Supply rail decoupling is
collector current contains harmonics of the input                achieved by bypassing RF currents to ground
frequency, since for a single-ended amplifier stage              with decoupling capacitors whose reactance is
212   AnalogElectronics

very low at the frequency involved, whilst placing         (ii) A VHF broadcast FM transmitter radiates
a high series impedance in the supply rail, in the         a signal modulated with a 1 kHz sine wave,
form of an inductance so as not to incur any               with 63 kHz deviation. What is the peak phase
voltage drop at DC. For a more detailed coverage           deviation?
of radio-frequency technology, see Ref. 8.           2.    In question l(ii), approximately how many
                                                           significant sidebands are there?
                                                     3.    In a small-signal common emitter RF ampli-
                                                           fier, what is the mechanism contributing to
1. Notes on the Theory of Modulation. J. R.                potential instability? How does the cascode
   Carson. Proc. LR.E. Vol. 10, p. 57. February            circuit circumvent the problem?
   1922.                                             4.    Describe two different methods used to ensure
2. Solid State Design for the Radio Amateur. Hay-          the stability of a single transistor common
   ward and DeMaw. 2nd printing 1986. p. 189,              emitter small-signal RF amplifier stage.
   American Radio Relay League Inc.                   5.   A small-signal class A RF amplifier produces
3. High Dynamic Range Transistor Amplifiers                an output o f - 1 0 d B m     for an input of
   Using Lossless Feedback, D. E. Norton, p. 53.            -20dBm. When two such signals are applied
   Microwave Journal. May 1976.                            simultaneously, the third-order intermodula-
4. Need a PIN-Diode Attenuator? R. S. Viles,               tion products at the output are at -50 dBm.
   Electronic Design 7, p. 100. March 29, 1977.            What is the amplifier stage's third-order inter-
5. Symmetric MOSFET Mixers of High Dynamic                 cept point?
   Range, R. P. Rafuse, p. 122. Session XI, 1968      6.   Draw the circuit diagram of a continuously
   International Solid State Conference.                   variable bridged Tee attenuator. Why, in
6. Single Balanced Active Mixer Using MOS-                 practice, is the attenuation range limited at
   FETs. E. S. Oxner, p. 292, Power FETs and               both the minimum and maximum extremes?
   Their Applications, 1982, Prentice-Hall.           7.   What measures are incorporated in some
7. The PW Imp 3-Waveband Receiver. I. Hick-                 Schottky Quad diode double balanced mixers,
   man, p. 41. Practical Wireless, May 1979.               to increase linearity and dynamic range?
   NOTE: 'Plessey' devices are now manufactured       8.   Draw a block diagram of an HF SSB trans-
   by GEC Plessey Semiconductors Ltd.                      mitter, showing clearly the stages in the pro-
8. Practical RF Handbook, 2nd edition, 1997, Ian            duction of the single sideband signal.
   Hickman, Butterworth-Heinemann.                    9.    Describe the operation of the ratio detector
                                                            for broadcast FM signals. How is the
                                                           necessary quadrature         reference voltage
 1. (i) Describe and contrast the distinguishing     10.    Compare and contrast the Hartley and
    features of amplitude modulation and fre-               Colpitts oscillators. What feature of the
    quency modulation, with particular reference            Clapp variant of the Colpitts oscillator
    to the phase of the sidebands relative to the           contributes to its superior frequency stability?

    9                            Signal sources

Signal sources play an important role in electronic       the diode. Even so, the improvement is inadequate
test and measurements, but their use is far from          for any purposes other than the cheapest and
limited to that. They form an essential part of           simplest stabilized power supply. Figure 9.1b
many common types of equipment. For example, a            shows how the performance of the regulator can
stabilized power supply needs an accurate DC              be notably improved by using the high drain slope
voltage source as a reference against which to            resistance of a junction FET in place of the
compare its output voltage. Many pieces of elec-          resistor. Unfortunately an FET is a lot dearer
tronic equipment incorporate an audio-frequency           than a resistor. Two-lead FETs with the gate and
signal source as an essential part of their oper-         source internally connected as shown are available
ation, from the mellifluous warble of a modern            as 'constant current diodes' and work very well;
push-button telephone to the ear-shattering squeal        unfortunately they are even more expensive than
of a domestic smoke detector. And RF sources -            FETs, which themselves have always commanded
oscillators- form an essential part of every radio        a price ratio relative to small-signal bipolar tran-
transmitter and of virtually every receiver. So let's     sistors of about five to one. If an FET is used, the
start with the DC signal source or voltage refer-         problem of the usual 5:1 spread in Idss can be
ence circuit.                                             alleviated by including a source bias resistor, as in
                                                          Figure 9.1c, or even by adjusting it for a given
                                                          drain current as in Figure 9.1d.
Voltage references
                                                             Zener diodes have been much improved over the
The traditional voltage reference was the Weston          years. Earlier types left one with the difficult choice
standard cell, and these are still used in calibration    of going for lowest slope resistance - which was
laboratories. However, in most electronic instru-         found in devices with a rating of about 8.2 V - or
ments nowadays, from power supplies to digital            for lowest temperature coefficient (TC or
voltmeters (DVMs), an electronic reference is used        'tempco'), then found in 5.1 V devices. With
instead.                                                  modern devices such as the Philips BZX79.series,
   A Zener diode exhibits a voltage drop, when            lowest TC and lowest slope resistance occur for the
conducting in the reverse direction, which is to a        same voltage rating device, i.e. + 0 . 4 % / ~ and
first approximation independent of the current            10 ~ at 5 mA respectively in the BZX79 C6V2 with
flowing through it, i.e. it has a low slope resistance.   its 6.2 V •     voltage rating. A point to watch out
Thus if a Zener diode is supplied with current via a      for is that the measurement of a Zerier diode's
resistor from say the raw supply of a power supply        slope resistance is usually an adiabatic measure-
(Figure 9.1a), the voltage variations across the          ment. This means that a small alternating current
Z e n e r - b o t h AC due to supply frequency ripple     is superimposed upon the steady DC and the
and DC due to fluctuations of the mains voltage -         resulting alternating potential is measured. The
will be substantially less than on the raw supply,        frequency of the AC is such that the diode's
provided that the value of the resistor is much           temperature does not have time to change in
greater than the diode's slope resistance. In prac-       sympathy with each cycle of the current. If now
tice, this means that about as many volts must be         there is a change in the value of the steady DC
'thrown away' across the resistor as appear across        component of current through the diode, there will
214   Analog Electronics


                                       ~ N channel

                (a)                    (b)                     (c)                     (d)
Figure 9.1   Zener DC voltage references, simple and improved (reproduced by courtesy of New Electronics).

be an accompanying instantaneous small change             Semiconductor, which is used in series with a
in voltage 8V due to 8I, the change in current            resistor or constant current circuit, just like a
flowing through the slope resistance Rs, followed         Zener diode. This 1.2 V reference device is avail-
by a slower change of voltage due to the TC as the        able in 1% or 2% selection tolerance, operates
operating temperature of the diode changes. This          over a current range of 10pA to 20mA, and
clearly highlights the benefit of a range of diodes       features a dynamic impedance of 1 f~; the suffix
where the minimum slope resistance and TC can             X version features a TC at 100 ~A of less than 30
be had in one and the same device.                        PPM/~ A 2.5V device, the LM385-2.5, is also
   Returning to Figure 9.1a, this arrangement can         available. Other commonly available reference
provide a stabilization ratio Vraw/Vreg of about          voltage ICs come in various output voltages,
100:1 or 1%, whereas the FET aided version                including 5.0V, 10.0 V and 10.24 V.
improves on this by a factor of about 30, depend-
ing on the FET's slope resistance. However, a
useful if not quite so great improvement can be                              ~aw
provided by the arrangement of Figure 9.2.1 Here
the diode current is stabilized at a value of
approximately 0.6/R2, since the PNP transistor's
 Fbe changes little with change of emitter current.
Consequently, if Fraw increases, most of the re-                                        R2
sultant increase in current through R1 is shunted
via the collector to ground rather than through the
Zener diode. Where a modest performance, about
 10 times better than Figure 9.1a, is adequate, the
circuit of Figure 9.2 offers a very cheap solution.
Where substantially better performance is re-
quired, a voltage reference IC is nowadays the
obvious choice. These are available from most
manufacturers of linear ICs and operate upon              Figure 9.2 Inexpensive improved Zener voltage
the bandgap principle. A typical example is the           references      (reproduced   by   courtesy   of   New
micropower two-lead LM385-1.2 from National               Electronics).
                                                                                        Signal sources   215

                                                        It results in a greater angle between the two
Non-sinusoidaI waveform generators
                                                        changing voltage levels at the point at which
Sources of AC signals can be divided into two           regeneration occurs, and this makes that instant
main categories: sine wave generators, and gen-         less susceptible to influence by external or internal
erators of non-sinusoidal waveforms. The latter         circuit noise. Thus the frequency of oscillation is
can be subdivided again into pulse generators and       more stable, a worthwhile improvement since the
other types. Pulse generators provide pulses of         frequency purity of astable oscillators generally is
positive- or negative-going polarity with respect       very much poorer than that of sinusoidal oscilla-
to earth or to a presettable DC offset voltage. The     tors using an LC resonant circuit. In the latter the
pulse repetition frequency, pulse width, amplitude      stored energy is much greater than any circuit
and polarity are all adjustable; on some pulse          noise, which consequently has less effect. However,
generators, so too are the rise and fall times.         a circuit such as Figure 9.5a contains two time
Commonly also the output may be set to provide          constants, both of which play a part in determin-
'double pulses', that is pulse pairs with variable      ing the frequency. The circuit of Figure 9.3b will
separation, and a pulse delay with respect to a         provide a 10:1 variation of frequency for a 10:1
prepulse, which is available at a separate output       variation of the resistance R forming part of the
for test and synchronization purposes. Pulse gen-       frequency determining time constant CR. The
erators of this type are used mainly for test           same applies to the circuit of Figure 9.5a only if
purposes in digital systems, so they are not con-       more than one resistor is varied in sympathy. Thus
sidered further here. So let's press straight on and    the circuit of Figure 9.5a is more attractive in fixed
look at those 'other types'.                            frequency applications or where a tuning range of
   Non-sinusoidal or astable waveform generators        less than an octave is required. For wide frequency
may be categorized as operating in one of two           applications, as in a function generator providing
modes, both of which are varieties of relaxation        sine, triangular and square output waveforms, it is
oscillator. As the name implies, the oscillation        not uncommon to opt for the economy of single
frequency is determined by the time taken by the        resistor control.
circuit to relax or recover from a positive extreme        Figure 9.5b shows a popular and simple astable
of voltage excursion, towards a switching level at      oscillator circuit. There is only a single path
which a transient occurs. The transient carries the     around the circuit, for both the positive and the
output voltage to a negative extreme and the            negative feedback. At any time (except during the
circuit then proceeds to relax towards the switch-      switching transients) only one of the two tran-
ing level again, but from the opposite polarity. On     sistors conducts, both tail currents being supplied
reaching it, the circuit switches rapidly again,        via the 1K resistor or from the + 15 V rail.
finishing up back at the positive extreme.                 The circuit of Figure 9.6 works on a slightly
   The two modes are those in which differentiated      more sophisticated principle than the circuits of
(phase advanced) positive feedback is combined          Figures 9.3 and 9.4, where the feedback voltage
with broad band negative feedback on the one            relaxes exponentially. 2 It uses the Howland current
hand, and types in which broad band positive            pump, a circuit discussed earlier (Figure 7.6), to
feedback is combined with integrated (phase re-         charge a capacitor, providing a linearly rising
tarded) negative feedback on the other. Figures 9.3     ramp. When this reaches the trigger level of half
and 9.4 show both discrete component and IC             the supply rail voltage (at the non-inverting input
versions of these two types respectively. The circuit   of the comparator), the trigger level, and the
operation should be clear from the circuit dia-         voltage drive to the current pump, both reverse
grams and waveforms given.                              their polarity, setting the voltage on the capacitor
   There is no reason why such an oscillator should     charging linearly in the opposite direction. The
not use differentiated positive feedback and inte-      frequency is directly proportional to the output of
grated negative feedback, as in Figure 9.5a;            the current pump and hence to the setting of the
indeed, there is a definite advantage in so doing.      10 K potentiometer, which can be a multiturn type
216   Analog Electronics

                                        "'5V                                               1/6 CD4069
                                                                       1/6 CD4069                        .+ 15 V

                                                                                   R                    nF
                                         20K                     10 M U         100K

           681                           i0K


        +14.8 V--


         +10.5 V

          1!]o                                               +7.5 V-
                                                                            -              "7~. .....
            -1          B
                    I       s S                                  0V--                  I/
                    ~,s~ IS                                                            I


                                                               +15 V - -

Figure 9.3 Astable (free-running) circuits using differentiated positive feedback and fiat (broad band) negative
  (a) Cross-coupled astable circuit. The dashed line shows the 0 V level at which the discharge at point C is aiming
when it reaches the switching level.
  (b) Astable circuit using CMOS inverters. The waveform at B is similar to that at C except that the excursions
outside the 0 V and + 15 V supply rails have been clipped off by the device's internal gate protection diodes.

with a ten-turn digital dial. With the values shown          low-impedance triangular and square wave out-
the circuit provides five frequency ranges from 0 to         puts.
1 Hz up to 0 to 10 kHz, with direct read-out of                Most function generators provide a sine wave
frequency. Each range determining capacitor has              output of sorts. The popular 8038 function gen-
an associated 4K7 preset resistor associated with            erator IC includes an on-chip shaping stage to pro-
it, enabling the full-scale frequency to be set up for       duce a sine wave output by shaping the triangle
each range, even though ordinary 10% tolerance               waveform. This operates purely on a waveform
capacitors are used. The circuit provides buffered           shaping basis and thus works equally well at any
                                                                                              Signal sources   217

                                                                    +15 V





                                                            +15V       '

        +10.4--J                                                                          B

           +1.0                                  ~~--'-
           +0.5       B                ...//~"

                  J       -""     /"-, /
                                -'T,,    !--
                                  l "'.-..1 B

                                                            7.5 V      ~    -     -   -

Figure 9.4 Astable (free-running) circuits using broad band positive and integrated (delayed) negative feedback.
  (a) Cross-coupled astable circuit.
  (b) Astable circuit using CMOS inverters.

frequency. An alternative scheme is to use an               approximation to a sine wave is to use an amplifier
integrator: a triangular (linear) waveform is inte-         which runs gently into saturation on each peak of
grated to a parabolic (square law) waveform which           the triangular waveform. Unlike the integrator
forms a passable imitation of a sine wave,the total         method, where the sharp point at the peak of the
harmonic distortion being about 3.5%. However,              triangle wave becomes a slope discontinuity at the
the disadvantage of the integrator approach is that         zero crossing point of the pseudo-sine wave, it is
the output amplitude varies inversely with fre-             difficult with the aperiodic shaping method to
quency, unless the value of the integrator's input          avoid some residual trace of the point at the
resistor is varied to compensate for this.                  peak of the sine-shaped waveform. A scheme
   An     aperiodic (non-frequency dependent)               which has been used to avoid this is to slice off
method of shaping a triangular wave into an                 the peaks of the triangular wave before feeding it
218   Analog Electronics

                                                                                    +10V~              -~+15    V
                   Op           ~a 1 nF
                                                                                        1K~       lwj
                           ,.. A

         100K                               1K         1K

                                                                                         B               C
                ~ 7 lnF
                                                                                                         24 K

                                                            +15V                      I mAtappr~         1 mA15 V

                                                            -15 V
             +7.5V ~                             .. +15 I

                                          " - ~
                                   + 3 . 5~""".
                       I   ./         I ",              /      ."

                                                                          +1.4 V             ~
              -7.5 V                                                      --0.6 V
                                      (a)                                 +9.4 V - - ~ ~ _
                                                                          +7.4 V

Figure 9.5 Other types of astable circuit.
  (a) Astable circuit using both differentiated positive and integrated negative feedback. Aiming potentials of
points B and C prior to switching shown dashed.
  (b) The Bowes, White or emitter coupled astable does not have separate positive and negative feedback paths,
so differing from the oscillators of Figures 9.3, 9.4 and 9.5a.

to the shaping circuit. 3 In the reference cited, by                   Some function generators are capable of pro-
choosing the optimum degree of preclipping and                      ducing other waveforms besides the usual square/
of non-linearity of the shaping amplifier gain,                     triangle/sine waves. A popular waveform is the
distortion as low as 0.2% is achieved at low                        sawtooth and its close cousin the asymmetrical
frequencies (a times ten improvement on the                         triangle (see Figure 9.7). This figure also indicates
results usually achieved by this method). The                       how a stepwise approximation to any arbitrary
shaping amplifier is implemented in an IC using                     waveform can be produced by storing the data
a 1 GHz device process, resulting in good conver-                   values corresponding to say 256 successive sam-
sion of triangular waveforms to sine waves at                       ples of the waveform over one whole cycle in a
frequencies up to 100 MHz.                                          read only memory (ROM), and then reading
                                                                                                      Signal sources   219

                                                                          buffered low-impedance
                                                                                 *V/2 triangular
                                                                             L V~V       output
                                      lOOk       lOOk            ~/T           vF[21.F1
                                                                                   auxiliary square
                                                                                    wave outpul

                                     T0o;oT T ;::QQO)0
                                     T.T.,T.T,T O0
                                         TTc_            c
                                     4.71JF- 47-nF - 470pF/r~/

Figure 9.6 Function generator using a Howland current pump. The five 4K7 preset potentiometers enable the
maximum frequency of the ranges to be set to 1 Hz to 10 kHz exactly; range capacitors C can thus be inexpensive
10% or even 20% tolerance types. If 10K resistor R is a ten-turn digital dial potentiometer, it will indicate the output
frequency directly. + V and - V supplies must be equal, but frequency is independent of the value of V.
(Reproduced by courtesy of New Electronics.)

them out sequentially to a digital-to-analog con-                 of the next waveform sample. The output currents
verter (DAC). In this way it is possible to                       of the two DACs are summed to give a smoothly
reproduce natural sounds which have been re-                      changing voltage output from the opamp. The
corded and digitized, for example the sound of a                  generation of a sine wave by this means is illus-
diapason or reed pipe from a real pipe organ, as                  trated in Figure 9.9, but any arbitrary waveform
is done in some electronic organs. The step nature                can be produced once the appropriate values are
of the output will correspond to very high-                       stored in ROM. In practice, both the new D A C
frequency harmonics of the fundamental, which                     values are simply applied at the same instant that
in the organ application may well be beyond the                   the sawtooth waveforms fly back to their starting
range of hearing, but where necessary the steps                   values: any 'glitch' in the output voltage, if appre-
can be smoothed off with a low-pass filter. This                  ciable, can be smoothed out with a little integrat-
can still have a high enough cut-off frequency to                 ing capacitor across the summing opamp's
pass all the harmonics of interest in the output                  feedback resistor, which in Figure 9.8 is internal
waveform.                                                         to the DAC.
   Another way of achieving a smooth, step-free                      An interesting application of this is for writing
output waveform is to make use of the multiplying                 data on the screen of a real-time oscilloscope. Such
capability of a DAC. The output current from a                    an oscilloscope uses the electron beam to write the
D A C is equal to the input bit code times the                    traces under control of the X and Y deflection
reference voltage input. Figure 9.8 shows two                     plates, but it does not produce a raster scan like a
multiplying DACs with reverse sawtooth wave-                      TV display, so some other means is needed if
forms applied to their reference inputs so that as                information such as control settings is to be
the output of the P D A C decreases, that of the Q                displayed on the screen. Two completely separate
D A C increases. Sample values are fed to the                     but complementary voltage waveform generators
DACs at the same rate as the sawtooth frequency.                  such as Figure 9.9 can be used to produce the
When the output of the P D A C reaches zero, its                  appropriate X and Y deflection voltages to write
input code is changed to that currently present at                alphanumeric data on the screen, the appropriate
the input of the Q DAC. Immediately after this the                D A C data being stored in ROM. This scheme is
sawtooth waveforms fly back to their initial values,              used on many makes of oscilloscope. When the
so that the output from the Q DAC is now zero,                    display read-out is on, it is possible under certain
and its input bit code is promptly changed to that                conditions to observe short breaks in the trace
220   AnalogElectronics






Figure 9.7 Generalized triangle waveform and universal waveform generator.
  (a) $awtooth and asymmetrical triangle waveforms; both are generally provided by the more versatile type of
function generator. The sawtooth and the triangular wave (Figure 9.6) can both be considered as limiting cases of
the asymmetrical triangular wave.
  (b) Simple ROM waveform generator (reproduced by courtesy of Electronic Engineering).

where the beam goes away temporarily to write the          tuned circuit oscillators and relaxation oscillators,
read-out data.                                             and in some cases not much better than the latter.
                                                           To measure the distortion of a high-fidelity audio
                                                           power amplifier, one needs, in addition to a
Sine wave generators
                                                           distortion meter, a sine wave source of excep-
Turning now to sine wave generators, let's look            tional purity. Not only must the source's distor-
first at audio-frequency generators. These gener-          tion be exceedingly low, but its frequency stability
ally do not use LC tuned circuits to determine the         must be of a very high order. This is because the
frequency, and therefore have a degree of fre-             usual sort of distortion meter works by rejecting
quency stability intermediate between that of              the fundamental component of the amplifier's
                                                                                                                     Signal sources             221

                        P DAC ref. input                        l.
                        0~,!~     ~                    Vref Rib Outl~                -      1~
                                                                      Outl)Irrbz"                                 oVo

                                                        Digital input P
                        Q DAC ref. input

                        0                                             2,/_11~
                                                       Vref Q DAC OUtout

                                           Digital input Q
Figure 9.8 Interpolating DACs (reproduced by courtesy of Electronic Engineering).

                                Y9            Sinusoid output                                                               )utput waveform


                                              ~   12                                                                                           =, t

                                      bAC Q Output waveform

                                                                          0 ,Y| I Y2 I Y3 I Y4 I Y5 I Y6 I Y7 I Y8 I Y9 I YI0~YI1 DACPContents

                                        --"                     ~ t   IY]I Y2 , Y3.. Y4 , Y5 , Y6 , Y7 . Y8 , Y9 , Y ] 0 , Y ] ] , 0 [DACQ Contents

Figure 9.9 Waveform synthesis (reproduced by courtesy of Electronic Engineering).

output with a narrow notch filter, so that the                        it in the notch long enough to take a measure-
harmonics, residual noise and hum can be meas-                        ment. However, even if its drift is negligible, it
ured. Their level relative to the total output                        may exhibit very short-term frequency fluctua-
signal, expressed as a percentage, is the total                       tions. Thus it will 'shuffle about' in the notch,
harmonic distortion (THD) or, more strictly, the                      resulting in a higher residual output than if its
total residual signal if noise and hum are sig-                       frequency were perfectly steady, as it tends to
nificant. Clearly, if the frequency of the sine wave                  peep out first one side of the notch and then the
generator drifts it will be difficult to set and keep                 other.
222   Analog Electronics

                                                        .   .    .   .   .   .   .       .                  _ lo)-219"

                                            Vector representing pure sine wave of frequencyfHz

                              o ~,..l


                                                                                     f                                 log frequency
                         Pure sine wave of frequencyfHz represented in the frequency domain

                                                                                                        F          F

                        Sine wave with AM and FM noise sidebands (A, F), grossly exaggerated
                                                                Peak level [ 4"Width actually less than a
                                                                               millionth of the centre frequency

                                                                                             ~~.    tBroadband noise floor, more

                                        <                        J                                   han 120 dB below peak level

                                                                                                                         log frequency
                                               Corresponding frequency domain representation
Figure 9.10 Sine waves.
  (a) Ideal pure sine wave.
  (b) Real-life sine wave.

   Now this is simply an explanation in the time                                                   at frequencies very close to that of the sine wave,
domain of something which can equally well be                                                      falling rapidly in amplitude as the frequency
explained in the frequency domain. Figure 9.10a                                                    difference increases. The FM noise sidebands are
shows an ideal sinusoidal signal, whilst Figure                                                    the manifestation in the frequency domain of
9.10b shows, much exaggerated for clarity, a prac-                                                 slight phase variations which were noted as fre-
tical sine wave, warts and all. In addition to the                                                 quency shuffle in the time domain and which are
ideal sine wave there are close-in noise side-bands                                                shown as FM sidebands in Figure 9.10b. There are
of two sorts, AM and FM. These represent energy                                                    also AM sidebands corresponding to slight ampli-
                                                                                       Signal sources    223

tude variations in the sine wave, and these also will    phous semiconductor whose resistance falls
contribute to the residual. The residual may be          rapidly with increasing temperature; the negative
considered as being responsible for it being im-         feedback via the thermistor/resistor arm therefore
possible to say exactly where the tip of the vector      increases, and the bridge approaches balance. At
in Figure 9.10b is at any time; it will be somewhere     an output voltage of about 3 V peak to peak, the
in the much exaggerated 'circle of uncertainty'          dissipation in the thermistor, with the circuit
shown. (Note that noise sidebands, both AM               values shown, is approaching the rated maximum,
and FM, are also found either side of the output         corresponding to a temperature of the pellet inside
frequency of an LC oscillator and even of a crystal      its evacuated glass envelope of 125~ and the
oscillator; it is just that in those cases they are      output amplitude is stabilized. Oscillators operat-
restricted by the high Q of the frequency determin-      ing on this principle are commercially available
ing components to a very much narrower frac-             from many manufacturers, such are their popular-
tional bandwidth about the centre frequency.) In a       ity. The oscillator can even be made to cover the
well-designed audio oscillator, the energy in the        frequency range 10 Hz to 10 MHz, although it is
noise sidebands which is not rejected by the notch       not then possible to optimize the circuit for the
of the distortion meter is always lower in level than    lowest possible distortion in the audio-frequency
the energy of the harmonics.                             range.
   Figure 9.11 shows an audio oscillator using the          The main problems with a thermistor stabilized
popular Wien bridge configuration. In Figure             Wien bridge oscillator are amplitude bounce and
9.11a you can see the principle of the thing. By         poorish distortion. The former is due to the
using the idea of extremums - replacing a capa-          thermistor: it is found that on changing frequency,
citor by an open-circuit at 0 Hz and by a short-         the amplitude of the output oscillates up and down
circuit at infinite frequency- there will clearly be     several times before settling to a steady value.
no signal at B at these frequencies. It turns out (the   Running the thermistor near its maximum per-
sums are not difficult, have a go) that at the           mitted dissipation helps to minimize this. The
frequency f = 1/27cRC the amplitude at B is one-         other problem is due to the limited selectivity of
third of that at A and the two waveforms are in          the Wien bridge, which does little to reduce any
phase. At other frequencies the attenuation is           distortion in the maintaining amplifier, and (at
greater and the waveforms are out of phase. If           frequencies below 100Hz) to the finite thermal
the bridge is just out of balance sufficiently to        time constant of the thermistor.
provide the necessary input to the maintaining              The Wien bridge oscillator shown in Figure
amplifier, then the latter will drive the bridge at      9.11b uses a two-gang variable resistor to vary
an amplitude adequate to produce the said input.         both resistors of the frequency determining net-
If this sounds like a specious circular argument, it     work simultaneously. This keeps constant the
is: in the practical circuit of Figure 9.11b the         attenuation through the network at the zero
necessary degree of bridge imbalance is provided         phase shift frequency. It can also provide a 10 to
by a thermistor. The usual type is an R53, which         1 frequency tuning range for a 10 to 1 resistance
has a cold resistance of 5K (or 5 x 10 3 f~, hence       variation, as can be deduced from the formula for
the type number). At switch-on, the bridge is            the frequency quoted above. There are numerous
unbalanced by much more than is necessary, so            sine wave oscillator circuits which provide fre-
that the positive feedback via the CR network            quency variation using only a single variable
exceeds the negative feedback via the thermistor/        resistor, but in these the frequency ratio obtained
resistor combination. Therefore the circuit com-         is only equal to the square root of the resistance
mences to oscillate at the frequency at which the        variation. 4,5
phase shift and attenuation of the CR network is            An improved audio oscillator can be based on
least. As the amplitude of the oscillation builds up,    the stat