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					FEEDBACK AMPLIFIERS AND OSCILLATORS


                     for
             Electronic Circuits



                     by
              Prof. Michael Tse



              September 2004
Contents
Feedback
         Basic feedback configuration
         Advantages
         The price to pay
Feedback Amplifier Configurations
         Series-shunt, shunt-series, series-series, shunt-shunt
         Input and output impedances
         Practical Circuits with loading effects
         Compensation
         Op-amp internal compensation
Oscillation
         Oscillation criteria
         Sustained oscillation
         Wein bridge, phase shift, Colpitts, Hartley, etc.



                       C.K. Tse: Feedback amplifiers and           2
                                  oscillators
Basic feedback configuration
The basic feedback amplifier consists of a basic amplifier and a feedback network.


                  +           e
         si                               A                       so
       input       –                                           output
                                        basic amplifier

   Careful!!             sf
                                           f

                                  feedback network

  A = basic amplifier gain
  f = feedback gain

                          C.K. Tse: Feedback amplifiers and                    3
                                     oscillators
Characteristics
                                                                 The input is subtracted by a
           +                                                     feedback signal which is part of the
 si                     e       A                       so       output, before it is amplified by the
input          –                                     output      basic amplifier.
                               basic amplifier
                   sf                                                  so = Ae = A(si - s f )
                                    f
                                                                 But, since sf = f so, we get
                            feedback network
                                                                       so = A(si - fso )

                                                                 Hence, the overall gain is
                                                                             so   A
      si                         Ao                   so              Ao =      =
                                                                             si 1+ Af

                                                                                        1
                                                                 If Af >> 1,     Ao ª
                                                                                        f
                                        C.K. Tse: Feedback amplifiers and                          4
                                                   oscillators
Simple viewpoint

                           +            e
                    si                          A                   so
                  input        –                                  output
                                               basic amplifier
                                   sf
                                                    f

                                            feedback network


If A is large, then e must be very small in order to give a finite output.
So, the input si must be very close to the feedback signal sf .
That means sf ≈ si .

But, sf is simply a scaled-down copy of the output so.
                          so 1
Hence, f so = si or          ª
                          si   f

                               C.K. Tse: Feedback amplifiers and             5
                                          oscillators
Obvious advantage
If the feedback network is constructed from passive elements having
stable characteristics, the overall gain becomes very steady and
unaffected by variation of the basic amplifier gain.

Quantitatively, we wish to know how much the overall gain Ao
changes if there is a small change in A.

Let assume A becomes A + dA. From the formula of Ao, we have

                         dAo Ê dA ˆÊ 1 ˆ
                            = Á ˜Á       ˜
                          Ao Ë A ¯Ë1+ Af ¯

Obviously, if Af is large, then dAo/Ao will be reduced drastically.
                                                                  1
Feedback reduces gain sensitivity! In fact, the gain is just Ao ª   .
                                                                  f

                         C.K. Tse: Feedback amplifiers and               6
                                    oscillators
                                                                 so
Another advantage
Suppose the basic amplifier is
distortive. So, the output does
                                              output
                                                                       A
not give a sine wave for a sine
                                                                                   e
wave input.



But, with feedback, we see that
the gain is about 1/f anyway,                                         input
regardless of what A is (or as
long as Af is large enough).
                                                                 so
This gives a very good property
                                                           1/f
of feedback amplifier in terms
                                                                                  si
of eliminating distortion.


                        C.K. Tse: Feedback amplifiers and                      7
                                   oscillators
     Other advantages
     •    Improve input and output resistances (to be discussed later).
     •    Widening of bandwidth of amplifier (to be discussed later).
     •    Enhance noise rejection capability.

         ni
                                                                      ni
si              A             so           +                          +
                                    si              e
                                                        A’                     A            so
                                   input                              +
                                               –                                           output
                                                          basic amplifier
     Signal-to-noise ratio:                        sf
         È so ˘ È si ˘                                                     f
         Í ˙=Í ˙                                               feedback network
         Î no ˚ Î ni ˚                                                             È so ˘  È si ˘
                                         Signal-to-noise ratio improves!           Í ˙ = A¢Í ˙
                                                                                   Î no ˚  Î ni ˚

                                   C.K. Tse: Feedback amplifiers and                              8
                                              oscillators
The price to pay


Of course, nothing is free!

Feedback comes with reduced gain, and hence you may need to add a pre-
amplifier to boost the gain.

Also, wherever you have a loop, there is hazard of oscillation, if you don’t
want it.

Later, we will also see how we can use feedback to create oscillation
deliberately.




                         C.K. Tse: Feedback amplifiers and                 9
                                    oscillators
Terminologies

Basic amplifier gain = A

Feedback gain = f
                                       A     1
Overall gain (closed-loop gain) =          ª
                                     1+ Af   f

Loop gain (roundtrip gain) = Af                Some books use T to denote Af.




                       C.K. Tse: Feedback amplifiers and                         10
                                  oscillators
Feedback amplifiers


What is an amplifier?              si         A             so



Signals can be voltage or current.

General model for voltage amplifier:
                                                  Ro
                       +                                        +
                                             +
                       vin             Rin   –   Avin           vo
                       –                                        –

                                 voltage amplifier



                        C.K. Tse: Feedback amplifiers and             11
                                   oscillators
Models of amplifiers
                          Ro                              iin                         io
+                                        +                              Aiin
                     +
vin            Rin   –   Avin            vo                           Rin        Ro
–                                        –

            voltage amplifier                                      current amplifier



      iin                 Ro                                                          io
                                         +          +                   Aiin
                     +                              vin               Rin        Ro
               Rin   –   Aiin            vo
                                         –          –


      transresistance amplifier                                  transconductance amplifier




                               C.K. Tse: Feedback amplifiers and                             12
                                          oscillators
Feedback amplifier configurations
Voltage amplifier
                         +            e
                 si                           A                       so
                input        –                                      output
                                             basic amplifier
                                 sf
                                                  f
    voltage                                                                  voltage
                                          feedback network



To subtract voltage from
                                                      To copy voltage, we should use
voltage, we should use series
                                                      parallel (shunt) connection
connection
           +                                               •         +
           vi
           –
                                            A                        vo
                                                                     –
                                                               •
                        – vf +                                      Hence, series-shunt feedback
                                 C.K. Tse: Feedback amplifiers and                         13
                                            oscillators
Series-shunt feedback (for voltage amplifier)

                                                       Ro
                                                                   +
            +          +                        +                  vo
            vi         ve            Ri         –   Ave
                       –                                           –
            –




                                 +
                                 –        fvo




                                                       vo    A
  Overall gain (closed-loop gain) :             Ao =      =
                                                       v i 1+ Af



                           C.K. Tse: Feedback amplifiers and             14
                                      oscillators
Series-shunt feedback (for voltage amplifier)
To find the input resistance, we consider the ratio of vi and ii, with output opened.


                                     ii                                Ro
                                                                                  +
                              +            +                      +               vo
                              vi           ve          Ri         –   Ave
                                           –                                      –
                              –
        vi    vi               RIN
RIN =      =
        ii v e /Ri
           v e + fv o                              +
     = Ri                                          –        fvo
               ve
     = Ri (1+ Af )

The input resistance has been enlarged by (1+Af). This is a desirable
feature for voltage amplifier as a large input resistance minimizes loading
effect to the previous stage.
                           C.K. Tse: Feedback amplifiers and                       15
                                      oscillators
Series-shunt feedback (for voltage amplifier)
 To find the output resistance, we consider shorting the input source and calculate
 the ratio of vo and io.
                                        Ro               io          First, we have ve = – fvo.
 +                                                             +     Also,
             +
             ve        Ri
                                  +                            vo            v o - Av e v o + Afv o
 vi                               –   Ave                             io =             =
 –
             –                                                 –                 Ro          Ro
                                                                     Hence,
                                                          ROUT
                                                                                  vo   Ro
                                                                       ROUT =        =
                   +
                            fvo
                                                                                  io 1+ Af
                   –




The output resistance has been reduced by (1+Af). This is a desirable
feature for voltage amplifier as a small output resistance emulates a better
voltage source for the load.
                                  C.K. Tse: Feedback amplifiers and                              16
                                             oscillators
Series-shunt feedback (for voltage amplifier)
Summary of features                                       Equivalent model

                     A     1                                                      Ro
Closed-loop gain =       ª                                                       1+ Af
                   1+ Af   f
                                               +                            +         Av i     +
Input resistance = Ri ( 1 + Af )              vi           Ri ( 1 + Af )    –
                                                                                     1+ Af
                                                                                              vo
                                               –                                               –
                       Ro
Output resistance =
                      1+ Af
                                             NOTE: We did not consider loading effect of the
                                             feedback network, i.e., we assume that the feedback
                                             network is an ideal amplifier which feeds a scaled-down
                                             copy of the output to the input.


                                                                    +
                                                                    –           ∞

                                                                  feedback network

                               C.K. Tse: Feedback amplifiers and                                17
                                          oscillators
Feedback amplifier configurations
Transresistance amplifier
                        +            e
                 si                          A                       so
                input       –                                      output
                                            basic amplifier
                                sf
                                                 f
    current                                                                 voltage
                                         feedback network



To subtract current from
                                                     To copy voltage, we should use
current, we should use shunt
                                                     parallel (shunt) connection
(connection) connection
           ii                                             •         +
                                           A                        vo
                                                                    –
                                                              •

                                                                   Hence, shunt-shunt feedback
                                C.K. Tse: Feedback amplifiers and                        18
                                           oscillators
Shunt-shunt feedback (for transresistance amplifier)

            ii          ie                         Ro
                                                              +
                                            +                 vo
                                 Ri         –   Aie
                                                              –




                                      fvo




                                                   vo   A
  Overall gain (closed-loop gain) :         Ao =      =
                                                   ii 1+ Af



                         C.K. Tse: Feedback amplifiers and          19
                                    oscillators
Shunt-shunt feedback (for transresistance amplifier)
To find the input resistance, we consider the ratio of vi and ii, with output opened.


                                     ii         ie                       Ro
                                                                                       +
                              +
                                                                    +                  vo
                              vi                         Ri         –   Aie
                                                                                       –
                              –
         v i Riie
 RIN =      =                  RIN
         ii   ii
              ie
      = Ri
          ie + fv o                                           fvo

          Ri
      =
        1+ Af
The input resistance has been reduced by (1+Af). This is a desirable
feature for transresistance amplifier as a small input resistance ensures
better current sensing from the previous stage.
                           C.K. Tse: Feedback amplifiers and                       20
                                      oscillators
Shunt-shunt feedback (for transresistance amplifier)
 To find the output resistance, we consider opening the input source (putting ii = 0)
 and calculate the ratio of vo and io.
                                                                        First, we have ie = – fvo.
   ii = 0     ie                        Ro             io
                                                                   +    Also,
                                  +                                             v o - Aie v o + Afv o
                      Ri          –   Aie                          vo    io =            =
                                                                   –                Ro         Ro
                                                                        Hence,
                                                            ROUT
                                                                                     vo   Ro
                                                                          ROUT =        =
                                                                                     io 1+ Af
                           fvo




The output resistance has been reduced by (1+Af). This is a desirable
feature for transresistance amplifier as a large small resistance emulates a
better voltage source for the load.
                                 C.K. Tse: Feedback amplifiers and                                    21
                                            oscillators
Shunt-shunt feedback (for transresistance amplifier)
Summary of features                                      Equivalent model

                     A     1                                            Ro
Closed-loop gain =       ª                                             1+ Af
                   1+ Af   f                    ii
                                                                        Aii    +
                     Ri                                     Ri     +
Input resistance =                                                 –
                                                                       1+ Af
                                                                               vo
                   1+ Af                                  1+ Af                –
                       Ro
Output resistance =
                      1+ Af




Similar, we can develop the feedback configurations for
transconductance amplifier and current amplifier.
         Transconductance amplifier: series-series feedback
         Current amplifier: shunt-series feedback

                              C.K. Tse: Feedback amplifiers and                 22
                                         oscillators
Series-series feedback (for transconductance amplifier)
                                                            io
          +               +                       Ro
          vo              ve         Ri
                          –                      Ave
          –


                                                             io

                                 +
                                 –        f io



                                            io      A
 Overall gain (closed-loop gain) :        Ao =  =
                                            v i 1+ Af
 Input resistance:                    RIN = Ri (1+ Af )           Desirable!

 Output resistance:                  ROUT = Ro (1+ Af )           Desirable!



                         C.K. Tse: Feedback amplifiers and                      23
                                    oscillators
Shunt-series feedback (for current amplifier)
              ii          ie                                io
                                                  Ro
                                  Ri
                                                Aie


                                                            io

                                       f io




 Overall gain (closed-loop gain) :              io   A
                                        Ao =       =
                                                ii 1+ Af
 Input resistance:                                Ri
                                       RIN =
                                                1+ Af       Desirable!
 Output resistance:                              Ro
                                     ROUT     =             Desirable!
                                                1+ Af
                         C.K. Tse: Feedback amplifiers and                24
                                    oscillators
Practical feedback circuits (with loading effects)
In practice, the input source has resistance and the feedback network has
resistance.

Example: shunt-shunt feedback
                                 ie                        Ro
                                                                  +
        ii                                        +               vo
                                          Ri      –   Aie
                                                                  –




                                                fvo



 What are the effects on the gain, input and output resistances?


                        C.K. Tse: Feedback amplifiers and                25
                                   oscillators
Systematic analysis using 2-port networks
The best way to analyze feedback circuits with loading effects is to use two-port
models.
For shunt-shunt feedback, input and output sides are both parallel connected.
Thus, the loading can be combined by summing the conductances. Also, voltage
is common at both sides. So, y-parameter is best.

The first step is to put everything in y-parameter:


                                                                                 +
                                       +
                ii          yi         vi           y11                y22       vo
                                       –                  y21vi                  –


                                             1                               2


                                                                   y22f
                                                   y11f       y21fvo

                           C.K. Tse: Feedback amplifiers and                      26
                                      oscillators
Systematic analysis of shunt-shunt feedback using
y-parameter

                                                           +
                   +
   ii       yi     vi          y11               y22       vo
                   –                 y21vi                 –


                        1                              2


                                             y22f
                              y11f      y12fvo



In order to use the standard results, we have to convert
this model to the standard form (slide 19).


                   C.K. Tse: Feedback amplifiers and             27
                              oscillators
Systematic analysis of shunt-shunt feedback using
y-parameter
                        y11f                                   y22f
                                                                      +
                   +
   ii      yi      vi              y11               y22              vo
                   –                     y21vi                        –


                               1                           2



                                            y12fvo



One step closer…



                   C.K. Tse: Feedback amplifiers and                        28
                              oscillators
Systematic analysis of shunt-shunt feedback using
y-parameter
                  y11f                                   y22f
                                                                +
         +
   ii    vi       yi         y11               y22              vo
         –                         y21vi                        –


                         1                           2



                                      y12fvo



One more step closer…



                 C.K. Tse: Feedback amplifiers and                    29
                            oscillators
Systematic analysis of shunt-shunt feedback using
y-parameter

                           y11+y11f+yi    y22f +y22
                                                          +
          +
   ii     vi                                              vo
          –                       y21vi                   –


                       1                              2



                                     y12fvo



Yet another step closer…



                  C.K. Tse: Feedback amplifiers and             30
                             oscillators
Systematic analysis of shunt-shunt feedback using
y-parameter
                   in conductance (S)             in resistance (Ω)

                           y11+y11f+yi 1/(y22f +y22)
                                                                      +
          +
   ii                                         +                       vo
          vi                                  –
          –                      -y 21v i                             –
                               y 22f + y 22
                       1                               2
                                                                      Use Thevenin

                                        y12fvo



Yet another step closer…



                  C.K. Tse: Feedback amplifiers and                            31
                             oscillators
Systematic analysis of shunt-shunt feedback using
y-parameter
                    in conductance (S)          in resistance (Ω)

                   ie       y11+y11f+yi 1/(y22f +y22)
                                                                         +
          +
   ii                                     +                              vo
          vi                              –
          –                                                              –


                        1                               2                           -y 21ie
                                                                    ( y 22f   + y 22 )( y11 + y11f + y i )


                                       y12fvo



Finally, we get the same standard form.



                   C.K. Tse: Feedback amplifiers and                                          32
                              oscillators
Systematic analysis of shunt-shunt feedback using
y-parameter

We can simply apply the standard results:
                                                        -y 21                 -y 21
Basic amplifier gain            A=                                           =
                                      ( y 22f   + y 22 )( y11 + y11f + y i ) y oT y iT
Feedback gain                  f = y21f

                                        A    1  1
Overall (closed-loop) gain     Ao =         ª =
                                      1+ Af  f y12f

                                                        1
Input resistance              RIN =
                                      (y11 + y11f      + y i )(1+ Af )

                                                      1
Output resistance            ROUT =
                                       (y 22f + y 22 )(1+ Af )
                        C.K. Tse: Feedback amplifiers and                                 33
                                   oscillators
Appropriate 2-port networks for analyzing
feedback circuits

For shunt-shunt feedback, use y-parameter.
For shunt-series feedback, use g-parameter.
For series-series feedback, use z-parameter.
                                                           WHY?
For series-shunt feedback, use h-parameter.


The procedure is essentially the same as in the previous shunt-shunt case.




                        C.K. Tse: Feedback amplifiers and                34
                                   oscillators
General procedure of analysis

1.   Identify the type of feedback.

2.   Use appropriate 2-port representation.

3.   Lump all loading effects in the basic amplifier, giving a modified
     basic amplifier.

4.   Apply Thevenin or Norton to cast the model back to the standard
     form (without loading).

5.   Apply standard formulae to find A, f, RIN and ROUT.




                         C.K. Tse: Feedback amplifiers and               35
                                    oscillators
Example
                                     Rf



                                 –
                                     a
              is                 +                 +
                                                   vo     RL
                                                   –




Type of feedback:           shunt-shunt
Appropriate 2-port type:    y-parameter

So, the first step is to represent the circuit in y-parameter networks.




                       C.K. Tse: Feedback amplifiers and                  36
                                  oscillators
Example


                         –
                              a
          is             +                 +
                                           vo     RL
                                           –




                         Rf




               C.K. Tse: Feedback amplifiers and        37
                          oscillators
    Example
    Converting to y-parameter


              –                  Ro            +
                          +
    is        vi    Ri    –                    vo           RL
              +               avi              –




Note: this
goes to the              Rf
–ve input
of A.                                                                           y11f        y22f
                                                                 y12fvo


                                                                      1             -1             1
                                                             y11f =        y12f =        y 22f =
                                                                      Rf            Rf             Rf



                              C.K. Tse: Feedback amplifiers and                                 38
                                         oscillators
Example
Converting to y-parameter


          –                                  Ro                  +
                                       +
is        vi                 Ri        –                         vo       RL
          +                                avi                   –




                                      y11f         y22f
                 y12fvo

                        1                  -1               1          REMEMBER:
               y11f =             y12f =          y 22f =
                        Rf                 Rf               Rf         y11f and y22f are conductance!


                                    C.K. Tse: Feedback amplifiers and                            39
                                               oscillators
Example
Casting it to standard form


         –                              Ro                  +
                                  +
is       vi       Ri || R f       –
                                                 Rf||RL     vo
         +                            avi                   –




               y12fvo


                       1               -1              1
              y11f =          y12f =         y 22f =
                       Rf              Rf              Rf



                               C.K. Tse: Feedback amplifiers and   40
                                          oscillators
Example
Casting it to standard form


         –                          Ro || R f || RL     +
                              +
is       vi     Ri || R f     –     a(R f || RL )       vo
         +                                        vi    –
                                   Ro + R f || RL




                            -v o
                            Rf




                             C.K. Tse: Feedback amplifiers and   41
                                        oscillators
Example
Finally, we get the standard form
          ie

          –                        Ro || R f || RL   +
                              +
is        vi    Ri || R f     –                      vo
          +                        Aie               –



                                                          Using Thévenin theorem,
                                                                                R f || RL
                                                          Aie = av i
                            -v o
                                                                       (R   f   || RL ) + Ro

                            Rf
                                                          A = -a
                                                                 (R || R )(R || R )
                                                                       i         f       f       L

                                                                   (R || R ) + R
                                                                           f         L       o

                                                                    Ri R 2 RL
                                                                         f                 1
                                                          A = -a
                                                                   (Ri + R f ) ( R f RL + Ro R f + Ro RL )


                             C.K. Tse: Feedback amplifiers and                                        42
                                        oscillators
Example
Apply standard results:
                                                           Ri R 2 RL
                                                                f                1
Basic amplifier gain (transresistance)          A = -a
                                                         (Ri + R f ) ( R f RL + Ro R f + Ro RL )

Feedback gain:                                        -1
                                               f =
                                                      Rf

                                                        A    1
Overall (closed-loop) gain                     Ao =         ª = -R f                 if Af >> 1
                                                      1+ Af  f

                                                      Ri || R f
Input resistance                             RIN =
                                                      1+ Af
                                                      Ro || R f || RL
Output resistance                          ROUT =
                                                        1+ Af



                          C.K. Tse: Feedback amplifiers and                                   43
                                     oscillators
Frequency response
Gain and bandwidth                                             Suppose the basic amplifier
                                                               has a pole at p1, i.e.,
        +            e                                                              ALF
 si                         A(jw)                    so                A( jw ) =
                                                                                      jw
input       –                                     output                           1+
                            basic amplifier                                            p1

                sf                                          20log10|A| (dB)
                              f

                         feedback network                   ALF
                                                                              slope = –20dB/dec




                                                                                                  w
                                                                       p1




                                    C.K. Tse: Feedback amplifiers and                              44
                                               oscillators
Frequency response
Gain and bandwidth                              Hence, we see that the overall gain has a
                                                pole at
The overall (closed-loop) gain is                          pc = p1(1 + fALF)
                                                and the low-frequency gain is lowered
                 A( jw )                        to                ALF
    Ao ( jw ) =                                         Ao,LF =
               1+ A( jw ) f
                                                                1+ fALF
                     ALF
             =
               Ê   jw ˆ
               Á1+    ˜ + fALF
               Ë   p1 ¯                                20log10|A| (dB)
                       È               ˘
                 ALF Í        1        ˙
                                                       ALF               basic amplifier
             =         Í               ˙
               1+ fALF Í1+     jw      ˙                                        feedback amplifier
                       Í               ˙
                       Î p1 (1+ fALF ) ˚             Ao,LF

                                                                                               w
                                                                  p1       pc

                            C.K. Tse: Feedback amplifiers and                              45
                                       oscillators
Stability of feedback amplifier
Definition: A feedback system is said to be stable if it does not
oscillate by itself at any frequency under a given circuit condition.

Note that this is a very restrictive definition of stability, but is
appropriate for our purpose.

Therefore, the issue of stability can be investigated in terms of the
possibility of sustained oscillation.




                   feedback circuit
                                                    sustained oscillation at certain frequency




                          C.K. Tse: Feedback amplifiers and                             46
                                     oscillators
Why and how does it oscillate?
The feedback system oscillates because of the simple fact that it has a
closed loop in which signals can combine constructively.

Let us break the loop at an arbitrary point along the loop.

                    +
              si                     A                            so
            input       –                                       output


                                      f
                                                    B      B’
Signal at B, as it goes around the loop, will be multiplied by f and A, and
also –1.
                                  SB’ = – A f SB


                            C.K. Tse: Feedback amplifiers and                  47
                                       oscillators
Why and how does it oscillate?
Clearly, if SB’ and SB are same in magnitude and have a 360o phase
difference, then the closed loop will oscillate by itself.

Oscillation criteria:

1.           Af = 1

2.           Af = ±180o           This is known as the Barkhausen criteria.

The idea is
If the signal, after making a round trip through A and f, has a gain of 1
and a phase shift of exactly 360o, then it oscillates. But, in the negative
feedback system, there is already a 180o phase shift. Therefore, the
phase shift caused by A and f together will only need to be 180o to
cause oscillation.

                          C.K. Tse: Feedback amplifiers and                    48
                                     oscillators
The loop gain T
An important parameter to test stability is the loop gain, usually
denoted by T.
                                 T = Af

          |T| (dB)

                                       crossover frequency
                                       (where the gain is 1)
                                 wo
      0dB                                    w
             f
                                             w


        fT

                                                 If fT = –180o, OSCILLATES!


                         C.K. Tse: Feedback amplifiers and                 49
                                    oscillators
Phase margin
Phase margin is an important parameter to evaluate how stable the
system is.
                  Phase margin fPM = –180o – fT

         |T| (dB)

                                     crossover frequency
                                     (where the gain is 1)
                               wo
      0dB                                  w
            f
                                           w


        fT
     –180o
                         phase margin fPM (the larger the better)

                       C.K. Tse: Feedback amplifiers and             50
                                  oscillators
Compensation
Compensation is to make the amplifier more stable, i.e., to increase fPM.
REMEMBER: We should always look at T, not A or Ao.

        |T| (dB)

                                              crossover frequency
                                              (where the gain is 1)
                                       wo
    0dB                                                         w
                   p1        p2
          f
                                                                w


       fT
    –180o
                        phase margin fPM (how to increase it?)



                        C.K. Tse: Feedback amplifiers and                   51
                                   oscillators
Method 1: Lag compensation
Add a pole at a low frequency point. The aim is to make the crossover point
appear at a much lower frequency. The drawback is the reduced bandwidth.

Compensation                      |T| (dB)                  crossover frequency
                                                            after compensation
function Gc is
                                                                   crossover frequency
               1
Gc ( jw ) =                                                        before compensation
                jw
            1+
                pa            0dB                                                          w
                                        pa       p1   p2
                                    f
                                                                                           w
        before compensation
        after compensation


                              –180o
                                                                          phase margin fPM
                                             phase margin fPM             before compensation
                                             after compensation
                                C.K. Tse: Feedback amplifiers and                           52
                                           oscillators
Method 2: Lead compensation
Add a zero near the first pole. The aim is to reduce the phase shift and hence
increase the phase margin and keep a wide bandwidth. But the drawback is the
more difficult design.
Compensation               |T| (dB)
function Gc is                                        crossover frequency   crossover frequency
               jw                                     before compensation
            1+                                                              after compensation
               za
Gc ( jw ) =
               jw    0dB               za
            1+                              p1   p2
                                                                                    w
               pa
                           f
                                                                                    w
     before compensation
     after compensation


                    –180o
                                   phase margin fPM
                                                                      phase margin fPM after
                                   before compensation
                                                                      compensation

                               C.K. Tse: Feedback amplifiers and                          53
                                          oscillators
Op-amp stability problem
The op-amp has a high DC gain, and hence at crossover it is likely that the
phase shift is significant. The worst-case scenario is when the feedback gain is
1 (maximum for passive feedback). We call this unity feedback condition, and
use this to test the stability of an op-amp.

Under unity-gain feedback condition, the loop gain T = Af = A, because f = 1.
                 |A| (dB)

                                                 op-amp frequency response




                                           p1          p2

               –90o                                             phase margin too small
              –180o

                            C.K. Tse: Feedback amplifiers and                      54
                                       oscillators
Op-amp internal compensation
Usually, op-amps are internally compensated. The technique is by lag
compensation, i.e., adding a pole at low frequency such that the phase margin
can reach at least 45o.

Suppose we add a low-frequency dominant pole at pa. If we can put pa such
that p1 (original dominant pole) is at crossover, then the phase margin is about
45o.               |A| (dB)
                                                   op-amp frequency response
                                                   before compensation

                                                                op-amp frequency response
                                                                after compensation

                      pa                 p1          p2

               –90o                                                  phase margin too small
              –180o
                           phase margin ≈ 45o
                             C.K. Tse: Feedback amplifiers and                           55
                                        oscillators
Op-amp internal compensation
Typically, pa is about a few Hz, say 5 Hz. Then, we have to create a pole at
such a low frequency.

First, consider the input differential stage of an op-amp. One way to add the
pole is to put a capacitor between the two collectors of the differential stage.


                                                   Equivalent model:
 RL                RL                                                                    next stage
                               to next stage
                                                      rπ                ro//RL        2C           RIN
          C


                                                   The dominant pole is
                                                                       1
                                                         pa =                      = 2p (5)
                                                              2(ro || RL || RIN )C

                                                   We can find C from this equation.

                            C.K. Tse: Feedback amplifiers and                                  56
                                       oscillators
Op-amp internal compensation
If we use the previous method of inserting a C between collectors of the
differential stage, the size of C required is very large, as can be found from
                   1                             Using this method, C can be as large as hundreds
   pa =                          = 2p (5)        of pF, which is too large to be implemented on
          2(ro || RL || RIN )C
                                                 chip. NOT practical!


Better solution: Use                                                           to active load
Miller effect.                                                             C
                                            RL                 RL
                                                                                                output stage
Miller effect can expand
capacitor size by a factor
of the gain magnitude.
So, we may put the
capacitor across the input
and output of the main
                                                                         main gain stage
gain stage in order to use
                                                                         CE stage
Miller effect. In this way,
C can be much smaller,
say a few pF.
                                      C.K. Tse: Feedback amplifiers and                                57
                                                 oscillators
    Example: Op-amp 741 internal compensation
                                  +Vcc
                                                                                       Data:
    Q1 differential Q2                   Q13B         Q13A
         input stage                                                                   DC gain = 70 dB
+                             –
                                                                                       Poles:
                                                                        Q14               30 kHz
    Q3                 Q4                                                                500 kHz
                                     +Vcc                                     output
                                                                                          10 MHz
                                                                        Q20

             –VEE                        Q16                  Q23
                                                Q17
    Q5                 Q6
                            main gain stage
                                  CE stage

                                  –VEE



                                     C.K. Tse: Feedback amplifiers and                           58
                                                oscillators
Example: Op-amp 741 internal compensation
Unity-gain feedback (worst case stability problem): T = A




                                                          p1 = 30 kHz




                                                  Bad stability
                                                  because of the
                                                  substantial phase
                                                  shift!

                       C.K. Tse: Feedback amplifiers and                 59
                                  oscillators
Example: Op-amp 741 internal compensation
Compensation trick (based on lag compensation approach):

• Introduce a low-frequency pole at pa such that p1 is at crossover.
• This ensures the phase angle at crossover = –135º. Hence, PM = 45º.

             |A| (dB)



                                                          op-amp frequency response
                                                          after compensation

                  pa                p1

          –90o
          –180o
                       phase margin ≈ 45o

                           C.K. Tse: Feedback amplifiers and                           60
                                      oscillators
Example: Op-amp 741 internal compensation
Graphical construction method




                                                            p1 = 30 kHz

                    pa




                                                    Bad stability
                                                    because of the
                                                    substantial phase
                                                    shift!

                         C.K. Tse: Feedback amplifiers and                 61
                                    oscillators
  Example: Op-amp 741 internal compensation

  Exact calculation of pa :


                                               0 - 70          -70
               slope = –20 dB/dec      =                  =
70 dB                                      log p1 - log pa 4.477 - log pa


                                                         Hence, pa = 9.5 Hz
        pa                      p1 = 30 kHz




                              C.K. Tse: Feedback amplifiers and                62
                                         oscillators
Example: Op-amp 741 internal compensation




               After compensation, the
               phase margin is 45º.




               C.K. Tse: Feedback amplifiers and   63
                          oscillators
Example: Op-amp 741 internal compensation
Question: How to create the 9.5 Hz pole with a reasonably small C ?
Solution: Take advantage of Miller effect to boost capacitance.
                                        +Vcc

          Q1 differential Q2                   Q13B         Q13A
      +        input stage          –
                                     Cc                                  Q14
          Q3                 Q4
                                           +Vcc                                output
                                                                         Q20

                   –VEE                        Q16                 Q23
                                                      Q17
          Q5                 Q6
                                  main gain stage
                                        CE stage

                                        –VEE

                              C.K. Tse: Feedback amplifiers and                          64
                                         oscillators
    Example: Op-amp 741 internal compensation
                                   +Vcc
                                                                                        Given:
     Q1 differential Q2                                                                 Ro17 = 5 MΩ
                                           Q13B         Q13A
+         input stage          –
                                                                                        Ro13 = 720 kΩ
                                   Cc                                                   Ri23 = 100 kΩ
                                                                        Q14
     Q3                 Q4
                                        +Vcc                                   output
                                                                        Q20

              –VEE                        Q16                   Q23
                                                  Q17
     Q5                 Q6                                                    Gain of CE stage:
                             main gain stage
                                   CE stage                                   ACE = Gm[Ro17||Ro13||Ri23]

                                   –VEE                                       Miller-effect capacitor
                                                Gm = 6 mA/V                   CM = Cc (ACE + 1)
                                                                                 = 518.74 Cc
                                     C.K. Tse: Feedback amplifiers and                             65
                                                oscillators
    Example: Op-amp 741 internal compensation
                                   +Vcc
                                                                   Given:
     Q1 differential Q2                                            Ri16 = 2.9 MΩ
                                           Q13B         Q13A
+         input stage          –
                                                                   Ro4 = 10 MΩ
                                   Cc                              Ro6 = 20 MΩ

     Q3                 Q4                                        Equivalent ckt:
                                        +Vcc
                                                                    Ro4||Ro6


              –VEE                        Q16                                    Ri16           CM
                                                  Q17
     Q5                 Q6
                             main gain stage
                                   CE stage                        pa = 1 / 2π CM [Ro4||Ro6|| Ri16]
                                                                   and CM = 518.74 Cc
                                   –VEE
                                                                   Hence, Cc = 15 pF

                                     C.K. Tse: Feedback amplifiers and                            66
                                                oscillators
Oscillation
In designing feedback amplifiers, we want to make sure that oscillation does
not occur, that is, we want stable operation.

However, oscillation is needed to make an oscillator. As shown before, the
criteria for oscillation in a feedback amplifier are

1.   Loop gain magnitude | T | = 1
2.   Roundtrip phase shift fT = ±180o

Thus, the same feedback structure can be used to make an oscillator. In other
words, we construct a feedback amplifier, but try to make it satisfy the above
two criteria.

In practice, T is a function of frequency, and the above criteria are
satisfied for one particular frequency. This frequency is the oscillation
frequency.



                           C.K. Tse: Feedback amplifiers and                     67
                                      oscillators
Oscillator principle
As T = Af, we can deliberately create phase shift in A or f.
                                                       NOTE: Since this model is a
        +                                              negative feedback, we need the total
 si                  A(jw)                     so      phase shift of A(jw) and f(jw) to be
input       –                               output     180o at the frequency of oscillation.
                     basic amplifier                    If a positive feedback is used, we
                                                       need the total phase shift to be 360o.
                      f(jw)
                                                       |A(jw) f(jw)| (dB)
                  feedback network



                                                                                    wo           w

                                                                                                 w

                                                       –180o


                              C.K. Tse: Feedback amplifiers and                              68
                                         oscillators
Sustained oscillation
There are two problems! How does oscillation start? And how can oscillation
be maintained?

First, there is noise everywhere! So, signals of all frequencies exist and go
around the loop. Most of them get reduced and do not show up as oscillation.
But the one at the oscillation frequency starts to oscillation as it satisfies the
Barkhausen criteria.

If | T | is slightly bigger than 1, oscillation amplitude will grow and go to
infinity. But if | T | is slightly less than 1, oscillation subsides. The question is
how to maintain oscillation with a constant magnitude.

We need a control that changes | T | continuously. Typically, this is done by a
nonlinear amplitude stabilizing circuit, for example, an amplifier whose gain
drops when its output increases, and rises when its output decreases.




                             C.K. Tse: Feedback amplifiers and                          69
                                        oscillators
The Wien bridge oscillator

                      R2                              Model:
         R1
                  –

                  +                                                         A
                                                               +

Zp
                      C    R                                                    Zs
     C        R
                                                                       Zp
                                Zs


                                       R2
 Basic amplifier gain       A = 1+
                                       R1
 Feedback gain                        -Z p                        R              1+ jwCR
                               f =               where Z p =           and Z s =
                                     Zs + Z p                  1+ jwCR             jwC


                               C.K. Tse: Feedback amplifiers and                            70
                                          oscillators
The Wien bridge oscillator
Oscillation frequency
Note that we define the standard feedback structure with negative feedback. So,
the loop gain is                  Ê     ˆR
                                     -Á1+ 2 ˜
                                       Ë R1 ¯
                      T( jw ) =
                                     Ê        1 ˆ
                                3 + jÁwCR -     ˜
                                     Ë      wCR ¯

Applying the oscillation criteria, we can find the oscillation frequency and the
resistor values as follows:
                                     R
                T( jw ) = 1 fi 1+ 2 = 3 fi R2 = 2R1
                                     R1
                                            1                1
                fT = ±180 o fi w oCR =             fi wo =
                                         w oCR              CR

We can choose R2/R1 to be slightly larger than 2, say 2.03, to start oscillation.

                            C.K. Tse: Feedback amplifiers and                        71
                                       oscillators
The Wien bridge oscillator
Frequency response viewpoint
Suppose the amplifier has a fixed gain of A. The feedback network, however,
has a bandpass frequency response.

                                                        WHY OSCILLATE?
                       A                                Clearly, the roundtrip gain will be 1
 +
                                                        for f = fo if the basic amplifier has a
                                                        gain of 3.
                 |f|
                                                        The world is noisy. Signals of all
           1/3                                          frequencies exist everywhere!

                                freq
                                                        But signals at all frequencies except fo
                 ff                                     will be reduced after a round trip.
                       fo
                                                        Only signals at fo will have a
           + 90
                                                        roundtrip gain of 1.
                                freq
           –90                                          Hence, the oscillation frequency is fo.
                                                        From the filter structure, we can find
                                                        that fo is equal to 1/2πCR.

                            C.K. Tse: Feedback amplifiers and                                      72
                                       oscillators
The Wien bridge oscillator                                                                  +15V

Amplitude control                                                                              3k
If we choose R2/R1 = 2.03, then amplitude may                                          D2
grow. We have to stabilize the amplitude. The
following is an amplitude limiter circuit.
                                                           10k           20.3k                 1k
Diode D1 (D2) conducts when vo reaches its                               –
                                                                     A                        vo
positive (negative) peak.
                                                                         +
Just when D1 conducts, we have vA = vB.
                                                                                               1k
     10k     A 20.3k
                             vo                                              16n 10k          B
                        1k                              16n 10k
                        B                                                                      3k
                        3k                                                     D1

                                                                                            –15V
                    –15V
        3             1
-15 +     (v o + 15) = v o    fi     v o = 9 V.     Similar procedure applies for the negative peak.
        4             3                            So, the amplitude is 9 V.
                                  C.K. Tse: Feedback amplifiers and                                 73
                                             oscillators
 The phase shift oscillator
 This circuit matches exactly our negative
 feedback model. The basic amplifier gain is R2/R1,
 and the feedback network is frequency dependent.

                                                  For the feedback network, we want to find
                                                  the frequency at which the phase shift is
                            R2                    exactly 180o. At this frequency, if the
              R1
                        –                         roundtrip gain is 1, oscillation occurs. Note
                                                  that the negative feedback already gives 180o
                        +                         phase shift.
                                                      |f|
                                                      1
                    C       C     C                  29
                                                                               freq

               R'   R        R                                      fo
                                                       ff

R1 || R'= R                        f               180o
                                                                               freq


                                 C.K. Tse: Feedback amplifiers and                            74
                                            oscillators
The phase shift oscillator                                                  C        C   C

From the filter characteristic of the feedback network,
we know that the phase shift is 180o at fo, where its gain              R   R        R
is 1/29.

So, oscillation starts at fo if A ≥ 29. This means we need        |f|
to have R2 ≥ 29R1.
                                                                   1
We can prove that                                                 29
                                                                                             freq
                        1
                fo =                                                            fo
                     2p 6CR                                        ff

                                                              180o
                                                                                             freq

Note that the leftmost resistor in the feedback filter is R’ (not R). But R’//R1 is
exactly R. This will adjust the loading effect of the basic amplifier and make the
overall filter circuit easier to analyze since it is then simply composed of three
identical RC sections.


                               C.K. Tse: Feedback amplifiers and                               75
                                          oscillators
Resonant circuit oscillators
A general class of oscillators can be constructed by a pure reactive π-feedback network.

                                           For a voltage amplifier implementation, this structure
                                           can be modelled as a series-shunt feedback circuit:
                A

                                                     +                      Ro
      jX1               jX2                                           +
                                                     vi          Ri   –
                                                     –                    Avi


                jX3

 pure reactive π-feedback network                                     jX3
                                                      –
                                                      vi     jX1                jX2
                                                      +




                              C.K. Tse: Feedback amplifiers and                                     76
                                         oscillators
Resonant circuit oscillators
Analysis:
                                             -A( jX1 )( jX 2 )
The loop gain is T( jw ) =
                               j ( X1 + X 2 + X 3 ) Ro + jX 2 ( jX1 + jX 3 )
                                               AX1 X 2
                           =
                               j ( X1 + X 2 + X 3 ) Ro - X 2 ( X1 + X 3 )

For oscillation to start, we need T = –1.
                                                                      +                    Ro
                                                                                     +
Thus, the oscillation criteria become                                 vi        Ri   –
                                                                      –                  Avi
                 X1 + X 2 + X 3 = 0
                        AX1
                                =1
                      X1 + X 3

In practice, we may have
                                                                                     jX3
(a) X1 and X2 are capacitors and X3 is inductor.                       –
                                                                       vi      jX1             jX2
OR                                                                     +

(b) X1 and X2 are inductors and X3 is capacitor.


                                       C.K. Tse: Feedback amplifiers and                              77
                                                  oscillators
Colpitts oscillator
When X1 and X2 are capacitors and X3 is inductor, we have the Colpitts oscillator.

In this case, we have
                 1                   1
        jX1 =           and jX 2 =
                jwC1               jwC2
                   jX 3 = jwL3

                                                            +                           Ro
                                                                                  +
From X1 + X 2 + X 3 = 0                                     vi          Ri        –
                                                            –                         Avi
the oscillation frequency can be found:

                          1
           wo =
                       Ê C1C2 ˆ
                    L3 Á         ˜
                       Ë C1 + C2 ¯
                                                                                  L3
                                                             –
                                                             vi              C1        C2
                                                             +




                                     C.K. Tse: Feedback amplifiers and                        78
                                                oscillators
A practical form of Colpitts oscillator
The basic amplifier can be realized by a common-emitter amplifier.

The loop gain is
                                 1/sC1
               T(s) = Gm ZT
                               sL + 1/sC1


where      1              1          1                                                   ZT
              = sC2 +           +
           ZT                1    ro || Rc
                      sL +
                            sC1
Putting s = jw, and applying the Barkhausen criterion:
                           1
              wo =
                         Ê CC ˆ
                      L3 Á 1 2 ˜
                         Ë C1 + C2 ¯

               Gm C2 (ro || Rc )      C1 Gm Rc ro
                                 >1 fi   >                    for oscillation to start.
                     C1               C2 Rc + ro


                                   C.K. Tse: Feedback amplifiers and                           79
                                              oscillators
Hartley oscillator
When X1 and X2 are inductors and X3 is capacitor, we have the Hartley oscillator.

In this case, we have
         jX1 = jwL1 and jX 2 = jwL2
                           1
                  jX 3 =
                         jwC3

                                                            +                       Ro
                                                                              +
From X1 + X 2 + X 3 = 0                                     vi          Ri    –
                                                            –                     Avi
the oscillation frequency can be found:

                          1
             wo =
                    C3 ( L1 + L2 )

For both the Colpitts and Hartley oscillators, the           –                C3
gain of the amplifier has to be large enough to               vi          L1              L2
ensure that the loop gain magnitude is larger than           +
1.



                                     C.K. Tse: Feedback amplifiers and                         80
                                                oscillators

				
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