Phase Locked Loop PLL

Document Sample
Phase Locked Loop PLL Powered By Docstoc
					Chapter X. Phase-Locked Loops (PLL)

 Figure 10-1. Phase-locked loop applications.
The Phase Detector
        A simple analog phase detector is shown in Figure 10-2a. Consider the phase
detector as a simple switch, as illustrated in Figure 10-2b. The signal with frequency
fo simply opens and closes the (diode) switch. If f i  f 0 , then the circuit behavior is
that of a mixer producing the sum and difference frequencies. The capacitors shown
are chosen to bypass fi, fo, and fi + fo, and therefore only the beat (fi - fo) signal is seen
at vd. After the loop is locked, fo will be exactly equal to fi .

        A phase difference between the two input signals results in a dc voltage Vd,
which is proportional to the phase difference, ei - o .

Figure 10-2. (a) Analog phase detector. (b) Simpified model.

A mixer performs the mathematical function of multiplication. Thus for
sinusoidal inputs,
                vd  A sin  i   i   2 cos 0 t   0 
                     A sin i   i    0 t   0   A sin i   i    0 t   0 
        When phase-locked, o = i, the second harmonic term, Asin[2ot +o +i],
is filtered out, leaving
                Vd = Asin(i - o)                                             (10-1)
This voltage is directly proportional to the input signal amplitude and the phase error
e if the signal amplitude is held constant. Indeed, for small e, this transfer function
is linear as seen in Figure 10-3.

    Figure 10-3. Analog phase detector characteristic-output voltage
                     versus input phase difference.

Figure 10-4. Phase detector waveforms.

        Figure 10-4 helps to show the results graphically as oscilloscope
measurements. When the signals are out of phase by 90 o as in part (a), a zero dc
output results; if the phase is slightly advanced as in part phase as in part (b), a small
negative dc output is produced; and when the signals are exactly in phase as in part
(c), the result is a dc output proportional to the fi signal level-exactly the kind of
signal that is needed for a lock indication in telephone touchtone decoders or AGC in
coherent receivers.
Phase Detector Gain
         The PD characteristic is a continuous sinusoid repeating every 2 radians.
Also, during the tracking mode, operation is limited to the portion of the curve
between +/2 where |e| < /2. For sinusoidal inputs it is clear from Figure 10-6 that
the slope of the phase detector characteristic curve,
                          Vd = Asine                                            (10-2)
is not constant. In fact, it rises with a maximum slope at e = 0, and levels off to a
slope of zero (no gain) at e = /2 radians.

           The peak voltage A is the volts-per-radian gain of this phase detector
because the tangents to the peak and the PD curve at e = 0 intersect at one radian, as
seen in Figure 10-6. Therefore the gain of the analog phase detector is
                                   volts 
                            k  A                                               (10-3)
                                   radian 

Figure 10-6. Sinusoidal characteristic of analog phase detector.

           If the input signals are both square-waves, the phase detector characteristic
will be linear, as illustrated in Figure 10-7. The gain of this circuit is constant over the
range of input e = +/2 and is given by k = Vd/e = A/(/2). That is,
                                   2 A  volts 
                            k                                                 (10-4)
                                     radian 
         Figure 10-7. Phase comparator characteristic for square-wave

Figure 10-8 Integrated circuit balanced detector.

        Figure 10-8 is typical of balanced integrated circuit implementations. This
circuit is also used as a balanced AM modulator for producing double-sideband /
suppressed-carrier signals and consists of differential amplifiers. The oscillator input
polarity determines which differential pair conducts, while the signal input
determines whether RC1 or RC 2 receives the current. The output voltage will be the
difference between i1 RC1 and i2 RC 2 .
Digital Phase (Timing) Comparators
       Digital phase detectors can be realized using an exclusive-OR (Figure 10-9)
or an edge-trggered set-reset flip-flop (RS-FF) circuit. The exclusive-OR output Y is
low when both inputs are high or low; otherwise Y is high, indicating “or.” The
output is smoothed (integrated) to produce Vd. The exclusive-OR requires
symmetrical squarewave inputs, which may become a system problem, whereas the
edge-triggered RS-FF works well with pulses.

Figure 10-9. Digital implementation of phase detector using an
exclusive-OR gate.

As illustrated for the circuit of Figure 10-10, the RS-FF phase detector can produce a
linear PLL over a full e range of 2 rad, which is twice that for the other phase
detectors. The problem with using digital phase detectors in sensitive communication
receiver applications is in the difficulty of filtering the sharp impulses and their
harmonics to prevent radio-frequency interference (RFI).

Figure 10-10. Digital implementation of phase detector using a
set-reset flip-flop.

Figure 10-11. Operational amplifiers increase PLL loop gain.

        The second loop component is an amplifier commonly referred to as the dc
amp. Its function sis to increase the loop gain by amplifying the phase detector output
voltage. Figure 10-11 shows three voltage amplifiers and their gain parameter
k A  AV (volts out/volts in). The bandwidth of the dc amp must be very high

compared to the loop bandwidth or loop instability will result-even to the point of
oscillation due to excessive phase shift around the loop, which would produce
positive (regenerative) feedback.
Voltage-Controlled Oscillator (VCO)

Figure 10-12. Tuning diode control of free-running multivbrator.

        The frequency of the free-running multi-vibrator circuit of Figure 10-12 is
controlled by the variable reactance of D1 and D2. In IC implementations, D1 and D2
are realized by reverse-biased collector junctions. It should be noted that the control
voltage must not exceed VE + 0.5V + ve, where ve is the positive peak of the oscillator
signal across RE and 0.5V causes forward bias of the silicon diodes. The input-output
characteristic for the VCO is shown in Figure 10-13.

                         Figure 10-13. VCO characteristic.
Locking the loop
        Start with switch S 1 open (Figure 10-14) and a signal generator with
frequency fi connected to the input. With fi not equal to the free-running frequency
(fFR), the phase detector will produce the sum and defference frequencies. The loop
(low-pass) filter filters out the sum frequency (fi + fFR), fi, and fFR, while the difference
(fi - fFR)-the beat between the signal generator and VCO-is allowed to pass through.
The beat is amplified and seen as V0 on an oscilloscope. As the generatir frequency is
varied to bring fi closer to fFR, the beat frequency gets lower and lower. This is
illustrated in Figure 10-15.

Figure 10-14. PLL block diagram.

       Figure 10-15. Beat-frequency output at Vo with loop open. The
     input generator frequency is being varied from fi < fFR to fi > fFR.

        In Figure 10-14, with the VCO input grounded and Vo = 0, measurements will
show that fi = fFR. However, if f i  f FR , then the beat is observed at Vo. When the
switch is closed, the beat-frequency signal at Vo will cause the VCO frequency fo to
change. If the voltage is large enough (high loop gain) and the filter bandwidth
wide enough, then the VCO will be deviated from fFR and lock at the instant that fo =
fi . The amount by which the VCO frequency must be changed is f = fi - fFR. The
time required for the loop to lock depends on the type of loop and loop dynamics.
For the simplest PLL with no loop filter, this acquisition time is on the order of 1/kv
seconds. Also, the range of fi over which the loop will lock, the lock range, is equal to
the hold-in range for the simple PLL.

Locked Loop: The Tracking Mode

When the loop is locked we know that fo = fi. Only a phase difference between the
signal and the VCO can exist. This phase difference ei - o is called the static (dc)
phase error. eis the input to the phase detector when the loop is locked and is
required in order for the phase detector to produce a dc output voltage Vd which,
when amplified by the dc amplifier, will produce exactly enough Vo to keep the VCO
frequency deviated by f. If fi increases, then f increases and e must increase in
order to provide for more Vo to keep the VCO tracking fi. The definition of locked is
that fi = fo and the loop will track any change in fi. Any subsequent shift of i or o
will be tracked-out so that only e remains.
Hold-In Range

        The range of frequencies for fi over which the loop can maintain lock is
called hold-in range. Assuming that the amplifier does not saturate and the VCO has a
wide frequency range, the phase detector characteristic limits the hold-in range. It
should be clear from the phase detector characteristics (Figures 10-6 and 10-7) that,
as the static phase error increases due to increasing fi, a limit for Vd is reached beyond
which the phase detector cannot supply more voltage for VCO correction. The phase
detector simply cannot produce more than A volts. The total range of Vd is ±A = 2A,
so that the total range of e is  radians. From Equation 10-7, the minimum to
maximum input frequency range, fi(max) - fi(min) = fH, will be
                        fH = kL
fH = kv / 2                                     (10-9)
The edge-triggered R-S flipflop phase comparator of Figure 10-9 can provide twice
this, fH = kv.
Loop Gain and Static Phase Error
The locked PLL is seen in Figure 10-16. The phase comparator develops an output
voltage Vd in response to a phase difference between the reference input and the VCO.
The transfer gain k has units of volts/radian of phase difference. The amplifier shown
is wideband with a voltage gain of kA volts/volt (dimensionless). Thus, Vo = kAVd.

Figure 10-16. PLL in tracking mode (locked).

The VCO free-running frequency is fFR. The VCO frequency fo will change in
response to an input voltage change. The transfer gain ko has units of kHz/V. The loop
gain for this system is simply the gain of each block multiplied around the loop, thus
                   kL = k.kA.ko                           (10-6)
                             .      .
The units of kL are (V/rad) (V/V) (kHz/V) = kHz/rad.
Assume that a signal with frequency fi is an input to the phase detector, and the loop
is locked. If the frequency difference before lock was f = fi - fER, then a voltage Vo
=f / ko is required to keep the VCO frequency equal to fi. So the phase comparator
must produce Vd = Vo/kA = f / kokA, and the static phase error ei - o must be
eVd/k. Combining gives e= f/kokAk = f/kL. This is a fundamental equation for
the PLL in phase lock;
                          e= f / kL.                                      (10-7)
In many computations the loop gain must be in units of radians/second rather than in
kHz/radian. The conversion is made using 2 radians/cycle. Hence, loop gain is also
given by
                          kv = 2kkAko                                       (10-8)
in unit of sec or radians/second.
         Figure 10-17 provides enough information to analyze the static behavior of a
phase-locked loop.
1. Determine kA for the op-amp.
2. Calculate the loop gain in units of sec-1 and in dB (at  = 1 rad/s).
3. With S1 open as shown, what is observed at Vo with an oscilloscope?
4. When the loop is closed and phase-locked, determine
(a) the VCO output frequency,
(b) the static phase error at the phase comparator output, and
(c) Vo (is this rms, pk-pk, or what?).
5. Determine the hold-in range fH.
6. Determine A, the maximum value of Vd.

Figure 10-17. Example PLL.

1. k A  R f / R1   1  4 k/1k + 1 = 5.

2. k L  k k A k 0  0.1 V/rad x 5 x (-30 kHz/V) = -15x103(Hz/rad). Then, kv = 15x103

      cycles/s-rad x (2 rad/cycle) = 94.3 k sec-1, and kv(dB) = 20 log kv =
      20log(94.3x103) = 99.5dB at 1 rad/s.
3. Vo will be a sinusoidally varying voltage with a frequency of |fi – fFR| = 10 kHz.
This assumes that a very small capacitor internal to the phase comparator filters out fo,
fi and fo + fi.

4. (a). When the loop is locked, fo = fi = 100 kHz by definition of locked, and only a
phase difference can exist between the input signal and VCO. This phase difference
e is the loop-error signal (static phase error) which results in Vd at the detector output
and, when amplified by kA, provides enough voltage Vo to make the VCO frequency
be exactly equal to fi.
    (b). The free-running frequency of the VCO is 110 kHz. In order for the VCO to
    equal 100 kHz, the VCO input voltage must be Vo = (100 kHz -110 kHz)/ko = 10
    kHz/(-30 kHz/V) = 0.33Vdc. Then, because kA = 5, Vd must be Vd = 0.33V/5 =
    0.0667V. Finally e = Vd/k= 0.0667V/0.1V/rad = 0.667 rad. Once again, we have
    derived the basic relationship, e = f/kL = (fi - fER)/kL = -10 kHz/(-15x103 Hz/rad)
    = 0.667 rad.
(c). The input to the phase detector (loop-locked) was determined from e = f/kL =
0.667 rad. Since Vd = ke, we have Vd = 0.1 V/rad x 0.667 rad = 0.0667Vdc. Now, we
are assuming Zin of the op-amp is much larger than R of the loop filter, so there is no
voltage drop across R. The input to the op-amp is 0.0667Vdc, so that Vo = kAVd = 5 ×
0.0667Vdc = 0.33Vdc. This is enough to keep the VCO at 100 kHz when in fact its rest
frequency is 110 kHz.
5. The question is, when the loop is locked, how much can fi change in frequency
before the loop just cannot provide enough Vo to keep the VCO at fo = fi? Assuming
that the VCO and dc amplifier don't saturate, we look at the phase detector
characteristic. Clearly Vd can increase with e until Vd ---> Vmax = A, at which point e
= /2. Beyond this, Vd decreases for increasing static phase error, and the phase
detector simply cannot provide more output voltage to continue increasing fo, and the
loop breaks lock. The total hold-in range is +2, or  rad. The frequency difference
between these break-lock points will be fH = e(max) x kL = x 15 kHz/rad = 47.1
6. At the frequency where e = /2, we have Vd(max) = A. Therefore Vd = ke = 0.1
V/rad × π/2 rad = 0. 157Vdc.
When a PLL has locked to an input signal, the VCO will follow slow changes in the
input signal frequency fi. Suppose fi increases by an amount fi. In order for the loop
to remain locked (fo - fi), the VCO voltage must increase by Vo = fi/ko. This voltage
change is produced by the amplified change in Vd, which is produced by an increased
phase difference, e = 2fi/kv.
As a specific example, suppose that an FM signal with carrier frequency fi is
modulated to an index of mf = 4 by a 1-kHz sinusoid. The carrier frequency will be
deviated above and below fi by an amount fi = mffm = 4 ×1 kHz = 4 kHz pk. If this
FM signal is the input to a PLL with a VCO gain of ko = 10 kHz/V and loop
bandwidth 1 kHz, then the VCO input voltage Vo will be a 1-kHz sinusoid with a
peak amplitude of Vo = fi/ko = (4 kHz pk)/(10 kHz/V) = 400 mV pk.

To get a quantitative idea of the loop noise margin, consider the results of Example
10-3 as seen in Figure 10-23b. The output voltage Vo is 2V for a transmitted MARK.
How high can Vo rise on a noise transient caused by a deviation of the MARK
frequency or circuit variations of Vo before the loop breaks lock? The static phase
error when fi = fM = 2 kHz is e = f/kL = (2 kHz - 3.5 kHz)/(1.19 kHz/rad) = 1.26 rad.
However, for typical phase detector's, the loop will break lock if e exceeds /2 =
1.57 rad. Consequently loop transients that would cause e to increase by 0.31 rad
will result in a loss of lock. In terms of voltages, Vd(max) = ke(max) = (0.3184
V/rad) × (1.57 rad) = 0.5V and Vo(max) = 5 ×0.5 = 2.5V. Since Vo(MARK) = 2V and
we-have calculated Vo(max) = 2.5V, we see that the noise margin for Vo will be
Vo(NM) = 0.5 Vpk. This can result from noise in the PLL itself, from the noise input
signal-amplitude if no limiter precedes the PLL or from an input signal frequency
deviation (due to noise) of fi(NM) = ko x Vo(NM) = (0.75 kHz/v) x (0.5Vpk) = 375
Hz peak noise.
A PLL with ko = -0.75 kHz/V, fFR = 3.5 kHz, k = 0.3184 V/rad, and kA = 5 is used as
an FSK demodulator. The input signal has fS = 4 kHz, fM =2 kHz, and the modulation
is shown in Figure 10-23a. As seen, the baud rate is 1333 bits/s and the data is …1 0
0… Sketch accurately the PLL output Vo(t).

For fi = fM = 2 kHz, Vo = fo/ko = (fi–fFR)/ko = (2 kHz – 3.5 kHz)/(-0.75 kHz/V) = +2V.
For fi = fS = 4 kHz, Vo = (4 kHz – 3.5 kHz)/(-0.75 kHz/V) = -0.67V. The loop time
constant is  =1/kv = 1/(0.75 kHz/V)(0.3184 V/rad)(5)(2 rad/cycle) = 1/7502 = 0.
133 ms. It takes 0.133 ms for Vo to rise from -0.67V to 63% of the total voltage range
2V - (-0.67V) = 2.67V. 63% of 2.67V is 1.69V, so at time , Vo = 1.69V - 0.67V =
1.02V (see the plot of Vo in Figure 10-23b).

Figure 10-23. FSK input (a) and demodulated output (b) of PLL.
     Figure 8.3-6. PLL frequency multiplier.

Suppose a double-conversion SSB receiver needs fixed LO frequencies at 100 kHz
(for synchronous detection) and 1.6 MHz (for the second mixer), and an adjustable
LO that covers 9.90 -- 9.99 MHz in steps of 0.01 MHz (for RF tuning). The
custom-tailored synthesizer in Fig. 8.3-7 provides all the required frequencies by
dividing down, multiplying up, and mixing with the output of a 10-MHz oscillator.

Figure 8.3-7. Frequency synthesizer with fixed and adjustable output.
Linearized PLL Models and FM Detection
Suppose that a PLL has been tuned to lock with the input frequency fc, so f = 0.
Suppose further that the PLL has sufficient loop gain to track the input phase
(t) within a small error , so . These suppositions constitute the basis for the
linearized PLL model in Fig. 8.3-8a, where the LPF has been represented by its
impulse response h(t).
Since we’ll now focus on the phase variations, we view(t) as the input
“signal” which is compared with the feedback “signal”

to produce the output y(t). We emphasize that viewpoint by redrawing the linearized
model as a negative feedback system, Fig. 8.3-8b. Note that the VCO becomes an
integrator with gain 2Kv while phase comparison becomes subtraction.

Figure 8.3-8. Linearized PLL modes. (a) Time domain;
(b) phase; (c) frequency domain.
Direct Frequency Synthesizers
Figure 10-37 shows a high-stability, 64-kHz master reference oscillator followed
(horizontally) by a comb generator, which is a circuit used to produce a pulse rich in
harmonics of the 64-kHz input signal. A harmonic-selector filter controlled by tuning
logic is tuned to the desired harmonic and rejects all other spurious outputs. If the
synthesizer consisted only of this group of blocks, the resolution would be 64 kHz
because the output can be switched only to the various harmonics of the 64-kHz
master reference.
In order to improve the resolution and thereby achieve a finer separation between the
possible Output frequencies, a divide-by-16 circuit is used with a comb generator and
selector filter to produce 4-kHz frequency steps. The selected frequencies from the
upper and lower harmonic-select filters are mixed to produce sum and difference
frequencies, and the output (switchable) filter passes the desired output frequency.
The resolution, or smallest possible discrete frequency step, is now seen to be 4 kHz.
For instance, with N1 = 2, N2 = 2, and the output filter passing, the mixer sum, then fo
= 128 kHz + 8 kHz = 136 kHz. The next higher output frequency would be 140kHz.

     Figure 10-37. Direct synthesizer.
A multicrystal, direct-synthesis scheme for producing the transmit carrier and two
receiver local oscillators for a 23-channel citizens band transceiver is shown in Figure
10-38. This synthesizer is a 6-4-4 crystals/oscillator scheme (a 6-4-2 scheme is also
used) and, when tuned to emergency CB channel 9 (27.065 MHz carrier frequency),
crystals 3, 7, and 11 are used. With the receiver oscillator off, crystals 3 and 11
produce the transmit carrier: 37.700 - 10.635 = 27.065 MHz. With the receiver
oscillator on and the transmit oscillator off, the first and second local oscillators for
this double-conversion receiver are produced by the synthesis and receiver oscillators
as follows: 37.700 MHz from the synthesis oscillator with Xtal-3 is the 1st-LO
frequency. Thus, the 1st-IF frequency is 37.700 - 27.065 MHz = 10.635 MHz, so that
FM receiver IF transformers can be used. The receiver oscillator with Xtal-7(10.180
MHz) is the 2nd-LO, and the 2nd-IF frequency is 10.635 MHz - 10.180 MHz = 455
kHz, so that AM receiver IF transformers can be used.

    Figure 10-38. Frequency synthesis (6-4-4) for 23-channel CB
Phase-Locked Synthesizers
The most frequency used technique for frequency synthesis is the indirect method
utilizing a voltage-controlled oscillator in a programmable PLL. The simplest system
is the one-loop synthesizer of Figure 10-39, consisting of a digitally programmable
divide-by-N circuit used to divide the VCO Output frequency for comparison with a
stable reference source.

     Figure 10-39. Phase-locked frequency synthesizer.

The digitally programmable frequency-divider output, fo/N, is determined by the
value of N selected by the user and is compared to the reference signal in the phase
detector (PD). When the loop is locked for a specific value of N, then fo/N = fref by
definition of phase-locked; therefore the synthesizer output is
fo = Nfref                                      (10-26)

The divider can be a simple integer divider such as the 74192 programmable
up-down counter, or noninteger divider systems such as the fractional-N method
(producing fo = (N+1/M)fref) and the two-modulus prescaler circuit of Figure 10-44,
the MC12012, for example, using a technique called pulse swallowing. For our
purposes, only integer dividers are considered.

        The loop gain for the simple PLL synthesizer of Figure 10-39 is
it is important to realize that the frequency-divider circuit reduces the loop gain so
that the other loop components need to have relatively higher gain than the
conventional PLL. A more troublesome design problem, however, is that, as N
changes, so does the loop gain. There are linearizer circuit to ameliorate this problem.
Figure 10-40 Microprocessor-controlled LO synthesizer for TV.

Figure 10-40 shows the use of a very high frequency prescaler in a one-loop
synthesizer used for push-button TV channel selection. The VHF local oscillator (LO)
frequency is greater than 100 MHz, so high frequency emitter-coupled logic (ECL)
dividers are used to prescale the VHF signal below 1 MHz, where low-cost TTL or
CMOS technology can be used. The prescaler will reduce the resolution by an
amount equal to the prescale division ration P. Hence,
                   Resolution = Pfref                               (10-28)
with a prescaler.
The microprocessor-controlled VHF LO synthesizer of Figure 10-40 has a phase
comparator with k = 1 V/rad and an output impedance of 3.5 kΩ. Determine the
1. fref.
2. N for the TV to receive channel 5 (fLO = 123.000 MHz).
3. The synthesizer frequency resolution.
4. Loop gain and value of capacitor to compensate the loop to  = 0.5 and have the
VCO frequency within 10 % of its specified value in less than 10 ms after selection
of channel 5. (Assume that the maximum frequency step at the phase detector is
within the loop bandwidth.)
5. What value must the VCO sensitivity be?
1. fref = fXO/3580=3583.5 kHz/3580 = 1.00098 kHz.
2. , therefore N = 123.000 MHz/[(256)(1.00098 kHz)] = 480.
3. With the prescaler, the resolution will be Pfref = 256fref = 256.25 kHz. To prove
this, change the programmable divider to N+1 = 481, and compare the new fo to the
                  fo(N + 1) =1.00098 kHz × 256 × 481 =123,256.67 kHz
                 fo(N) = 1.00098 kHz × 256 × 480 = 123,000.42 kHz
Resolution: fo(N + 1) – Nfo = 256.25 kHz.
4. With the assumption stated, the loop will lock up unaided (without frequency
   sweep circuitry); hence, we use the universal overshoot and ringing curves of
   Figure 10-27. Vo must stay within relative values 0.90 and 1.1 (+10 %) on the  =
   0.5 curve. This is satisfied by nt = 4.6. With t = ts = 10 ms, we need the loop to
   have n = 4.6/10 ms = 460 rad/s. Since  = 0.5  c / k v , then c = kv for  = 0.5.

   Also  n   c kv , so that n = c = kv = 460 rad/s.

A capacitor is placed across the phase detector output (Ro = 3.5 kΩ) to form the
lag-compensation network. C = 1/cRo = 1/(460 × 3500) = 0.62 μF.
5. kv = 2kko/N = 460 rad/s. Therefore, ko is required to be ko = Nkv/2k = (256 x
480) x (460) / 2π(1V/rad) = 9 MHz/V. This figure is not unrealistic for 123 MHz
Translation Loops and Multiple-Loop Synthesizers
One technique used to reduce a high-frequency VCO output to reasonable
frequencies without a prescaler, and to provide a frequency offset, is shown with in
the dashed area of Figure 10-41. Figure 10-41 is the receiver block diagram for a PLL
synthesized 40-channel citizen band transceiver with delta tuning for fine-frequency
adjustments. The mixer and 35.42-MHz crystal oscillator translate the VCO
frequency range from 37.66-38.10 MHz down to 2.24-2.68 MHz for input to the
programmable divider. Notice that the reference oscillator also provides the second
LO for the double-conversion receiver.

Figure 10-41. PLL frequency synthesizer for a 40-channel CB
Multiple-loop synthesizers combine all of the techniques discussed thus far. The
addition of more loops increases the resolution and frequency coverage; the
individual loops also act as tracking filters to reduce unwanted mixer products and
spurious output components. Ideally, double-balanced mixers are used throughout.
Figure 10-42 illustrates the basic multiple-loop synthesizer for n loops. If the mixer
outputs are filtered to pass the difference frequency and the output frequency of each
VCO is lower for higher n, the synthesizer output frequency can shown to be
                      Resolution =                           (10-30)

Figure 10-42. An n-loop, multiloop synthesizer.
A variation of the multiple-loop synthesizer shown in Figure 10-43 is used in a Cubic
Communications HF-1030 AM and single-sideband receiver. The figure is incomplete
but it is clear that the resolution is 10 Hz, and 3 million discrete local oscillator
frequencies can be synthesized. The oven temperature control of the crystal-reference
oscillator for the portable HF-1030 allows the unit to maintain frequency stability
specifications of 1 ppm/month and less than 1 Hz/°C for environmental temperature

Figure 10-43. Multiloop main synthesizer for HF-1030
communications receiver.