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PHASE LOCKED LOOP APPLICATIONSFAST CAPTURE EXHIBITEDBY FIRST ORDER LOOPLINEAR ANALYSIS FOR LOCK CONDITION -FREQUENCY TRACKINGWhen the PLL is in lock, the non-linear capture transientsare no longer present. Therefore, under lock condition, thePLL can often be approximated as a linear control system(see Figure 8-4) and can be analyzed using Laplacetransform techniques. I n this case, it is convenient to usethe net phase error in the loop (Os - eo) as the systemvariable. Each of the gain terms associated with th,e blockscan be defined as follows:Figure 8-3EFFECT OF THE LOW PASS FILTERIn the operation of the loop, the low pass filter serves adual function: First, by attenuating the high frequencyerror components at the output of the phase comparator,it enhances the interference-rejection characteristics;second, it provides a short-term memory for the PLL andensures a rapid recapture of the signal if the system isthrown out of lock due to a noise transient. The low passfilter bandwidth has the following effects on systemperform ance:a.)b.)c.)d.)The capture process becomes slower, and thepUll-in time increases.The capture range decreases.Interference-rejection properties of the PLLimprove since the error voltage caused by aninterfering frequency is attenuated further bythe low pass filter.The transient response of the loop (the responseof the PLL to sudden changes of the inputfrequency within the capture range) becomesunderdamped.conversion gain of phase detector (volt/rad)transfer characteristic of low pass filteramplifier voltage gainveo conversion gain (rad/volt-sec)Note that, since the veo converts a voltage to a frequencyand since phase is the integral of frequency, the veofunctions as an integrator in the feedback loop.The open loop transfer function for the PLL can be writtenasKv F(s)T(s) = --swhere Kv is the total loop gain, i.e., Kv = KoKdA. Usinglinear feedback analysis techniques, the closed loop transfercharacteristics H(s) can be related to the open loopperformance asT(s)H(s) =--1 + T(s)and the roots of the characteristic system polynominal canbe readily determined by root-locus techniques.From these equations, it is apparent that the transientperformance and frequency response of the loop is heavilydependent upon the choice of filter and its correspondingtransfer characteristic, F (s).The last effect also produces a practical lim itation onthe low pass loop filter bandwidth and roll-off characteristicsfrom a stability standpoint. These points will beexplained further in the following analysis.LINEARIZED MODEL OF THE PLLAS A NEGATIVE FEEDBACK SYSTEMFigure 8-4The simplest case is that of the first order loop whereF(s) = 1 (no filter). The closed loop transfer function thenbecomesT(s)Kvs + KvThis transfer function gives the root locus as a function ofthe total loop gain Kv and the corresponding frequencyresponse shown in Figure 8-5a. The open loop pole at theorigin is due to the integrating action of the veo. Notethat the frequency response is actually the amplitude ofthe dtfference frequency component versus modulatingfrequency when the PLL is used to track a frequencymodulated input signal. Since there is no low pass filter inthis case, sum frequency components are also present onthe phase detector output and must be filtered outside ofthe loop if the difference frequency component (demodulatedFM) is to be measured.13
PHASE LOCKED LOOP APPLICATIONSROOT LOCUS AND FREQUENCYRESPONSE PLOTS OF FIRST ANDSECOND ORDER LOOPS-----Fh)0----0+jw-6----.-..... 6R,F~".R,C , l+jw-6 .6-, ,,-Figure 8-5aA1\\\ w, wThe stability problem can be eliminated by using a lag-leadtype of filter, as indicated in Figure 8-5c. This type of afilter has the transfer function1 + 72sF(s) =----1 + (71 + 72)swhere 72 = R2C and 71 = R1C, By proper choice of R2,this type of filter confines the root locus to the left halfplane and ensures stability. The lag-lead filter gives afrequency response dependent on the damping, which cannow be controlled by the proper adjustment of 71 and 72'In practice, this type of filter is important because it allowsthe loop to be used with a response between that of thefirst and second order loops and it provides an additionalcontrol over the loop transient response. If R2 = 0, the loopbehaves as a second order loop and if R2 = 00, the loopbehaves as a first order loop due to a pole-zero cancellation.Note, however, that as first order operation is approached,the noise bandwidth increases and interference rejectiondecreases since the high frequency error components in theloop are now attenuated to a lesser degree.-jwROOT LOCUSFREQUENCY RESPONSEFigure 8-5bSECOND ORDER PLL RESPONSEWITH SIMPLE LAG FILTERWith the addition of a single pole low pass filter F(s) of theform+jwF(s) =where 71 = R1C, the PLL becomes a second order systemwith the root locus shown in Figure 8-5b. Here, we againhave an open loop pole at the origin because of theintegrating action of the VCO and another open loop pole-1at a position equal to- where 71 is the time constant of71the low pass filter._jwOne can make the following observations from the rootlocus characteristics of Figure 8-5b.a.)b.)As the loop gain K v increases for a given choiceof 71, the imaginary part of the closed looppoles increase; thus, the natural frequency ofthe loop increases and the loop becomes moreand more underdamped.If the filter time constant is increased, the realpart of the closed loop poles becomes smallerand the loop damping is reduced.As in any practical feedback system, excess shifts ornon-dominant poles associated with the blocks within thePLL can cause the root loci to hend toward the right halfplane as shown by the dashed line in Figure 8-5b. This islikely to happen if either the loop gain or the filter timeconstant is too large and may cause the loop to break intosustained oscillations.FREQUENCY RESPONSEFigure8-5cIn terms of the basic gain expressions in the system, thelock range of the PLL W L can be shown to be numericallyequal to the dc loop gain2wL = 41TfL = 2Kv'Since the capture range wL denotes a transient condition,it is not as readily derived as the lock range. However, anapproximate expression for the capture range can bewritten as14
Page 2 and 3: TABLE OF CONTENTSSECTION TITLE PAGE
Page 4: SECTIONINTRODUCTIONPHASE LOCKED LOO
Page 7 and 8: USER'S QUICK-LOOK GUIDE TO SIGNETIC
Page 10: SECTIONTHE PHASE LOCKED LOOP PRINCI
Page 13: PHASE LOCKED LOOP APPLICATIONSIt is
Page 17 and 18: PHASE LOCKED LOOP APPLICATIONSINTEG
Page 19 and 20: ~PHASE LOCKED LOOP APPLICATIONSFUNC
Page 21 and 22: PHASE LOCKED LOOP APPLICATIONSFREQU
Page 24 and 25: PHASE LOCKED LOOP APPLICATIONSGENER
Page 26 and 27: PHASE LOCKED LOOP APPLICATIONSPROBA
Page 28 and 29: ~PHASE LOCKED LOOP APPLICATIONSCapt
Page 30 and 31: PHASE LOCKED LOOP APPLICATIONSTRANS
Page 32 and 33: PHASE LOCKED LOOP APPLICATIONSThe V
Page 34 and 35: PHASE LOCKED LOOP APPLICATIONSINTER
Page 36 and 37: PHASE LOCKED LOOP APPLICATIONSIt is
Page 38 and 39: PHASE LOCKED LOOP APPLICATIONSWhen
Page 40 and 41: PHASE LOCKED LOOP APPLICATIONSSIMPL
Page 42 and 43: PHASE LOCKED LOOP APPLICATIONSSCHEM
Page 44 and 45: PHASE LOCKED LOOP APPLICATIONSNARRO
Page 46 and 47: PHASE LOCKED LOOP APPLICATIONSTwo e
Page 48: PHASE LOCKED LOOP APPLICATIONSd.)e.
Page 52 and 53: PHASE LOCKED LOOP APPLICATIONSFM IF
Page 54 and 55: PHASE LOCKED LOOP APPLICATIONSPHASE
Page 56 and 57: PHASE LOCKED LOOP APPLICATIONSTRANS
Page 58 and 59: PHASE LOCKED LOOP APPLICATIONSbaud
Page 60 and 61: PHASE LOCKED LOOP APPLICATIONSFigur
Page 62 and 63: PHASE LOCKED LOOP APPLICATIONSinput
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PHASE LOCKED LOOP APPLICATIONfTRIAN
Page 66 and 67:
PHASE LOCKED LOOP APPLICATIONSRF-FM
Page 68 and 69:
PHASE LOCKED LOOP APPLICATIONSVHF O
Page 70 and 71:
PHASE LOCKED LOOP APPLICATIONSPRECI
Page 72 and 73:
PHASE LOCKED LOOP APPLICATIONSFSK D
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PHASE LOCKED LOOP APPLICATIONSThe c
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PHASE LOCKED LOOP APPLICATIONSFLUTT
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PHASE LOCKED LOOP APPLICATIONSSALES
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PHASE LOCKED LOOP APPLICATIONSINTER
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