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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