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324 Analog and Digital Filter Design




                 loop Filters

                       In  both  frequency multiplier and  FM  signal  demodulator  applications, the
                       purpose of the loop filter is to average the phase detector output voltage. It must
                       do this while allowing the system to respond to changes in the reference signal’s
                       frequency. The phase detector will output some spurious signals at the reference
                       frequency. The filter must remove these signals before they reach the oscillator.
                       Otherwise the oscillator will be modulated unnecessarily and produce jitter at
                       its output. The loop filter is thus a critical part of  the phase-locked loop circuit.

                       There are two simple loop filters: a first-order lowpass CR network and a lead-
                       lag network. The lead-lag network is similar to the CR network except that it
                       has a resistor in series with the shunt capacitor. These are illustrated in Figure
                        13.4. The CR network is simple and its performance is not dependent on the
                       value of capacitor or resistor, but is determined by the product CR. The lead-
                       lag network  gives the  designer more  control  over  the  Performance, such as
                       damping, cutoff, and natural frequency.  It does, however, have the disadvantage
                       of  limited stopband attenuation.


                                               CR Network            Lead-Lag Network
                                                 R






                 Figure 13.4
                 Simple Loop Filters


                       The design equations for an RC network follow. The filter cutoff  frequency is
                       simply:

                             wLp = lIRC, in rads

                       The natural  frequency of  the phase-locked loop depends on the phase com-
                       parator  gain, K$, and the oscillator gain, KO, as well as the cutoff  frequency.
                       The natural frequency is the rate  of  oscillation that  occurs when the phase-
                       locked loop’s reference input signal is suddenly changed (assuming that it is not
                       over damped):

                             m,? = d(K$. KO. mLd, this is also in radls.
                       The transfer function for an RC network is given by:
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