NIST Technical Note 1337: Characterization of Clocks and Oscillators

12 FREQUENCY AND TIME MEASUREMENT225cINPUT ~-.,t;--4-j,>--"---0 OUTPUTFIG. 12-21 Circuit diagram of the most common loop filter for a second-order phaselockedloop. Resistor R 1 is required for stable operation. Capacitor C provides the lowfrequencygain needed to reduce the phase errors of the first-order loop.FIG. 12-22 Bode plot for the loop filter of Fig. 12-21.Thus, if we know the behavior of Geq(jw), then we can relate the measuredspectrum of the voltage at the output of the phase detector or at the varaC10rtuner to the sum of the spectral densities of the phase noise of the twooscillators.The loop filter is often chosen to be a pure gain. The resulting first-orderloop has a significant drawback: the two oscillators are offset from quadratureby a phase shift proportional to their open-loop frequency difference. Inorder to maintain system calibration, the operator must remove the frequencyoffset from time to time. This problem can be eliminated by using asecond-order loop. Figure 12-21 illustrates one loop filter that can be used toachieve the desired frequency response. The transfer function of this filter isF(s) == (1 + S!2)!S!I' (12-74)where !z == RzC and !I = RIC. Figure 12-22 shows the Bode plot of thefrequency-response function of this filter. Substitution of Eq. (12-74) into Eq.(12-69) yields the open-loop frequency-response function. w~ + 2j(w n wGeq{jw) == - l' (12-75)w-whereand(12-76)(12-77)TN-95