NIST Technical Note 1337: Characterization of Clocks and Oscillators

226 SAMUEL R. STEIN10010FIG. 12-23 Bode plOI ofthe open-loop frequency-response function ror a phase-locked loophaving the loop filter of Fig. 12-21. Parameters were chosen to illustrate a stable condition.The first requirement to be satisfied by the loop parameters is that theclosed loop be stable. Since the transfer function G.q(s) has no poles or zerosfor s > 0, a sufficient requirement for the phase-locked loop to be stable isthat the slope of the Bode plot of IGcqUw)! be less steep than -12 dB/octave atthe point where IGcqUw)1 = 1. The Bode plot of /G.qUw)1 is shown in Fig.12-23 for a case where the loop operation is stable.I t is desirable for the loop to be nearly critically damped, that is, s= 1. Atcritical damping the natural frequency of the loop is related to '2 bywwn.~=t = 2/'2' (12-78)Under the same conditions the unity gain frequency is(12-79)The second requirement to be satisfied by the phase-locked loop is relatedto the accuracy with which spectral-density measurements can be made.Substitution of Eq. (12-75) into Eq. (12-72) yieldsK~W4S~)CJ) = (2 2)2 + 4- 2 1 2 [S4>.e'(W) + S",Jw)]. (12-80)w - W n ~ W W nSince the proportionality factor has a high pass response, it is possible to usean essentially constant calibration to relate Sva(w) and S.p(w). For example, ifwe require that(12-81)with no more than 1O~~ error for all Fourier frequencies greater than2n rad/sec, then for the critically damped loop the requirement on '2 is'2 > 1.4 sec.TN-96