. The open-loop gain of the CT-LTI approximation then becomes A(s)= (4.68) The factor wo/2 in front arises from the sampling-PFD model (4.35). Fig. 4.13 shows a typical gain characteristic. It contains three poles (the first two at DC) and one zero. This characteristic will be used for all further numerical computations. Using (4.61) and (4.62), we find the PLL's open loop gain HTM to equal A(s-jwo) G(s)=V(s).W(s) 1 1 1 (4.69) A(s+jwo) 4.2 PHASE-LOCKED LOOP ANALYSIS Substituting this into (4.66) and (4.67), the input-output HTM H(s) is found to equal A(s-jwo) (4.70) A(s+jwo) Here, (4.71) is the PLL's effective open-loop gain. The HTM element ,((s), modeling the closed- loop transfers from baseband to baseband, is given by (4.72) The latter approximation, corresponding to traditional LTI analysis, is valid as long as A(s) >> (s+jmwo). If this no longer holds, e.g. for feedback loops with large unity-gain frequency WUG, time-varying effects become important and the other terms of X(s) must be taken into account. Note that (4.70) also allows us to study signal transfers to frequency bands other than baseband. Fig. 4.14 illustrates the impact of an increasing WUG/WO on Ho,o(s). The solid lines are obtained by evaluating (4.72) while the marks follow from spectral analysis of time- marching simulation results in Matlab/Simulink. They are within 2%. This shows the accuracy ofthe methods here developed. Note, however, that evaluating (4.72) is only a matter of seconds while it takes several minutes for the time-marching simulations to complete. 4.2.6.2 Stability behavior Observing Ho,o(s) in Fig. 4.14, it is seen that the effective bandwidth shifts to the right as WUG/WO increases. Also, peaking at the passband's edge becomes worse. Fig. 4.15explains this behavior in terms of the effective open-loop gain X(s). The increase of the closed-loop bandwidth corresponds to an increase of WUG,eff, the (ef- fective) unity-gain frequency of X(s). More important is the phase margin of X(s) which is rapidly degrading for increasing