phase margin is already 9% worse than predicted by CT-LTI analysis. This makes clear that CT-LTI analysis does not give reliable results when WUG/WO grows large. Expressions for this degarding phase margin are obtained by assuming that, for w near WUG, X(jw) is dominated by A(jw), i.e. << |A(jw)|. With = phase then approximately equals (4.73) 4.2.6 Example 1: PLL with sampling PFD W/WUG Figure 4.14: HTM element H0,0(s) capturing baseband-to-baseband sig- nal transfers for a PLL with sampling-PFD. The element is plotted for WUG/WO = 0.01, 0.1 and 0.15. The solid lines are obtained by evaluating (4.70). The marks are extracted from Matlab/Simulink simulations. "UG!" 0 Figure 4.15: The normalized effective unity-gain frequency (upper plot) and the effective open-loop phase margin (lower plot) versus WUG/WO as obtained by means of the methods developed in this work. In the lower plot, the dashed horizontal line indicates the phase margin as predicted by CT-LTI analysis. 4.2 PHASE-LOCKED LOOP ANALYSIS A(jw) |A(jw-jwo) |A(jw) |A(jw-jwo) A(jWuGj-wo) A(jw UG j-w0) Figure 4.16: Impact of the position of WUG on the stability of PLLs with sampling-type PFDs. (a) For low WUG, , aliasing terms have only a low impact on the phase margin which is then largely determined by the phase margin of A(jw). (b) As WUG increases, the impact of the aliasing grows, degrading the phase margin. Note that in the figures above, for sake of simplicity, we approximated (joo+ jmoo) for W near WUG. The second term In the expression above explains the problems that arise when the unity-gain frequency of A(jw) approaches wo/2. As illustrated in Fig. 4.16, for an increasing WUG, the second term in (4.73) starts to yield a significant contribution on top of the phase shift generated by A(jw). This pushes the total phase shift of the open-loop gain 2(jw) towards - resulting in a less stable feedback loop. 4.2.6.3 Noise folding in PLLs Stability, however, is not the only thing that is going bad as WUG increases. For fast feedback loops, phase noise beh