ts that are obtained by means of such an approximation for a PLL with a VCO that has a constant ISF, i.e. T(t) = . = For the PFD, we select the mixing-PFD model specified by the HTMs (4.29)-(4.31). After a little algebra, we then find (4.55) Vns)001 (4.56) Substituting (4.55) and (4.56) into (4.54) finally yields (4.57) Observing the input-output relation above, we note that the HTM element that models the phase transfers from baseband to baseband, i.e. (4.58) corresponds with the closed-loop transfer function as it is obtained by means of tradi- tional CT-LTI analysis. However, (4.57) also predicts a transfer of high-frequency in- put phase variations (varying at about twice the carrier frequency) to the low-frequency phase content at the output. These transfers are due to the fact that input phase vari- ations at twice the carrier frequency contribute at DC component at the output of the mixing PFD. They are captured by the HTM element ,((s) (4.59) = 1 Ho,o(s) 1 (4.60) 4.2 PHASE-LOCKED LOOP ANALYSIS and are unaccounted for by traditional CT-LTI analysis. However, as (4.60) shows that H0,2(s) is of the same order of magnitude as Ho,o(s), one must be careful in neglecting them. 4.2.5.2 The continuous-time LTI PLL model and its limitations The example above highlights the link between the traditional CT-LTI PLL model and the time-varying PLL models developed in this chapter: CT-LTI PLL approximation: The closed-loop PLL transfer function model obtained by means od traditional CT-LTI analysis corresponds to the HTM element Ho,o(s) of a rank-one approximation to (4.45). The conditions for the contnuous-time LTI model to yield valid results are therefore twofold: 1. The approximations (4.49) and (4.51) must hold for P a (rank-one) vector, i.e. for R = 0. Since the accuracy of the approximations drop as s -> wo/2, the LTI model is only valid at sufficiently low frequencies. 2. Since the LTI model only captures transfers from baseband to baseband, the inputs, e.g. Oin(t), should not contain any high-frequency component