as a whole. However, when the settling behavior of the VCO is fast as compared to that of the PLL, we can make Note that this linearized model is valid since a PLL in lock prevents the VCO phase from drifting. As discussed in [Dem98, Dem00a], phase drifting renders a linearized VCO model invalid. 4.2.4 HTM-based PLL building block models use of results from oscillator phase noise theory [Dem98, Dem00a, Haji98]7 to obtain a more compact VCO model. This model accounts for the long-term phase behavior while neglecting all VCO transients. Again, it is only valid when the VCO setling behavior is sufficiently fast compared to that of the PLL as a whole. This condition, however, is satisfied for most practical PLL designs8. Let U(t) be the input, produced by the loop filter, that steers the VCO. Furthermore, we assume that U(t) can be decomposed in a large-signal DC component U0, a com- ponent (t) that captures the small periodic operating-point variations around U0 and a small-signal contribution u(t) induced, for example, by the small-signal input phase variation Oin(t). In summary, we assume (4.38) Since both Uop(t) and u(t) are small, we can use the results presented in [Dem98, Dem00a] to show that the VCO's small-signal phase variation ((t) is governed by9 (4.39) Here, T(t) is the VCO's impulse sensitivity function (ISF) associated with the input source u(t). The ISF is periodic with a period equal to that of the VCO (before fre- quency division). Hence, if the frequency-divider divides by N, this means that the pe- riod of F(t) equals T/N where T is the period of the PLL input signal Xin n(t). For most VCOs, the ISF associated with the control signal u(t) is nearly constant, i.e. T(t) 22 . In that case, (4.39) reduces to the traditional LTI model (4.12) with = Kvco. In a stable PLL, the control signal u(t) is such as to keep the VCO phase deviations ((t) small, i.e. NO/T << 1. It is therefore justified to linearize (4.39) near ((t) = 0. This results in the closed-form expression (4.40) (4.41) These results are als