 oscillator): This example derives the aver- aged equations that describe the behavior of the amplitude, phase and common-mode level of the voltage-controlled oscillator in Fig. 5.9. Its intent is to illustrate the proce- dure to do SO and to provide some numerical results that demonstrate the accuracy of the behavior predicted by these averaged equations. Note that further numerical results are presented in section 5.8. In previous examples, the behavior of the oscillator's state vector p = [ A Z ] superscript(r) was shown to be governed by (5.79)-(5.81). The right-hand sides of these equations are 2-periodic in T. Hence, applying the ideal lowpass averaging operator (5.90) cor- responds to selecting the DC component of their Fourier series expansion with respect to T. This is equivalent to the use of the averaging operator (5.89) for TM = 2 The averaged equations associated with (5.79)-(5.81) are then found to equal (5.92) (5.93) (5.94) Note that none of the averaged right-hand sides depends on the phase 0. This is due to the periodic dependence on T of all right-hand side terms in (5.92)-(5.94) combined with the fact that 0 always appears in conjunction with T as T + 0. Fig. 5.20 demonstrates the accuracy of the results obtained from (5.92)-(5.94) by com- paring the predicted amplitude and common-mode startup behavior with results ob- tained by a direct (SPICE-like ¹² solution of the original circuit equations (5.28)-(5.32). The computations are performed for an oscillator with Q 2 8. The results in Fig. 5.20 clearly show that (5.92)-(5.94) accurately predicts the behavior of the oscillator's av- eraged amplitude and common-mode behavior. Moreover, the lower plot in Fig. 5.20 Superscript(2)In this text, a "SPICE-like" solver corresponds either to a Runge-Kutta method or an algorithm based on numerical differentiation formulas. Figure 5.20: Oscillator startup behavior as respectively obtained from the av- eraged equations (5.92)-(5.94) (thick slow-varying envelopes/line) and from the original equations (5.28)-(