, (E.20) Since we neglected all transistor capacitances, both IM,a() and f(.) are memoryless operators. Hence, g (A,t) can be expanded as g(A,T)= = 8&(A)cos(k (E.21) (E.22) Here, 8(k) represents the Kronecker delta function. Given (E.18)-(E.19), we extract the equations that govern the (dominant) slow-varying changes of A and 0. This is accomplished by applying the ideal lowpass averaging operator Mideal[.] (see (5.90) in section 5.6 of chapter 5) to both right-hand sides in (E.18) and (E.19). This yields dt Mideal (E.23) (E.24) as the equations governing the oscillator's averaged (slow-varying) dynamics. E.3 CONCLUSIONS E.2 Phase behavior near operating point many practical problems, the magnitude of the the injected current u(T) lies well below that of the currents flowing through the transistors, e.g. when u(T) models some injected noise current. In that case, the right-hand side of (E.23) is dominated by the first two terms. If the oscillator is stable, these terms determine a (stable) operating- point amplitude Aop, equal to the (non-zero) solution of (E.25) Because of the stability of this operating point, disturbances, like the current u(T), only cause small amplitude deviations to occur. Hence, for most practical purposes we can take A(t) 22 Aop. The phase behavior near this operating point is then (approximately) governed by (E.26) where Iosc,op = is the amplitude of the operating-point current circulating in the resonant tank. If we now compare (E.26) to (5.108) in section 5.7 of chapter 5, we find that the ISF corresponding to the source u(T) (approximately) equals (E.27) This concludes our analysis of the harmonic oscillator in Fig. E.1. E.3 Conclusions This appendix has presented expressions for the averaged amplitude and phase behav- ior of the harmonic oscillator in Fig. E. 1. This, amongst others, resulted in an approx- imate expression for the oscillator's ISF with respect to a disturbing signal u(T) that is directly injected into the resonant tank Bibliography [Abra88] R. Abraham, J. Marsden and