king underlies a number of phase and frequency modulators. It is also the working principle for systems of coupled oscillators. In what follows, we check whether (5.107) correctly predicts injection locking to occur. 5.7.2 Practical application: oscillator injection locking Figure 5.23: SPICE simulation result of a harmonic oscillator injected with a sinewave. The oscillator's output signal (solid line) phase-locks onto the injected signal (dashed line). Both the oscillator and the injected signal are normalized to have their amplitudes (nearly) equal to 1. If the resonant tank of the harmonic oscillator in Fig. 5.22(a) has a sufficiently high Q-factor, then it is derived in appendix E that (5.110) r(t) sin(t) (5.111) Here, Vd = V1 - V2 is the differential voltage over the resonant tank's capacitor while A is a is the oscillator's fixed operating point amplitude. Furthermore, Iosc = CAWO = A/Loo is the amplitude of the current circulating in the resonant tank. Substituting (5.109) and (5.111) into (5.107) yields (5.112) In order for the right-hand side of this equation to be small, i.e. in order for the results from the perturbation analysis to be valid, it must hold that Iu/losc << 1/Q where Q is the quality factor of the resonant tank. Hence, the injected current must be much smaller than the tank current. If this is satisfied, then the equation (5.112) governs the behavior of the harmonic oscillator's phase changes induced by the injection of a sinusoidal current. In order to solve the phase equation (5.112), we apply the averaging framework dis- cussed in section 5.6. Use of the ideal lowpass averaging operator (5.90) and the as- sumption that Aw/wo << 1, yields (5.113) 5.7 OSCILLATOR PHASE (NOISE) ANALYSIS as the equation governing the averaged phase behavior ((). This equation in further simplified by introducing (5.114) If we substitute this into (5.114), then we obtain (5.115) This equation models fairly accurately what happens when a harmonic oscillator is injected with a small sinusoidal current. What a