bout injection locking? Well, if (5.116) then the right-hand side of (5.115) contains a stable fixed point Yop = arccos (5.117) By means of (5.114), we then find - arccos (5.118) As a result, the oscillator's effective oscillation frequency equals (5.119) Hence, the effective (averaged) oscillation frequency locks onto that of the injected sinewave. Here, (5.116) provides an expression for the locking range, i.e. the range of input-frequencies for which frequency locking is accomplished. Also, when in lock, the phase difference between the oscillation signal and the injected signal remains constant at /2 - arccos Again, note the simplicity whereby these symbolic expressions are derived by means of the methods here presented. Fig. 5.24 plots the averaged phase difference between the oscillation signal and the in- jected signal as solved from (5.113) for Aw = 0. The initial phase difference gradually disappears as the oscillator locks onto the injected sinewave. 5.7.3 Averaging in the presence of random perturbations The previous example illustrates how averaging facilitates solving the phase equation (5.107) (or equivalently (5.105)) for deterministic input signals. Such a simplification is also possible when the injected signals are random in nature. In what follows, we discuss oscillator phase noise theory. Phase noise involves an oscillator's random phase shifts due to injection of noise in the circuit. As illustrated in Fig. 5.25, this results, for 5.7.3 Averaging in the presence of random perturbations Figure 5.24: The (averaged) difference between the phase of the harmonic oscillator and that of the injected signal versus (normalized) time. Here, the injected signal runs at the oscillator's free-running frequency (Aw = 0). eu(t) Figure 5.25: Phase noise involves an oscillator's random phase shifts due to injection of noise in the circuit. This results, for instance, in an ever increas- ing uncertainty on the positions of the zero crossings. 5.7 OSCILLATOR PHASE (NOISE) ANALYSIS instance, in an ever increasin