Injection locking/pulling analysis of oscillators under fractional excitation frequency

A method to analyze the injection locking and injection pulling of arbitrary oscillators under small excitation is proposed. The most general case when the excitation frequency is close to a rational fraction of the oscillator fundamental is considered. The phase differential equation for an arbitrary periodic excitation is derived. Expressions to evaluate characteristics of the oscillator in injection locking and injection pulling modes are obtained. The characteristics include the locking range, the waveforms of the phase and the instantaneous frequency, the dependencies of the beat frequency and output spectrum on the excitation frequency. The circuit example is presented to illustrate the application of proposed approach.