A General Theory of Injection Locking and Pulling in Electrical Oscillators—Part I: Time-Synchronous Modeling and Injection Waveform Design

B. Hong, A. Hajimiri

A general model of electrical oscillators under the influence of a periodic injection is presented. Stemming solely from the autonomy and periodic time variance inherent in all oscillators, the model’s underlying approach makes no assumptions about the topology of the oscillator or the shape of the injection waveform. A single first-order differential equation is shown to be capable of predicting a number of important properties, including the lock range, the relative phase of an injection-locked oscillator, and mode stability. The framework also reveals how the injection waveform can be designed to optimize the lock range. A diverse collection of simulations and measurements, performed on various types of oscillators, serve to verify the proposed theory.