A General Theory of Injection Locking and Pulling in Electrical Oscillators--Part I: Time-Synchronous Modeling and Injection Waveform Design
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.
© 2019 IEEE. Manuscript received November 17, 2018; revised February 20, 2019; accepted March 25, 2019. Date of current version July 23, 2019. This paper was approved by Associate Editor Pietro Andreani. This work was supported by the Air Force Office of Scientific Research (AFOSR) under MURI Grant FA9550-16-1-0566. The authors would like to thank P. Khial and R. Fatemi of the California Institute of Technology (Caltech) for technical discussions, M. Gal-Katziri and A. White of Caltech for extensive assistance with measurements, and M. Gal-Katziri for his design of the comparator used as the Schmitt trigger in the implementation of the fabricated Bose oscillator.