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Published August 2019 | metadata_only
Journal Article

A General Theory of Injection Locking and Pulling in Electrical Oscillators--Part II: Amplitude Modulation in LC Oscillators, Transient Behavior, and Frequency Division


A number of specialized topics within the theory of injection locking and pulling are addressed. The material builds on our impulse sensitivity function (ISF)-based, time-synchronous model of electrical oscillators under the influence of a periodic injection. First, we show how the accuracy of this model for LC oscillators under large injection is greatly enhanced by accounting for the injection's effect on the oscillation amplitude. In doing so, we capture the asymmetry of the lock range as well as the distinct behaviors exhibited by different LC oscillator topologies. Existing LC oscillator injection locking and pulling theories in the literature are subsumed as special cases. Next, a transient analysis of the dynamics of injection pulling is carried out, both within and outside of the lock range. Finally, we show how our existing framework naturally accommodates locking onto superharmonic and subharmonic injections, leading to several design considerations for injection-locked frequency dividers (ILFDs) and the implementation of a low-power dual-modulus prescaler from an injection-locked ring oscillator. Our theoretical conclusions are supported by simulations and experimental data from a variety of LC, ring, and relaxation oscillators.

Additional Information

© 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. They would also like to thank A. Fikes of Caltech for his instrumental and diligent assistance with the measurement of the divider presented in Section VIII. His efforts included preparing the PCB, wirebonding the chip, and setting up all of the measurement equipment. Finally, they would like to thank B. Abiri of Caltech for his design of the LC QVCO used in Section VIII.

Additional details

August 19, 2023
August 19, 2023