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Published January 2019 | metadata_only
Journal Article

A Phasor-Based Analysis of Sinusoidal Injection Locking in LC and Ring Oscillators


A new perspective into the locking behavior of LC and ring oscillators is presented. By decomposing a sinusoidal injection current into in-phase and quadrature-phase components, exact expressions for the amplitude and phase of an injection-locked LC oscillator which hold for any injection strength and frequency are derived and confirmed by simulation. The analysis, which can be naturally extended to an arbitrary LC resonator topology, leads to a rigorous understanding of the fundamental physics underlying the locking phenomenon. Furthermore, an investigation of the different necessary and sufficient conditions for injection locking to occur is carried out, leading to a more precise notion of the lock range. The ring oscillator is also analyzed in an analogous fashion, resulting in simple yet accurate closed-form expressions for the fractional lock range in the small-injection and long-ring regimes; the expressions are validated by simulations of single-ended inverter-based ring oscillators in 65-nm CMOS. The mathematics behind how the injection modifies the phase delay contributed by each stage in the ring is discussed. A corollary that generalizes the small-injection lock range to any feedback-based oscillator topology is established. Conceptual and analytical connections to the existing literature are reviewed.

Additional Information

© 2018 IEEE. Manuscript received February 27, 2018; accepted July 10, 2018. Date of publication August 17, 2018; date of current version December 6, 2018. 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, both of Caltech, for technical discussions.

Additional details

August 19, 2023
August 19, 2023