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

Abstract

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.