A study of locking phenomena in oscillators

TLDR

An external signal of similar frequency alters an oscillator’s amplitude and frequency. Assuming circuit time constants are small compared to a beat cycle, the authors derive a differential equation for the oscillator phase over time. The derived equation explains pull‑in transients, distorted beat notes, and, via a simple mechanical pendulum model, illustrates the full spectrum of locking phenomena.

Abstract

Impression of an external signal upon an oscillator of similar fundamental frequency affects both the instantaneous amplitude and instantaneous frequency. Using the assumption that time constants in the oscillator circuit are small compared to the length of one beat cycle, a differential equation is derived which gives the oscillator phase as a function of time. With the aid of this equation, the transient process of "pull-in" as well as the production of a distorted beat note are described in detail. It is shown that the same equation serves to describe the motion of a pendulum suspended in a viscous fluid inside a rotating container. The whole range of locking phenomena is illustrated with the aid of this simple mechanical model.