A Study of Locking Phenomena in Oscillators

TLDR

External signals of similar frequency modulate an oscillator’s amplitude and frequency. The study aims to describe the pull‑in transient and distorted beat generation using the derived equation. A differential equation for the oscillator phase is derived under the assumption that circuit time constants are much shorter than a beat cycle. The equation also models a pendulum in viscous fluid within a rotating container, illustrating 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.