An Exact Solution of Injection-Phase-Locking

Abstract

Recent advances of solid-state device technology of generating microwave power from low voltage dc power in one step have renewed interest in the study of injection locking. The solid state millimeter-wave devices have many potential applications such as parametric amplifier pumps, transponder sources, local and self-test oscillators, and high bit rate millimeter-wave repeater systems. The purpose of this article is to solve the nonlinear differential equation of injection locking. Using the method of Riccati's equation, an exact solution has been obtained which is much simpler and more explicit than that shown by Mackey. This article shows the tracking and acquisition behavior of the loop for different initial phase offsets and for different ratios of initial frequency offset D to loop gain B. It also compares the transient and steady-state responses with the exact solution and the linear approximation solution. This article concludes that the difference of steady-state responses obtained from the exact solution and the linear approximation solution will be greater if the D/B ratio is greater.