Analytical equations for predicting injection locking in LC and ring oscillators

Abstract

In this paper, we present a analytical method for predicting injection locking in the LC and ring oscillators. Our method is very convenient for analyzing the injection locking phenomenon in oscillators, since it doesn't require the Q factor, as the Adler equation (Adler, 1946) does. We show that our analytical solution is equivalent with Adler's equation for LC oscillators. Moreover, our method can predict injection locking in ring oscillators, where Adler's equation fails since the operating mechanism is different. We demonstrate the proposed techniques on LC and ring oscillator circuits, comparing results from our analytical equations against full SPICE-level circuit simulations. Numerical experiments show our methods are able to predict injection locking in oscillators with great accuracy.