Analytical equations for predicting injection locking in LC and ring oscillators

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.