Nonlinear Analysis of Differential LC Oscillators and Injection Locked Frequency Dividers

Abstract

A comprehensive nonlinear analysis of autonomous and periodically forced fully-differential, negative-resistor LC oscillators is presented. Through nonlinear transformations in the state space, it is shown that oscillators within this class exhibit qualitatively similar dynamical behavior in terms of their limit cycles and bifurcation curves, at least within an open region containing the origin. The case of autonomous, complementary BJT oscillators is used to validate the qualitative analysis and demonstrate a general approach of how to numerically extend the bifurcation curves away from the equilibrium point and determine the oscillatory conditions. When external periodic force is present, we focus on the special case of periodically multiplicatively-forced fully-differential, negative-resistor, LC oscillators and use Harmonic Balance techniques to derive analytical expressions estimating the locking range in the weak injection regime. The results are used to calculate the locking range of a harmonically forced complementary BJT oscillator yielding explicit expressions closely aligned with experimental measurements, thus verifying the validity of the analysis.