Small-Signal Analysis of Oscillators Using Generalized Multitime Partial Differential Equations

Standard small-signal analysis methods for circuits break down for oscillators because small-input perturbations result in arbitrarily large-output changes, thus invalidating fundamental assumptions for small-signal analysis. In this paper, we propose a novel oscillator ac approach remedying this situation, thus restoring validity and rigour to small-signal analysis of oscillators. Our approach centers around a novel general equation formulation for circuits that we term the Generalized Multitime Partial Differential Equations (GeMPDE). While this formulation is broadly applicable to any kind of circuit or dynamical system, we show that it has unique advantages for oscillators in that small-input perturbations now lead to small output ones, thus making small-signal analysis valid. A key feature of our approach is to solve for bivariate-frequency variables with the help of novel augmenting-phase-condition equations. Unlike prior oscillator-analysis methods, which require special handling of the phase mode, our GeMPDE-based small-signal analysis provides both amplitude and frequency characteristics in a unified manner and is applicable to any kind of oscillator described by differential equations. We obtain speedups of 1-2 orders of magnitude over the transient-simulation approach commonly used today by designers for oscillator-perturbation analysis