Steady State Analysis of Crystal Oscillator Circuits Using a Self-consistent Carleman Linearization

Abstract

Crystal oscillators provide a precise signal with low phase noise in comparison to other oscillator designs. Therefore, they are commonly used for high-end applications. However, due to the high quality factor Q slow transient behavior occur. Since a sufficient small time step for a rather long simulation time is necessary, the transient simulation is very time consuming. In order to decrease the simulation time, only the envelope of the signal can be calculated. However, in this case the time behavior of the steady state cannot be determined. Alternatively, a harmonic balance simulation, which is performed in the frequency domain, can be used in order to determine the steady state behavior. In this contribution, an alternative approach in the time domain based on the Carleman linearization is presented. In combination with a self-consistent technique, which can be used for the calculation of a Poincare map, the steady state behavior of weakly damped oscillators can be determined directly by solving a real fix point problem without performing a transient simulation. Therefore, a fast steady state analysis can be performed. The proposed method is illustrated by means of a crystal Colpitts oscillator.