Stability and Resolution Analysis of a Phase-Locked Loop Natural Frequency Tracking System for MEMS Fatigue Testing

Abstract

An analysis of a nonlinear control system that was used to track the natural frequency of a MEMS resonator is presented in this paper. A phase-locked loop system is used to track the natural frequency of the resonator due to fatigue of the spring element. A model for the control system is established and the system behavior is analyzed using an averaging method. The analysis provides a quantitative criterion for selecting the control gain to achieve stability. Simulation results are shown to be in agreement with the theoretical analysis. Tracking accuracy under the presence of Brownian noise and capacitive position sensing noise is also analyzed by using a variance propagation equation for the nonlinear dynamic system utilizing a first-order Taylor series approximation. The theoretically estimated resolution is also found to be in good agreement with simulation results.