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

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 esti- mated resolution is also found to be in good agreement with simulation results. @DOI: 10.1115/1.1514658# 1. Large strains in the test samples can only be achieved at resonance. The MEMS resonators are lightly damped and have high quality factors Q (;100 in air and in the range of 10 4 -1 0 5 in vacuum!. Thus, the large displacements needed to produce sufficient deformation can only be achieved at resonance, i.e., at a frequency very close to the natural fre- quency of a lightly damped system. 2. The natural frequency decay is used to determine the change of the stiffness of the test samples due to the evolution of fatigue damage. During cyclic loading, grains in the test samples may be damaged, causing the effective stiffness to decrease, and, in turn, the natural frequency of the resonator to decay. Therefore, the characteristics of the fatigue process can be studied by observing the evolution of the natural frequency. Phase-locked loops ~PLLs! have been previously used to track the natural frequency of a vibrating resonator in MEMS fatigue testing experiments @3-5#. However, rigorous analysis of the sta- bility and performance of this control scheme has not been re- ported in the literature. The principal objective of this study, there- fore, is to quantitatively analyze the stability and resolution of this PLL control scheme. A fifth-order nonlinear dynamic model of the entire system including the PLL, resonator, and driving and sens- ing circuits is presented, and the nonlinear dynamic system is examined by developing a simplified ~although still nonlinear! system of dynamic equations. It is demonstrated that excitation at the natural frequency yields a stable closed orbit of the original system, provided a certain criterion for the control gain is satis- fied. Simulation results for the original system that verify the va- lidity of the analysis are discussed, and the tracking accuracy under the presence of Brownian noise and capacitive position sensing noise is interpreted in light of the obtained results. The tracking resolution is estimated by approximating the nonlinear variance propagation equation by a first-order Taylor series. The natural frequency tracking resolution at steady state is obtained by solving a Lyapunov equation.