Behavioral modeling and testing of a CMOS-MEMS parametric resonator governed by the nonlinear Mathieu equation

Modeling and simulation of complex phenomena in environments that emulate end applications demonstrate the effectiveness of MEMS composable design methodologies. A CMOS-MEMS nonlinear resonator featuring parametric excitation driven by an oscillating voltage applied across the non-interdigitated comb fingers has been targeted as a demonstration vehicle. This paper reports the schematic-based parameterized behavioral modeling and vibration testbed of parametric resonators governed by the nonlinear Mathieu equation. The linear and cubic stiffness of the electrostatic force and the folded flexure are modeled by Verilog-A MEMS behavioral models. The transition frequencies and the jump amplitudes are characterized optically by sweeping the frequency bi-directionally. The observed parametric resonance, occurring near excitation frequency of twice the resonant frequency, is verified by perturbation solution of Mathieu equation, and validated by circuit-level behavioral model simulation matching to 0.6%. The dynamic behavior and its dependence on system parameters are also analyzed.