Multiple-Degree-of-Freedom Modeling and Simulation for Seismic-Grade Sigma–Delta MEMS Capacitive Accelerometers

Abstract

The high-order mechanical resonances of the sensing element in a high-vacuum environment can significantly degrade the noise and distortion performance of seismic-grade sigma–delta MEMS capacitive accelerometers. However, the current modeling approach is unable to evaluate the effects of high-order mechanical resonances. This study proposes a novel multiple-degree-of-freedom (MDOF) model to evaluate the noise and distortion induced by high-order mechanical resonances. Firstly, the MDOF dynamic equations of the sensing element are derived using the principle of modal superposition and Lagrange’s equations. Secondly, a fifth-order electromechanical sigma–delta system of the MEMS accelerometer is established in Simulink based on the dynamic equations of the sensing element. Then, the mechanism through which the high-order mechanical resonances degrade the noise and distortion performances is discovered by analyzing the simulated result. Finally, a noise and distortion suppression method is proposed based on the appropriate improvement in high-order natural frequency. The results show that the low-frequency noise drastically decreases from about −120.5 dB to −175.3 dB after the high-order natural frequency increases from about 130 kHz to 455 kHz. The harmonic distortion also reduces significantly.

A Study on the Harmonic Distortion of Seismic-Grade Sigma-Delta MEMS Accelerometers Using a Multiple Degree-of-Freedom Model

“…Detailed expressions of the other parameters, including the stiffness, inertial forces, and rest electrostatic force, etc., are all presented in the previous study [23].…”

“…Based on the MDM, the sigma-delta system for simulating the harmonic distortion of accelerometers is established using Simulink, as shown in Figure 2. As in the previous study [23], the system consists of an MEMS sensing element, a displacement-voltage converter, a zero-order holder, a lead compensator, a third-order sigma-delta modulator, and an electrostatic force block. The only difference is that the simulating system for the harmonic distortion system contains parasitic capacitance mismatch.…”

“…In this section, the nonlinear relationship between the input acceleration and output signal is derived based on the MDM established in our previous study [23]. By ignoring the acceleration and velocity terms of Equation (19) presented in [23], the steady-state equation of the sensing element is expressed as follows:…”

“…and Fes1 and Fes2 denote the electrostatic forces applied to the static fingers belonging to C1 and C2, respectively. The expressions of Qp and the electrostatic forces givenin the previous study [23] are rewritten here The expressions of Q p and the electrostatic forces givenin the previous study [23] are rewritten here…”

“…Studies focusing on harmonic distortion are all based on the single degree-of-freedom model (SDM) of the sensing element. In our previous study [23], a multiple degree-offreedom model (MDM), including finger flexibility, was established to analyze the noise. However, in this case, harmonic distortion was not investigated in detail.…”