Macro-Modeling Technique of Capacitive MEMS Based on the Implicit Condensation Method and Lagrange Principle

System-level macro-models play a crucial role in the design and optimization of microelectromechanical systems (MEMSs). However, accurately capturing coupling effects and nonlinearities is currently a significant challenge. To address the issues, a hybrid macro-modeling technique is developed by a combination of implicit condensation and conservative system Lagrange methods. For nonlinear electrostatic force, the electrostatic potential energy term in the Lagrange equation of the conservative system is obtained by fitting the capacitance function after a series of static simulations. Regarding stiffness estimation, polynomial stiffness terms are derived through implicit condensation method rather than relying on strain energy sampling, which is applicable to a variety of in-plane structures. Instead of calculating complex analytical formulations of squeeze and sliding film damping, computational fluid dynamics (CFDs) numerical simulation is employed to determine the modal damping ratios, avoiding analytical estimation of complex geometries and enhancing the modeling automation. Using double-end fixed beam and capacitive accelerometer as examples, the effectiveness of the proposed macro-modeling technique is illustrated. Compared with full finite element (FE) simulation, the overall error of the macro-model is less than 1.6%. System-level simulation and experimental results indicate that the error of accelerator sensitivity is less than 5.0%, excluding the effects of parasitic capacitance. The technique benefits from being non-intrusive, which means that it can be implemented with most commercial FE software without requiring access to FE core code.