Fringe-Field Actuation of MEMS Cantilever Resonators: Capacitance Model for Frequency-Amplitude Response of Parametric Resonance
Authors: Martinez Miguel A. , Caruntu Dumitru I. 
Published: November 2024

Abstract
The paper investigates the frequency-amplitude response of micro-electromechanical system (MEMS) cantilever beams, which are actuated by the fringe field at parametric resonance. In this design, a micro cantilever beam (flexible electrode) and a ground plate (fixed electrode) are placed in parallel and connected through an AC voltage. The electrostatic force is due to electric field lines within the volume directly between the cantilever and the ground plate. The parallel-plate capacitance is directly related to the overlapping area of the two electrodes. However, to accurately compute the capacitance of the two electrodes, the effects due to the fringing field must be accounted for. The fringe field or fringe effect involves the bending of electric field lines around the edges of the cantilever. In this case, the fringe field lines go from the ground plate to the top and side walls of the cantilever beam. For the present study, the cantilever beam is actuated by the fringe effect only. This is modeled by having a hole in the ground plate directly below the cantilever beam. The hole allows the electrostatic force (electric field directly below cantilever) to be neglected. An empirical formula is used to approximate the fringe capacitance. This formula considers contributions from the top and sidewalls of the cantilever beam. Note that, the electrostatic term of the capacitance formula is isolated and neglected so that the fringe field becomes the only actuating force. This work focuses on parametric resonance, in which the frequency of the AC voltage applied is near the natural frequency of the cantilever beam. Two Reduced Order Models (ROMs) are developed from the partial differential equation of motion. The Method of Multiple Scales (MMS), a perturbation method, is applied to analytically solve the formulated model. A ROM using one mode of vibration is solved using MMS. The frequency-amplitude response (bifurcation diagram) is then predicted by MMS and ROM. The fringe actuation forces are expanded in Taylor series and the terms up to the 3rd and 5th powers are retained. MMS predicts three solutions: a zero-amplitude (trivial) solution and two non-zero solutions which consist of two branches, one stable and the other unstable. The stable and unstable branches intercept the x-axis at the super- and sub-critical bifurcation points, respectively. Moreover, a reduced order model to include two modes of vibration (five-term ROM) is derived and numerically solved. The equations obtained from the five-term ROM are then implemented into AUTO-07p, a continuation and bifurcation software for ODEs, to generate a frequency response. The frequency responses from MMS and 2T ROM are then compared with each other. In addition, the response due to fringe actuation is compared with other approaches that include electrostatic actuation only, as well as electrostatic and fringe combined. The study also explores the influence of damping, fringe, and detuning frequency on the frequency-amplitude response.