Modeling Methodology for a CMOS-MEMS Electrostatic Comb

A methodology for combined modeling of capacitance and force in a multi-layer electrostatic comb is demonstrated in this paper. Conformal mapping-based analytical methods are limited to 2D symmetric cross-sections and cannot account for charge concentration effects at corners. Vertex capacitance can be higher than 30% of the total capacitance in a single-layer 2 μm thick comb with 10 μm overlap. Furthermore, analytical equations are strictly valid only for perfectly symmetrical finger positions. Fringing and corner effects are likely to be more significant in a multi-layered CMOS-MEMS comb because of the presence of more edges and vertices. Vertical curling of CMOS-MEMS comb fingers may also lead to reduced capacitance and vertical forces. Gyroscopes are particularly sensitive to such undesirable forces, which therefore, need to be well-quantified. In order to address the above issues, a hybrid approach of superposing linear regression models over a set of core analytical models is implemented. Design of experiments is used to obtain data for capacitance and force using a commercial 3D boundary-element solver. Since accurate force values require significantly higher mesh refinement than accurate capacitance, we use numerical derivatives of capacitance values to compute the forces. The model is formulated such that the capacitance and force models use the same regression coefficients. The comb model thus obtained, fits the numerical capacitance data to within 3%and force to within 10%.The model is experimentally verified by measuring capacitance change in a specially designed test structure. The capacitance model matches measurements to within 10%. The comb model is implemented in an Analog Hardware Description Language (AHDL) for use in behavioral simulation.