A Highly Accurate Closed-Form Model for Pull-in Voltage of Circular Diaphragms under Large Deflection

A simple easy to use highly accurate closed-form model to determine the pull-in voltage of electrostatically actuated circular diaphragms subject to large deflection has been developed. The model takes account of the nonlinear stretching of the diaphragm during large deflection, effects of residual stress, bending stress, and the effects of fringing field capacitances. At first a linearized uniform model of the nonlinear non-uniform electrostatic force has been derived that includes the effects of the fringing field capacitances. This model is then used in conjunction with the load-deflection model of a circular diaphragm subject to large deflection to derive the closed-form model for the pull-in voltage based on the condition that at unstable equilibrium, the electrostatic force is just balanced by elastic restoring force. The model has been verified by extensive 3-D electromechanical finite element analysis (FEA) using IntelliSuite. The model is in excellent agreement with 3-D electromechanical FEA with a maximum deviation of 1.6% for a wide range of residual stress and pull-in voltage values. The model leads to an integrated design strategy to optimize the electrical and mechanical design variables for MEMS-based capacitive type sensors having circular diaphragms. The model can be used to determine pull-in voltages of MEMS capacitive type pressure sensors, capacitive micromachined ultrasound transducers (CMUT) for medical diagnostic imaging, MEMS based microphones, touch mode pressure sensors, and other application areas where electrostatically actuated circular diaphragms are used.