Analytical Deflection Profiles and Pull-In Voltage Calculations of Prestressed Electrostatic Actuated MEMS Structures

This paper presents a zeroth and first order perturbative analysis of prestressed electrostatic actuated Microelectromechanical systems (MEMS). Perturbation theory is used to calculate the deflection profile of various MEMS structures, from which the pull-in voltage is estimated using the weighed residual method, where both a Galerkin expression and a Dirac delta function have been used as weight functions. A prestressed circular Capacitative Micromachined Ultrasonic Transducer (CMUT) is used as the main example in this paper. This device is modeled as a circular clamped plate subjected to an electrostatic pressure. The calculated pull-in voltage has been compared with experimental data of highly prestressed CMUTs, where a relative error between −36% and −8% is observed for a model that does not include stress. The model that includes the residual stress lowers the range of the relative error to values between −5% and 21%. To improve the accuracy of the pull-in voltage estimate Richardson extrapolation has been calculated from the zeroth and first order estimates. The pull-in voltage models are compared with a Finite Element Model (FEM), where an overestimation, in the high stress regime, of 10% is observed for the zeroth order model, less than 5% for the Galerkin method and 3% for the Richardson extrapolation.