A methodology and model for the pull-in parameters of electrostatic actuators

This paper presents a generalized model for the pull-in phenomenon in elect rostatic actuators with a single input, either charge or voltage. The pull- in phenomenon of a general electrostatic actuator with a single input is re presented by an algebraic equation referred to as the pull-in equation. Thi s equation directly yields the pull-in parameters, namely, the pull-in volt age or pull-in charge and the pull-in displacement. The model presented her e permits the analysis of a wide range of cases, including nonlinear mechan ical effects as well as various nonlinear, nonideal, and parasitic electric al effects. In some of the cases, an analytic solution is derived, which pr ovides physical insight into how the pull-in parameters depend upon the des ign and properties of the actuator. The pull-in equation can also yield rap id numerical solutions, allowing interactive and optimal design. The model is then utilized to analyze analytically the case of a Duffing spring, prev iously analyzed numerically by Hung and Senturia, and captures the variatio ns of the pull-in parameters in the continuum between a perfectly linear sp ring and a cubic spring. Several other case studies are described and analy zed using the pull-in equation, including parallel-plate and tilted-plate ( torsion) actuators taking into account the fringing field capacitance, feed back and parasitic capacitance, trapped charges, an external force, and lar ge displacements.