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

This paper presents a generalized model for the pull-in phenomenon in electrostatic actuators with a single input, either charge or voltage. The pull-in phenomenon of a general electrostatic actuator with a single input is represented by an algebraic equation referred to as the pull-in equation. This equation directly yields the pull-in parameters, namely, the pull-in voltage or pull-in charge and the pull-in displacement. The model presented here permits the analysis of a wide range of cases, including nonlinear mechanical effects as well as various nonlinear, nonideal, and parasitic electrical effects. In some of the cases, an analytic solution is derived, which provides physical insight into how the pull-in parameters depend upon the design and properties of the actuator. The pull-in equation can also yield rapid numerical solutions, allowing interactive and optimal design. The model is then utilized to analyze analytically the case of a Duffing spring, previously analyzed numerically by Hung and Senturia, and captures the variations of the pull-in parameters in the continuum between a perfectly linear spring and a cubic spring. Several other case studies are described and analyzed using the pull-in equation, including parallel-plate and tilted-plate (torsion) actuators taking into account the fringing field capacitance, feedback and parasitic capacitance, trapped charges, an external force, and large displacements.