Dynamic deflection of electrostatic MEMS

An idealized electrostatic MEMS device, consisting of a thin dielectric elastic membrane supported above a rigid ground plate, is analyzed. The upper surface of the membrane is coated with a thin conducting film. When a voltage V is applied to the thin conducting film, the membrane deflects towards the ground plate and if V is increased beyond a critical value V* (the pull-in voltage), touchdown occurs at finite time creating a so-called "pull-in instability". The mathematical model lends to a nonlinear parabolic problem for the dynamic deflection of the thin dielectric membrane. By allowing for a spatially varying permittivity profile, it is shown that the pull-in instability can be delayed until larger values of V* and that greater pull-in distances can be achieved. Applying various analytical and numerical techniques, finite touchdown time and a refined description of MEMS touchdown behavior are also discussed.