This study is devoted to finding the precise pull-in voltage/position of a micro-device formed by two parallel charged plates. Pull-in is a phenomenon where the electrostatic force induced by the applied voltage across two plates of the device exceeds the elastic, restoring force exerted by the deformed plates, leading to a contact between the two plates. To offer a precise prediction of the pull-in, a dynamic model in the form of a partial differential equation (PDE) is established based on the equilibrium among plate flexibility, residual stress and distributed electrostatic forces. The Galerkin method is employed to decompose the established PDE into discrete modal equations. By considering lower order modes and solving them, one arrives at a prediction of plate deflection in terms of the applied bias voltage. Approximating the solved deflection by a fifth-order series and full-order numerical integration, the pull-in position and voltage are successfully approximated. The pull-in position in terms of center deflection of the deformed plate is found to be 48% of the air gap between the plates, which presents a better estimation than the commonly used one-third of the gap derived by all past studies based on a less realistic one-dimensional lump model. A closed form of the pull-in voltage is derived to offer design guidelines for the device prior to production. The aforementioned theoretical findings are finally validated by finite element and experimental studies on a MEMS device of parallel charged micro-plates designed and fabricated in the laboratory.

We study the low-density limit of the Nagel-Schreckenberg model with a large speed limit. Three distinct regions in the fundamental diagram are identified. Analytical approximations are obtained. The dependence on three parameters ͑speed limit v max , random braking p, and lattice size L͒ is discussed. The relation to finite-size effect is also discussed.