Closed-form solutions of the parallel plate problem

Closed-form solutions to the parallel plate problem have been derived for design of electrostatic devices that employ the parallel plate. With dimensionless height and force introduced to simplify the nonlinear parallel plate problem, a simple cubic equation implying behavior of the height of the movable plate corresponding an applied voltage has been derived and theoretically solved to provide closed-form solutions of the movable plate height, effective stiffness, resonant frequency, capacitance and their sensitivities to voltage. The theoretical height agreed well with experimental data obtained from a surface-micromachined parallel plate. When the applied voltage approaches the pull-in voltage, the height of the movable plate reaches 2/3 of the initial height, the effective stiffness and resonant frequency go to zero and the capacitance becomes 3/2 times the initial capacitance. These closed-form solutions can be used to analyze and design micro- and nano-devices employing electrostatic parallel plates.