Nonlocal problems in MEMS device control

Perhaps the most ubiquitous phenomena associated with electrostatically actuated MEMS devices is the `pull-in' voltage instability. In this instability, when applied voltages are increased beyond a certain critical voltage there is no longer a steady-state configuration of the device where mechanical members remain separate. This instability severely restricts the range of stable operation of many devices. Here, a mathematical model of an idealized electrostatically actuated MEMS device is constructed for the purpose of analyzing various schemes proposed for the control of the pull-in instability. This embedding of a device into a control circuit gives rise to a nonlinear and nonlocal elliptic problem which is analyzed through a variety of asymptotic, analytical, and numerical techniques. The pull-in voltage instability is characterized in terms of the bifurcation diagram for the mathematical model. Variations in various capacitive control schemes are shown to give rise to variations in the bifurcation diagram and hence to effect the pull-in voltage and pull-in distance.