Self-consistent simulation and modelling of electrostatically deformed diaphragms

Design of MEMS devices which apply electrostatic forces to deformable structures is complicated by the fact that as the structure deforms, the charges redistribute, thereby modifying the mechanical loads. This paper reports three different levels of model used to simulate such structures: a very simple lumped parallel-plate-capacitor-plus-spring model, a one-dimensional numerical model based on beam and/or plate theory with loads derived from an incrementally parallel-plate capacitor, and a fully self-consistent three-dimensional numerical simulation combining finite-element structural modeling with multipole-accelerated boundary element capacitance analysis. The relative merits of each model are described, and several specific examples are examined. The lumped model has the advantage of simplicity plus a full analytic description, which permits assessment of stability of solutions. The one-dimensional numerical model, in the specific cases examined, agrees with the full 3-D model, and is computationally far less costly However, the simple structures examined here have nearly parallel conductors which remain in the small-deflection regime up to the pull-in voltage. For non-parallel conductors, and for structures which involve large deflections, the full 3-D model will be required. Furthermore, the full 3-D model is extremely valuable in identifying the limitations of the beam-theory simulation, some of which could be critical to a design.