Pull-in solutions to MEMS model of parallel plate capacitor

Abstract

In the MEMS model of a parallel plate capacitor, pull-in instability occurs when the voltage exceeds the threshold value. This phenomenon represents a performance limit in most MEMS devices, whereas microscale switches and accelerometers operate by this. Although single-degree-of-freedom (SDOF) spring–mass models are commonly employed to forecast static and dynamic pull-in, precise time-domain solutions cannot be represented by elementary functions. Current analyses typically either linearize the governing nonlinear ordinary differential equation or rely on numerical simulations and semi-analytic solutions. This thesis presents an implicit analytical pull-in solu tion for the pull-in time of the undamped, constant-voltage single-degree-of-freedom model, expressed clearly in relation to complete elliptic integrals. We proceed by developing two complementary Puiseux-series expansions for the transient trajectory. The first expansion is derived using the Lagrange inversion theorem in relation to the touchdown point, while the second is aligned with the zero-initial-condition deriva tives through a truncated series and a linear system. Both series illustrate the final approach to collapse through the use of fractional exponents, shedding light on the singularity structure of the solution. The use of high-precision solvers for numerical integration demonstrates a strong alignment between the implicit formula and the exact trajectory, with L2 errors remaining below 1e − 9 throughout the entire pa rameter range of K > 1 8. The Puiseux series approximation achieves an accuracy of less than one percent for K ≥ 05, and when used together, they offer a practical and clear approximation throughout the entire transient period. This thesis combines elliptic-integral solutions with convergent series in the neighborhood of the pull-in, providing a complete analytical toolkit for rapidly predicting the pull-in dynamics in standard MEMS capacitors. This advancement allows for significant reductions in computational costs and improves our understanding of nonlinear collapse, which is crucial for effective device design and optimization.