Curvature Dependent Electrostatic Field in the Deformable MEMS Device: Stability and Optimal Control

Abstract

The recovery of the membrane profile of an electrostatic micro-electro-mechanical system (MEMS) device is an important issue because, when applying an external voltage, the membrane deforms with the consequent risk of touching the upper plate of the device (a condition that should be avoided). Then, during the deformation of the membrane, it is useful to know if this movement admits stable equilibrium configurations. In such a context, our present work analyze the behavior of an electrostatic 1D membrane MEMS device when an external electric voltage is applied. In particular, starting from a well-known second-order elliptical semi-linear di erential model, obtained considering the electrostatic field inside the device proportional to the curvature of the membrane, the only possible equilibrium position is obtained, and its stability is analyzed. Moreover, considering that the membrane has an inertia in moving and taking into account that it must not touch the upper plate of the device, the range of possible values of the applied external voltage is obtained, which accounted for these two particular operating conditions. Finally, some calculations about the variation of potential energy have identified optimal control conditions.

Curvature-Dependent Electrostatic Field as a Principle for Modelling Membrane-Based MEMS Devices. A Review

“…Moreover, , regardless of the deformation of the membrane, is always locally orthogonal to the straight line tangent to the membrane at the point considered (see Figure 1 b) [ 27 ]. Thus, we consider proportional to the curvature of the membrane [ 16 , 17 , 20 , 26 ]: where is the curvature of the deformed membrane and is the proportionality function between and .…”

“…In Reference [ 26 ] authors studied whether the movement of the membrane in 1D geometry, when V is applied, admits stable equilibrium configurations. Furthermore, since the membrane has an inertia while moving and considering that it should not touch the upper plate, in [ 26 ], the range of possible values of V in 1D geometry was achieved. Finally, using concepts based on the variation of potential energy stored in the device, optimal control conditions were obtained.…”

“…Finally, using concepts based on the variation of potential energy stored in the device, optimal control conditions were obtained. In this section, we will present the main results obtained in [ 26 ] and some original results regarding 2D geometry.…”

“…In particular, by combining the physics of the problem with important results of differential geometry [ 25 ] analytical results have been obtained in terms of the existence and uniqueness of the solution in some cases [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ] and numerical solutions in the absence of ghost solutions in other cases [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ]. However, the conditions that guarantee existence and uniqueness are often independent of the electromechanical properties of the membrane material, making the studied models unattractive from an industrial point of view [ 16 , 20 , 24 , 26 ]. All these works start from the physical observation that the electrostatic field, , on the membrane is orthogonal to the tangent line to the membrane at the point considered.…”

“…Moreover, the greater the , the more the membrane will deform, thus, it appeared legitimate in these works to consider proportional to the curvature K of the membrane. This made it possible to obtain, both in 1D and 2D geometries with circular symmetries, the second order semi-linear elliptic differential models that, although not explicitly resolvable, easily allow algebraic conditions to be obtained, ensuring the existence and uniqueness of the solution [ 16 , 20 , 26 ]. Further, regarding 1D geometry, this approach allowed for a differential model wherein any singularities, typical of models known in literature, do not appear explicitly [ 16 ].…”