Abstract
A particular 1D II-order differential semi-linear elliptic model for electrostatic membrane MEMS devices, which is well-known in the literature, considers the amplitude of the electric field locally proportional to the membrane’s geometric curvature, which contains a term involving the fringing field according to Pelesko and Driscoll’s theory. Thus, in this paper, we will begin from this elliptical model, of which the uniqueness condition for the solution does not depend on the electromechanical properties of the membrane’s constituent material. In particular, after analyzing the model’s advantages and disadvantages, we present a new uniqueness condition for the solution depending on the properties listed above, which appears to be more important than the existence condition of the solution that is well-known in literature. Therefore, once the fringing field’s mode of action on the electrostatic force acting on the membrane is evaluated, suitable numerical techniques are used and compared to recover the membrane profile without ghost solutions and to propose an innovative criterion for selecting the membrane material, which depends on the electrical operative parameters and vice-versa. Finally, the possible industrial uses of the studied device are evaluated.
Highlights
A particular 1D II-order differential semi-linear elliptic model for electrostatic membrane MEMS devices, which is well-known in the literature, considers the amplitude of the electric field locally proportional to the membrane’s geometric curvature, which contains a term involving the fringing field according to Pelesko and Driscoll’s theory
Highlighting that there is no trace of terms taking into account the effects caused by the fringing field in the models studied in [19,21,23–25], the following 1D second-order differential semi-linear elliptic model with homogeneous boundary conditions has been studied according to the Pelesko–Driscoll theory [26]:
The study of a semi-linear elliptic II-order 1D differential model of a membrane MEMS device, previously described in the literature, in which |E| was shown to be locally proportional to the geometric curvature of the membrane, has been thoroughly explored here
Summary
A different situation occurs when considering an MEMS membrane device. Pel and fel assume that the membrane is at rest This hypothesis is fully justified because, for the most common membrane MEMS devices, d L. As shown in [1], the most accredited model for industrial electrostatic membrane MEMS devices is derived from the following IV-order differential dimensionless model with Dirichlet conditions, concerning two parallel metallic-plate electrostatic MEMS devices (with equal and finite thickness) in which the ground plate deforms towards the upper one: α∆2u(x) = β |∇u(x)|2dx + γ ∆u(x) +. V determines E in the device, which locally generates fel and, pel acts on the membrane, which deforms towards the upper plate, avoiding contact between them.
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