A wide range of microelectromechanical systems (MEMSs) and devices are actuated using electrostatic forces. Multiphysics modeling is required, since coupling among different fields such as solid and fluid mechanics, thermomechanics and electromagnetism is involved. This work presents an overview of models for electrostatically actuated MEMSs. Three-dimensional nonlinear formulations for the coupled electromechanical fluid–structure interaction problem are outlined. Simplified reduced-order models are illustrated along with assumptions that define their range of applicability. Theoretical, numerical and experimental works are classified according to the mechanical model used in the analysis.

https://iopscience.iop.org/article/10.1088/0964-1726/16/6/R01/pdf

Review of modeling electrostatically actuated microelectromechanical systems

A consistent one-dimensional distributed electromechanical model of an electrically actuated narrow microbeam with width/height between 0.5–2.0 is derived, and the needed pull-in parameters are extracted with different methods. The model accounts for the position-dependent electrostatic loading, the fringing field effects due to both the finite width and the finite thickness of a microbeam, the mid-plane stretching, the mechanical distributed stiffness, and the residual axial load. Both clamped–clamped and clamped-free (cantilever) microbeams are considered. The method of moments is used to estimate the electrostatic load. The resulting nonlinear fourth-order differential equation under appropriate boundary conditions is solved by two methods. Initially, a one-degree-of-freedom model is proposed to find an approximate solution of the problem. Subsequently, the meshless local Petrov–Galerkin (MLPG) and the finite-element (FE) methods are used, and results from the three methods are compared. For the MLPG method, the kinematic boundary conditions are enforced by introducing a set of Lagrange multipliers, and the trial and the test functions are constructed using the generalized moving least-squares approximation. The nonlinear system of algebraic equations arising from the MLPG and the FE methods are solved by using the displacement iteration pull-in extraction (DIPIE) algorithm. Three-dimensional FE simulations of narrow cantilever and clamped–clamped microbeams are also performed with the commercial code ANSYS. Furthermore, computed results are compared with those arising from other distributed models available in the literature, and it is shown that improper fringing fields give inaccurate estimations of the pull-in voltages and of the pull-in deflections. 1641

Electromechanical Model of Electrically Actuated Narrow Microbeams

A simple computationally efficient closed-form model has been developed to determine the pull-in voltage of a cantilever beam actuated by electrostatic force. The approach is based on a linearized uniform approximate model of the nonlinear electrostatic pressure and the load deflection model of a cantilever beam under uniform pressure. The linearized electrostatic pressure includes the electrostatic pressure due to the fringing field capacitances and has been derived from Meijs and Fokkema's highly accurate empirical expression for the capacitance of a VLSI on-chip interconnect. The model has been verified by comparing the results with published experimentally verified 3D finite element analysis results and also with results from similar closed-form models. The new model can evaluate the pull-in voltage for a cantilever beam with a maximum deviation of ±2% from the finite element analysis results for wide beams, and a maximum deviation of ±1% for narrow beams (extreme fringing field).

https://iopscience.iop.org/article/10.1088/0960-1317/15/4/012/pdf

A closed-form model for the pull-in voltage of electrostatically actuated cantilever beams

A comparison is made between various approximations of the line-to-ground capacitance problem in a VLSI environment. It is shown that with up-to-date dimensions, the simple parallel-plate model is no longer adequate. However, easy-to-use and fast-to-compute formulas exist that result in accurate and reliable capacitance values. >

/pdf/line-to-ground-capacitance-calculation-for-vlsi-a-comparison-2uneuaay8x.pdf

Line-to-ground capacitance calculation for VLSI: a comparison

In a parallel multiwire structure, the exact spacing and size of the wires determine both the resistance and the distribution of the capacitance between the ground plane and the adjacent signal carrying conductors, and have a direct effect on the delay. Using closed-form equations that map the geometry to the wire parasitics and empirical switch factor based delay models that show how repeaters can be optimized to compensate for dynamic effects, we devise a method of analysis for optimizing throughput over a given metal area. This analysis is used to show that there is a clear optimum configuration for the wires which maximizes the total bandwidth. Additionally, closed form equations are derived, the roots of which give close to optimal solutions. It is shown that for wide buses, the optimal wire width and spacing are independent of the total width of the bus, allowing easy optimization of on-chip buses. Our analysis and results are valid for lossy interconnects as are typical of wires in submicron technologies.

/pdf/maximizing-throughput-over-parallel-wire-structures-in-the-hj67w86va5.pdf

Maximizing throughput over parallel wire structures in the deep submicrometer regime

https://repository.tudelft.nl/islandora/object/uuid%3A1dde3262-a4f7-4335-80a3-3c90a202e1fa/datastream/OBJ/download

J.T. Fokkema

Papers

VLSI circuit reconstruction from mask topology