# A two-dimensional electrostatic model of interdigitated comb drive in longitudinal mode

## TL;DR

This paper develops a homogenization model for the electrostatic behavior of interdigitated comb drives in longitudinal mode, providing a rigorous mathematical framework to analyze electrostatic forces in microelectromechanical systems.

## Contribution

It introduces a two-scale convergence-based homogenization model for in-plane electrostatic forces in comb drives with many fingers, considering specific geometric scaling.

## Key findings

- Derived a limit model for electrostatic forces in comb drives
- Established strong convergence results for the homogenization process
- Provided a mathematical framework for analyzing longitudinal electrostatic interactions

## Abstract

A periodic homogenization model of the electrostatic equation is constructed for a comb drive with a large number of fingers and whose mode of operation is in-plane and longitudinal. The model is obtained in the case where the distance between the rotor and the stator is of an order $\varepsilon ^{\alpha }$, $\alpha \geq 2$, where $\varepsilon$ denotes the period of distribution of the fingers. The model derivation uses the two-scale convergence technique.\ Strong convergences are also established. This allows us to find, after a proper scaling, the limit of the electrostatic force applied to the rotor in the longitudinal direction.