Analytical Model for Comb-Capacitance Fringe Fields

Analytical expressions for electric potential and electric fringe fields in regions above the fingers of MEMS (microelectromechanical systems) comb capacitances are derived using potential-theoretic methods. The formulas are valid for the following: (1) a comb geometry exhibiting a large number of identical fingers and (2) a finger geometry where the gap between fingers is small compared to the height of the fingers and the finger overlap. For these conditions, symmetries that are inherent to the comb geometry can be exploited fruitfully to set up a properly defined Dirichlet problem formulation for the potential which can be solved for explicitly, yielding a series expansion for the electrostatic potential and electric field components. The accuracy of the approximated analytical solutions, obtained by truncating the series expansions to contain only a finite number of terms, is compared with the results obtained from finite element simulations of the electrostatic potential and electric field. From the analytic result, an approximation to the levitation force acting on the upper finger surfaces is derived. A formula expressing the mean length of the fringe electric field lines emanating from the upper finger surfaces into the ambient space is presented.