Deflection of an Eccentric Tooth of a Comb Drive in an Electrostatic Field

The elastic deflection of a comb drive tooth in an electrostatic field is considered. The tooth can be symmetrically located between two rigid teeth of the matching comb, in which case the problem reduces to a pure bifurcation problem for which the critical voltage can be determined. Alternatively, due to an approximate straight-line mechanism, the tooth can have a uniform initial lateral displacement and a smooth curve of equilibria is found which has a limit point, after which pull-in occurs. An assumed deflection shape and a series expansion of the electrostatic capacity yield the deflection curves for the case with a uniform initial lateral displacement. This shows that pull-in occurs at a voltage that is reduced by a factor that is about proportional to the two-third power of the relative lateral initial displacement. The theoretical results have been experimentally tested. The results show a qualitative agreement, but the experimental deflections are larger and the pull-in voltages are lower. These differences can be explained from neglected fringe fields and deviations from the nominal shape. INTRODUCTION Electrostatic comb drives, first demonstrated by Tang et al. [1], are common as actuation elements in microelectromechanical Address all correspondence to this author. systems. They consist of two arrays of interlocking teeth or fingers, each connected to a base. By maintaining a voltage difference between them, a longitudinal force is generated that is almost independent of the actuator displacement over a fairly large range of motion. The lateral force is ideally zero, but as the electric field gives rise to a negative lateral stiffness, comb drives can suffer from instability if the support stiffness is not sufficiently high, which can result in a sudden pull-in [2]. Normally, the instability occurs globally, where a comb is pulled in as a whole, but also locally, individual teeth can be pulled in, while the base remains in place. The case of a tooth centrally located between two rigid teeth has been considered by Elata and Leus [3], who analytically derived the critical voltage. Some experiments on buckling were performed later [4], which confirmed the theory. The combs are usually guided by an elastic straight line mechanism. In some designs, the approximate nature of the guidance causes the teeth to have a lateral or rotational offset. For instance, in a tilted folded flexure with a length of 1000μm and a tilt angle of five degrees, the lateral displacement is about 20 nm and the rotation about 0.7 mrad for a longitudinal displacement of 100μm [5]. The precise values depend strongly on the specific guidance used and the configuration. In this paper, only the influence of a uniform lateral offset is considered, where it is assumed that the rotation is small due to a symmetry in the configuration. The influence of a rotation can be investigated in a similar way as presented for the lateral offset. Moreover, the analysis is 1 Copyright c © 2013 by ASME