AC transfer function of electrostatic capacitive sensors based on the 1-D equivalent model: application to silicon microphones

Parallel plate electrostatic transducers can be described with the one-dimensional (1-D) lumped model. The one-dimensional approximation based on the elastic, the damping and the inertial force is extended with the electrostatic force (due to the electrical biasing) to model the behavior of electrostatic actuators. In case of sensors, the effect of the external excitation has to be also included. The final equation describing the dynamic behavior of the sensor can only be solved numerically avoiding a compact solution. In this paper the perturbation method applied to solve the equations describing parallel plate capacitive sensors is presented. A compact expression is obtained and applied to model silicon microphones. For the sake of comparison, the silicon microphone is also modeled with the well-known analog equivalent electric circuit, which is extended to take into account the resistor used to bias the microphone. It is shown in which conditions both modeling techniques give equivalent results. However, in front of the traditional equivalent electric circuit, the model based on mass, spring constant and damping coefficient allows taking into account the pull-in instability. Assessment of the modeling method is carried out by experimental measurements on a silicon microphone and previous experimental results reported in the literature. A very good agreement between theory and measurements is obtained.