Mechanically coupled clamped circular plate resonators: modeling, design and experimental verification
Mechanically coupled resonators usually require a mechanical coupling element; this introduces additional complexity to the picture. We propose a novel modeling and design approach, followed by experimental verification, for mechanically coupled clamped circular plate resonators in which no additional coupling element is present. In our study, the flexural mode clamped circular plate resonators overlap to an extent, and their clamps at the overlap region are removed to generate a freely moving coupling boundary between the resonators. The practical measure of the overlapping is small enough to preserve the characteristics of each resonator. This result enables modeling the coupled resonators based on the clamped circular plate resonator model. A physics-based lumped element equivalent circuit model is developed where dimensions, bias voltage, and material properties are controllable variables. Each of the model parameters is expressed as the corresponding single resonator model parameter multiplied by a function of the amount of overlap. Analytical derivations and finite element method simulations are used to extract the dependency of the model parameters on the amount of overlap. Closed-form expressions for center frequency, bandwidth, and termination impedance of the coupled resonator are derived using the developed model. A design procedure is introduced to determine dimensional parameters and bias voltage. The proposed coupled-resonator offers up to 5% fractional bandwidth. For a typical design using a polysilicon plate with 100 nm gap height, the ratio of the termination impedance to the center frequency is calculated to be 158 Ω MHz−1. This result indicates that on-chip intermediate frequency filters can be implemented at center frequencies up to several 100 MHz using this type of coupled resonators. A coupled resonator is designed and realized for a proof of concept demonstration. The measurement results of the coupled resonator show good agreement with the equivalent circuit model simulations.