Two-Port Electromechanical Model for Bulk-Piezoelectric Excitation of Surface Micromachined Beam Resonators

A small-signal model that describes the energy exchange between surface micromachined beams and bulk-lead zirconium titanate (PZT) actuators attached to the silicon substrate is presented. The model includes detection of acoustic waves launched from electrostatically actuated structures on the surface of the die, as well as their actuation by bulk waves generated by piezoelectric ceramics. The interaction is modeled via an empirical equivalent circuit, which is substantiated by experiments designed to extract the model parameters. The equivalent model is valid for cases where the beam resonance frequency is much smaller than the thickness mode resonance of the PZT/silicon stack. In this paper, the resonance frequency of the beams ranges between 200 and 300 kHz. As energy transfer between bulk-PZT and electrostatic actuated beam resonators must be reciprocal for small signals, this paper uses the extracted equivalent model to describe the physical sources of error that account for discrepancies in reciprocity.