An equivalent network representation of a clamped bimorph piezoelectric micromachined ultrasonic transducer with circular and annular electrodes using matrix manipulation techniques

An electric circuit model for a clamped circular bimorph piezoelectric micromachined ultrasonic transducer (pMUT) was developed for the first time. The pMUT consisted of two piezoelectric layers sandwiched between three thin electrodes. The top and bottom electrodes were separated into central and annular electrodes by a small gap. While the middle electrode was grounded, the central and annular electrodes were biased with two independent voltage sources. The strain mismatch between the piezoelectric layers caused the plate to vibrate and transmit a pressure wave, whereas the received echo generated electric charges resulting from plate deformation. The clamped pMUT plate was separated into a circular and an annular plate, and the respective electromechanical transformation matrices were derived. The force and velocity vectors were properly selected using Hamilton's principle and the necessary boundary conditions were invoked. The electromechanical transformation matrix for the clamped circular pMUT was deduced using simple matrix manipulation techniques. The pMUT performance under three biasing schemes was elaborated: 1) central electrode only, 2) central and annular electrodes with voltages of the same magnitude and polarity, and 3) central and annular electrodes with voltages of the same magnitude and opposite polarity. The circuit parameters of the pMUT were extracted for each biasing scheme, including the transformer ratio, the clamped electric impedance, and the open-circuit mechanical impedance. Each pMUT scheme was characterized under different acoustic loadings using the theoretically developed model, which was verified with finite element modeling (FEM) simulation. The electrode size was optimized to maximize the electromechanical transformer ratio. As such, the developed model could provide more insight into the design, optimization, and characterization of pMUTs and allow for performance comparison with their cMUT counterparts.