Plate equations and equivalent circuit for micromachined piezoelectrically actuated flextensional transducers

Abstract

In the absence of analytical expressions for the equivalent circuit parameters it is difficult to calculate the optimal design parameters and dimensions and to chose suitable materials. The influence of coupling between flexural and extensional deformation and coupling between structure and acoustic volume on the dynamic response of piezoelectrically actuated flextensional transducer is analyzed using three analytical methods: classical plate theory, Mindlin plate theory, and simple variational procedure. The transducer design is based on a flextensional transducer that excites the axisymmetric resonant modes of a clamped circular plate. It is constructed by depositing a thin piezoelectric annular plate onto a thin, edge clamped, circular plate. An ac voltage is applied accross the piezoelectric material to set the compound membrane into flexural vibration. The devices have a range of operating resonance frequencies starting from 450 kHz up to 4.5 MHz. The device is manufactured by silicon surface micromachining and implemented in the form of two-dimensional arrays. Individual elements are made of thin silicon nitride membranes covered by a coating of piezoelectric zinc oxide. Classical thin plate theory, Mindlin plate theory, and variational methods are applied to derive two-dimensional plate equations for the transducer, and to calculate the coupled electromechanical field variables such as mechanical displacement. These methods use classical kinematic relations for a plate and a variational equation for the coupled electromechanical field to reduce three-dimensional field equations to two-dimensional plate equations. As a result, three different exact solutions to corresponding systems are obtained. An equivalent circuit of the transducer is also obtained from these solutions.