Analytic model for 1-D axisymmetric piezoelectric transducers

An analytic model has been developed for multilayer axisymmetric (cylindrical) structures in which at least one of the layers is piezoelectric. The structures are assumed to be infinitely long so that the corresponding mathematical model reduces to a 1-D problem with only radial dependence. This work draws a direct parallel to existing, widely used 1-D models for piezoelectric transducers with planar geometry. The main challenge is in deriving the 3-port impedance matrix representation of the piezoelectric layer in cylindrical coordinates. This has been accomplished starting from first principle. This derivation also gives the general solution for the 2-port representation of non-piezoelectric cylindrical layers. The complete solution to a cylindrical transducer composed of a multi-layer structure can be solved by applying the usual transmission matrix formulation. Numerical models have been developed to predict the open-circuit and short-circuit resonance frequencies of general N-layer cylindrical piezoelectric transducers, and the associated electromechanical coupling coefficients. Transmit output and receive sensitivity of cylindrical transducers can also be calculated. Modeling of an unloaded PZT-8 cylinder clearly shows the lowest order hoop resonance mode of the piezoelectric layer, and the much higher frequency thickness vibration modes.