A GENERAL APPROXIMATED 2-DIMENSIONAL MODEL FOR PIEZOELECTRIC ARRAY ELEMENTS

In this paper we describe an approximate two-dimensional model of a single piezoelectric array element. Substantially, the element is a two-dimensional structure whose vibrations can be described by two coupled differential wave equations with coupled boundary conditions. We take as a solution of this system two orthogonal wave functions which depend only on one axis, corresponding to the propagation direction, and which satisfy the boundary conditions only in an integral form. This solution makes it necessary to neglect the piezoelectric coupling in the transverse direction; nevertheless this approximation does not substantially affect the computed results because the transverse elastic coupling is much stronger than the piezoelectric one. With our model, the external behavior of the element in the frequency domain can be described by a 5x5 matrix from which all the transfer functions of the element can be easily computed. We compared our results with those of the classical Mason-Sittig one dimensional model [1], finding, as expected, a small difference in the principal thickness resonance frequency and the presence of a lateral mode. To verify the results obtained with our model, we computed the frequency spectrum of the element varying the width/thickness ratio. We found a good agreement with the low branch of the spectrum computed by the coupled mode theory, which Kino and DeSilets [2] showed to be a good representation of the experimental situation. Finally, we computed, in the frequency domain, the dynamic electromechanical coupling coefficient k(U), evaluating the internal energy distribution of the element and applying the definition reported by Berlincourt [3]. We compared the values at resonance with the appropriate quasistatic coupling factors k'(33) and k'(31), and with the effective coupling factor k(epsilon ff)computed by the same model.