A new model for the thickness shear resonators with spherical contours: application to bevelled devices

Contoured resonators are mostly employed to obtain very low loss devices employed to generate very stable frequencies. The partially contoured (bevelled) resonators are also used for filtering. Their design remains still somewhat empirical, so that a better theoretical understanding of such devices is very desirable. In a preliminary study, experimental determinations of the mode shapes were performed using an important set of bevelled resonators, with designs ranging from nearly biconvex ones to nearly plane ones having only a very small bevel at their edge. Several conclusions were drawn from these experiments. The present model of the contoured and the partially contoured resonators is based upon the Tiersten theory of the transversely varying essentially thickness modes. We propose a new method of resolution of the Tiersten partial derivative equations based upon the algebraical solutions obtained for the same problem in the case of a very close geometry (non spherical contour respecting the lateral anisotropy of the plate). These simple solutions are of a very large generality since they contain nearly all the previously known ones. They bring a better understanding of the different kinds of thickness shear contoured resonators. In this model, for the plane electroded regions (bevelled resonators) we use solutions of the Tiersten equations which are basically series expansions in the exact solutions for a case where the electrodes and the plate have elliptical geometries respecting the lateral anisotropy. For the contoured regions, we use similarly a representation of the shear displacement consisting in an expansion in the eigen solutions obtained for the above mentioned non-spherical contours. The continuity conditions between the different regions and the boundary conditions at the edge are expressed at a discrete number of points to fit nearly or more exactly, the actual geometry of the resonators presenting a spherical contour and a circular geometry. The set of linear equations so obtained constitute an homogeneous linear system whose determinant must vanish to have a solution. This condition is a frequency equation which is solved to find the eigen frequencies and the eigen modes. The numerical implementation of this model makes use, for the contoured regions, of the confluent hypergeometric (Whittaker) functions [M_k,m(r²/2) or W_k,m(r²/2)] which are bounded at zero or at infinity, and of the Jn(r) Bessel functions for the flat region. The convergence of the solution with the discretization appears to be extremely fast, and a very high accuracy seems to be obtained. The comparison with the experimental results indicates a very good agreement. Several important features observed during the experimental study were confirmed and explained by the calculations