Equivalent electromechanical representation of trapped energy transducers

Trapped energy resonators and transducers have attained a considerable importance in quartz crystal technology both as single-frequency resonators for the control of crystal oscillators and as drivers for the monolithic crystal filter which appears likely to have a wide use as a channel filter for the separation of voice frequency channels for long-distance carrier systems, microwave radio, and submarine cable systems. It is the purpose of this paper to derive the equations for trapped energy resonators of the thickness-shear and thickness-twist types and to calculate the ratios of capacitances for straight crested waves. It turns out that the ratios are lower (coupling higher) than are observed in practice. It appears that this difference is connected with the finite width of the plate which causes the motion at the edge of the plate to be somewhat smaller than the motion in the center of the plate. While no exact solution has been obtained for the finite plate, an approximation is made which is in good agreement with the experiment. The resonator on a plate is a symmetrical device, whereas a transducer for driving a monolithic filter is a dissymmetrical device since it is driving different impedances on its two boundaries. To represent this dissymmetry requires a distributed network representation which is somewhat similar to that found for a plane longitudinal or shear wave except that the propagation constant for a trapped wave replaces that of the plane wave. The representation also requires a transformer whose transformation ratio is a function of the frequency and two negative element terms. By transformations the negative elements can be made to disappear. These together produce an equivalent circuit whose values depend on the ratio of the electrode length to the crystal thickness.