Semi-analytic analysis of coupled thickness-shear and thickness-twist in monolithic crystal filters

The wave equation describing the displacement of rotated Y-cut quartz trapped-energy resonators and two-pole monolithic filters with rectangular electrodes and operating in fundamental and overtones of coupled thickness-shear and thickness-twist vibrations, due to H.F. Tiersten (1977), is treated by a finite-element method. Rectangular elements are used for the surface of the plate both in the electroded and unelectroded regions, and displacements at the corners of elements are taken as unknown. Natural boundary conditions imply that, in several cases the solution obtained differs from the analytical solution. The method can be applied to the treatment of resonators and complex monolithic structures having N resonators (with N ⩾2), including the effect of size and position of resonator tabs. The analysis allows the design of monolithic filters both on AT-cut quartz and rotated Y-cut berlinite. Optimum designs for unwanted mode attenuation can be found