Rigorous two-dimensional equations for the analysis of contoured crystal resonators

A generalization of previous techniques is used to deduce rigorous two-dimensional equations for a piezoelectric plate resonator with arbitrary contour and crystallographic orientation. Methods are proposed for constructing approximate equations, involving only a finite set of mode amplitudes, from the general two-dimensional equations. The effects of the various mode coupling terms are considered for AT- and SC-cut resonators of both plano-convex and biconvex type. Approximate solutions are deduced for simple contoured plate geometries by means of a finite-element approach. The degree of generality offered by the finite-element method is required for the solution of the equations. It also allows realistic boundary conditions to be used at the plate edges and provides improved estimates of resonator Q-factor