An analysis of doubly rotated quartz resonators utilizing essentially thickness modes with transverse variation

Abstract

Closed‐form asymptotic expressions for the frequency–wavenumber dispersion relations in doubly rotated quartz plates vibrating in the vicinity of the odd pure thickness frequencies are derived from the equations of linear piezoelectricity and the associated boundary conditions on the major surfaces. The usual assumptions of small piezoelectric coupling and small wavenumbers along the plate are made and it is supposed that the pure thickness frequencies are sufficiently different that one pure thickness wave is dominant at a time. In the treatment the mechanical displacement is decomposed along the eigenvector triad of the pure thickness solution to facilitate the asymptotic analysis. The fact that the wavenumbers along the plate are restricted to be small significantly reduces the complexity of the equations without neglecting any transformed elastic constants. The resulting asymptotic dispersion equation enables the construction of a scalar differential equation describing the transverse behavior of essentially thickness modes of vibration in doubly rotated quartz plates. The scalar equation is applied in the analysis of both trapped energy resonators with rectangular electrodes and contoured crystal resonators using established procedures. In particular, calculations performed for the contoured SC cut and a number of other doubly rotated orientations are shown to be in excellent agreement with experiment. Since the differential equation for each harmonic family depends on the order of the harmonic and in the general doubly rotated case contains mixed derivatives in the plane of the plate, a different transformation is required for each harmonic family to obtain the coordinate system in which the mixed derivatives do not appear and, hence, the equation is separable. An interesting consequence of this transformation is that since the nodal planes of the anharmonics of each harmonic family of the contoured SC‐cut quartz resonator are oriented along the transformed coordinate system for that harmonic family, they are oriented differently for each harmonic family.