An Analysis of Contoured SC-Cut Quartz Crystal Resonators

Abstract

A previous treatment of SC-cut quartz trapped energy resonators is extended to the case of plates with slowly varying thickness. As in the earlier treatment, the approximate equations are referred to rotated coordinate axes obtained from the eigenvector triad of the pure thickness solution for the SC-cut. The approximate dispersion equation describing the mode shape along the surface of the SC-cut, which was used in the recent treatment of the trapped energy resonator, is extended to include the influence of certain relatively large transformed elastic constants that were neglected in the earlier work. The extended equation is considerably more accurate than the earlier one. A scalar differential equation describing the mode shapes along the surface of the contoured SC-cut for the family associated with each odd harmonic is obtained from the extended dispersion equation for the flat plate. The scalar equation is applied in the analysis of contoured SC-cut quartz crystal resonators, and an approximate lumped parameter representation of the admittance, which is valid in the vicinity of a resonance, is obtained. The analysis holds for the fundamental and odd harmonic and anharmonic overtone thickness modes of interest.