Analysis of Doubly-Rotated Contoured Quartz Crystal Resonators

摘要

The analysis presented is an extension of the recent treatment of the doubly-rotated contoured SC-cut quartz resonator, in which a number of relatively small transformed elastic constants were neglected. In this work no transformed elastic constants are neglected and only the fundamental assumptions of small piezoelectric coupling and small wavenumbers along the plate are made. The resulting equations are applicable to any potentially practical contoured quartz resonator independent of orientation. In particular, calculations performed for the SC-cut reveal that the earlier treatment, which neglected certain relatively small terms, was quite accurate in the prediction of the resonant frequencies. However, since the differential equation for each harmonic family depends on the order of the harmonic and in this more general treatment 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. Keywords include: Piezoelectric Resonators, Contoured Resonators, Doubly-Rotated, Linear Vibrations, Plate Vibrations, Forced Vibrations, Quartz Resonators, Harmonic Overtones, Anharmonic Overtones, Mode Spacing, Anisotropic, Piezoelectric Plates, Dispersion Relations, Asymptotic, Transformed Constants, and Energy Trapping. (Reprints).