Analysis of Spurious Characteristics of Rectangular AT-Cut Quartz Resonators Using Plate Equations and a One-Dimensional Finite Element Method
Many spurious vibrations can be supported in rectangular AT-cut quartz resonators. These vibrations generally belong to the modes with phase reversals along both the X and Z' axes. The two-dimensional coupled vibrations of thickness-shear and thickness-flexure have been previously analyzed. In this paper, the coupled vibrations of face-shear and face-flexure are also analyzed. The analysis of face-shear and face-flexure is based on the Mindlin's plate equations and the plane-stress approximation. A one-dimensional finite element technique is effectively applied to solve the two equations, in which the coupling with the thickness-shear and thickness-flexure are omitted. The change of resonant frequencies are calculated as a function of Z'-length, and compared with the experiments. The results shows that almost all the spurious resonances in rectangular AT-cut quartz resonators belong to the thickness-flexure, face-shear and face-flexure. It is also shown that the plane-stress approximation is effective to predict the frequencies of the coupled vibrations of face-shear and face-flexure.