Reduction of spurious responses in a thickness-twist mode resonator made of crystal class 32 by tilting edges

Rotated-Y-cut quartz resonators vibrating in a thickness-twist (TT) mode have been widely used in oscillators, filters and sensors. The TT mode is a pure shear mode with the displacement parallel to the X axis. If a plate is isotropic, side surfaces normal to the plate and parallel to the X axis are traction free and hence can be cut without disturbing the vibration to make a rectangular strip. This is not the case of anisotropic quartz, and such surfaces cause spurious responses due to coupling with other modes. Tilt angle can be calculated from the exact solution for a semi-infinite parallelogram strip at a certain temperature. Optimum combinations of width vs thickness ratio, cut angle, tilt angle and the length vs width ratio have been determined. One of the authors developed a method for analyzing the characteristics of a strip resonator with tilted edges and a finite length. This paper presents two further works, one for other crystals and the other for doubly-rotated-Y-cuts. Gallium orthophosphate (GPO) and the langasite family (LGS, LGN and LGT) belong to the same crystal class 32 as quartz and useful rotated-Y cuts have been found. Much improvement in spurious response reduction over rectangular strips is obtained.