Reduction of Spurious Responses of a Strip Resonator with Tilted Edges

In 1970, Mindlin obtained an exact solution of thickness twist modes in an infinite strip made of rotated Y cut of quartz, neglecting piezoelectric effects, by tilting side surfaces parallel to the X axis slightly off normal to the major planes. This angle is called Mindlin's angle. It means that excellent resonator characteristics of an infinite plate can be retained in a miniature strip resonator without excess spurious responses. Based on this concept, billions of miniature AT strip resonators have been produced. Similar designs have been extended to resonators of such crystals as gallium orthophosphate (GaPCH) and Langasite family (LGS, LGN and LGT) in crystal class 32. In Mindlin's exact solution, major spurious responses are due to face shear modes. Electrodes on major surfaces still piezoelectrically excite them, although their elastic coupling with thickness shear modes is null. Their distribution of piezoelectrically induced charge along the Z axis constitutes a harmonic cosine series. Hence suitable choice of the width and the shape of electrodes can eliminate piezoelectric excitation of face shear modes. In particular, if the area of electrodes along the X axis is chosen to be proportional to a cosine distribution of a certain mode, responses of all other modes can be eliminated. Vormer applied a similar design to an X cut quartz resonator vibrating in length-extensional modes. For a high frequency resonator, which has numerous spurious responses due to other modes, it is more practical to reduce not all but a few nearby responses. Adjusting of the width of a rectangular electrodes or a combination of several rectangular electrodes, which are easier to fabricate than a cosine electrode, can do this. In a real resonator with a finite length along the X axis, there exist spurious responses due to thickness shear modes propagating along the X axis. No exact solution has been available. Approximate solutions and direct experimental observation of distribution of induced charge, however, suggested that the distribution of induced charge along the X axis of these spurious responses also approximately constituted a harmonic cosine series. Hence a similar design of electrodes can be applied to reduce responses of these modes.