Contents I. Introduction
Thickness-shear and thickness-twist vibrations in a partially electroded piezoelectric plate can be confined to the area under and close to the electroded region of the plate. This phenomenon is called energy trapping [1]. Energy trapping has been known and used for a long time in piezoelectric resonators. Quartz crystal is a material with very weak piezoelectric coupling. Therefore, in the calculation of resonant frequencies and analysis of energy trapping in quartz resonators, the small piezoelectric coupling is often neglected, and a pure elastic study is usually sufficient [2], [3]. Energy trapping is then shown to be due to the inertia of the electrode mass. For materials with relatively strong piezoelectric coupling, it has been shown that both electrode mass inertia and piezoelectric coupling contribute to energy trapping [4], [5]. Contoured plates with varying thicknesses have been used to achieve strong energy trapping. The study of energy trapping in these resonators has been of continuing research interest, e.g., as in [6]–[11]. It is well known that the manufacturing process of contoured resonators involves technical challenges. On the other hand, it is not difficult to deposit electrodes with varying thicknesses on a crystal resonator such that the resonator with electrodes resembles the geometry of a contoured resonator. This offers an alternative for producing resonators with strong energy trapping. Since the electrode material is usually heavier than quartz, effective energy trapping may be achieved by relatively thin electrodes with varying thicknesses. These have been recently verified by an elastic analysis of a quartz thickness-shear mode resonator using two-dimensional plate equations [12]. In addition to electrode thickness, electrode shapes have also been considered for design optimization [13]. The results in [12] on nonuniform electrodes are for the case of symmetric electrodes.
