A variational analysis of a new shape face shear mode quartz crystal resonator formed by an etching method

Variational analysis, the frequency-temperature characteristics, and the equivalent electrical circuit constants of a face-shear-mode quartz crystal resonator supported at ends, formed by an etching method are described. A vibration analysis is performed in which the equations of motion of the resonator, including both the vibrational and supporting portions, are derived using and energy method. A frequency equation is obtained by applying Galerkin's method to solve the resulting partial differential equations and using displacement obtained from experiments on motional inductance. The resonant frequency, frequency-temperature behavior, and electrical characteristics of the resonator are analyzed by utilizing the frequency equation and displacement. The calculated results are shown to be in good agreement with the measured values. >