Temperature dependence of the resonant frequency of electroded contoured AT-cut quartz crystal resonators

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Abstract
A system of approximate equations recently obtained for the determination of thermal stresses in electroded piezoelectric plates is applied to contoured AT-cut quartz crystal resonators. The changes in resonant frequency resulting from the thermally induced biasing stresses and strains, the temperature derivatives of the fundamental elastic constants of quartz, and the very important temperature dependence of the thickness piezoelectric constant for the AT cut are determined from an equation for the perturbation of the eigenfrequency of the piezoelectric solution due to a bias. The changes in resonant frequency with temperature are calculated for the fundamental and some of the harmonic overtone thickness modes, which were obtained in a recent analysis of overtone modes in contoured crystal resonators. The influences of both the contouring and the electrode size are clearly exhibited.