Effects of non-homogeneous thermal stresses on the frequency-temperature behavior of AT-cut quartz resonators

The frequency-temperature (f-T) behavior of a quartz resonator can be predicted quite accurately if the resonator crystal is under an ideal stress-free condition. Under such a condition, the steady state temperature change then induces a homogeneous field of thermal strains in the crystal. The ideal thermal stress-free condition is however seldom achieved practically. In practical devices, thermal stresses are often present with temperature changes. We study the effects of non-homogeneous thermal stresses on the f-T behavior of AT-cut quartz resonators by employing a novel method of superposing the results from three existing methods for calculating (1) thermal stresses, (2) acceleration effects, and (3) f-T curves under a homogeneous thermal strain condition. We assume that for a steady state temperature change, the crystal resonator undergoes not only a homogeneous thermal strain field but also a nonhomogeneous thermal stress field. We present numerical results compared with experimental results for an AT-cut quartz resonator mounted on glass. The difference between the thermal expansion coefficients of glass and quartz give rise to the thermal stresses.