Investigation into temperature variation of equivalent-circuit parameters of AT-cut quartz crystal resonators

Despite the abundance of published information on the variation of resonant frequency with temperature of AT-cut quartz crystals, no work has been reported on the temperature variation of the equivalent-circuit elements. Measurements of the change in motional capacitance with temperature are reported in the paper. Results showed that motional capacitance increased linearly over the temperature range ¿¿ 10 to 70¿¿C. This increase was nearly three orders of magnitude greater than the change in crystal resonant frequency. Clearly, a corresponding decrease in motional inductance must occur to yield the observed frequency/temperature behaviour of AT-cut crystals. The apparent change in angle of cut which occurs with capacitive loading of a crystal was found to be a direct consequence of the large temperature coefficient of motional capacitance. Using this information it is now possible to calculate the apparent angle of cut for various load factors, and to predict the temperature behaviour of crystal oscillators more precisely.