RESUMO
The accurate calculation of vibration frequency is essential in design of circular quartz crystal resonators which are the core elements of high-precision microbalances used for testing and measurement. Currently, the prediction of thermal effects on frequency through an analytical analysis is still in its developmental stage, mainly due to the complexity of solving the 3-D equations with the consideration of asymmetric structure of resonators and electrodes along with material anisotropy. By using a scalar differential equation for vibrations with the eigen-displacement of thickness mode, the eigen-frequency of a plano-convex AT-cut circular quartz crystal plate with asymmetric electrodes is determined. Furthermore, the temperature effect in the scalar differential equation is successfully obtained by incorporating the incremental thermal field theory into the 1-D analysis. The theoretical results agree well with the experimental data. The combination of the thermal field and the thickness model for circular quartz crystal resonators can realize a full analysis of thermal properties in vital applications such as high-sensitivity microbalance sensors.