The Analysis And Calculation Of Electrical Circuit Parameters Of Piezoelectric Acoustic Resonators

As an electronic component for RF circuit applications, the usual requirements of quartzcrystal resonators are specified as electrical parameters such as the resistance, capacitance,inductance, and quality factors for the analysis of the oscillating circuits. This clearly shows a gapbetween the analyses of resonators with the plate theory for vibrations, because the solutions wecan obtain are usually given in vibration frequency, displacements, and mode couplings which arealmost useless to electrical engineers. Thus, we have to calculate the electrical parameters whichcan be used for compared with results from product testing and measurement. It turns out that themajor parameters can be calculated with the consideration of material viscosity which is related tothe resistance and quality factor through the energy loss in the vibrations. But the results showthat the parameters from the theoretical calculation with material viscosity which is from earliermeasurements are much smaller than the actual measurements, and it is suspected that the viscosityfrom the bonding process of electrodes and other factors are not considered but they are significant.For this reason, we calculate electrical circuit parameters of quartz crystal resonators through theconsideration of equivalent viscosity with complex elastic constants with the purpose of find anoptimal viscosity coefficient and reasonable estimation of circuit parameters.We have derived the relations between the frequency of thickness-shear vibration mode andequivalent viscosity coefficient with three-dimensional elasticity equations. The results show thatthe effects of dissipation become more pronounced as the frequency of resonators increases.Then, we analyses free vibration of quartz crystal plate with consideration of equivalent viscosityfactor with first-order Mindlin plate equations. Dispersion curves and frequency spectrum arecomputed and plotted. There are no longer any pure real, no pure imaginary of wavenumberswith real frequency, and the same as frequency with the real length to thickness ratio. Butdissipation is slightly influence to dispersion relation and frequency spectra when we selected theseeffective data. Electrical circuit parameters such as capacitance ratio, motional inductance,motional resistance and quality factor are calculated by the analysis of vibrations of electrodedquartz crystal plate with consideration of equivalent viscosity and piezoelectric effect. Thoseresults from the simplified resonators model of AT-cut quartz crystal are very useful to the designof new product, especially the model of partly electroded of quartz crystal plate. Last but not least,we study the frequency-temperature relations of thickness-shear and flexural vibrations ofrectangle quartz crystal resonators.We do believe the theoretical approach in this study will provide the first step for precisereasonable estimation of electrical circuit parameters and avoid those intervals which performance of circuit parameters is not ideal, finally help the optimal design of better quality product.Moreover, this method can also be extended to design other piezoelectric acoustic resonators.