Abstract
It is often desirable to represent characteristics of a piezoelectric resonator by an equivalent circuit which has the same impedance as the resonator at a frequency range of interest. This is especially required when one apply the highly developed theory of LC filters to the design of piezoelectric filters. A resonator with high electromechanical coupling can be represented by a clamped capacitance (C_D), shunted by a number of motional arms as shown in Fig. 1. Each motional arm represents a resonance and consists of an inductance and a capacitance in series. Although this circuit representation is exact, tedious calculations are necessary to evaluate its impedance, and also its form is inconvenient for use in filter design. For a resonator with low coupling, a simplified equivalent circuit consisting of a parallel capacitance (C_0) and only one motional arm permits a good approximation. In fact, this equivalent circuit has been widely used for quartz crystal resonators. The consists of the motional arm can be determined in the same manner as the case of a low coupling resonator. A problem arise how to determine the parallel capacitance, if one is going to adopt the simplified circuit for an approximation. This is because the parallel capacitance C_0 now consists not only of the clamped capacitance C_D but also of the effects from other neglected motional arms and becomes highly frequency dependent. Hence a capacitance measurement at any one frequency will yield an ambiguous result. For example the use of the capacitance value at a very low frequency as the parallel capacitance yields a poor approximation of resonator impedance. This paper shows that the simplified equivalent circuit is still a reasonable approximation for a high coupling resonator, provided that parallel capacitance is so chosen that the circuit yields the exact antiresonance frequency. Since the antiresonance frequency can be easily measured, all the constants of the proposed circuit can be experimentally determined even when an exact solution is not available. This approximation was compared with known exact solutions for both piezoelectrically stiffened and unstiffened modes of vibrations. The accuracy was evaluated in two different ways. At first, the difference of the susceptance as a function of frequency was calculated. Fig. 3 shows the results for: (a) stiffened mode; (b) unstiffened extensional mode; and (c) unstiffened radial mode of a circular plate. The coupling factor (k) is a parameter. Next, the error of the shift of the antiresonance frequency due to load capacitance was calculated. Fig. 4 shows the results for: (a) stiffened mode; and (b) unstiffened extensional mode. Both results show that the proposed simplified circuit is accurate enough for most practical applications. As an example, the frequency characteristics of a lattice filter as shown in Fig. 5 were calculated based on the exact solution and on the present approximation. The two results shown in Fig. 6 were almost identical. As a comparison, a curve for the case that the low frequency capacitance is chosen as the parallel capacitance C_0 is included in the figure. It poorly approximates in contrast with the above results. A resonator in practice has various sources of loss. Its effects can be well approximated by the insertion of a series resistance in the motional arm, as it has been done for low coupling resonators. The authors with to thank Dr. S. Nishikawa for his careful reading of this manuscript.
