Determination of electromechanical coupling factors of low Q piezoelectric resonators operating in stiffened modes

The electromechanical coupling coefficient is usually determined from the relative spacing of the frequencies of resonance and antiresonance. The conventional formula is derived from equations describing the electrical behavior of an ideal piezoelectric resonator in the absence of losses. In this paper, the influence of the intrinsic material losses on the shift of the resonance/antiresonance frequencies, and therefore on the accuracy of the standard formula to determine k, is analyzed. The exact one-dimensional model of the piezoelectric resonator vibrating in a pure stiffened mode, with a rigorous account of the internal mechanical losses, has been taken as a reference, instead of the frequently used lumped approximate equivalent circuit (Butterworth-Van Dyke). It is shown that the coupling coefficient determined from the frequencies f(s) and f(p) is less than the intrinsic coupling coefficient, and that the error increases for highly attenuating materials with weak electromechanical coupling. The error due to the effect of attenuation, which increases with the decrease of the product Q(m)k of the resonator intrinsic parameters, has been systematically evaluated and plotted for 0.5<Q (m)k<15. The accuracy and the field of application of the conventional formula to determine the coupling factor by a non-iterative method is therefore established.