An accurate equivalent circuit for the unloaded piezoelectric vibrator in the thickness mode

An equivalent circuit model for the unloaded piezoelectric vibrator in the thickness mode is presented. The model contains two branches, the motional branch and the static branch, like the lossless resonator model, but the circuit elements are generalized by making each a complex constant. The mechanical, dielectric and piezoelectric losses associated with the vibrator are accounted for by the imaginary components of the circuit elements. The model produced impedance curves that closely matched the impedance calculated by using equations derived from vibration theory and the data measured for lead zirconate titanate and PVDF-TRFE co-polymer samples. The calculation of the circuit parameters from the complex elastic, dielectric and piezoelectric material constants is straightforward and the model accurately fits both the baseline dielectric behaviour and the piezoelectric resonance around and below the fundamental resonance. Conversely, when the complex circuit parameters are known, the complex material constants can be derived by straightforward calculations without any loss of information.