Accurate equivalent circuits for unloaded piezoelectric resonators

Equivalent circuit models for unloaded piezoelectric vibrators are presented. The new circuit model includes terms which take into account the dielectric and piezoelectric loss as well as the mechanical loss found in the standard Van Dyke´s model. The new model contains two branches, the motional branch and the static branch, like the Van Dyke´s model. However there is no resistance element in the motional branch and the losses (mechanical, dielectric, piezoelectric) associated with the vibrator are represented as imaginary components of the remaining circuit elements (C₀, C₁, L₁). The model produced impedance curves that very closely matched the impedance calculated by using the equation derived from vibration theory and data from lead zirconate titanate and PVDF copolymer samples. The calculation of the circuit parameters C₀, C₁, and L₁ from the complex elastic, dielectric and piezoelectric material constants is straightforward and the model accurately represents the impedance spectra about the fundamental resonance even when the mechanical Q of the resonator is as low as 2. Conversely, If the circuit parameters are known, the material constants including losses can be derived by straightforward calculations without the loss of any information