Characterization of a thickness-shear mode quartz resonator with multiple nonpiezoelectric layers

This paper describes a one‐dimensional analysis of a piezoelectric resonator with multiple nonpiezoelectric layers of arbitrary thickness and complex shear modulus. This analysis shows under what conditions the simple Butterworth–Van Dyke (BVD) lumped‐element equivalent circuit can be used to extract the properties of these layers, and provides a theoretical basis for doing so under these conditions. The method presented here is based on the physics of wave propagation, but uses transmission line techniques to transform the equations and boundary conditions into a simple string of 2×2 matrix multiplications, which easily accommodate any number of layers. An analysis of the AT‐cut quartz/polymer/liquid composite resonator is presented in detail because of its importance in sensor and electrochemical quartz crystal microbalance applications. The analysis shows that adding the polymer and liquid impedances in the motional arm of a BVD circuit is not valid near the polymer film resonance, but is appropriate for a thin, rigid polymer film in contact with a liquid.