Equivalent-circuit model for the thickness-shear mode resonator with a viscoelastic film near film resonance

We derive a lumped-element, equivalent-circuit model for the thickness-shea r mode (TSM) resonator with a viscoelastic film. This modified Butterworth- Van Dyke model includes in the motional branch a series LCR resonator, repr esenting the quartz resonance, and a parallel LCR resonator, representing t he film resonance. This model is valid in the vicinity of film resonance, w hich occurs when the acoustic phase shift across the film is an odd multipl e of pi/2 rad. For low-loss films, this model accurately predicts the frequ ency changes and. damping that arise at resonance and is a reasonable appro ximation away from resonance. Elements of the parallel LCR resonator are ex plicitly related to film properties and can be interpreted in terms of elas tic energy storage and viscous power dissipation. The model leads to a simp le graphical interpretation of the coupling between the quartz and film res onances and facilitates understanding of the resulting responses. These res ponses are compared with predictions from the transmission-line and Sauerbr ey models.