Composite piezoelectric resonator 1D modeling with loss

Maximizing the benefit of applying piezoelectric materials in composite structures, which are a mix of piezoelectric and elastic elements, requires analytical modeling. In this paper, we describe two main classes of resonators and how to model using network equivalent including material losses. These include extensional devices where the layer areas are parallel to the direction of wave travel as found in a variety of thickness or length mode transducers (BC. Stress T₁ = T₂ at boundary) and transverse mode resonators (BC. Strain S₁ = S₂ at boundary) where the major layer area is perpendicular to the wave direction as is found in radial mode or length thickness mode composite resonators. In order to illustrate these models, we will use Mason's equivalent circuits with elastic, dielectric and piezoelectric loss. It is noted that in order to maintain consistency with the linear equations of piezoelectricity and the wave equation care is required when applying complex loss coefficients to the models. Although these techniques are applicable to other models, we use Mason's network equivalent circuit because it is intuitive for composite modeling due to the frequency independent turns ratio. Also, because of the independence of the equations describing the elastic layers from the adjoining layer properties which aid in understanding the boundary conditions to impose on the layer boundaries. We will present network models for a variety of applications with loss and show how to calculate their impedance spectra, interface stress, strain velocity and displacement spectra.