Weak nonlinearities in piezoelectric transducers. Equivalent circuits

When the amplitude of a thickness transducer is high enough, nonlinear terms must be added as a perturbation to both constitutive equations of the material. Electrical and acoustical fields with multiple frequencies will appear when a single harmonic voltage is applied to the transducer. This can be modeled by a series of equivalent circuits (one for each harmonic frequency) which include some elements (voltage and current sources) that depend nonlinearly on the currents at all circuits. The nonlinear elements of the circuit at a given frequency can be obtained if we know the functions Φ(x) and Ψ(x), which are the perturbations of the constitutive equations. Two equivalent circuits are developed. In the first one, a symmetric transducer is considered, radiation losses are modeled by an impedance, only one port (electrical) is considered and all elements are reduced to their electrical equivalent. In the second one, which is a modified Mason circuit, two mechanical ports are considered, so mechanical boundaries may be non-symmetrical, and uses a transformer as an electromechanical coupling, so the mechanical circuit is described in terms of force and velocity