Multilayer transducer transfer matrix formalism

A complete formulation of direct and inverse 4 x 4 transfer matrices for parallel and series, electrically connected, mechanically stacked, 1-D thickness mode multilayer piezoelectric transducers is presented. Complex coefficients account for the mechanical, dielectric, and piezoelectric losses. The direct or inverse 4 x 4 transfer matrix transfers quantities at the top surface into their values at the bottom surface or vice versa, respectively. The 4 x 4 transfer matrices derive from the 3 x 3 transfer matrices, which follow from the 3 x 3 matrix for the general three-port. For both parallel and series connections, the 3 x 3 and 4 x 4 direct and inverse transfer matrices are interrelated through transformation symmetries; also, the inverse matrix can be obtained from the direct matrix by changing the sign of both the piezoelectric coefficient and the explicitly occurring complex variable, j. For the electrically parallel-connected case, explicit voltage orientation reversals occur at successive piezoelectric layers. Cascading the 4 x 4 matrices yields the sum of the currents through the piezoelectric layers for the electrically parallel-connected case and the sum of the voltage differences across the layers for the electrically series-connected case. The resultant matrices are calculated for the cascading of n identical piezoelectric layers connected in parallel and series.