Multilayer transducer transfer matrix formalism

A complete formulation of direct and inverse 4 /spl times/ 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 /spl times/ 4 transfer matrix transfers quantities at the top surface into their values at the bottom surface or vice versa, respectively. The 4 /spl times/ 4 transfer matrices derive from the 3 /spl times/ 3 transfer matrices, which follow from the 3 /spl times/ 3 matrix for the general three-port. For both parallel and series connections, the 3 /spl times/ 3 and 4 /spl times/ 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 /spl times/ 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.