Complex material coefficients and energy ratios for lossy piezoelectric materials

This correspondence reviews complex material coefficients of piezoelectric materials and their influence on the energy ratios in coupled systems. In lossless systems, it is shown for the length extensional (LE 33) mode in a C_infin material that there are at least 4 energy ratios that will produce the same coupling value. In addition, in the (LE 33) mode there are at least 2 different experimental conditions to define an energy ratio that produces the same coupling. With the introduction of loss, these 2 experiments and the 4 energy ratios diverge and no longer produce the same coupling. It is shown that the instantaneous ratio of coupled to input energy or the time average of this ratio, if a numerical value is desired, is the appropriate energy ratio in a dissipating system.