Extension to the definition of quasistatic material coupling factor to include losses

In general the coupling factor is a dimensionless coefficient, defined as a particular combination of the dielectric, elastic, and piezoelectric coefficients that may be useful for the internal energy conversion description in piezoelectric materials. In order to extend the definition of the quasistatic coupling factor as ratio of energies to dynamic conditions and to lossy materials, its current definition and its derivation are reviewed. It is shown that this parameter can be computed as ratio of energies also in dynamic conditions, and the factors obtained in the static and the dynamic case are simply related by a proportionality coefficient. The coupling factor is computed as the square root of the ratio between the converted (from mechanical to electrical or vice versa) and the total energy involved in a transformation cycle for lossy materials in quasistatic conditions, obtaining a complex quantity related to the complex material parameters taking the losses into account. In order to apply this definition to the element vibrating around its resonance frequency, the kinetic is considered as the total energy and the electrical potential as the converted energy. The obtained result is a complex quantity related to the complex material coupling factor by means of the same proportionality coefficient of the case without losses. Finally, it is shown that both the material and the dynamic coupling factors still can be considered as real parameters for real lossy materials. It also is shown that the obtained results do not depend on the wave propagation direction (longitudinal or transverse).