An analytical model of a phononic crystal with a piezoelectric defect for energy harvesting using an electroelastically coupled transfer matrix

Phononic crystal (PnC)-based piezoelectric energy harvesting (PEH) has been recently introduced to amplify the input energy fed into PEH devices. If a defect is imposed into a PnC, as a result, defect bands are created within the band gap. At a defect band frequency, elastic waves can be localized inside the defect. By attaching a PEH device to the defect, output electric power can be significantly enhanced. However, since the attachment of the PEH device leads to the shifting of the defect band frequency, the output electric power can drop significantly if an excitation frequency does not align with the shifted defect band frequency. Thus, this study aims to develop an electroelastically coupled analytical model of a PnC with a piezoelectric defect; the proposed model considers the inertia and stiffness effects. An electroelastically coupled transfer matrix is derived by simultaneously solving a mechanical equation of motions and an electrical circuit equation, with the help of Green's function. The proposed electroelastically coupled analytical model can be used to predict defect band frequencies and defect mode shapes (based on a transfer matrix method; TMM) and output electric power (based on an S-parameter method; SPM) across an external electrical resistance. Results of the proposed electroelastically coupled analytical model are in very good agreement with those of a finite element method (FEM). The contributions of the proposed electroelastically coupled analytical model are two-fold: (1) this is the first attempt to propose analytical model of a PnC with a piezoelectric defect for energy harvesting; and (2) the model allows physical parameterization of the electroelastically coupled transfer matrix in an explicit manner.