## 2 Theoretical Analysis of Cavity-Based WPT Systems
### WPT Model Decomposition
Thus, the generalized scattering matrix for a waveguide section containing the Tx dipole antenna, whose rear end is terminated into a conducting metallic plate, can be found as follows:  
\[\mathbf{S}^{\text{TX-DP}}=\mathbf{S}_{11}^{\text{TX-DIP}}+\mathbf{S}_{12}^{ \text{TX-DIP}}\mathbf{T}^{\text{TX-PT}}\Big{(}\mathbf{I}^{\text{DR}}-\mathbf{ S}^{\text{PLATE}}\mathbf{T}^{\text{TX-PT}}\mathbf{S}_{22}^{\text{TX-DIP}}\mathbf{T}^{ \text{TX-PT}}\Big{)}^{-1}\mathbf{S}^{\text{PLATE}}\mathbf{T}^{\text{TX-PT}} \mathbf{S}_{21}^{\text{RX-DIP}}, \tag{2}\]  
where \(\mathbf{S}_{11}^{\text{TX-DIP}}\) and \(\mathbf{S}_{22}^{\text{TX-DIP}}\) are the generalized reflection matrices for mode Groups 1 and 2 of the Tx antenna's dipole sub-model, respectively, \(\mathbf{S}_{21}^{\text{TX-DIP}}\) and \(\mathbf{S}_{12}^{\text{TX-DIP}}\) are the generalized transmission matrices governing the transmission of the waves from mode Group 1 (the lumped port exciting the TEM wave in the feed line + waveguide Port 1 at the front end of the waveguide section) to mode Group 2 (Port 2 defined at the section's rear end) of the same sub-model, and vice versa, \(\mathbf{T}^{\text{TX-PT}}\) is a diagonal matrix whose order is equal to the number of modes at Port 2 and the \(n\)-th diagonal entry of the matrix is given by \((\mathbf{T}^{\text{TX-PT}})_{n,n}=\mathrm{e}^{-\tau_{n}d_{\text{TX-PLATE}}}\), \(d_{\text{TX-PLATE}}\) and \(\gamma_{n}\) are the distance between the rear edge of the Tx dipole antenna reflector and the rear-end of the waveguide and the propagation constant of the \(n\)-th waveguide mode, respectively, and \(\mathbf{I}^{\text{DR}}\) is the identity matrix of the same order as \(\mathbf{T}^{\text{TX-PT}}\). Similarly, one can obtain the generalized scattering matrix for the receiving antenna backed by another metallic plate (the front end of the waveguide in the Ansys model or the closet bottom wall in the real-life closet used in the experiments):  
\[\mathbf{S}^{\text{RX-DP}}=\mathbf{S}_{11}^{\text{RX-DIP}}+\mathbf{S}_{12}^{ \text{RX-DIP}}\mathbf{T}^{\text{RX-PT}}\Big{(}\mathbf{I}^{\text{DR}}-\mathbf{ S}^{\text{PLATE}}\mathbf{T}^{\text{RX-PT}}\mathbf{S}_{22}^{\text{RX-DIP}} \mathbf{T}^{\text{RX-PT}}\Big{)}^{-1}\mathbf{S}^{\text{PLATE}}\mathbf{T}^{ \text{RX-PT}}\mathbf{S}_{21}^{\text{RX-DIP}}, \tag{3}\]  
where \(\mathbf{S}_{11}^{\text{RX-DIP}}\) and \(\mathbf{S}_{22}^{\text{RX-DIP}}\) are the generalized reflection matrices for mode Groups 1 and 2 of the Rx dipole's sub-model, respectively, \(\mathbf{S}_{21}^{\text{RX-DIP}}\) and \(\mathbf{S}_{12}^{\text{RX-DIP}}\) are the generalized transmission matrices describing the transmission of the waves from Group 1 (lumped port + Port 1 defined at the front end of the waveguide section containing the Rx antenna) to port Group 2 (Port 2 defined at the rear end) of the Rx dipole's sub-model, and vice versa, \(\mathbf{T}^{\text{RX-PT}}\) is a diagonal matrix whose order is equal to the number of waveguide modes at Port 2 and whose \(n\)-th diagonal entry of the matrix is given by \((\mathbf{T}^{\text{RX-PT}})_{n,n}=\text{e}^{-\gamma_{n}d_{\text{RX-PLATE}}}\), and \(d_{\text{RX-PLATE}}\) is the distance between the Rx dipole antenna and the front end of the waveguide. The matrix describing the reflection of modes from the conducting plate \(\mathbf{S}^{\text{PLATE}}\) may be chosen to be equal to the negative of the identity matrix, corresponding to a PEC (perfectly electrically conducting) plate. Some numerical studies conducted by the authors revealed that such an approximation may significantly compromise the amplitude and phase response accuracy. Therefore, in this study, the reflection of the conducting plate is treated as a surface with imposed impedance boundary conditions. Although this model is also approximate, it dramatically reduces calculation errors. The surface impedance is calculated based on the closet walls' estimated conductivity and relative magnetic permeability.