## 2 Theoretical Analysis of Cavity-Based WPT Systems
### WPT Model Decomposition
Likewise, one can calculate the other three submatrices of \(\mathbf{S}^{\text{YAG}}\) :  
\[\mathbf{S}^{\text{YAG}}_{22}=\mathbf{S}^{\text{DIR}}_{22}+\mathbf{S}^{\text{DIR }}_{21}\mathbf{T}^{\text{DIP-DIR}}\left(\mathbf{I}^{\text{DR}}-\mathbf{S}^{ \text{TX-DP}}_{22}\mathbf{T}^{\text{DIP-DIR}}\mathbf{S}^{\text{DIR}}_{11} \mathbf{T}^{\text{DIP-DIR}}\right)^{-1}\mathbf{S}^{\text{TX-DP}}_{22}\mathbf{T}^ {\text{DIP-DIR}}\mathbf{S}^{\text{DIR}}_{12}, \tag{5}\]  
\[\mathbf{S}^{\text{YAG}}_{21}=\mathbf{S}^{\text{DIR}}_{21}\mathbf{T}^{\text{DIP-DIR }}\left(\mathbf{I}^{\text{DR}}-\mathbf{S}^{\text{TX-DP}}_{22}\mathbf{T}^{ \text{DIP-DIR}}\mathbf{S}^{\text{DIR}}_{11}\mathbf{T}^{\text{DIP-DIR}} \right)^{-1}\mathbf{S}^{\text{TX-DP}}_{21}, \tag{6}\]  
\[\mathbf{S}^{\text{YAG}}_{12}=\mathbf{S}^{\text{TX-DP}}_{12}\mathbf{T}^{\text{ DIP-DIR}}\left(\mathbf{I}^{\text{DR}}-\mathbf{S}^{\text{DIR}}_{11}\mathbf{T}^{ \text{DIP-DIR}}\mathbf{S}^{\text{TX-DP}}_{22}\mathbf{T}^{\text{DIP-DIR}} \right)^{-1}\mathbf{S}^{\text{DIR}}_{12}, \tag{7}\]  
where \(\mathbf{S}^{\text{DIR}}_{22}\) is the reflection matrix for Port 2 of the director sub-model, \(\mathbf{S}^{\text{DIR}}_{21}\) and \(\mathbf{S}^{\text{TX-DP}}_{12}\) are the matrices describing mode transmission from Port 1 to Port 2 of the same sub-model, and from Port 2 to Port 1, respectively, \(\mathbf{S}^{\text{YAG}}_{22}\) is the reflection coefficient matrix at Port 2 (waveguide port) of the transmitting Yagi-like antenna, and \(\mathbf{S}^{\text{YAG}}_{21}\) and are the transmission coefficient matrices from Port 1 (lumped port) to Port 2 and vice versa, respectively. The final step in the WPT system scattering parameter derivation process is the identification of the four crucial S parameters. These parameters relate the amplitudes and phases of the TEM waves at the generator and the receiving end of the system. Unlike the Yagi-like antenna, where the generalized scattering matrix was derived from the parts characterizing the antenna, the resulting S data are a set of four scalar quantities in this case. This is because the Tx and Rx ends operate in a deep TEM regime, where the feed line dimensions are significantly smaller than the wavelength at the operating frequency. The contribution of higher-order modes under these conditions is negligible. As a result, the overall S parameters of the WPT system can be expressed in terms of scattering matrices \(\mathbf{S}^{\mathrm{YAG}}\) and \(\mathbf{S}^{\mathrm{RX-DP}}\) as follows:  
Figure 6.0  
Figure 6.1  
Figure 6.2  
Figure 6.3  
where \(\mathbf{S}^{\mathrm{RX-DP}}_{11}\) is the reflection coefficient matrix at Port 1 of the receiving dipole antenna, \(\mathbf{S}^{\mathrm{RX-DP}}_{21}\) and \(\mathbf{S}^{\mathrm{RX-DP}}_{12}\) are the transmission matrices from Port 1 to Port 2 of the dipole, and vice versa, respectively, \(\mathbf{S}^{\mathrm{RX-DP}}_{22}\) is the generalized reflection matrix at Port 2, \(\mathbf{T}^{\mathrm{Tx-Rx}}\) is a diagonal matrix of order two whose \(n\)-th diagonal entry is \(\left(\mathbf{T}^{\mathrm{Tx-Rx}}\right)_{n,t}=\mathrm{e}^{-\gamma_{n}d_{ \mathrm{Tx-to-Rx}}}\), and \(d_{\mathrm{Tx-to-Rx}}\) is the distance between the Tx and Rx antennas.