## 2 Theoretical Analysis of Cavity-Based WPT Systems
### WPT Model Decomposition
Of course, the proposed model is by no means complete in terms of various subtle features present in the real-life closet serving as the waveguide, such as metallic  
Figure 5: The Ansys HFSS model constructed to determine the generalized scattering matrix relating the waveguide mode amplitudes and phases at both ends (ports) of a waveguide section containing a dipole antenna model and the amplitude and phase of the dipole feed line’s TEM wave. Figure 6: The Ansys HFSS model for finding the generalized scattering matrix relating the waveguide mode amplitudes and phases at both ends of a waveguide section containing the Yagi-like antenna’s director (**a**) and the same model with a port and the relevant mode integration lines highlighted (**b**). anchors, a small gap between the metal shelf and the closet door, and non-flatness of the closet door. Once the generalized matrices describing the behavior of the dipole antenna and its director are calculated for various combinations of their parameters, they are exported to MATLAB files to be used by MATLAB routines developed for computing the overall S parameters of the WPT system by combining the sub-model matrices. The director has only one variable parameter--its length, whereas the dipole antenna has three variable parameters: reflector's length, dipole's length, and dipole's feed line length. To further facilitate the derivation of the generalized S matrix of sub-model combinations, the generalized matrices can be decomposed into several lower-order sub-matrices relating the mode amplitudes and phases of the incident and reflected waves at a pair of ports. Thus, each submatrix may be interpreted either as a generalized reflection or a transmission matrix. For example, for some arbitrary two-port structure operated in multimode regime, the submatrix matrix \(\mathbf{S}_{11}\) of the structure's generalized scattering matrix, \(\mathbf{S}\), relates the amplitudes of the incident and reflected modes at Port 1 and, therefore, may be interpreted as the extension of the reflection coefficient at Port 1. Likewise, \(\mathbf{S}_{12}\) is a submatrix of \(\mathbf{S}\) relating the amplitudes of modes excited at Port 1 to those arriving at Port 2. Hence, this matrix may be regarded as the generalized transmission matrix relating amplitudes of the modes emanating from Port 2 to those reaching Port 2. Submatrix \(\mathbf{S}_{12}\) is, in fact, the generalized transmission matrix from Port 2 to Port 1, and the last submatrix \(\mathbf{S}_{22}\) is a generalization of the reflection coefficient at Port 2, as it relates the incident and reflected mode amplitudes and phases at the port. Also, for notational convenience and to simplify the treatment of generalized matrices, the modes belonging to several ports may be treated as a single mode group. For example, the TEM mode of the lumped port and the waveguide Port 1 modes can be combined into a single group to find the S matrix for a dipole loaded waveguide section terminated in a conducting plate at the rear end. In such a case, the generalized reflection and transmission matrices will relate the amplitudes of the modes of different groups instead of ports.