## 2 Theoretical Analysis of Cavity-Based WPT Systems
### WPT Model Decomposition  
To facilitate the analysis of the WPT system at hand, the original system's model is simplified so that the resulting model is a rectangular metallic chamber containing a pair of antennas. The chamber's dimensions are identical to those of the closet. In contrast to the original mode, this model ignores the effects of the metallic shelf and the non-uniformity of the door's surface on the fields. Since one of the three linear closet dimensions (the height) is much larger than the other two (the width and the depth), it is more advantageous from the theoretical analysis point of view to treat the chamber as a rectangular waveguide terminated into conducting plates (chamber's top and bottom sides) on both sides. The waveguide-based treatment of the fields within the chamber is chosen in favor of the conventional resonant cavity analysis to considerably facilitate this study's aim of gaining insight into the mechanisms responsible for the high PTE region expansion due to the additional conducting strip in the Tx antenna design. Specifically, the number of waveguide modes needed to approximate the fields adequately is much smaller than the number of resonant cavity modes. The most computationally expensive step of the WPT system design is the optimization process, which requires calculating the WPT system's overall power reflection and transmission coefficient data for a large number of parameter values. To circumvent this issue, a simplified system's model is decomposed into three sub-models. Then, each of these models is treated separately. The first sub-model contains the dipole antenna in an otherwise empty waveguide section whose length is slightly greater than the length of the dipole antenna substrate. Both ends of the waveguide sections are treated as waveguide ports. Since the number of propagating modes at the operating frequency is seven, at least modal fields corresponding to this set of modes must be used to approximate the fields on the surface of each port. In many cases, this approximation ensures satisfactory result accuracy; however, in some specific cases, the result accuracy may not be satisfactory. The reason this way of approximating the fields sometimes fails is that the evanescent waveguide modes still affect the field pattern at the port location even though they decay exponentially with the distance from a discontinuity (dipole antenna or Yagi-antenna's director). The rate at which the fields of modes whose cut-off frequency is not much higher than the operating one decay is not sufficiently large to ignore their effect on the port fields. To mitigate this issue, the number of modal fields for approximating the port field is increased to 12. Numerical studies carried out by the authors show that using 12 waveguide modes guarantees excellent agreement between the fast model decomposition-based method and the direct full-wave analysis of the closet-based WPT model at hand. The WPT system sub-model involving the dipole antenna without the director constructed for the generalized S matrix calculation using full-wave software Ansys HFSS 2023 R1 is depicted in Figure 5. The blue arrow indicates the location of the reference plane defined for the waveguide port at the rear end of the section. The relevant reference plane is positioned at the tip of the arrow, i.e., it is aligned with the rear edge (reflector side) of the dipole antenna. The second sub-model involves the transmitting antenna director only. The director strip can be separated from the dipole model due to a relatively large distance between them in the optimal antenna configuration, as will be shown next in this paper. The WPT system sub-model involving the Yagi-like transmitting antenna director built for the evaluation of the relevant generalized S matrix using Ansys HFSS 2023 R1 is illustrated in Figure 6.  
The last sub-model is the receiving dipole antenna, which is constructively identical to the receiving one with the director removed but, in general, may have different dimensions of its elements.