## 2 Theoretical Analysis of Cavity-Based WPT Systems
### Analysis of the WPT System's Sub-Models  
Now, we examine the sub-models containing a dipole antenna to understand better the interaction between different waveguide modes and the antenna itself. For this purpose, the generalized scattering parameters relating the amplitudes and phases of the waveguide modes and the incident TEM mode in the antenna feed line must be calculated and analyzed. Notably, not all seven propagating modes are responsible for the power transfer between the generator and the receiver (RF-DC) converter. Specifically, the modes have different field patterns in the cross-sectional planes of the waveguide, resulting in different degrees of interaction between them and the antenna dipole. In this interaction, the electric field vector plays a crucial role, as the strength of the current induced on the dipole is solely determined by the electric field strength and direction at the location of the dipole. To determine which modes will interact strongly with the dipole and, therefore, contribute to the energy transfer, let us first provide the conditions under which a sufficiently strong voltage can be induced across the ends of the dipole arms. The main conditions are as follows:  
* A mode must have a relatively large electric field component along the dipole for the current to be induced in it;
* The electric field intensity must be asymmetric about the waveguide central plane, which is perpendicular to the dipole antenna plane and runs parallel with the waveguide;
* The electric field intensity must be large at the dipole location--the dipole must be located within an electric field maximum, or two maxima with opposite field directions, and the same field strength must occur at the dipole arms. Only modes simultaneously satisfying the above conditions are strongly coupled to the TEM mode from the generator via the feed line. Given the dimensions of the metallic closet and the operating frequency (865.5 MHz) from the seven propagating modes, the above conditions are fulfilled only for three modes: TE\({}_{01}\), TM\({}_{21}\), and TE\({}_{21}\). Specifically, the TE\({}_{01}\) mode has only one electric field vector component parallel to the antenna dipole. Its direction is uniform, and the peak electric field is in the middle of the waveguide, where the antenna should be placed to achieve the highest PTE. The other two modes, TM\({}_{21}\) and TE\({}_{21}\), are degenerate and exhibit a maximum at the central part of the antenna dipole. Similar to the TE\({}_{01}\) mode, the electric field intensity of these is asymmetric relative to the antenna symmetry plane. It is worth noting that these modes have non-uniform field distribution along the dipole, leading to weaker coupling between them and the TEM wave. First, consider the sub-model consisting of the dipole antenna without the director located in an empty section of a rectangular waveguide. The cross-sectional dimensions of the waveguide are identical to those of the closet used in the real-life WPT system. Both ends of the waveguide section are terminated in a perfectly matched load ensured in Ansys HFSS by using two waveguide ports defined at the ends of the waveguide section. Figure 7 shows the amplitudes of the waveguide modes and the reflected feed line's TEM wave when the sub-model under consideration is excited by an incident TEM wave whose amplitude is unity. Under this assumption, the amplitudes are equal to the absolute values of the corresponding generalized S parameters, which, in turn, may be interpreted as coupling coefficients between the incident TEM wave and the waveguide modes evaluated at different values of the dipole length. As can be seen in the absence of the conducting plate terminating the rear end of the waveguide (the end of the waveguide closest to the dipole antenna reflector), the antenna-excited waveguide modes can propagate in both directions despite the presence of the reflector intended to suppress the radiation in the undesired direction.