## 2 Theoretical Analysis of Cavity-Based WPT Systems
### Analysis of the WPT System's Sub-Models
In the real-life WPT system employed in the present study, the top of the closet is made of the same metal as the side walls; therefore, the conducting plate terminating the waveguide end, which corresponds to the closet's top wall, may be treated as such. As follows from the above discussion, higher-order modes are not sufficiently strongly reflected by the reflector, resulting in a variation in the field pattern with the gap size between the Tx antenna reflector and the terminated waveguide end. The coupling coefficients between the forward-propagating modes and the TEM wave and the TEM wave reflection coefficient are presented in Figure 8a. As can be seen, the amplitude of the TE\({}_{01}\) mode is larger than without the conducting plate terminating the waveguide. This amplification is due to the superposition of the forward- and backwards-propagating TE\({}_{01}\) modes with equal or at least almost equal phases. The phase difference is determined by the feed line length of the dipole antenna, as the backwards-propagating TE\({}_{01}\) mode is largely reflected from the reflector. However, this is not the case with the other two modes--as evidenced by Figure 8a, their amplitudes are only slightly affected by the presence of the conducting plate. The phases of the coupling coefficients (generalized S parameters) and that of the TEM line reflection coefficient plotted as functions of the transmitting antenna dipole length are presented in Figure 8b. The mode phases demonstrate gentle variation with the dipole length at the length values far from the \"resonant\" one, where the maximum power coupling between the TX antenna and the waveguide at hand is observed. In the vicinity of the resonant value, which in this case is about \(L_{\text{dipole}}=62.5\) mm, the phase curve slopesfor all modes become noticeable higher. The TEM line reflection coefficient shows a similar behavior, except that it exhibits a jump of about \(160^{\circ}\) at the resonant value of \(L_{\text{dipole}}\). The perfect matching between the transmitting and the receiving antenna (no reflection back to the generator) could be achieved if the amplitudes of the three power-carrying modes upon the arrival of the receiving antenna were preserved and the relative phases were the negatives of those radiated by the Tx antenna. Namely, the phasors of the modes arriving at the Tx antenna must be the complex conjugates of those emanating from the Rx antenna. This case is the ideal one--a goal to strive for. The main factor limiting the PTE of such an ideal hollow waveguide-based WPT system would be the attenuation experienced by the modes while travelling from one antenna to the other. The generalized S matrix relating the mode amplitudes and phases at the Yagi-like antenna model's ports (the faces perpendicular to the wave propagation direction) can be deduced by combining the S matrices of the dipole antenna and the director, respectively. The effect of the empty lossy wall waveguide section joining the sub-models can be evaluated analytically since, for a homogeneously filled rectangular waveguide, there are closed-form relations between the mode amplitudes at the ends of the section. Only the propagation constants for the considered modes need to be found to obtain the relations, which can be done analytically for rectangular waveguides. Alternatively, one can find the mode phase constants and the modal fields analytically by assuming that the waveguide walls are perfectly conducting; then, the effect of the wall losses on the wave propagation expressed in terms of mode attenuation constants may be quantified using either the power loss model [33; 34], or the boundary impedance perturbation model [35; 36]. The power loss model has been shown to fail to provide adequate accuracy close to the cut-off frequency. The boundary impedance perturbation method, in turn, provides a reliable estimation of the complex mode propagation constant.