## 2 Theoretical Analysis of Cavity-Based WPT Systems
### Analysis of the WPT System's Sub-Models
Equivalently, the reflector does not ensure total reflection--some portion of the EM energy still reaches the rear end of the waveguide. This means that, in practice, the distance between the top wall of the metallic closet will affect the transmitting dipole antenna's performance due to wave reflections. Additionally, Figure 7 reveals that for a sufficiently long reflector, the amplitude of the main (TE\({}_{01}\)) wave propagating towards the rear end of the waveguide is the smallest among the six waveguide modes displayed. This indicates that the backwards-propagating  
Figure 7: The coupling coefficients between the TEM line mode and the three power-carrying waveguide modes (TE\({}_{01}\), TM\({}_{21}\), and TE\({}_{21}\)) against the transmitting dipole length, calculated at \(L_{\text{feed}}=70\) mm and \(d_{\text{dip-dir}}=145\) mm without the PEC terminating plate behind the dipole antenna. The waveguide modes propagating in the desired direction are indicated by (+), whereas those propagating in the opposite direction are indicated by (–). component of the mode TE\({}_{01}\) experiences very strong reflection from the conducting strip intended to suppress energy from flowing in the opposite direction, thus giving rise to a pronounced standing wave pattern due to multiple reflections from the waveguide terminations and the antennas themselves. In brief, in the case of the TE\({}_{01}\) mode, the reflector behaves as required, sending the TE\({}_{01}\) mode power in the desired direction. As regards the other two modes responsible for the power exchange between the Tx and Rx (ensuring power delivery to the receiver) and having identical phase velocities, the amplitude of the TM\({}_{21}\) mode is nearly twice as high as that of its degenerate counterpart TE\({}_{21}\), as apparent in Figure 7. Interestingly, the amplitude of the backwards-propagating TM\({}_{21}\) mode is just about 30% smaller than that of the forward-propagating TE\({}_{21}\) mode. This result reveals that the interaction between the TM\({}_{21}\) mode and the reflector is appreciably weaker than in the case of the TE\({}_{01}\) mode, meaning that this mode will contribute significantly to the appearance of the PTE minima and maxima along the waveguide. Moreover, the shape of the PTE versus the Tx-to-Rx distance curve is likely to be sensitive to the distance between the dipole antenna end (reflector's edge) and the top wall of the closet (rear end of the waveguide). Also, it is observed that the TE\({}_{21}\) mode demonstrates the weakest coupling to the TEM feedline mode. However, the forward-to-backwards-propagating mode amplitude ratio for the TE\({}_{21}\) mode is almost equal to that of TM\({}_{21}\). In view of poor backwards-propagation suppression of the higher order power transferring modes TM\({}_{21}\) and TE\({}_{21}\) by the reflector, it is of paramount importance to examine the effect of the conducting wall placed at the rear side of the waveguide on the amplitudes of the modes sent towards the receiving antenna. Similar to the previous case, in place of the receiving antenna, perfect impedance matching is ensured for each of the considered modes at the unterminated end of the waveguide. It should be noted that the reflection coefficient (S\({}_{11}\)) of the transmitting antenna characterizes the quality of matching between the antenna and the waveguide (WPT system under study). In the case shown in Figure 7, an almost perfect matching is achieved at resonance, which occurs when \(L_{\text{dipole}}=62.5\) mm and \(L_{\text{feed}}=70\) mm. These antenna parameter values ensure almost zero reflection from the transmitting antenna, implying that all power produced by the generator is converted into the energy stored in the three waveguide modes that are responsible for power transfer to the intended receiver. Now consider the effect of a conducting plate used to terminate the rear end of the waveguide.