## 2 Theoretical Analysis of Cavity-Based WPT Systems
### Analysis of the WPT System's Sub-Models
However, it requires evaluating surface integrals, which may be cumbersome when treating a waveguide with complex cross-section geometry. Therefore, calculating the propagation constants using full-wave simulation tools is more favorable as it gives sufficiently accurate values, provided appropriate boundary conditions are imposed (see Table 1). The Ansys HFSS simulation data presented in Table 1 show that of the three power transferring modes, the TM\({}_{21}\) exhibits the highest attenuation, while the TE\({}_{01}\) mode has the lowest. Figure 8: The coupling coefficients between the TEM line mode and the three power-carrying waveguide modes (TE\({}_{01}\), TM\({}_{21}\), and TE\({}_{21}\)) (**a**) and their phases relative to that of the incident TEM wave against the transmitting dipole length (**b**), calculated for a waveguide section terminated into a conducting plate at the rear end containing a dipole antenna with \(L_{\text{feed}}=70\) mm. Another model whose behavior is also of theoretical interest is the one involving the dipole antenna augmented with a conducting strip (director), as it might shed some light on the effect of the antenna director on the field distribution within the resonant cavity. The mode coupling coefficients and the TEM wave reflection coefficient are plotted in Figure 9. Note that, like in the first case, there is no termination at both waveguide ends. The figure reveals that the coupling with the forward-propagating mode TM\({}_{21}\) is virtually unaffected by the presence of the director. The same applies to the forward-propagating TE\({}_{21}\) mode. Meanwhile, the amplitude of the forward-propagating TE\({}_{01}\) mode is about 30% smaller than without the director. In contrast, the amplitudes of all the backwards-propagating modes are considerably larger. The results show that from the three power-transferring waveguide modes, only the TE\({}_{01}\) mode exhibits strong interaction with the additional conducting strip; furthermore, the director reflects the TE\({}_{01}\) mode and converts a significant portion of its power into the other two modes. The last case to be examined is a Yagi-like antenna composed of a dipole antenna and a conducting strip (copper strip), referred to throughout this paper as a director, by analogy with the conventional Yagi-Uda antennas. The director is running parallel to the dipole and separated from it by a certain distance denoted by \(d_{\rm dip-dir}\). Hence, the conventional dipole antenna may be viewed as a special case of the Yagi-like antenna with a director length equal to 0. Figure 10a shows the most practically significant generalized S parameters (coupling coefficients) calculated for the Yagi-like Tx antenna. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode** & **Attenuation** & **Phase** & **Mode** & **Attenuation** & **Phase** \\  & **Constant [s\({}^{-1}\)]** & **Constant [s\({}^{-1}\)]** & **Constant [s\({}^{-1}\)]** & **Constant [s\({}^{-1}\)]** \\ \hline TE10 & 0.0212 & 16.9227 & TE21 & 0.0747 & 6.2534 \\ TE01 & 0.0279 & 14.5773 & TE30 & 7.7375 & 0.0968 \\ TM11 & 0.0567 & 13.0302 & TE02 & 11.7587 & 0.0900 \\ TE11 & 0.0399 & 13.0133 & TE32 & 13.2780 & 0.0911 \\ TE20 & 0.0404 & 12.5117 & TE41 & 13.3381 & 0.0327 \\ TM21 & 0.1477 & 6.3300 & TE22 & 13.4590 & 0.0973 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Waveguide mode parameters. Figure 9: 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 \(L_{\rm feed}=70\) mm and \(d_{\rm dip-dir}=145\) mm. Interestingly, the resonant value of \(L_{\text{dipole}}\) is the same as in the case of a single dipole without a conducting plate at the rear port. Comparing the results with those of Figure 8a, one finds that the director does not affect the amplitudes of the TE modes, whereas that of the TM\({}_{21}\) mode slightly increases.