## 4 Conclusions  
A theoretical design of a four-plate parallel DCPT coupler applicable to mobile robot-level wireless charging is provided. This design includes the PEC design, incorporating the coupler's loss resistance, and the derivation process of the scattering matrix for each network. The terms of the scattering matrix for each network were expressed in impedance  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **Point1** & **Point2** & **Point3** & **Point4** & **Point5** \\  & **(\(-\)50 mm)** & **(\(-\)35 mm)** & **(0 mm)** & **(32.5 mm)** & **(50 mm)** \\ \hline S21\_max & 0.31 & 0.01 & 0.86 & 0.02 & 0.45 \\ \(\rm C_{M}\) (pF) & 0.04 & 0.08 & 0.52 & 0.07 & 0.04 \\ \(\rm C_{TX}\) (pF) & 3.79 & 3.89 & 3.5 & 3.9 & 3.86 \\ \(\rm C_{RX}\) (pF) & 1.7 & 1.74 & 1.38 & 1.75 & 1.72 \\ \hline \end{tabular}
\end{table}
Table 4: Summary of S21 peak and capacitance by y-axis distance sweep.  
Figure 8: S21 and mutual capacitance by Y-axis distance sweep.  
Figure 7: S21 and capacitance by x-axis distance sweep.  
form, allowing the overall scattering matrix of the proposed PEC to be defined through equation simplification. Valid mutual capacitance values were derived from the scattering matrix calculations of the PEC. This was verified by comparing with the results of 3D full-wave simulations. When the receiver's displacement was 0 mm, the transmission coefficient results from the PEC matched those from the 3D full-wave simulation.  
Additionally, changes in electromagnetic characteristics due to the misalignment of the transmitter and receiver of the four-plate parallel DCPT coupler were observed. The S21 and mutual capacitance were compared as the receiver moved between \(-50\) mm and \(50\) mm along the y-axis. Null-power regions, where the power transmission efficiency sharply decreased, occurred at two points: \(-35\) mm and \(32.5\) mm. The change in S21 showed a similar trend to the mutual capacitance. The parallel structure of the CPT coupler presented in this study has high mutual capacitance and strong coupling when the transmitter and receiver are aligned, but it is vulnerable to null-power regions.  
A theoretical design method for the relatively under-researched DCPT coupler is provided. In the future, theoretical analysis of equivalent circuit models could be an approach to analyzing external factors that induce changes in the electromagnetic characteristics of wireless power transfer couplers. Further research is needed to closely analyze and improve structural alternatives for null-power regions occurring in the four-plate parallel DCPT structure.  
Conceptualization, S.P. and H.B.; methodology, S.P.; software, H.B.; validation, H.B.; formal analysis, H.B.; investigation, H.B.; resources, S.P.; data curation, H.B.; writing--original draft preparation, H.B.; writing--review and editing, H.B.; visualization, H.B.; supervision, S.P.; project administration, S.P.; funding acquisition, S.P. All authors have read and agreed to the published version of the manuscript.  
This results was supported by the \"Regional Innovation Strategy (RIS)\" through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (MOE) (2021 RIS-004), and This work was supported by the Soonchunhyang University Research Fund.  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors upon request.  
The authors declare no conflicts of interest.