# A Three-Coil Constant Output Wireless Power Transfer System Based on Parity-Time Symmetry Theory
## 4 Experimental Verification
### Coupling Coil Structure Design and Simulation  
For the three-coil PT-WPT system, the coupling coil must meet specific criteria. Firstly, the transmitting coil should have a substantial transmission area to facilitate a large flux  
Figure 7: The control block diagram of the proposed closed-loop control method with rated output power. region. Additionally, the relay and receiving coils need to conserve space. This study employed square coils with rounded corners to reduce sharp current changes at right angles for both the relay and receiving coils, maintaining a high quality factor. To conserve space, a multilayer coil winding method was utilized to augment the coil's self-inductance. The system's structure is schematically depicted in Figure 8. The self-inductance of a multilayer coil is composed of the self-inductance of each layer, the strong positive mutual inductance between the coils, and the staggered connection between the coil layers, as depicted in Figure 9a,b. The self-inductance and mutual inductance between two adjacent resonant cavities can be expressed by Equations (19) and (20). \[L=\sum\
olimits_{i=1}^{n}L_{i}+2\sum\
olimits_{i,j=1,j\
eq j}^{n}M_{ij} \tag{19}\]  
where \(L_{i}\) denotes the self-inductance of layer \(i\) of the multilayer coil and \(M_{ij}\) denotes the mutual inductance of layer \(i\) and layer \(j\) coils. \[\left\{ \begin{array}{l}M_{T1}=\sum\limits_{i=1}^{n}M_{Ti}\\ M_{R1}=\sum\limits_{i,j=-1,j\
eq j}^{n}M_{ij}^{\prime}\end{array} \right. \tag{20}\]  
where \(M_{Ti}\) denotes the mutual inductance between the transmitting coil and the coil of layer \(i\) of the relay coil, and \(M_{ij}^{\prime}\) denotes the mutual inductance between layer \(i\) of the relay coil and layer \(j\) of the receiving coil. The relationship between the coil's internal resistance, its radius, number of layers, turns per layer, and turn spacing in the multilayer coil were examined. This analysis also considered the impact of increased self- and mutual inductance on the coupling coefficient and flux loss resulting from a significant variance in the number of coil turns. The relay and receiver coils shared the same design constraints, leading to the transmitting coil and multilayer coil's similar design requirements. \[\left\{ \begin{array}{l}|N_{T}-N_{m}|\leq 12\\ |a_{T}-a_{m}|\leq 10\;\mathrm{cm}\end{array} \right. \tag{21}\]  
\(N_{T}\) and \(N_{m}\) represent the numbers of turns of the transmitting and multilayer coils, respectively. \(a_{T}\) and \(a_{m}\) denote the distances from the center of the transmitting coil and multilayer coils to the outermost layer of the coil, respectively. Figure 8: Multilayer coil structure schematic. To satisfy the three-coil PT symmetry condition, it is crucial to disregard the multilayer coil's internal resistance and mutual inductance \(M_{TR}\), considering the coil's boundary constraint. \[\left\{ \begin{array}{l}r_{m}\leq\frac{1}{10}R_{Leq}\\ M_{TR}\leq\frac{1}{10}M_{Tr}\\ 0<N_{T}<25\\ 0<N_{m}<15\\ 2\;\text{cm}<l_{p}<18\;\text{cm}\\ 2\;\text{cm}<l_{s}<8\;\text{cm}\\ \end{array} \right. \tag{22}\]  
The optimized coil parameters are shown in Table 1.  
The magnetic field intensity distributions of the single-layer transmitting coil and the multilayer coil were studied using ANSYS 2022 R1 Maxwell finite element analysis software. Figure 10 illustrates the magnetic field intensity distributions of both coils under the same excitation. \begin{table}
\begin{tabular}{c c} \hline
**Structure Parameters** & **Parameters Values** \\ \hline Outside diameter of transmitting coil \(l_{p}\) & 14 cm \\ Number of turns of transmitting coil \(N_{T}\) & 11 \\ Multilayer coil outer diameter \(l_{s}\) & 5 cm \\ Number of turns of multilayer coils \(N_{m}\) & 8 \\ Winding width \(t_{w}\) & 1.5 mm \\ Turn spacing of coils \(t_{r}\) & 3.78 mm \\ Multilayer coil layer spacing \(t_{l}\) & 5.5 mm \\ \hline \end{tabular}
\end{table} 
Table 1: Design parameters of transmitting and multilayer coils. Figure 9: (**a**) Multilayer coil simulation diagram. (**b**) Connection of each layer.