# A Three-Coil Constant Output Wireless Power Transfer System Based on Parity-Time Symmetry Theory
## 2 Modeling and Analysis of the Three-Coil PT-WPT System  
### Circuit Model of the Three-Coil PT-WPT System  
Figure 1 displays a schematic of the proposed three-coil PT-WPT system employing capacitor series compensation. [PERSON] et al. [23] established a negative resistor using a full bridge inverter, achieving PT symmetry within the system. The proposed system exhibits controllable negative resistance owing to a DC-DC module based on a full bridge inverter, with \(-R_{T}\), \(L_{T}\), \(C_{T}\), and \(r_{T}\) representing the negative resistance, inductance, compensation capacitance, and internal coil resistance of the transmitting coil's resonant tank, respectively. \(L_{1}\), \(C_{1}\), and \(r_{1}\) represent the coil inductance, compensation capacitance, and internal coil resistance, respectively. \(L_{R}\), \(C_{R}\), \(r_{R}\), and \(R_{Leq}\) represent the inductance, compensation capacitance, internal coil resistance, and load AC equivalent resistance of the load coil resonant tank, respectively. \(M_{pq}(p,q=T,1,R,p=q)\) represents the coupling between the coils. The circuit model for a three-coil PT-WPT with an operating angular frequency \(\omega\) can be established by applying [PERSON]'s voltage law and circuit theory, and is mathematically expressed as follows: (1) Assuming that the distance between the receiving coil \(L_{R}\) and the transmitting coil \(L_{T}\) is considerably large, the cross-coupling effect is negligible (\(M_{TR}=0\)). Additionally, the internal resistance of the relay coil can be neglected, considering its high-quality factor (\(r_{1}=0\)). \[\left[ \begin{array}{ccc}-R_{T}+r_{T}\\ L_{T}\end{array} +j(\omega-\frac{\omega_{T}^{2}}{\omega})&j\omega k_{T1}\sqrt{ \frac{L_{1}}{L_{T}}}&0\\ j\omega k_{T1}\sqrt{\frac{L_{T}}{L_{1}}}&j(\omega-\frac{\omega_{1}^{2}}{ \omega})&j\omega k_{R1}\sqrt{\frac{L_{R}}{L_{1}}}\\ 0&j\omega k_{R1}\sqrt{\frac{L_{1}}{L_{R}}}&\frac{R_{Leq}+r_{R}}{L_{R}}+j( \omega-\frac{\omega_{R}^{2}}{\omega})\end{array}\right]\left[ \begin{array}{c} I_{T}\\ I_{1}\\ \dot{I}_{R}\end{array} \right]=0 \tag{1}\]  
Here, \(\omega_{T}\), \(\omega_{1}\), and \(\omega_{R}\) represent the intrinsic resonant angular frequencies of the transmit resonant tank, relay resonant tank, and receive resonant tank, denoted as \(\omega_{T}=1/\sqrt{L_{T}C_{T}}\), \(\omega_{1}=1/\sqrt{L_{1}C_{1}}\), and \(\omega_{R}=1/\sqrt{L_{R}C_{R}}\), respectively; \(k_{T1}\) and \(k_{R1}\) are the coupling coefficients between the adjacent coils, expressed as \(k_{T1}=M_{T1}/\sqrt{L_{T}L_{1}}\) and \(k_{R1}=M_{R1}/\sqrt{L_{R}L_{1}}\). Figure 1: Equivalent circuit of a three-coil PT-WPT system. In the context of a two-coil PT-WPT system [10], the transmitting and receiving coils necessitate identical intrinsic resonant frequencies (i.e., \(\omega_{T}=\omega_{R}\)). Similarly, in a three-coil PT-WPT system with a relay coil, specific conditions must be fulfilled:  
\[\left\{ \begin{array}{l}\omega_{T}=\omega_{1}=\omega_{R}=\omega_{0}\\ k_{T1}=k_{R1}=k\end{array} \right. \tag{2}\]  
where \(\omega_{0}\) is a custom parameter. To obtain a non-zero solution, the determinant of (1) must be zero, meaning that both the real and imaginary parts must be zero [22]. Therefore,  
\[\left\{ \begin{array}{l}\left[\frac{-R_{T}+r_{T}}{L_{T}}+\frac{R_{Leq}+r_{R}} {L_{R}}\right]\left[\left(1-k^{2}\right)\omega^{4}-2\omega_{0}^{2}\omega^{2}+ \omega_{0}^{4}\right]=0\\ \left[\left(1-2k^{2}\right)\omega^{4}-\left(2\omega_{0}^{2}+\frac{-R_{T}+r_{T }}{L_{T}}\times\frac{R_{Leq}+r_{R}}{L_{R}}\right)\omega^{2}+\omega_{0}^{4} \right]\left(\omega^{2}-\omega_{0}^{2}\right)=0\end{array} \right. \tag{3}\]  
The real part of the imaginary part of (3) can be resolved into the two polynomials \(\left(1-k^{2}\right)\omega^{4}-2\omega_{0}^{2}\omega^{2}+\omega_{0}^{4}=0\) and \(\left(\omega^{2}-\omega_{0}^{2}\right)=0\), which cannot be satisfied simultaneously.