## 2 Analysis of the Proposed LCCL-S Topology  
The proposed fully integrated LCCL-S topology is depicted in Figure 2. Where \(L_{p}\) and \(L_{s}\) adopt the unipolar coil as the transmitter and receiver coils, and \(L_{in}\) adopts the proposed extended DD coil as the compensation inductor, as shown in Figure 3. Specifically, the bipolar coil generates a magnetic field that travels from one side of the coil to the other, with the magnetic flux circulating between the two sides of the coil, forming a complete loop. \(\Phi_{in-p}\) is the magnetic flux passing through \(L_{in}\) from \(L_{p}\), then \(\Phi_{p-in}\) is the magnetic flux excited by \(L_{p}\) passing  
In the IPT system, the direct current (DC) to alternating current (AC) conversion is realized using a high-frequency full-bridge inverter, as shown in Figure 2, where \(U_{in}\) is the inverter AC output voltage, which can be expressed as follows:  
\[U_{in}=\frac{2\sqrt{2}U_{DC}}{\pi}. \tag{3}\]  
\(R_{eq}\) is the equivalent AC load resistance of \(R_{o}\), and the relationship can be represented as follows:  
\[R_{eq}=\frac{8R_{o}}{\pi^{2}}. \tag{4}\]  
The current and voltage in the circuit loop can be expressed by [PERSON]'s voltage law (KVL) as follows:  
\[ \begin{array}{l}U_{in}=(R_{i}+j\omega L_{in})I_{in}+\frac{1}{j\omega C_{p}}( I_{in}-I_{p})-j\omega M_{in-p}I_{p}+j\omega M_{in-s}I_{s}\\ 0=\frac{1}{j\omega C_{p}}(I_{in}-I_{p})-(R_{p}+j\omega L_{p}+\frac{1}{j\omega C _{f}})I_{p}+j\omega M_{p-s}I_{s}+j\omega M_{in-p}I_{in}\\ 0=j\omega M_{in-s}I_{in}-j\omega M_{p-s}I_{p}+(R_{s}+R_{eq}+j\omega L_{s}+ \frac{1}{j\omega C_{s}})I_{s}.\end{array}  \tag{5}\]  
The nondiagonal components of the impedance matrix of the fully integrated LCCL-S circuit can be expressed as follows:  
\[ \begin{array}{l}Z_{a}=R_{i}+j\omega L_{in}+\frac{1}{j\omega C_{p}},\,Z_{b}= j\omega M_{in-p}+\frac{1}{j\omega C_{p}},\,Z_{c}=j\omega M_{in-s},\\ Z_{d}=j\omega M_{p-s},\,Z_{e}=R_{p}+j\omega L_{p}+\frac{1}{j\omega C_{p}}+\frac {1}{j\omega C_{f}},\,Z_{f}=R_{s}+j\omega L_{s}+R_{eq}+\frac{1}{j\omega C_{s}}. \end{array}  \tag{6}\]  
Figure 3: (**a**) Transmitter coil with an extended DD coil and (**b**) receiver coil. Figure 2: Circuit topology of a fully LCCL-S compensated IPT system. In addition, according to \(ZI=U\), the impedance matrix can be written as follows:  
\[ \begin{bmatrix}\text{Z}_{a}&-Z_{b}&Z_{c}\\ Z_{b}&-Z_{e}&Z_{d}\\ Z_{c}&-Z_{d}&Z_{f}\end{bmatrix}\begin{bmatrix}I_{in}\\ I_{p}\\ I_{s}\end{bmatrix} = \begin{bmatrix}U_{in}\\ 0\\ 0\end{bmatrix} . \tag{7}\]  
From the above analysis, the inverter output current \(I_{in}\) and the transmitter and receiver coil currents \(I_{p}\) and \(I_{s}\) can be expressed as follows:  
\[ \begin{split} I_{in}&=\frac{U_{in}Z_{a}Z_{b}(Z_{b}^{2}(2-Z_{b} ^{2})-Z_{e}Z_{g}}{Z_{d}^{2}(Z_{e}Z_{g}(Z_{a}-1)-Z_{d}^{2}-Z_{b}^{2}Z_{g}-Z_{c}Z_ {d}^{2})+Z_{e}Z_{g}(Z_{e}(Z_{a}Z_{g}+Z_{e}^{2})-Z_{b}^{2}Z_{g}-Z_{g}Z_{c}Z_{d}},\\ I_{p}&=I_{in}\frac{Z_{b}Z_{e}Z_{e}-Z_{e}Z_{d}}{Z_{e}Z_{g}-Z_{d}^{2}}, \quad I_{s}=-I_{p}\frac{Z_{e}Z_{e}-Z_{e}Z_{g}}{Z_{d}Z_{e}-Z_{b}Z_{g}}.\end{split}  \tag{8}\]  
At the resonant frequency (\(f_{o}\)), the compensation topology should satisfy the resonance conditions, and it can be expressed as follows:  
\[C_{p}=\frac{1}{\omega^{2}L_{in}},C_{s}=\frac{1}{\omega^{2}L_{s}},C_{f}=\frac{1 }{\omega^{2}(L_{p}-L_{in})}. \tag{9}\]  
The rectifier input voltage can be simplified to the following:  
\[U_{out}=\frac{M_{p-s}}{L_{in}}U_{in}=\frac{2\sqrt{2}U_{dc}}{\pi}. \tag{10}\]  
Equation (10) clearly shows that the system maintains a constant output voltage independent of the load. Simultaneously, \(U_{out}\) can be adjusted by changing the size of \(L_{in}\) to satisfy the charging voltage requirements of the battery. The AC output power \(P_{out}\) can be expressed as follows:  
\[P_{out}=U_{out}l_{s}=\frac{i\omega(M_{in-s}I_{in}+M_{p-s}I_{p})2\sqrt{2}U_{dc} }{\pi R_{o}}. \tag{11}\]  
Combined with the designed magnetic coupling structure, the misalignment distance of the system increases.