## 2 Fundamentals of LCC-LCC Systems  
In this section, the WPT part of the system is discussed independently from the communication part. The top side of Figure 1 shows the LCC-LCC compensated WPT system [20, 21] with an active rectifier on the right side. The WPT system features a bidirectional power transfer function since both sides are symmetrically designed, as shown in the electric circuit. The components on the left (right) side are identified with notations with indexes starting with 1 (2). The WPT energy direction is signaled with the subordinated communication system. Therefore, an equivalent circuit of the WPT with two voltage sources helps in modeling the bidirectional WPT system, as shown in Figure 2.  
In the following paragraphs, the outside currents of the LCC-LCC-compensated WPT system are first analytically solved by applying the superposition law to the two voltage sources. Based on the closed-form equations of currents, the power of the source and sink can be obtained. Finally, the mathematical relationship between control parameters and system behaviors in terms of transmission power, energy direction, and efficiency are derived. The design frequency \(\omega_{\mathrm{d}}\) of a WPT system is usually selected in the 85 kHz band to ensure interoperability between systems. The LCC-LCC-compensated WPT system is usually designed at the frequency of \(\omega_{\mathrm{d}}\), and the calculations of compensation components are given as follows. Figure 1: System overview of the WPT system with the integrated communication components. Figure 2: Simplified circuit of LCC–LCC-compensated system for bidirectional WPT. \[C_{1\text{f}} =\frac{1}{L_{1\text{f}}\omega_{\text{d}}^{2}} \tag{1}\] \[C_{2\text{f}} =\frac{1}{L_{2\text{f}}\omega_{\text{d}}^{2}}\] (2) \[C_{1\text{s}} =\frac{1}{(L_{1}-L_{1\text{f}})\omega_{\text{d}}^{2}}\] (3) \[C_{2\text{s}} =\frac{1}{(L_{2}-L_{2\text{f}})\omega_{\text{d}}^{2}} \tag{4}\]  
The input inductances \(L_{1\text{f}}\) and \(L_{2\text{f}}\) are usually selected to be less than the coil's inductances, \(L_{1}\) and \(L_{2}\). Therefore, at the design frequency, the three branches have the same impedance, as shown in (5) and (6). \[\omega_{\text{d}}L_{1\text{f}} =\frac{1}{\omega_{\text{d}}C_{1\text{f}}}=\omega_{\text{d}}L_{1} -\frac{1}{\omega_{\text{d}}C_{1\text{s}}}\overset{\text{def}}{=}X_{1} \tag{5}\] \[\omega_{\text{d}}L_{2\text{f}} =\frac{1}{\omega_{\text{d}}C_{2\text{f}}}=\omega_{\text{d}}L_{2} -\frac{1}{\omega_{\text{d}}C_{2\text{s}}}\overset{\text{def}}{=}X_{2} \tag{6}\]  
The input impedance of the LCC-LCC-compensated network can be derived as follows:  
\[\underline{Z}_{\text{IN}}=\frac{X_{2}^{2}X_{2}^{2}}{\omega_{\text{d}}^{2}M^{ 2}\underline{Z}_{\text{load}}}, \tag{7}\]  
where the impedance of sink \(\underline{Z}_{\text{load}}\) is defined as  
\[\underline{Z}_{\text{load}}= \begin{cases}-\frac{U_{2}}{L_{2}}&\text{for forward transfer}\\ -\frac{U_{1}}{L_{1}}&\text{for reverse transfer.}\end{cases}  \tag{8}\]  
The fundamental and \(n\)th harmonic of the primary side voltage \(u_{1}(t)\) and secondary side voltage \(u_{2}(t)\) can be calculated using Fourier expansion:  
\[U_{1,2}(n)=\frac{2\sqrt{2}U_{1\text{LCC},2\text{DC}}}{n\pi}\sin(\frac{n\pi D_{ 1,2}}{2})\sin(\frac{n\pi}{2}), \tag{9}\]  
where \(n\) is the number of harmonics, and \(D_{1,2}\) is the duty cycles of the primary and secondary side in the range of \(0\leq D_{1,2}\leq 1\). Based on the impedance calculation in (5) and (6) and the fundamental voltages in (9), the primary and secondary currents \(\underline{I}_{1}\) and \(\underline{I}_{2}\) are derived as follows:  
\[\underline{I}_{1} =\omega_{\text{d}}^{2}C_{1\text{f}}^{2}R_{1}\underline{U}_{1}+j \omega_{\text{d}}^{3}MC_{1\text{f}}C_{2\text{f}}\underline{U}_{2} \tag{10}\] \[\underline{I}_{2} =\omega_{\text{d}}^{2}C_{2\text{f}}^{2}R_{2}\underline{U}_{2}+j \omega_{\text{d}}^{3}MC_{1\text{f}}C_{2\text{f}}\underline{U}_{1}, \tag{11}\]  
where \(R_{1,2}\) is the loss resistances in the coil paths.