## 4 Analysis of Bilateral LCCL Resonant Networks  
This chapter uses a bilateral LCCL compensation method. The IPT resonant circuit topology using this compensation method is shown in Figure 1.  
For such a high-order system, in order to facilitate the analysis of basic characteristics, the following two approximations need to be made: (1) ignoring the internal resistance of all passive components in the circuit; (2) using the fundamental analysis method--i.e., assuming that only the fundamental component in square wave \(v_{s}\) and \(v_{p}\) plays a major role in energy transmission. Thus, the bilateral LCCL-compensated IPT resonant network equivalent circuit of Figure 1 can be represented as shown in Figure 6. The physical quantity with the superscript in the figure indicates the derivation value of the secondary side to the primary side. In this circuit topology, the ratio of the magnetic coupler is defined as \(n=L_{1}/L_{2}\). The other physical quantities in Figure 6 are defined in Equation (2). \[ \begin{array}{rcl}L_{m}&=&k\cdot L_{1}\\ L_{1m}&=&(1-k)\cdot L_{1}\\ L^{\prime}_{2m}&=&(1-k)\cdot nL_{2}\\ C^{\prime}_{2}&=&C_{2}/n\\ C^{\prime}_{p}&=&C_{p}/n\\ L^{\prime}_{p}&=&n\cdot L_{p}\\ v^{\prime}_{p}&=&\sqrt{n}\cdot v_{p}\end{array}  \tag{2}\]  
In Equation (2), \(k\) represents the coupling coefficient of the primary and secondary coils. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multirow{2}{*}{\(v_{p}\) **\textgreater{} 0**} & \multicolumn{2}{c}{**Switches**} \\ \cline{3-4}  & & **Forward** & **Reverse** \\ \hline
1 & 1 & \(S_{A},S_{D}\) & \(S_{B},S_{C}\) \\
2 & 0 & \(S_{B},S_{C}\) & \(S_{A},S_{D}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Two operating states of the full-bridge converter. Figure 6: The re-coupling logic of the A phase drive signal. Although a high-order system usually has multiple resonant frequencies, the original intent of this chapter is to find a compensation topology suitable for fixed-frequency control to simplify the design of the control system. Therefore, the characteristics of the proposed topology over the entire frequency domain will not be discussed; there will be an analysis of the characteristics of the resonant frequency point that satisfies Equation (3) only. \[\omega_{0}^{-2}=L_{\text{s}}C_{s}=L_{p}C_{p}=(L_{1}-L_{\text{s}})C_{1}=(L_{2}-L _{p})C_{2} \tag{3}\]  
In Equation (3), \(\omega_{0}\) is the resonant angular frequency. For ease of analysis, the series of \(C_{1}\) and \(L_{1m}\) in Figure 6 can be represented by an equivalent inductance, which is Equation (4). \[L_{1\text{s}}=\frac{1}{j\omega_{0}}\cdot\left(\frac{1}{j\omega_{0}C_{1}}+j \omega_{0}L_{1m}\right)=L_{1m}-\frac{1}{\omega_{0}^{2}C_{1}} \tag{4}\]  
Substituting the corresponding expressions of Equations (2) and (3) into Equation (4) is (5). \[L_{1\text{s}}=L_{1}-\frac{1}{\omega_{0}^{2}C_{1}}-kL_{1}=L_{\text{s}}-kL_{1} \tag{5}\]  
Similarly, \(C^{\prime}_{2}\) and \(L^{\prime}_{2m}\) can also be equivalent to Equation (6). \[L^{\prime}_{2p}=L^{\prime}_{p}-kL_{1} \tag{6}\]  
After the above equivalent, according to the superposition theorem, the effects of the input voltage \(\dot{V}_{\text{s}}\) and the output voltage \(\dot{V}^{\prime}_{p}\) on the resonant network can be separately analyzed. First, consider the case where only the input terminal voltage \(\dot{V}_{s}\) is applied to the resonant network, as shown in Figure 7; the broken line in the figure indicates that no current flows through the branch. From Equations (2) and (5), we can get Equation (7). \[L_{1\text{s}}+L_{m}=L_{\text{s}}-k\cdot L_{1}+k\cdot L_{1}=L_{\text{s}} \tag{7}\]  
Obviously, Equation (7) means that the second cell from the left in Figure 5 can also form a resonance. Then, under the action of \(\dot{V}_{s}\), since \(C_{\text{s}}\) and \((L_{1\text{s}}+L_{m})\), \(L^{\prime}_{p}\) and \(C^{\prime}_{p}\) form a stable parallel resonance state at the resonant frequency, Equation (8) is inevitably caused.