# Multi-Objective Optimization of LCC-S-Compensated IPT System for Improving Misalignment Tolerance
## 2 Design and Analysis of LCC-S Compensation Topologies  
In the magnetic coupler, a large amount of magnetic flux leakage remains; thus, in order to improve the transfer power and system efficiency, compensated resonant circuits are introduced. The mutual inductance model (M-model) and transformer model (T-model) are usually used to describe the system used to compensate for the mutual inductance or leakage inductance of a loosely coupled transformer. The application of the M-model aims to simplify the circuit and reduce the number of variables by removing the leakage inductance and the ratio from the model. As shown in Figure 1a, in the M-model, \(L_{\rm P}\) and \(L_{\rm S}\) are self-inductances of the primary and secondary coils. \(U_{\rm P}\) and \(I_{\rm P}\) are the input voltage and current at the primary side of the loosely coupled transformer, and \(U_{\rm S}\) and \(I_{\rm S}\) are the output voltage and current at the secondary side. \(M\) stands for the mutual inductance between the transmitting and receiving coils. The induced voltage is generated in the primary and secondary side coils, which are denoted as \(-j\omega MI_{\rm S}\) and \(j\omega MI_{\rm P}\). The \(\omega\) parameter is the circuit operating frequency, which equals the switching frequency of the inverter voltage. In the T-type model of LCT, the magnetic couplers are replaced, and all the parameters are reflected to the primary-side. As shown in Figure 1b, \(L_{\rm P}\), \(L_{\rm S}\)\({}^{\prime}\) and \(L_{\rm M}\) are the reflected leakage inductances of the primary coil, secondary coil and mutual inductance, respectively. Although the two models have different forms, they can be converted to each other according to the two-port network equivalence. The circuit of the LCC-S-compensated IPT system is shown in Figure 2, which includes three parts: the voltage power supply \(U_{\rm d}\), the resonant tank and the couplers and load \(R_{\rm O}\). \(L_{\rm P1}\), \(C_{\rm P1}\), \(C_{\rm P2}\) and \(C_{\rm S}\) make up the LCC-S compensation topology. The MOSFETs \(S_{1}\)-\(S_{4}\) combine into the full bridge inverter, and \(D_{1}\)-\(D_{4}\) represent the rectifier diodes. \(C_{f}\) is the filter capacitor. Here, \(U_{\rm AB}\), \(I_{\rm AB}\), \(U_{\rm ab}\) and \(I_{\rm ab}\) are the root mean square (RMS) values of the fundamental harmonics of \(u_{\rm AB}\), \(i_{\rm AB}\), \(u_{\rm ab}\) and \(i_{\rm ab}\), respectively. When the phase-shifting angle is zero, the value of \(U_{\rm AB}\) can be calculated using (1) according to the first harmonic approximation (FHA). Similarly, the equivalent AC load resistance \(R_{\rm E}\) can be expressed as (2). \[U_{\rm AB} = \frac{2\sqrt{2}}{\pi}U_{\rm d} \tag{1}\] \[R_{\rm E} = \frac{8}{\pi^{2}}R_{\rm O} \tag{2}\]  
In the LCC-S topology, a simple series capacitor forms the receiving side compensation networks and the secondary impedance \(Z_{\rm sec}\) is expressed as  
\[Z_{\rm sec}=R_{\rm E}+\frac{1}{j\omega C_{\rm S}}+j\omega L_{\rm S} \tag{3}\]  
Figure 1: Analysis models of the loosely coupled transformer: (a) the mutual inductance model; (b) the transformer model. Figure 2: Circuit diagram of LCC-S-compensated IPT system. The inverter input impedance \(Z_{\rm in}\) of the compensate circuit and load is given as  
\[Z_{\rm in}=\left(\frac{(\omega M)^{2}}{Z_{\rm sec}}+\frac{1}{\mathrm{j}\omega C_{ \rm P2}}\right)\parallel\frac{1}{\mathrm{j}\omega C_{\rm P1}}+\mathrm{j}\omega L _{\rm P1} \tag{4}\]  
Based on the defined impedances of the circuit, the input current of the inverter \(I_{\rm AB}\) and the currents flowing into two coupling coils are derived as  
\[\left\{ \begin{array}{l}I_{\rm AB}=\frac{U_{\rm AB}}{Z_{\rm in}}\\ I_{\rm LP}=I_{\rm AB}(Z_{\rm in}-\mathrm{j}\omega L_{\rm P1})/\left(\frac{( \omega M)^{2}}{Z_{\rm sec}}+\frac{1}{|\omega C_{\rm P2}}\right)\\ I_{\rm LS}=I_{\rm LP}\frac{\mathrm{j}\omega M}{Z_{\rm sec}}\end{array} \right. \tag{5}\]  
Therefore, the output current \(I_{\rm RO}\) and the output voltage \(U_{\rm RO}\) can be expressed as  
\[\left\{