# Multi-Objective Optimization of LCC-S-Compensated IPT System for Improving Misalignment Tolerance
## 2 Design and Analysis of LCC-S Compensation Topologies
\begin{array}{l}I_{\rm RO}=\frac{2\sqrt{2}}{\pi}I_{\rm LS}=\frac{2 \sqrt{2}}{\pi}\frac{\mathrm{j}\omega M}{Z_{\rm sec}}I_{\rm LP}\\ U_{\rm RO}=\frac{\sqrt{2}\pi}{4}I_{\rm AB}=\frac{2\sqrt{2}}{\pi}\frac{\mathrm{ j}\omega M}{Z_{\rm sec}}I_{\rm LP}R_{\rm O}\end{array} \right. \tag{6}\]  
Different power transmission characteristics can be obtained using heterogeneous parameter compensation. Firstly, the M-type model is used to procure a load-independent voltage output in the compensation design for the LCC-S topology. A resonant capacitor connects in series to the receiving coil, which compensates for the self-inductance of the receiving coil. The primary LCC compensation circuits are typically designed to have zero phase angle (ZPA) input impedance or weak inductive input impedance for zero-voltage switching (ZVS) turn-on operation of MOSFETs. The primary LCC resonant topologies can supply the transmitting coil with a constant current, and a constant voltage on the secondary side coil is induced. After series resonance consisting of a secondary coil and compensation capacitor, the load can obtain excellent constant current output characteristics. The compensation design is as follows:  
\[\left\{ \begin{array}{l}C_{\rm PI}=1/\left(\omega^{2}L_{\rm PI}\right)\\ C_{\rm P2}=1/\left[\omega^{2}(L_{\rm P}-L_{\rm PI})\right]\\ C_{\rm S}=1/\left(\omega^{2}L_{\rm S}\right)\end{array} \right. \tag{7}\]  
For the CVO type LCC-S compensation, the output voltage can be expressed as  
\[U_{\rm RO}=\frac{M}{L_{\rm P1}}U_{\rm d} \tag{8}\]  
In order to obtain good performance of load-independent current output, the T-type model of LCT was used to design the LCC-S compensation topology parameters. Part of the primary-side compensation capacitor \(C_{\rm PI}\) is used to compensate for the inductance \(L_{\rm PI}\), and the other part is tuned to compensate for the reflected mutual inductance \(L_{\rm M}\). The specific compensation elements can be calculated from [16]. The values of \(C_{\rm PI}\), \(C_{\rm P2}\) and \(C_{\rm S}\) are designed by the following equations to achieve resonant at the switching frequency:  
\[\left\{ \begin{array}{l}C_{\rm P1}=\frac{1}{\omega^{2}L_{\rm PI}}+\frac{1}{ \omega^{2}nM}\\ C_{\rm P2}=\frac{1}{\omega^{2}(1-k)L_{\rm P}}\\ C_{\rm S}=\frac{1}{\omega^{2}(1-k)L_{\rm S}}\end{array} \right. \tag{9}\]  
where \(k\) is the coupling coefficient related to the mutual coupling of the couplers, defined as: \(k=M/\sqrt{L_{\rm P}L_{\rm S}}\), and \(n\) denotes the turn ratio of the loosely coupled transformer. Therefore, the constant output current \(I_{\text{RO}}\) through \(R_{\text{O}}\) can be calculated as  
\[I_{\text{RO}}=\frac{8}{\pi^{2}}\frac{nU_{\text{d}}}{\omega L_{\text{P1}}} \tag{10}\]  
Equations (8) and (10) imply that the IPT system has constant voltage/current output characteristics and that the system can achieve different voltage/current output levels by using \(L_{\text{P1}}\) according to requirements. However, the above compensation parameters are designed without considering coil misalignment. When the spatial position of the two coupling coils changes, the transmission power of the system will fluctuate drastically. The output voltage sensitivity with respect to the mutual inductance is derived as  
\[\frac{\partial U_{\text{RO}}}{\partial M}=\frac{8}{\pi^{2}}\frac{\text{j}R_{ \text{O}}\omega U_{\text{d}}}{Z_{\text{in}}Z_{\text{sec}}}\left(1-\frac{M}{Z _{\text{in}}}\left(2M\omega^{2}+\text{j}\frac{\left(\omega M\right)^{2}}{Z_{ \text{sec}}-\text{j}\omega M}\right)\right) \tag{11}\]  
From (11), the input impedance \(Z_{\text{in}}\) is also related to the mutual inductance coupling, resulting in analytical solutions becoming difficult to obtain. In this paper, the nominal output levels were chosen to be 100 V and 1 A. The minimum coupling coefficient was set to 0.25, and within almost 100% variation, the coupling coefficient was 0.5 with no misalignment. Due to their simplicity in terms of structure and component count, the planar circular coils were adopted in the analysis.