# Optimization Design of Planar Circle Coil for Limited-Size Wireless Power Transfer System
## 2 Design Variables and Objectives of Planar Circular Coil  
The circuit diagram of the S-S compensation topology for the WPT system is shown in Figure 1. AC power \(\mathbf{U_{AC}}\) represents the high-frequency voltage after the inverter, \(L_{P}\) and \(L_{S}\) are the self-inductance of the coils, resonant in series with the compensation capacitors \(C_{P}\) and \(C_{S}\). \(M\) is the mutual inductance between the two coils, \(R_{P}\) and \(R_{S}\) are AC resistance of coils, \(\mathbf{I_{P}}\) and \(\mathbf{I_{S}}\) are the current of the transmitting loop and the receiving loop, respectively, and \(R_{L}\) means the load resistance. According to Kirchhoff voltage law (KVL), the equations in Figure 1 is:  
\[\left\{ \begin{array}{c}\mathbf{U}_{AC}=\Bigg{(}R_{p}+j\omega L_{p}+\frac{1}{j \omega C_{p}}\Bigg{)}\mathbf{I}_{p}-j\omega M\mathbf{I}_{s}\\ j\omega M\mathbf{I}_{p}=\Bigg{(}R_{s}+j\omega L_{s}+\frac{1}{j\omega C_{s}} \Bigg{)}\mathbf{I}_{s}\end{array} \right. \tag{1}\]  
Then the expressions of \(\mathbf{I_{P}}\), \(\mathbf{I_{S}}\), and \(\mathbf{U_{L}}\) is obtained:  
\[\mathbf{I}_{P}=\frac{\mathbf{U}_{AC}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C _{S}}\Bigg{)}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j\omega C_{P}}\Bigg{)} \Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S}}\Bigg{)}+\omega^{2}M ^{2}} \tag{2}\]  
\[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]  
\[\mathbf{U}_{L}=\mathbf{I}_{S}R_{L}=\frac{j\omega M\mathbf{U}_{AC}R_{L}}{\Bigg{(}R_{P}+j \omega L_{P}+\frac{1}{j\omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+ \frac{1}{j\omega C_{S}}\Bigg{)}+\omega^{2}M^{2}} \tag{4}\]  
Combining (2), (3), and (4), the PTE and TP can be obtained as  
\[TP=|\mathbf{U}_{L}||\mathbf{I}_{S}||\cos(\varphi_{u_{L}}-\varphi_{i_{S}})|=|\mathbf{I}_{S }||\mathbf{I}_{S}|R_{L}=\frac{|\mathbf{U}_{L}||\mathbf{U}_{L}|}{R_{L}} \tag{5}\]  
\[PTE=\frac{TP}{|\mathbf{U}_{AC}||\mathbf{I}_{P}||\cos(\varphi_{u_{AC}}-\varphi_{i_{P}} )|} \tag{6}\]  
where \(\varphi_{uL}\), \(\varphi_{iS}\), \(\varphi_{uAC}\), and \(\varphi_{iP}\) are the phase differences of \(\mathbf{U}_{L}\), \(\mathbf{I}_{S}\), \(\mathbf{U}_{AC}\), and \(\mathbf{I}_{P}\), respectively, with respect to the reference sine. For measuring system performance, researchers generally use dimensionless indicators. Therefore, we define the indicators without dimension efficiency \(\eta\) and the voltage modulus gain \(K\) when the load resistance is given, the corresponding expressions are as follows:  
\[K=\frac{|\mathbf{U}_{L}|}{|\mathbf{U}_{AC}|} \tag{7}\]  
Figure 1: Circuit diagram of S-S compensation topology. \[\eta=PTE=\frac{|\mathbf{U}_{L}||\mathbf{I}_{S}||\cos(\varphi_{u_{L}}-\varphi_{i_{S}})|}{| \mathbf{U}_{AC}||\mathbf{I}_{P}||\cos(\varphi_{u_{AC}}-\varphi_{i_{P}})|} \tag{8}\]  
Then the performance of a WPT system can be described by \(\eta\) and \(K\). According to the expressions when the load resistance \(R_{L}\) is determined, the variables affecting \(\eta\) and \(K\) includes inverter frequency \(f\), circuit natural angular frequency \(\omega_{0}\), mutual inductance \(M\), resistances of coils \(R_{P}\) and \(R_{S}\), and self-inductance \(L_{P}\) and \(L_{S}\). Figure 2 shows the coupler structure of the WPT system in this paper. The aluminum plate and ferrite core can shield the electromagnetic field to reduce its influence on the underwater vehicle; simultaneously, the ferrites can increase mutual inductance between two coils. After winding the coil, it is sealed with epoxy resin to avoid displacement and contact with water. Due to the excellent motion performance of the underwater vehicle and the good centering condition of the coupler, the primary and secondary side structures in the design of this paper are the same, namely \(R_{P}=R_{S}=R\) and \(L_{P}=L_{S}=L\).