# Optimization Design of Planar Circle Coil for Limited-Size Wireless Power Transfer System
## 2 Design Variables and Objectives of Planar Circular Coil
The contour diagram Figure 5 shows that when the mutual inductance of the coils is large, \(K\) can be increased at the expense of decreasing \(\eta\) by adjusting the inverter frequency not equal to the natural frequency of the circuit. However, at the same efficiency \(\eta\), the maximum voltage modulus gain \(K\), meaning the maximum TP, must occur under resonant conditions, where \(f=f_{0}\). Therefore, to ensure that \(K\) and \(\eta\) keep good performance simultaneously, the circuit natural resonant frequency \(f_{0}\) should be equal to the inverter frequency \(f\). According to Equations (2) and (3), the impedance of the secondary side circuit \(\mathbf{Z_{S}}\) and the impedance reflected from the secondary side to the primary side \(\mathbf{Z_{ref}}\) is:  
\[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]3  
\[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]4  
Then the primary side input impedance \(\mathbf{Z_{I}}\) is:  
\[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]5  
When the WPT system works in the resonant state, the current and voltage on the primary side and secondary side are in the same phase, meaning that the imaginary parts of \(\mathbf{Z_{I}}\) and \(\mathbf{Z_{S}}\) are zero to maintain the purely resistive. The resonance conditions can be obtained by analyzing Equations (9) and (11):  
\[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]6  
\[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]7  
In the article, \(L_{P}=L_{S}=L\) and \(C_{P}=C_{S}=C\), then the inverter frequency \(f\) satisfies the condition:  
\[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]8  
Then the system efficiency \(\eta\) and voltage modulus gain \(K\) can be described as follows:  
\[\mathbf{I}_{S}=\frac{j\omega M\mathbf{U}_{AC}}{\Bigg{(}R_{P}+j\omega L_{P}+\frac{1}{j \omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+\frac{1}{j\omega C_{S} }\Bigg{)}+\omega^{2}M^{2}} \tag{3}\]9  
\[\mathbf{U}_{L}=\mathbf{I}_{S}R_{L}=\frac{j\omega M\mathbf{U}_{AC}R_{L}}{\Bigg{(}R_{P}+j \omega L_{P}+\frac{1}{j\omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+ \frac{1}{j\omega C_{S}}\Bigg{)}+\omega^{2}M^{2}} \tag{4}\]0 The partial contour diagram of \(K\) and \(\eta\). \[\mathbf{U}_{L}=\mathbf{I}_{S}R_{L}=\frac{j\omega M\mathbf{U}_{AC}R_{L}}{\Bigg{(}R_{P}+j \omega L_{P}+\frac{1}{j\omega C_{P}}\Bigg{)}\Bigg{(}R_{S}+R_{L}+j\omega L_{S}+ \frac{1}{j\omega C_{S}}\Bigg{)}+\omega^{2}M^{2}} \tag{4}\]1  
In summary, the design variables only include inverter frequency \(f\), mutual inductance \(M\) of the two coils, and coils' AC resistance \(R\). During the WPT system charging, the underwater vehicle will be mechanically connected to the base station to prevent the water flow from affecting the system's stability. At this time, the distance between the two coils remains unchanged, so the coils' mutual inductance \(M\) and coils' AC resistance \(R\) is only related to the coil structure. Figure 6 is the cross-sectional structure model of the planar coil. The coil diameter (\(2r\)) usually depends on the current.