# A Method to Reduce Eddy Current Loss of Underwater Wireless Power Transmission by Current Control
## 2 Circuit Analysis in the Air  
In the WPT system, the commonly used types of circuit compensation topology are series-series (SS), inductor-capacitor-inductor (LCL), double-sided inductor-capacitor-inductor (LCC-LCC), and LCC-P compensation topology. Because the LCC-LCC compensation topology has the characteristics of constant current output under the condition of constant voltage input and strong controllability, the magnitude and phase of the current in the circuits of the primary side and secondary side can be controlled by adjusting the parameters of the compensation components. Considering that this article studies the method of reducing eddy current loss using current control, the LCC-LCC compensation topology was taken as an example for analysis and research, and the equivalent circuit model in the air is shown in the Figure 2 below. U is a DC power supply, and u\({}_{\text{AB}}\) is the input voltage applied on the compensated coil. S\({}_{1}\)\(\sim\)S\({}_{4}\) are four power MOSFETs forming the inverter circuit in the primary side. u\({}_{\text{ab}}\) is the output voltage before the rectifier diodes. D1-D4 are four diodes forming the rectifier circuit in the secondary side. L\({}_{\text{P}}\) and L\({}_{\text{S}}\) are the self-inductances of the primary side coil and secondary side coil, and their internal resistances are R\({}_{\text{P}}\) and R\({}_{\text{S}}\), respectively. L\({}_{\text{P1}}\), C\({}_{\text{P2}}\), and C\({}_{\text{P3}}\) are the primary-side compensation inductor and capacitors. L\({}_{\text{S1}}\), C\({}_{\text{S2}}\), and C\({}_{\text{S3}}\) are the secondary-side compensation inductor and capacitors, respectively. M is the mutual  
Figure 1: Schematic diagram of WPT system and planar dual-coil model. Figure 2: Double-sided LCC compensation topology for WPT system. inductance between two coils. L\({}_{0}\) and C\({}_{0}\) are the filter inductor and capacitor, and R\({}_{\rm L}\) is the load resistance. In order to facilitate the circuit analysis, we transformed the inverter circuit before point A and point B on the primary side into following form as shown in Figure 3.  
U\({}_{\rm in}\) is the AC voltage source of U transformed by the inverter circuit, and R\({}_{\rm in}\) is the equivalent internal resistance of U\({}_{\rm in}\). After using the same method, we obtain the equivalent circuit of the rectifier circuit after point a and point b on the secondary side in Figure 4.  
The variable R\({}_{\rm et}\) in Figure 4 represents the equivalent resistance on the conductor after converting the rectifier circuit. In Figures 2-4, \(\mathbf{I}_{\rm in}\), \(\mathbf{I}_{\rm CP2}\), \(\mathbf{I}_{\rm IP}\), \(\mathbf{J}_{\rm out}\), \(\mathbf{I}_{\rm CS2}\), and \(\mathbf{I}_{\rm S}\) are the currents going through R\({}_{\rm in}\), C\({}_{\rm P2}\), L\({}_{\rm P}\), R\({}_{\rm L}\), C\({}_{\rm S2}\), and L\({}_{\rm S}\), respectively, and they are all in phasor form. This paper aims to study how to reduce the eddy current loss of the underwater WPT system by controlling the current on the primary-side coil and secondary-side coil without considering the influence of frequency. Therefore, in the following research process of this paper, the internal resistance of all inductors and capacitors in the compensation circuit, such as L\({}_{\rm P1}\), C\({}_{\rm P2}\), and L\({}_{\rm S1}\), are neglected for the simplicity of analysis. It is also stipulated  
Figure 4: Equivalent circuit, focusing on the rectifier circuit of the secondary side. Figure 3: Equivalent circuit, focusing on the inverter circuit of the primary side. that the phase of AC voltage source U\({}_{in}\), generated by the inverter circuit, is zero, and the frequency is f, which satisfies the following equation:  
\[f=\frac{\omega}{2\pi} \tag{1}\]  
The parameter \(\omega\) represents the operating angular frequency of the AC current \(I_{in}\). According to the circuit diagrams of Figures 2-4, based on Kirchhoff's law, the equation of the primary-side circuit can be expressed as  
\[\left[