# A Method to Reduce Eddy Current Loss of Underwater Wireless Power Transmission by Current Control
## 3 Circuit Analysis in Seawater and Methods Proposed  
In the underwater environment, due to the role of the seawater medium, not only does the electromagnetic field intensity around the coil decrease and the phase change, but the electric field intensity at the position of the coil also changes (ignoring the volume of the coil). The change in electric field intensity and phase in seawater will affect the mutual inductance, and it will cause the change in coupler impedance simultaneously [17], which will affect the working efficiency of the system. In this section, we will establish the equivalent circuit model of the underwater WPT system, and its power transmission characteristics will be analyzed so as to obtain the method to reduce eddy current loss and improve the energy transmission efficiency of the system. According to the circuit model in the air, the underwater equivalent circuit model can be obtained as shown in Figure 5.  
The difference with the air is that the self-inductance and equivalent resistance of the primary-side coil are \(L_{P_{S}}\) and \(R_{P_{S}}\), respectively, the self-inductance and equivalent resistance of the secondary-side coil are \(L_{S_{S}}\) and \(R_{S_{S}}\), and the mutual inductance between them is \(M_{sat}\), which can be expressed as  
\[M_{sea}=K_{sa}M_{air}e^{-j\theta_{diff}} \tag{18}\]  
\(M_{air}\) represents the mutual inductance coefficient of two coils in the air, \(K_{sa}\) represents the modulus ratio of the mutual inductance coefficient between the seawater medium and air medium, and \(\theta_{diff}\) represents the difference between the phase angle of the mutual inductance in the sea water medium and the air medium [18]. The equivalent resistance \(R_{P_{S}}\) and \(R_{S_{S}}\) represent the sum of the resistance of the primary-side coil and the secondary-side coil and the eddy current loss resistance \(R_{ecl}\) caused by its working in the underwater environment, as shown in Equation (19). In this paper, the eddy current loss resistance is defined as  
\[R_{ecl}=\frac{P_{ecl}}{I^{2}} \tag{19}\]  
\[\left\{ \begin{array}{l}R_{P_{S}}=R_{P}+R_{P_{ecl}}=R_{P}+\frac{P_{ecl}}{I_{P} }\\ R_{S_{S}}=R_{S}+R_{S_{ecl}}=R_{S}+\frac{P_{ecl}}{I_{S}}\end{array} \right. \tag{20}\]  
\[P_{P_{ecl}}=\int\limits_{V}\mathbf{E}_{P}\mathbf{J}dV=\int\limits_{V}\sigma( \mathbf{E}_{P}^{2}+\mathbf{E}_{P}\mathbf{E}_{S})dV \tag{21}\]  
\[P_{S_{ecl}}=\int\limits_{V}\mathbf{E}_{S}\mathbf{J}dV=\int\limits_{V}\sigma( \mathbf{E}_{S}^{2}+\mathbf{E}_{P}\mathbf{E}_{S})dV \tag{22}\]  
where \(\mathbf{E}_{P}\) and \(\mathbf{E}_{S}\) are the electric field intensity and \(\mathbf{J}\) is the current density. According to Equations (19)-(22) above, it can be seen that the eddy current loss resistance is related to the electric field generated by the two coils. However, compared with the air environment, the electric field generated by the coils underwater will change. In addition, the eddy current loss resistance will also change greatly for current excitation at different frequencies. The differences between them are shown in the Figures 6-8 below. Figures 6-8 above come from the simulation results of a single-plane coil using electromagnetic field simulation software. Figure 6 shows the AC resistance and eddy current loss resistance of the planar coil of the WPT system in the air and seawater, respectively, and Figures 7 and 8 show the comparison of its electric field intensity modulus and phase angle under different frequency excitations. It can be seen from the figure that in the air medium, the eddy current loss resistance is always zero, while in the seawater medium, with the increase in frequency, the eddy current loss resistance increases rapidly and is much larger than its own AC resistance. Moreover, from the calculation results, it can be seen that under different frequencies, the modulus of electric field intensity around the coil is almost the same in seawater and air, and the modulus in seawater is slightly less than that in air.