# A Method to Reduce Eddy Current Loss of Underwater Wireless Power Transmission by Current Control
## 3 Circuit Analysis in Seawater and Methods Proposed
However, it can be seen from Figure 8 that in seawater, the phase angle of electric field strength is less than 90\({}^{\circ}\) in air, and the farther the measurement point is from the coil, the smaller the phase angle. With the increase in frequency, the rate of phase angle reduction also increases, which shows that the emergence of eddy current loss causes the coil to produce the phase loss of the electric field. Figure 5: Double-sided LCC compensation topology for underwater WPT system. According to the analysis above, it should be modified appropriately when the traditional circuit model in the air is extended to an underwater environment. On the basis of [PERSON]'s law and Figure 5, the equation of the underwater WPT system can be expressed as  
\[\left[ \begin{array}{cccc}&R_{in}+j\omega L_{P1}&\frac{1}{j\omega C_{P2}}&0&0\\ &0&\frac{1}{j\omega C_{P2}}&R_{P_{S}}+j\omega L_{P_{S}}+\frac{1}{j\omega C_{P3}} &j\omega\mathbf{M}_{sea}\end{array} \right]\ \cdot\ \left[ \begin{array}{c}\mathbf{I}_{in}\\ \mathbf{I}_{C_{P2}}\\ \mathbf{I}_{P}\\ \mathbf{I}_{S}\end{array} \right]=\left[ \begin{array}{c}\mathbf{U}_{in}\\ 0\end{array} \right] \tag{23}\]  
Figure 6: Variation of AC resistance and eddy current loss resistance with frequency. Figure 7: Modulus of electric field intensity generated by planar coil in the (**a**) radial direction (**b**) axial direction. The resonance condition on the secondary side satisfies the following equation:  
\[\left\{ \begin{array}{c}j\omega_{0}L_{S1}+\frac{1}{j\omega\mathcal{C}_{S2}}=0\\ j\omega_{0}L_{S_{S}}+\frac{1}{j\omega\mathcal{C}_{S3}}+\frac{1}{j\omega\mathcal{C}_ {S3}}=0\end{array} \right. \tag{27}\]  
We can obtain the total impedance and reflection impedance at the resonance state of the primary side thusly:  
\[Z_{S}=\frac{\omega_{0}^{2}L_{S1}^{2}}{R_{ot}+R_{L}}+R_{S_{S}} \tag{28}\]  
\[\mathbf{Z}_{ref}=\frac{\omega_{0}^{2}\mathbf{M}_{seq}^{2}}{Z_{S}}=\frac{\omega _{0}^{2}K_{sat}^{2}M_{air}^{2}e^{-j(2\theta_{diff})}}{Z_{S}} \tag{29}\]  
Due to the effect of eddy current loss, the reflection impedance of the secondary side is not purely resistive, so the resonance condition of the primary side is changed compared with that in the air. It can be obtained that the resonance condition on the primary side is  
\[\left\{ \begin{array}{c}j\omega_{0}L_{P1}+\frac{1}{j\omega_{0}\mathcal{C}_{P2 }}=0\\ j\omega_{0}L_{P_{S}}+\frac{1}{j\omega_{0}\mathcal{C}_{P2}}+\frac{1}{j\omega_{0} \mathcal{C}_{P1}}+j\text{Im}(\mathbf{Z}_{ref})=0\end{array} \right. \tag{30}\]  
It can be calculated when both the primary side and secondary side resonate that  
\[Z_{P}=\frac{\omega_{0}^{2}L_{P1}^{2}}{R_{P_{S}}+\text{Re}(\mathbf{Z}_{ref})}+R _{in} \tag{31}\]  
Figure 8: Phase of electric field intensity generated by planar coil in the (**a**) radial direction (**b**) axial direction. \[\mathbf{I}_{in}=\frac{\mathbf{U}_{in}}{Z_{P}} \tag{32}\]  
\[\mathbf{I}_{P}=\frac{j\omega_{0}C_{P3}}{j\omega_{0}C_{P3}+j\omega_{0}C_{P3}- \omega_{0}^{2}C_{P2}C_{P3}(\mathbf{Z}_{ref}+R_{P_{S}}+j\omega L_{P})}\mathbf{I} _{in} \tag{33}\]  
\[\mathbf{I}_{S}=\frac{j\omega_{0}K_{sa}M_{air}(R_{L}+R_{ot})}{R_{S_{S}}(R_{L}+R_ {ot})-\frac{1}{\omega_{0}^{2}C_{S2}C_{S3}}}\mathbf{I}_{P} \tag{34}\]  
If we take \(I_{P}\) as the reference, \(I_{P}\) and \(I_{S}\) can be expressed as  
\[\mathbf{I}_{P}=I_{P}Z0^{\circ} \tag{35}\]  
\[\mathbf{I}_{S}=\frac{\omega_{0}K_{sa}M_{air}(R_{L}+R_{ot})}{R_{S_{S}}(R_{L}+R _{ot})-\frac{1}{\omega_{0}^{2}C_{S2}C_{S3}}}I_{P}\angle(90^{\circ}+\theta_{ diff}) \tag{36}\]  
From Equations (35) and (36), we can infer that the phase difference between \(\mathbf{I}_{P}\) and \(\mathbf{I}_{S}\) in the seawater is not equal to \(90^{\circ}\) compared with that in the air.