# Array Design and Transmitter Coils Selection for Underwater Wireless Power Transfer System
## 3 Coupling Mechanism Design and Array Switching Control Strategy
### Transmitting Array Switching Control Strategy Based on Receiving Coil Position Estimation
#### 3.2.1 Receiver Coil Position Estimation  
When the receiving coil is laterally offset relative to the power supply coil, the offset trajectory of the receiving coil and the mutual inductance change curve are shown in Figure 9. It can be seen from the diagram that when the receiving coil gradually moves away from the current power supply coil and is close to the adjacent unpowered coil, the mutual inductance between the power supply coil and it gradually decreases, while the mutual inductance between the unpowered coil and it gradually increases. When moving to a certain position, the two are equal. As the receiving coil continues to move, the mutual inductance between the unpowered coil and it is greater than the mutual inductance between the power supply coil and it. According to the change law of mutual inductance, it can be seen that by comparing the mutual inductance between the power supply coil and the receiving coil, the position of the receiving coil can be estimated; that is, the receiving coil is above the transmitting coil with the largest mutual inductance value. Using the same method, it can be verified that when the receiving coil has longitudinal displacement and rotational offset, the position of the receiving coil can be estimated by comparing the mutual inductance values, so as to perform optimal transmitting coil switching control. Because it is difficult to directly measure the mutual inductance during the operation of the system, it is necessary to indirectly compare the mutual inductance between the transmitting coil and the receiving coil by detecting the circuit parameters, and then estimating the position of the receiving coil. The equivalent circuit of the array UWPT system is shown in Figure 10.  
Figure 9: Receiving coil motion trajectory and mutual inductance change curve. (**a**) Receiving coil motion trajectory. (**b**) Mutual inductance change curve. In Figure 10, \(U_{(x,y)}\) is the voltage at both ends of the transmitting array coil, and \(Z_{(x,y)}\) is the reflected impedance of the receiving end. When the transmitting coil TX(\(x,y\)) is powered, it can be obtained:  
\[\left\{ \begin{aligned} &\dot{U}_{\text{a}}=\left(\dot{i}\omega L_{ \text{a}}+\frac{1}{\mu\omega\text{C}_{\text{a}}}+R_{\text{a}}+R_{\text{s}} \right)\dot{I}_{\text{a}}+j\omega M_{(x,y)}\dot{I}_{\text{b}}\\ & j\omega M_{(x,y)}\dot{I}_{\text{a}}+\left(R_{\text{b}}+j\omega L _{\text{b}}+\frac{1}{\mu\omega\text{C}_{\text{b}}}+R_{\text{L}}\right)\dot{I}_ {\text{b}}=0\end{aligned} \right. \tag{1}\]  
In (1), \(M_{(x,y)}\) is the mutual inductance between the power supply coil TX(\(x,y\)) and the receiving coil RX. Further, we can obtain:  
\[\dot{U}_{\text{a}}=\left(R_{\text{a}}+j\omega L_{\text{a}}+\frac{1}{j\omega \text{C}_{\text{a}}}+\frac{\omega^{2}M_{(x,y)}{}^{2}}{R_{\text{b}}+R_{\text{ L}}}\right)\dot{I}_{\text{a}} \tag{2}\]  
When the phase difference between \(U_{\text{a}}\) and \(I_{\text{a}}\) is \(\theta\), the expression of the reflection impedance \(Z_{(x,y)}\) at the receiving end is as follows:  
\[Z_{(x,y)}=\frac{\omega^{2}M_{(x,y)}{}^{2}}{R_{\text{b}}+R_{\text{L}}}=\frac{U _{\text{a}}\cos\theta}{I_{\text{a}}}-R_{\text{a}} \tag{3}\]  
When the system works in the resonant state, the system impedance angle \(\theta=0^{\circ}\), the expression of the reflection impedance \(Z_{(x,y)}\) at the receiving end can be simplified as follows:  
\[Z_{(x,y)}=\frac{U_{\text{a}}}{I_{\text{a}}}-R_{\text{a}} \tag{4}\]  
Here, \(Z_{(x,y)}\) is defined as the secondary-contributed reflection impedance, calculated as the total primary-side impedance (\(U_{\text{a}}/I_{\text{a}}\)) minus the primary coil's inherent resistance \(R_{\text{a}}\). This isolates the impedance component strictly due to the secondary-receiver coupling, enabling direct comparison of mutual inductance M(\(x,y\)) between different transmitting coils. Since \(Z_{(x,y)}\propto M_{(x,y)}{}^{2}\) (Equation (4)), comparing \(Z_{(x,y)}\) across adjacent coils directly  
Figure 10: Equivalent circuit diagram of array UWPT system.