# 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
(**a**) System equivalent circuit. (**b**) Equivalent circuit of transmitting. reflects differences in \(M_{(x,y)}\), which is critical for estimating the receiver's position and selecting the optimal transmitting coil. Since the unpowered coil in the transmitting array is in an open state and no current flows through, the voltage at both ends is induced by the current \(I_{\text{a}}\) of the power supply coil and the current \(I_{\text{b}}\) of the receiving coil. The voltage \(U_{(y+1,y)}\) at both ends of the unpowered coil TX(\(x\)+1, \(y\)) can be expressed by \(I_{\text{a}}\):  
\[ \begin{split} ll\dot{U}_{(x+1,y)}&=j\omega_{1}M_{ \text{TX}}\dot{I}_{\text{a}}+j\omega_{1}M_{(x+1,y)}\dot{I}_{\text{b}}\\ &=\left(j\omega M_{\text{TX}}+\frac{\omega^{2}M_{(x,y)}M_{(x+1,y )}}{R_{\text{b}}+R_{\text{L}}}\right)\dot{I}_{\text{a}}\end{split}  \tag{5}\]  
Among them, \(M_{(x+1,y)}\) is the mutual inductance between the unpowered coil TX(\(x\)+1, \(y\)) and the receiving coil, and \(M_{\text{TX}}\) is the mutual inductance between adjacent coils in the transmitting array. When the coil position in the transmitting array is fixed and the coil parameters are the same, \(M_{\text{TX}}\) is constant. According to (5), the reflection impedance \(Z_{(x+1,y)}\) at the receiving end is as follows:  
\[ \begin{split} llZ_{(x+1,y)}&=\frac{\omega^{2}M_{(x, y)}M_{(x+1,y)}}{R_{\text{b}}+R_{\text{L}}}\\ &=\sqrt{\left(\frac{U_{(x+1,y)}}{I_{\text{a}}}\right)^{2}- \omega^{2}M_{\text{TX}}{}^{2}}\end{split}  \tag{6}\]  
Similarly, the expressions of the receiving end reflection impedance \(Z_{(x,y+1)}\), \(Z_{(x,y-1)}\) and \(Z_{(x-1,y)}\) corresponding to the unpowered coils TX(\(x,y\)+1), TX(\(x,y\)-1) and TX(\(x\)-1, \(y\)) are as follows:  
\[ \begin{split} llZ_{(x,y+1)}&=\frac{\omega^{2}M_{(x, y)}M_{(x,y+1)}}{R_{\text{b}}+R_{\text{L}}}\\ &=\sqrt{\left(\frac{U_{(x,y+1)}}{I_{\text{a}}}\right)^{2}- \omega^{2}M_{\text{TX}}{}^{2}}\\  llZ_{(x,y-1)}&=\frac{\omega^{2}M_{(x,y)}M_{(x,y-1)} }{R_{\text{b}}+R_{\text{L}}}\\ &=\sqrt{\left(\frac{U_{(x,y-1)}}{I_{\text{a}}}\right)^{2}- \omega^{2}M_{\text{TX}}{}^{2}}\\  llZ_{(x-1,y)}&=\frac{\omega^{2}M_{(x,y)}M_{(x-1,y) }}{R_{\text{b}}+R_{\text{L}}}\\ &=\sqrt{\left(\frac{U_{(x-1,y)}}{I_{\text{a}}}\right)^{2}- \omega^{2}M_{\text{TX}}{}^{2}}\end{split}  \tag{7}\]  
From (5) to (9), it can be seen that in the expressions of \(Z_{(x,y)}\), \(Z_{(x+1,y)}\), \(Z_{(x,y+1)}\), \(Z_{(x,y-1)}\), and \(Z_{(x-1,y)}\), only the \(M_{(x,y)}\), \(M_{(x+1,y)}\), \(M_{(x,y+1)}\), \(M_{(x,y-1)}\), and \(M_{(x-1,y)}\) in the molecules are different, and they are proportional. Therefore, by comparing the size of \(Z_{(x,y)}\), \(Z_{(x+1,y)}\), \(Z_{(x,y+1)}\), \(Z_{(x,y-1)}\), \(Z_{(x,y-1)}\), and \(Z_{(x-1,y)}\), the size relationship of \(M_{(x,y)}\), \(M_{(x+1,y)}\), \(M_{(x,y+1)}\), \(M_{(x,y-1)}\), and \(M_{(x-1,y)}\) can be indirectly obtained, so as to estimate the position of the receiving coil and realize the optimal power supply coil switching control. The switching control approach uses reflection impedance \(Z_{(x,y)}\) as the metric because \(Z_{(x,y)}\propto M_{(x,y)}{}^{2}\), where \(M_{(x,y)}\) is the mutual inductance between coils. By comparing \(Z_{(x,y)}\) across adjacent coils, we indirectly compare \(M_{(x,y)}\), ensuring the coil with the strongest magnetic coupling (highest \(M_{(x,y)}\)) is selected. This method avoids direct mutual inductance measurement (complex in underwater environments) while maintaining accuracy. In summary, this paper proposes a transmit array switching control strategy based on receiving coil position estimation.