## 3 Performance Improvement of Multiple Receiver WPT
### Selective MRWPT System  
The equivalent circuit of multiple receiver selective MRWPT system is illustrated in Figure 4. For the analysis, the system is chosen with four Rx coils, four load coils, and single Tx coil. The Tx, Rx, and load coil parameters are denoted by the subscript of \(T\), \(R_{i}\), and \(L_{i}\) (\(i=1,2,3,4\)) respectively. The PTE of the particular system can be maximized when the operating frequency is same as the Tx, Rx and load coil resonant frequency. Using this principle, the selective power transfer to multiple receivers can be achieved by tuning the transmitter side capacitance to match the operating frequency to be same as the Rx and Load resonant frequency. The resonant frequency of the receiver loops \(f_{r1}\), \(f_{r2}\), \(f_{r3}\) and \(f_{r4}\) are calculated as  
\[f_{r1}=\frac{1}{2\pi\sqrt{L_{R1}C_{R1}}}=\frac{1}{2\pi\sqrt{L_{L1}C_{L1}}};f_{ r2}=\frac{1}{2\pi\sqrt{L_{R2}C_{R2}}}=\frac{1}{2\pi\sqrt{L_{L2}C_{L2}}}\]  
\[f_{r3}=\frac{1}{2\pi\sqrt{L_{R3}C_{R3}}}=\frac{1}{2\pi\sqrt{L_{L3}C_{L3}}};f_{ r4}=\frac{1}{2\pi\sqrt{L_{R4}C_{R4}}}=\frac{1}{2\pi\sqrt{L_{L4}C_{L4}}} \tag{8}\]  
Figure 4: Equivalent circuit of Selective MRWPT system.  
Figure 3: Three coil circuit response (**a**) Power transfer efficiency; (**b**) Power delivered to load.  
Even though the receiver devices are designed with different resonant frequencies and selected to transfer the power uniformly to individual load there exist a cross-coupling between Tx and unselected Rx coils. For the reason that of the non-resonant coupling effect, there exists an inter cross-coupling among the Rx coils. As the number of coils with different resonant frequency is increasing the cross-coupling also increases which reduces the PTE of the selected link. The simulated response of the selective MRWPT system is presented in Figure 5. The four Rx and load coils are designed with the resonant frequency of 1.0 MHz, 1.50 MHz, 2.0 MHz, 3.0 MHz with the circuit parameters mentioned in Table 1. Initially, the Rx1 coil is selected by tuning the Tx coil to 1 MHz, and the response of the system is shown in Figure 5a, the peak magnitude occurs at 1 MHz along with that the unselected load coils magnitude are in the range of \(-\)18 dB at their respective resonant frequency. Likewise, the resonant frequency of the Tx coils is tuned to 1.5 MHz, 2 MHz, and 3 MHz to match with the Rx2 to Rx4 resonant frequency as shown in Figure 5b-d. Since the passband frequency of the system has much overlapping with other resonant frequency there exist a cross-coupling effect in all the link. The effects of cross-coupling need to be minimized to transfer the power uniformly to selected load at high efficiency.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Tx** & **Rx1/Load1** & **Rx2/Load2** & **Rx3/Load3** & **Rx4/Load4** \\ \hline Frequency (MHz) & - & 1.0 & 1.5 & 2.0 & 3.0 \\ Inductance (\(\mu\)H) & 112.58 & 25.33 & 11.248 & 6.333 & 2.814 \\ Capacitance (nF) & - & 10 & 10 & 10 & 10 \\ Resistance (\(\Omega\)) & 50.5 & 0.5/50.5 & 0.5/50.5 & 0.5/50.5 & 0.5/50.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Circuit design parameters.  
Figure 5: S Parameter dB of the system without compensation (a) dB(S) when Rx1-on; (b) dB(S) when Rx2-on; (c) dB(S) when Rx3-on; (d) dB(S) when Rx4-on.