## 3 Performance Improvement of Multiple Receiver WPT
### Cross-Coupling Rejection  
Even though all the Rx coils are designed with different resonant frequency in selective MRWPT system, the cross coupling between the coil is unavoidable. Also under the dynamic environment, the multiple Rx coils position will keep on change, and there exists a cross-coupling between the coils. The wide range selection of different resonant frequency between the receiver is limited in some applications like medical implants and tiny electronics, which leads to resonant frequency overlapping between the Rx coils. Moreover, when the separation among the Tx and Rx coils are reducing below the critical value, frequency splitting will be present in the system which leads to the variation in the resonant frequency and increases the cross coupling between the coil. In order to overcome the effects of cross coupling and increase the PTE of the selective MRWPT system, the compensation is required at each load coil. The optimal value of the compensation reactance is calculated by applying KVL to Figure 6.  
\[\left[ \begin{array}{c}V_{T}\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{array} \right]=\left[ \begin{array}{cccccccccccc}Z_{TT}&Z_{TR1}&Z_{TR2}&Z_{TR3 }&Z_{TR4}&Z_{TL1}&Z_{TL2}&Z_{TL3}&Z_{TL4}\\ Z_{R1T}&Z_{R1R1}&Z_{R1R2}&Z_{R1R3}&Z_{R1R4}&Z_{R1L1}&Z_{R1L2}&Z_{R1L3}&Z_{R1L4 }\\ Z_{R2T}&Z_{R2R1}&Z_{R2R2}&Z_{R2R3}&Z_{R2R4}&Z_{R2L1}&Z_{R2L2}&Z_{R2L3}&Z_{R2L4 }\\ Z_{R3T}&Z_{R3R1}&Z_{R3R2}&Z_{R3R3}&Z_{R3R4}&Z_{R3L1}&Z_{R3L2}&Z_{R3L3}&Z_{R3L4 }\\ Z_{R4T}&Z_{R4R1}&Z_{R4R2}&Z_{R4R3}&Z_{R4R4}&Z_{R4L1}&Z_{R4L2}&Z_{R4L3}&Z_{R4L 4}\\ Z_{L1T}&Z_{L1R1}&Z_{L1R2}&Z_{L1R3}&Z_{L1R4}&Z_{L1L1}&Z_{L1L2}&Z_{L1L3}&Z_{L1L4 }\\ Z_{L2T}&Z_{L2R1}&Z_{L2R2}&Z_{L2R3}&Z_{L2R4}&Z_{L2L1}&Z_{L2L13}&Z_{L214}\\ Z_{L3T}&Z_{L3R1}&Z_{L3R2}&Z_{L3R3}&Z_{L3R4}&Z_{L3L1}&Z_{L3L1}&Z_{L3L3}&Z_{L3L4 }\\ Z_{LAT}&Z_{L4R1}&Z_{L4R2}&Z_{L4R3}&Z_{L4R4}&Z_{L4L1}&Z_{L4L1}&Z_{L4L3}&Z_{L4L4 }\\ \end{array} \right]\left[ \begin{array}{c}I_{T}\\ I_{R1}\\ I_{R2}\\ I_{R3}\\ I_{L1}\\ I_{L2}\\ I_{L2}\\ I_{L3}\\ I_{L4}\\ \end{array} \right] \tag{9}\]  
Since the Tx coil coupling with load coil is negligible as compared with Tx and Rx, the mutual inductance \(M_{TL1}\), \(M_{TL2}\), \(M_{TL3}\), \(M_{TL4}\), \(M_{LL1T}\), \(M_{L2T}\), \(M_{L3T}\) and \(M_{LAT}\) could be assumed as zero. Under the resonance condition the reactive impedance of the circuit is cancelled and only the resistance, of the coil is present. Resolving (8) the obtained value of Rx coil and load coil current is  
Figure 6: Equivalent circuit of Selective MRWPT system with Compensation. \[I_{i}=-\sum_{k=1,\ i\
eq k}^{4}\frac{j\omega_{i}{M_{T1}}{M_{ik}}{I_{T}}}{R_{i}+R_{ k}+R_{Li}};\ i=R_{k},\ L_{k} \tag{10}\]  
Using (10) the input port impedance can be calculated by substituting in the first row of (9)  
\[Z_{in}=R_{T}+\sum_{k=1}^{4}\frac{\omega^{2}{M_{T1}}^{2}}{R_{T}+R_{Li}};\ i=R_{k},\ L_{k} \tag{11}\]  
From (9) and (10) we can observe that the cross-coupling between the receiver loop coil is not many negligible variations will occur in the input current and input port impedance which leads to a reduction in the overall power transfer efficiency. The optimal value of the load reactance is calculated from (10) and (11)  
\[X_{L1}=-\sum_{k=R1,\_R4}\frac{\omega_{1}{M_{T1}}{M_{R1}}{R_{1k}}(R_{2}+R_{L2} )}{M_{Tk}(R_{k}+R_{L2})} \tag{12}\]  
\[X_{L2}=-\sum_{k=R1,\_R4}\frac{\omega_{2}{M_{T2}}{M_{R2}}(R_{4}+R_{L4})}{M_{Tk} (R_{k}+R_{L4})} \tag{13}\]  
\[X_{L3}=-\sum_{k=R1,\_R4}\frac{\omega_{3}{M_{T3}}{M_{R3}}(R_{6}+R_{L6})}{M_{Tk} (R_{k}+R_{L6})} \tag{14}\]  
\[X_{L4}=-\sum_{k=R1,\_R4}\frac{\omega_{4}{M_{T4}}{M_{R4}}(R_{6}+R_{L6})}{M_{Tk} (R_{k}+R_{L6})} \tag{15}\]  
where \(X_{L1}\), \(X_{L2}\), \(X_{L3}\) and \(X_{L4}\) are the additional inductance added to the load coils to compensate the cross coupling.