## 3 Performance Improvement of Multiple Receiver WPT
### Power Transfer Efficiency Improvement  
The PTE of the MRWPT system primarily influenced by the coupling, quality factor of the Tx and Rx coil. Quality factor of the coil reduces due to the increase in the source and load port resistance of the circuit, which can be minimized by adding a relay coil to the circuit. The three coil WPT circuit gives a better power delivered to the load and PTE as compared with the four coil WPT circuit [34].  
Figure 1: Possible dynamic receiver configuration. (**a**) Identical Rx’s; (**b**) Nonidentical Rx’s; (**c**) Dynamic Rx’s.  
However, for the selective power distribution using time division multiplexing requires to tune the Tx coil frequency which makes difficult to add a relay coil at the transmitter section. In this work the additional relay coil is used at the Rx side and called as the load coil. The equivalent circuit of three coil MRWPT system is shown in Figure 2, consist of Tx coil, Rx coil and load coil. \(\mathbf{R_{1}}\), \(\mathbf{R_{2}}\), \(\mathbf{R_{L}}\), \(\mathbf{L_{1}}\), \(\mathbf{L_{2}}\), \(\mathbf{L_{3}}\), \(\mathbf{C_{1}}\), \(\mathbf{C_{2}}\), \(\mathbf{C_{3}}\), \(\mathbf{Q_{T}}\), \(\mathbf{Q_{R}}\) and \(\mathbf{Q_{L}}\) are the Tx, Rx and load coil resistance, self-inductance, capacitance, quality factor respectively.  
The equivalent network model of the circuit is formulated by applying KVL to each coil  
\[\left[\begin{array}{c}V_{T}\\ 0\\ 0\end{array}\right]=\left[\begin{array}{ccc}Z_{11}&Z_{12}&Z_{13}\\ Z_{21}&Z_{22}&Z_{23}\\ Z_{31}&Z_{32}&Z_{33}\end{array}\right]\left[\begin{array}{c}I_{1}\\ I_{2}\\ I_{3}\end{array}\right] \tag{1}\]  
where \(I_{1}\), \(I_{2}\) and \(I_{3}\) are the current in Tx, Rx and load coil respectively. The self-impedance \(\left(Z_{ij},i=j\right)\), mutual impedance \(\left(Z_{ij},i\
eq j\right)\) and coupling coefficient \(k_{ij}\) of the system can be expressed as  
\[Z_{ij}=Z_{ji}=R_{i}+j\omega L_{i}+\frac{1}{j\omega\mathcal{C}_{i}};\ for\ i=j\]  
\[Z_{ij}=Z_{ji}=j\omega M_{ij}=j\omega K_{ij}\sqrt{L_{i}L_{j}}\ \ for\ i\
eq j \tag{2}\]  
\[k_{ij}=\frac{M_{ij}}{\sqrt{L_{i}L_{j}}} \tag{3}\]  
The RMS value of input, output power and efficiency of the three coil MRWPT system as  
\[P_{T}=\frac{1}{2}V_{T}I_{1} \tag{4}\]  
\[P_{OUT}=\frac{1}{2}{I_{3}}^{2}R_{L} \tag{5}\]  
After solving the Equations (1) and (5) using (2), the efficiency of the system is given as  
\[\eta_{3 coil}=\frac{k_{12}{}^{2}k_{23}{}^{2}Q_{T}Q_{R}{}^{2}Q_{L}}{(1+k_{12}{} ^{2}Q_{T}Q_{R}+k_{23}{}^{2}Q_{R}Q_{L})(1+k_{23}{}^{2}Q_{R}Q_{L})} \tag{6}\]  
The total amount of power delivered to the load coil can be found by multiplying efficiency with source power  
\[P_{L}=\frac{V_{T}{}^{2}k_{12}{}^{2}k_{23}{}^{2}Q_{T}Q_{R}{}^{2}Q_{L}}{2R_{1}(1 +k_{12}{}^{2}Q_{T}Q_{R}+k_{23}{}^{2}Q_{R}Q_{L}){}^{2}} \tag{7}\]  
The variation of system efficiency and PDL concerning \(k_{23}\) and \(d_{23}\) are shown in Figure 3a,b. It can be seen that by choosing an optimal values of \(k_{12}\) and \(k_{23}\) the overall system efficiency and PDL can be maximized.  
Figure 2: Equivalent circuit of three coil magnetic resonant based wireless power transfer (MRWPT) system.