## 2 Coupling Coefficient and Efficiency of a WPT System  
A WPT system usually is designed and optimized considering its efficiency, power transfer capability, and sensitivity to coil misalignments [14; 24; 25]. The efficiency is one of the key parameters of a high-power WPT system and can be analyzed analytically according to an equivalent scheme of the WPT system (Figure 1). The equivalent scheme is presented in Figure 2.  
The operation of the circuit is described analytically by the set of equations according [PERSON]'s voltage law:  
\[\left\{\begin{aligned} \mathbf{V_{p}}&=\mathbf{I_{p}}\left( \mathsf{i}\omega L_{p}-\frac{1}{\mathsf{i}\omega C_{p}}+R_{p}\right)+\mathbf{I_{s}} \mathsf{i}\omega M,\\ \mathbf{V_{m}}&=\mathbf{I_{s}}\left(\mathsf{i}\omega L_{s}- \frac{1}{\mathsf{i}\omega C_{s}}+R_{s}\right)+\mathbf{I_{s}}R_{AC},\end{aligned}\right. \tag{1}\]  
where \(\mathbf{V_{p}}\) is the primary (source) voltage, \(\mathbf{I_{p}}\) and \(\mathbf{I_{s}}\) are currents of the primary and secondary coils, \(R_{p},R_{s},\ L_{p},\ L_{s}\) are the resistances and inductances of the primary and secondary coils, \(C_{p},\ C_{s}\) are the capacitances of primary and secondary compensation capacitors, respectively, \(M\) is the mutual inductance, \(\omega\) is the angular frequency, and \(\mathbf{V_{m}}\) is the voltage induced in the secondary coil \(\mathbf{V_{m}}=-\mathsf{i}\omega M\mathbf{I_{p}}\), while \(R_{AC}\) is the load resistance, representing the EV battery, seen from the AC side of the secondary (receiving) coil (Figures 1 and 2) [26].  
\[R_{AC}=\frac{8}{\pi^{2}}R_{bat}=\frac{8}{\pi^{2}}\times\frac{V_{bat}}{I_{bat}}. \tag{2}\]  
Coil-to-coil efficiency (the inverter and rectifier losses excluded), expressed from the set of Equation (1) is [27]  
\[\eta=\frac{P_{L}}{P_{1}}=\frac{\omega^{2}M^{2}R_{AC}}{R_{p}\bigg{(}\left(R_{s} +R_{AC}\right)^{2}+\left(\omega L_{s}-\frac{1}{\omega C_{s}}\right)^{2}\bigg{)} +\omega^{2}M^{2}(R_{s}+R_{AC})}. \tag{3}\]  
As can be seen from this equation, coil-to-coil efficiency is a complex function of the coils' resistances, secondary side reactance (if not fully compensated), angular frequency, load resistance, and mutual inductance. Considering the relationship of the mutual inductance \(M\) and the coupling coefficient \(k\),  
\[k=M/\sqrt{L_{p}L_{s}}, \tag{4}\]  
the efficiency could be expressed as a function of the coupling coefficient:  
\[\eta=\frac{P_{L}}{P_{1}}=\frac{\omega^{2}k^{2}L_{p}L_{S}R_{AC}}{R_{p}\bigg{(} \left(R_{s}+R_{AC}\right)^{2}+\left(\omega L_{s}-\frac{1}{\omega C_{s}}\right) ^{2}\bigg{)}+\omega^{2}k^{2}L_{p}L_{S}(R_{2}+R_{AC})}. \tag{5}\]  
Figure 1: Circuit diagram of a typical Inductive Power Transfer (IPT) system (SS compensation topology).  
Figure 2: Equivalent scheme of ITP system.  
The optimal load is found by differentiation of the expression (5) with respect to \(R_{AC}\) when the secondary side inductance is fully compensated \(\omega L_{s}-\frac{1}{\omega C_{s}}=0\). The expression of the optimal load resistance is [28]  
\[R_{AC,opt}=R_{s}\sqrt{1+k^{2}Q_{p}Q_{s}}, \tag{6}\]  
where \(Q_{p}\) and \(Q_{s}\) are the quality factors of the primary and secondary coils, respectively. The maximum achievable coil-to-coil efficiency according to [28] is  
\[\eta_{max}=\frac{k^{2}Q_{p}Q_{s}}{\left(1+\sqrt{1+k^{2}Q_{p}Q_{s}}\right)^{2}}. \tag{7}\]  
The batteries represent the dynamic loads. Therefore, the resistance \(R_{AC}\) depends on the operating point of a charging circuit. The DC/DC converters together with control circuits are applied to match the optimal one [29]. They assure the maximum efficiency with respect to the load in the particular state. The maximum achievable coil-to-coil efficiency, corresponding to the optimal load, depends on the coupling coefficient and the quality factors of the coils (7). To achieve the largest efficiency, these parameters must be maximized [30; 31; 32; 33].