## 3 Equivalent Circuits and Matching Network  
To achieve maximum efficiency there should be reactive energy compensation of the transmitting and receiving parts. The IPT device can be regarded as quasi-linear quadrupole and may be presented in the form of equivalent circuits. When using a standard T-equivalent circuit, all three inductances are compensated individually, according to Figure 5.  
Figure 4: Coil losses for different \(k\) and \(Q\). Figure 5: Compensation topologies of T-equivalent circuit. Self-inductances of the two coils (when the coils are not connected to other circuits) are equal to  
\[L_{1}=w_{1}^{2}g,\ L_{2}=w_{2}^{2}g, \tag{5}\]  
where \(g\)--magnetic permeance of the coil. When IPT is connected to the power source and the load, the elements values from Figure 5 can be calculated by the following equations:  
\[L_{N1}=L_{1}-Mw_{1}/w_{2}=L_{1}\Big{(}1-k\sqrt{g_{C2}/g_{C1}}\Big{)} \tag{6}\]  
\[L_{M2}=Mw_{2}/w_{1}=L_{2}k\sqrt{g_{C1}/g_{C2}}, \tag{7}\]  
\[L_{2N}=L_{2}-Mw_{2}/w_{1}=L_{2}\Big{(}1-k\sqrt{g_{C1}/g_{C2}}\Big{)}, \tag{8}\]  
\[k=M/\sqrt{L_{1}L_{2}} \tag{9}\]  
\[C_{1}=1/\Big{[}\alpha^{2}L_{1}\Big{(}1-k\sqrt{g_{C2}/g_{C1}}\Big{)}\Big{]} \tag{10}\]  
\[C_{2}=1/\Big{[}\alpha^{2}L_{2}\Big{(}1-k\sqrt{g_{C1}/g_{C2}}\Big{)}\Big{]} \tag{11}\]  
Figure 4:0  
where \(g_{C1}\) and \(g_{C2}\) are magnetic permeances of the transmitting and receiving coils. Equivalent circuits with two inductances allow us to compensate the IPT with a minimum number of capacitors. In accordance with the connecting points of the compensating capacitor at the secondary side and the load, there are two possibilities: series or parallel matching (Figure 6). The following equations are valid for the elements of both equivalent circuits in Figure 4:1  
Figure 4:2  
Figure 4:3  
i.e., \(n=k\) when \(w_{1}=w_{2}\)  
Figure 4:4  
Considering \((r_{2}/\omega_{0}L_{2})^{2}\ll 1,\ r_{2}/R_{L}\ll 1\) and  
Figure 4:5 then:  
Figure 4:6  
It is reasonable to assume that, at a certain frequency (the frequency for which IPT is designed) of supply voltage--\(\omega_{0}\) and \(Z\) value, which corresponds to the combination of parameters \(k_{0}\), \(L_{10}\), \(L_{20}\) (\(k=k_{0}\), \(L_{10}=L_{1}(k_{0})\), \(L_{20}=L_{2}(k_{0})\)), the values of matching elements are determined according to Equations (13)-(16). Figure 4:7 Matching of the IPT equivalent circuit with two inductances through serial capacitor at the primary side capacitor at the secondary side: (**a**) parallel and (**b**) series. With magnetic coupling \(k_{0}\), the efficiency can be calculated by the following expression:  
Figure 4:8  
where \(Q_{1}\) and \(Q_{2}\) are qualitative factors of IPT coils, including resistances \(r_{1}\) and \(r_{2}\), which define losses in the primary and secondary sides, i.e.,  
Figure 4:9  
To obtain efficiency greater than \(\eta_{0}\) it is necessary to fulfill the following condition:  
Figure 5:0  
With \(Q_{1}=Q_{2}=Q\), there follows:  
Figure 5:1  
This equation provides information about the minimum \(Q\) value and, consequently, the maximum values of \(r_{1}\) and \(r_{2}\), so that the current efficiency be greater than \(\eta_{0}\). According to Equations (19) and (20), it could be proved that there is maximum value of the efficiency--\(\eta_{0 MAX}\), only when \(R_{L}>\omega_{0}L_{20}\), i.e., at optimum load quality factor--\(Q_{L\ OPT}\). Figure 5:2  
In the case of serial compensation, IPT is most effective for \(k_{0}\) (respectively frequency \(\omega_{0}\)) and optimal load resistance, according to the following equation:  
Figure 5:3  
Physical interpretation of circuits in Figures 5 and 6 is different. The analysis outcomes and the relative values of equivalent inductances are summarized in Table 1 in the following order:  
* row 2--relative value of the magnetic permeances \(g(k)\)/\(g(k0)\) in the range of magnetic coupling \(0.2\leq k\leq 0.6\).