# Analytical Solution and Experimental Validation of the Electromagnetic Field in an IPT System
## 2 Mutual-Inductance Model of the IPT System  
The system structure is given in Figure 1. The alternating current from the source is first converted into a direct current through an AC/DC converter and then into a high-frequency alternating current through another AC/DC converter. Simultaneously, the transmitting coil \(Coil_{p}\) converts the electrical energy into electromagnetic energy, and the receiving coil \(Coil_{S}\) absorbs the energy from the near-zone electromagnetic field, converting it into electrical energy. The electrical energy is then conveyed through a rectifier to charge the battery. The coupling between the primary and secondary systems is defined as mutual inductance \(M\).  
Figure 1: Inductive power transfer (IPT) system structure.  
The mutual inductance model is shown in Figure 2. The equivalent series impedances on \(Coil_{P}\) and \(Coil_{S}\) are \(R_{P}\) and \(R_{S}\), respectively. The load impedance is \(R_{L}\). In this paper, the series-series compensation network is used in the IPT system, and compensating capacitors in the primary and secondary side are \(C_{P}\) and \(C_{S}\), respectively. The primary side is powered by a voltage source \(U\) and the primary circuit current and the secondary circuit current are \(I_{P}\) and \(I_{S}\), respectively. According to the KVL equation, the following equations can be obtained:  
\[[R_{P}+j[\omega L_{P}-1/\omega C_{P}]]I_{P}-j\omega MI_{S}=U, \tag{1}\]  
\[-j\omega MI_{P}+[R_{S}+R_{L}+j[\omega L_{S}-1/\omega C_{S}]]I_{S}=0, \tag{2}\]  
When the primary and secondary circuits are resonant, \(\omega~{}=~{}\omega_{0}=~{}2\pi f_{0}=~{}1/\sqrt{L_{S}C_{S}}=1/\sqrt{L_{P}C_{ P}}\). The system efficiency and load power are derived as follows [16]:  
\[\eta=\frac{\left(\omega_{0}M\right)^{2}R_{L}}{\left(R_{S}+R_{L}\right)\left[R _{P}[R_{S}+R_{L}]+\left[\omega_{0}M\right]^{2}\right]}, \tag{3}\]  
\[P_{L}=\frac{I_{p}^{2}(\omega_{0}M)^{2}R_{L}}{\left(R_{S}+R_{L}\right)^{2}}, \tag{4}\]  
Figure 3 shows a schematic of the electromagnetic field around the primary and secondary coil. The sine alternating current in the primary side coil generates the time-harmonic electromagnetic field. We assume that the time-harmonic electric field induced at the position of \(Coil_{S}\) is \(E_{P_{0}}\). Then, the integral of \(E_{P_{0}}\) on the secondary loop is equal to the induced voltage \(U_{S}\). Combined with Figures 2 and 3, the following expression is obtained:  
\[U_{S}=j\omega MI_{P}=-\oint_{Coll_{S}}E_{P_{0}}dI_{S}, \tag{5}\]  
where \(dI_{S}\) is a directional micro-element on the secondary side coil. In this paper, the bold symbols represent space vectors.  
Figure 3: Schematic of field distribution around the primary and secondary coil.  
Figure 2: An IPT system model by mutual inductance.  
Similarly, the corresponding time-harmonic electric field \(E_{S_{0}}\) will be generated around \(Coil_{P}\) by the induced current in the secondary coil. The corresponding induced voltage is \(U_{P}=j\omega MI_{S}=-\oint_{\mathrm{Coil_{P}}}E_{S_{0}}\cdot dl_{P}\). Therefore, the working principles of an IPT system can be understood from the perspective of an electromagnetic field. In Section 3, [PERSON]'s equations are used to calculate the electric fields generated by the primary and secondary coil in detail, and the derivation process is easy to understand for researchers engaged in IPT.