# Helmholtz Coils Based WPT Coupling Analysis of Temporal Interference Electrical Stimulation System
## 2 Principles and Methods Analysis
### Wireless Power Transfer System Based on Helmholtz Coils
#### 2.2.2 Circuit Analysis of a WPT System with Helmholtz Coils  
The law of electromagnetic induction shows that when the current in the transmitting coil changes, the resulting magnetic chain couples to the receiving coil and generates an induced electric potential in the receiving coil [30]. The free-space energy transfer system is mapped into a circuit model according to electromagnetic correlations. The approach to analyzing the characteristics of magnetically coupled resonant systems from the point of view of circuit principles is universal. The basic idea can be formulated as follows: firstly, a system of Kirchhoff equations for the equivalent circuit is established from the circuit topology, and the mathematical expressions for the transmission efficiency and corresponding parameters are obtained after solving the system of equations. Finally, the transmission characteristics of the MCR-WPT system are theoretically analyzed with practice [16; 31]. This is similar to a transformer model, with the difference being that the magnetic circuit is air rather than ferromagnetic material. An equivalent circuit model of a WPT system, consisting of two transmitting coils with one receiving coil, is shown in Figure 2.  
\(V_{S}\) is the power supply to the transmitting coil, \(R_{S}\) is the internal resistance of the power supply; \(R_{1}\) and \(R_{2}\) are the parasitic resistances of transmitting coil 1 and transmitting coil 2 respectively, \(R_{3}\) is the resistance of the receiving coil, and \(R_{L}\) is the load resistance. \(L_{1}\) and \(L_{2}\) are the inductances of the transmitting coil, and \(L_{3}\) is the receiving coil inductance. \(M_{12}\) is the mutual inductance between the two transmitting coils, while \(M_{13}\) and \(M_{23}\) are the mutual inductances between the receiving and transmitting coils. Moreover, because the resonant system works at high frequencies, the parasitic capacitance of the inductor coil cannot be ignored. The coil's series resonant compensation capacitance and the inductor coil's parasitic capacitance are equivalent to \(C_{1}\), \(C_{2}\) and \(C_{3}\). In addition, let the operating angular frequency of the system be \(w\), and the resonant angular frequencies of the transmitting and receiving coils are the same \(w_{0}\). From [PERSON]'s equation, it can be seen that there is magnetic coupling and electrical coupling between the coils. However, due to the large inductance of the coils, the magnetic coupling is much stronger than the electrical coupling. Therefore, the electric field coupling can be ignored in the analysis, and only the magnetic coupling between the coils is considered. In order to simplify the analysis, this paper makes the same simplification, noting that \(R_{T}=R_{1}+R_{2}+2R_{S}\), \(L_{T}=L_{1}+L_{2}+2M_{12}\). To simplify the calculation, assume that the currents in the transmitting coils are the same, i.e., \(l_{1}=l_{2}\). In analogy to a conventional two-coil circuit, the Kirchhoff circuit equation can be written as follows, where \(l_{3}\) is the current in the receiving coil. \[\left\{ \begin{array}{c}\left[R_{T}+j\omega L_{T}+\frac{1}{j\omega}(\frac{1}{ C_{1}}+\frac{1}{C_{2}})\right]I_{1}+j\omega(M_{13}+M_{23})I_{3}=2V_{s}\\ j\omega(M_{13}+M_{23})I_{1}+(R_{3}+R_{L}+j\omega L_{3}+\frac{1}{j\omega C_{3}})I _{3}=0\end{array} \right. \tag{7}\]  
The calculation formula for the inductance of a circular coil is:  
\[L=\mu_{0}rN^{2}[\ln{(\frac{8r}{a})}-2] \tag{8}\]  
where \(\mu_{0}=4\pi\times 10^{-7}H/\)m is the magnetic permeability in the vacuum, \(N\) is the turn number of the coil, \(r\) is the radius of the coil, and \(a\) is the wire diameter. Usually, since the receiving coil is located in the middle of the transmitting coil in a three-coil system, the coupling between the transmitting coils is very weak and can be neglected, i.e., \(M_{12}=0\).