# 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  
#### 2.2.1 Magnetic Fields Generated by Helmholtz Coils  
Research has shown that magnetic field non-uniformity in MCR-WPT systems can lead to resonant frequency splitting phenomena [25]. This leads to an impedance mismatch between the transmitting and receiving coils and system detuning, which in turn affects the energy transmission of the system [26]. Therefore, for the MCR-WPT system, a uniform magnetic field is essential to ensure the stability of the energy transmission efficiency in the operating area. Research has shown that using Helmholtz coils is an adequate solution for obtaining a uniform but time-varying magnetic field over a large area [27; 28]. Helmholtz coils are a pair of identical, parallel, and co-axial circular current-carrying coils, with current flowing in the same direction through both coils and a distance between them equal to the coil radius. According to the superposition principle, the spatial magnetic field distribution is the vector sum of the magnetic flux densities produced by the two energized coils [29]. The magnetic field outside the two coils gradually weakens, but a wide range of uniform magnetic fields can be generated near the midpoint of the two axes. The distribution of the magnetic field generated by the Helmholtz coils was analyzed by the following analytical method. First, assume that the radius of the two identical coils is \(R\), the distance between the coils is equal to the radius, the number of turns is \(N\), and the currents in the two coils are in the same direction with magnitudes \(I_{1}\) and \(I_{2}\), respectively. Next, establish a right-angle coordinate system in space as shown in Figure 1, with the coaxial line of the two coils as the z-axis. The coordinate origin \(O\), x-axes, and y-axes are all located in the plane of symmetry between the centers of the two coils. Assuming that point \(P(x,y,z)\) is any point in space, the magnetic flux density \(\mathbf{B_{P}}\) at point \(P\) can be regarded as a vector superposition of the flux densities generated by the two coils individually. First, calculate the magnetic field generated by a single coil at point \(P\). As shown in Figure 1, the magnetic flux density \(\mathbf{B}\), generated by a single n-turn coil at any point in space, is given by the Bio-Savart law as:  
\[\mathbf{B}=\frac{n\mu_{0}}{4\pi}\oint_{I}\frac{\mathrm{Id}I\times\mathbf{r}}{r^{3}} \tag{1}\]  
Figure 1: Spatial structure of the Helmholtz coils. According to the principle of vector superposition, the magnetic flux density vector \(B_{P}\) of the Helmholtz coils at any point \(P\) in space is known to be:  
\[\mathbf{B_{P}}=\frac{n\mu_{0}}{4\pi}(\oint\limits_{l_{1}}(\frac{I_{1}\mathrm{d}I_{1} \times\mathbf{r}_{1}}{r_{1}}^{3})+\oint\limits_{l_{2}}\frac{I_{2}\mathrm{d}I_{2} \times\mathbf{r}_{2}}{r_{2}}) \tag{2}\]  
where \(\theta\) is the angle between the current element \(I\mathrm{d}I\) and the positive direction of the x-axis. Assuming that the current element \(I\mathrm{d}I\) is located in the upper side space of the \(xoy\) plane, its corresponding spatial location coordinate is: \((R\cos\theta,R\sin\theta,0.5R)\). The three components of the x, y, and z directions of the current element \(I\mathrm{d}I\) are: \((-IR\sin\theta\mathrm{d}\theta,IR\cos\theta\mathrm{d}\theta,0)\). The vector \(\mathbf{r}\) in space pointing from the current element to the point \(P\) is: \((R\cos\theta-x,\)\(R\sin\theta-y,0.5R-z)\). Therefore:  
\[I\mathrm{d}I\times\mathbf{r}=\left| \begin{array}{ccc}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ I\mathrm{d}I_{x}&I\mathrm{d}I_{y}&I\mathrm{d}I_{z}\\ r_{x}&r_{y}&r_{z}\end{array} \right|=\left| \begin{array}{ccc}\mathbf{i}&\mathbf{j}& \mathbf{k}\\ -IR\sin\theta\mathrm{d}\theta&IR\cos\theta\mathrm{d}\theta&0\\ R\cos\theta-x&R\sin\theta-y&0.5R-z\end{array}