# Helmholtz Coils Based WPT Coupling Analysis of Temporal Interference Electrical Stimulation System
## 3 Analysis of Calculation and Experimental Results
### Calculation in Free Space  
According to Equations (4)-(6) presented in Section 2.2.1, the strength of the magnetic field generated by Helmholtz coils at any point in space can be calculated by the analytical method. In this section, the electromagnetic model of the proposed system is modeled in the AD/DC module of the finite element analysis software. Then, a numerical calculation is performed to verify the device's uniformity of magnetic field distribution. Figure 4 shows the electromagnetic model of the proposed system in the calculation software.  
In the coordinate system shown, the receiving coil is located at the origin of the coordinate axis. The two transmitting coils are placed parallelly and symmetrically to the receiving coil. The coils are set up with copper material. According to the definition of Helmholtz coils, the distance between the two transmitting coils is h = R = 5 mm. The inductance of the transmitting coils is \(L_{1}=L_{2}=18.12\) uH, and the inductance of the receiving coil is \(L_{3}=19.93\) uH in the steady state field calculation. Due to the resonant relationship of the circuit, their corresponding compensation capacitances for resonance at 490 kHz transmission frequency are \(\bar{C}_{1}=C_{2}=5823\) pF and \(C_{3}=5293\) pF, respectively. The parameters of the transmitting and receiving coils in the numerical calculation are shown in Table 2.  
The human sciatic nerve stimulation frequencies used in clinical treatment protocols are essentially in the range of 1-20 Hz. The stimulation current amplitude is mostly concentrated in the range of 0.2-0.8 mA, with the actual applied intensity based on patient tolerance [44]. Therefore, this paper uses 10 Hz as the stimulation frequency of the target nerve in the calculation. For the selection of the coil excitation frequency, this paper refers to the application band set by the International Telecommunication Union (ITU) for implantable medical devices [45]. This selection also combines the standards for ultra-low power animal implantable devices (ULP-AID) from the European Telecommunications Standards Institute (ETSI) published for electromagnetic compatibility and the radio spectrum [46]. Therefore, the excitation frequencies of the transmitting coils are set to 490 kHz and 490 kHz + 10 Hz, respectively. In addition, the current size in both coil 1 and coil 2 is set to 1 A. The distribution of the magnetic flux density in the _yoz_ plane in the calculation results is shown in Figure 5. This shows that there is a considerable area near the axis of the Helmholtz coils where the magnetic flux density is very uniform.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Coils** & **Parameters** & **Symbol** & **Value** \\ \hline \multirow{5}{*}{Transmitting Coils} & Number of turns & \(N_{1},N_{2}\) & 10 \\  & Radius & \(R_{1},R_{2}\) & 5 cm \\  & Inductance & \(L_{1},L_{2}\) & 18.12 uH \\  & Capacitances & \(C_{1},C_{2}\) & 5823 pF \\  & Distance between coils & \(h\) & 5 cm \\ \hline \multirow{5}{*}{Receiving Coil} & Wire diameter & \(d\) & 0.2 mm \\  & Outer diameter & \(D_{\mathrm{OT}}\) & 10 mm \\ \cline{1-1}  & Inner diameter & \(D_{\mathrm{IN}}\) & 6 mm \\ \cline{1-1}  & Inductance & \(L_{3}\) & 19.93 uH \\ \cline{1-1}  & Capacitances & \(C_{3}\) & 5393 pF \\ \hline \hline \end{tabular}
\end{table}
Table 2: The symbols and values of the parameters for the numerical calculation in free space.  
Figure 4: Helmholtz coils–based model for a temporally interfering WPT system.