# Helmholtz Coils Based WPT Coupling Analysis of Temporal Interference Electrical Stimulation System
## 2 Principles and Methods Analysis
### Differential Frequency Electrical Stimulation System Based on Temporal Interference  
#### 2.3.1 Temporal Interference Generates Low-Frequency Envelope Modulated Magnetic Fields  
The low-frequency envelope modulated magnetic field generated by temporal interference allows for both the low-pass characteristics of nerve cells and the high penetration of the magnetic field in human tissue. Due to the low magnetoresistance of human tissue at high frequencies, the applied high-frequency magnetic field signal easily penetrates human tissue. In the region where no interference occurs, given the low-pass properties of the nerve cell membrane and fatigue, the stimulation signal at the kHz level is outside the range of frequencies that the neuron can receive. It does not cause neurons to release action potentials and generate an effective stimulation response. However, in the area where the interference occurs, there is a low-frequency envelope modulated magnetic field with a frequency of \(\Delta f\). The neurons can receive the stimulus signal at that frequency and then respond to it. This also reflects the selective nature of temporal interference electrical stimulation. Activation occurs only in the target area to produce the desired therapeutic effect while avoiding the activation of non-target areas which could cause undesirable side effects. This function was successfully demonstrated in biological experiments. When stimulated with 2 kHz and 2.01 kHz currents by electrodes placed immediately above the mouse head, neuronal activation occurs only in the hippocampus, and there is no significant neural activity in the overlying cortex [39]. Figure 3 shows the principle of generating a low-frequency envelope modulated electromagnetic wave by using temporal interference. Two coils are placed at a fixed distance and angle in space. The high-frequency alternating current of kHz level with frequencies \(f_{1}\) and \(f_{2}=f_{1}+\Delta f\) are then passed through the two independent coils, respectively. According to the principle of electromagnetic induction, electromagnetic waves of the same frequency as the excitation current are generated in space. At the same time, due to the temporally interfering principle of electromagnetic waves, two high-frequency magnetic field signals of different frequencies produce a temporal interference phenomenon in the middle region. The two are modulated to produce a differential frequency magnetic field with a frequency of \(\Delta f\) (the frequency difference between the excitation signals of the transmitting coils). This low-frequency magnetic field is in the frequency range where the nerve can be activated. Figure 3: Waveform of the low-frequency envelope modulated signal produced by temporal interference. The mathematical expression for two sinusoidal magnetic field signals of different frequencies and amplitudes is as follows, where \(\varphi_{1}\) and \(\varphi_{2}\) are the initial phases of the two signals \(B_{1}\) and \(B_{2}\), respectively. \[B_{1}(t)=B_{1}\sin(2\pi f_{1}+\varphi_{1}) \tag{15}\]  
\[B_{2}(t)=B_{2}\sin(2\pi f_{2}+\varphi_{2}) \tag{16}\]  
The magnetic field formed after the interference is:  
\[ \begin{array}{ll}B(t)&=B_{1}(t)+B_{2}(t)\\ &=B_{1}\sin(2\pi f_{1}t+\varphi_{1})+B_{2}\sin(2\pi f_{2}t+\varphi_{2})\\ &=\sqrt{{B_{1}}^{2}+{B_{2}}^{2}+2B_{1}B_{1}\cos[2\pi\Delta ft+(\varphi_{2}- \varphi_{1})]}\cdot\sin\left[2\pi(\frac{f_{1}+f_{2}}{2})t+\varphi\right]\end{array}  \tag{17}\]  
Among them:  
\[\varphi=\varphi_{1}+\arctan\frac{B_{2}\sin[2\pi\Delta ft+(\varphi_{2}-\varphi_ {1})]}{B_{1}+B_{2}\cos[2\pi\Delta ft+(\varphi_{2}-\varphi_{1})]} \tag{18}\]  
The above derivation shows that the interference results in a kHz magnetic field signal with a frequency of \((f_{1}+f_{2})/2\), with an envelope modulation amplitude size of \(\sqrt{{B_{1}}^{2}+{B_{2}}^{2}+2B_{1}B_{2}\cos 2\pi\Delta ft}\) when the initial phase angle of the two signals is the same. Its envelope is pulsed with the frequency difference of \(\Delta f\) between the two signals.