# Helmholtz Coils Based WPT Coupling Analysis of Temporal Interference Electrical Stimulation System
## 1 Introduction  
Complex regional pain syndrome type I (CRPS-I) is a chronic neuropathic pain syndrome, secondary to limb injury, which occurs in up to 48.8% of patients who have had a stroke or fracture [1]. It mainly presents clinically with symptoms of intractable and variable pain, malnutrition, and functional impairment [2, 3]. The clinical treatment of CRPS-I is primarily conservative in terms of medication. However, due to the long treatment period, high dosage, and side effects, the results are not satisfactory. In recent years, interventional therapies have been widely used to treat neuropathic pain. In order to fill the gap in spinal cord electrical stimulation, [PERSON] et al. proposed a CRPS-I therapy, using an implantable device to stimulate the sciatic nerve directly, and clinically validated its effectiveness [4]. Compared to spinal cord stimulation (SCS), this method limits the stimulation area to the foot and has no additional impact on the non-working area, enabling an efficient and focused treatment of intractable pain [4]. Currently, it is common forimplantable devices to be battery powered [5; 6]. Although the power density of batteries is increasing, the limitations of the implantation space leave the batteries facing the dilemma of running out of power [7]. Studies have shown that over 60% of pacemakers need to be replaced for battery reasons within 5-12 years [8]. In addition, the second surgery that accompanies the battery replacement places a double financial and physical burden on the patient, making battery power a significant limitation in the long-term operation of implantable devices. Nevertheless, the wireless power transmission of implantable devices can solve the above problems [9; 10]. It enables an indefinite implantation in the body without facing the problem of running out of power [11]. Traditional wireless power transfer (WPT) devices usually have a transmission frequency in the MHz range [12; 13]. However, as the frequency of the applied stimulus is mostly in the tens to hundreds of hertz, the device needs to be equipped with a frequency conversion circuit to transform the signal from high to low frequencies [14]. Both this signal processing module and the battery component take up additional volume in the implanted device, hindering the miniaturization of the size of the implantable devices. Therefore, optimizing the implantable stimulation systems, while ensuring effectiveness, is a new research direction for CRPS treatment. This paper uses the principle of magnetic field superposition based on temporal interference (TI), eliminating the need for processing circuits that convert high-frequency signals to low-frequency signals in conventional WPT. This provides a new idea for the miniaturization of implantable devices. A comparison of the characteristics of the different power supply methods for implantable devices is shown in Table 1.  
Different circuit topologies have different transmission characteristics for magnetically coupled resonant wireless power transfer (MCR-WPT) systems. Coupling analysis of the proposed system was necessary to visually analyze and improve the configuration and spatial distribution of the coupling coils and thus stabilize the energy transfer efficiency of the whole system. The main methods for the quantitative analysis of the operating principle of the MCR-WPT were the coupled-mode theory and the circuit theory [15]. However, the coupled-mode theory was too theoretical and not sufficiently intuitive for optimizing the parameters of a system [16]. Therefore, this paper abstracts the system using the circuit model and uses the basic theory of resonant circuits to elaborate and analyze the transmission laws of the system. According to the Hodgkin-Huxley (HH) theory of neurophysiology and the McIntyre-Richardson-Grill (MRG) model, the excitability of nerve cells results from the different permeability of the cell membrane to sodium, potassium, and oxygen ions [17; 18; 19].