Simulation of Eddy Current Suppression and Efficiency Recovery in Mining MCR-WPT Systems Based on Explosion-Proof Slotting

Abstract

To meet safety regulations in underground coal mines, wireless power transfer (WPT) systems must house both the transmitter and receiver within explosion-proof enclosures. However, eddy currents induced on the surfaces of these non-ferromagnetic metal enclosures significantly hinder magnetic flux coupling, thereby reducing transmission efficiency. This paper proposes a slotting technique applied to explosion-proof enclosures to suppress eddy currents, along with the integration of magnetic flux focusing materials into the coils to enhance coupling. Simulations were conducted to compare three system configurations: (i) a WPT system without enclosures, (ii) a system with solid (unslotted) enclosures, and (iii) a system with slotted enclosures. The results show that solid enclosures reduce efficiency to nearly zero, whereas slotted enclosures restore efficiency to 90% of the baseline system without enclosures. Joule heating remains low in the slotted explosion-proof enclosures, with energy losses of 2.552 J for the transmitter enclosure and 2.578 J for the receiver enclosure. A conservative first-order estimation confirms that the corresponding temperature rise in the enclosure surfaces remains below 50 °C, which is well within the 150 °C limit stipulated by the Chinese National Standard GB 3836.1-2021 (Explosive Atmospheres—Part 1: Equipment General Requirements). These findings confirm effective eddy current suppression and efficiency recovery without compromising explosion-proof safety. The core innovation of this work lies not merely in the physical slotting approach, but in the development of a precise equivalent circuit model that fully incorporates all mutual inductance components representing eddy current effects in non-ferromagnetic explosion-proof enclosures, and its integration into the overall MCR-WPT system circuit.

1. Introduction

Amid global energy transition towards green and digital solutions, intelligent coal mines face critical challenges in powering electrical equipment safely [1,2]. Traditional wired charging in underground coal mines poses a risk of electric sparks in gas-rich environments, and battery replacement is prohibited in explosion-proof devices [3,4]. Figure 1 shows a typical charging chamber and an explosion-proof enclosure used in coal mines. Wireless power transfer (WPT), as a non-contact power supply method, offers a promising solution [5,6]. If successfully implemented underground, WPT could eliminate the need for wired cables for electrical equipment. Moreover, it would resolve the persistent issues that prevent battery replacement in sealed explosion-proof equipment and prohibit the underground charging of battery-based devices. This approach would not only streamline maintenance and management procedures for electrical equipment in underground coal mines but also significantly enhance the technical safeguards for mine safety.

Magnetic coupled resonant wireless power transfer (MCR-WPT), based on the principle of magnetic resonance, operates in the near-field domain [7,8,9,10]. In this configuration, both the transmitting and receiving coils must operate at a self-resonant or resonant state [11,12,13,14]. Currently, MCR-WPT has become a major research focus within the WPT field [15,16,17,18], as it enables efficient medium-distance power transfer with minimal electromagnetic radiation [19,20]. Its transmission range can reach up to several meters, making it highly suitable for complex environments such as underground coal mines. Therefore, a comprehensive and in-depth study of MCR-WPT systems for mining applications is necessary.

To meet safety requirements in underground environments, the MCR-WPT system must comply with explosion-proof standards, which require both the transmitter and receiver to be housed in separate explosion-proof enclosures [21]. Most of these enclosures are made of non-ferromagnetic, high-stiffness metals such as stainless steel. During energy transmission, however, eddy currents induced on the enclosure surfaces act as a shield, degrading magnetic flux coupling and near-field resonance, and significantly reducing the transmission efficiency. Researchers worldwide have conducted numerous studies to overcome these challenges and advance WPT technology for underground mining applications.

For example, reference [22] proposed a WPT system for electrical equipment in underground coal mines, considering the impact of gas and dust on electrical safety. The authors derived a method for determining the optimal resonant frequency by comparing the frequency characteristics of the impedance modulus with a pure mutual inductance model, ensuring full-resonance operation. Reference [23] established a mobile charging model using MCR-WPT to recharge wireless sensor nodes underground. Reference [24] introduced a full-resonance compensation scheme to address detuning caused by improper parameter configuration, which leads to non-purely resistive input impedance and reduced power supply efficiency. In response to complex underground scenarios, reference [25] developed an improved four-coil MCR-WPT system based on SS topology to enhance power transmission capacity.

Previous studies on mine WPT have focused on frequency optimization for gas and dust environments, mobile charging models for sensors, resonance compensation schemes, and four-coil SS-topology systems. However, none have addressed eddy current suppression in metal enclosures—a major obstacle to practical deployment [26,27,28,29,30,31]. Given the high safety risks and ethical requirements in explosive atmospheres, high-fidelity theoretical modeling and simulation represent an essential and mandatory step prior to any physical experimentation in this field. Our model study provides indispensable theoretical guidance and risk assessment for the design of future safe experiments. Unlike previous work, our study focuses on a slotting technique that complies with explosion-proof standards, achieving 90% efficiency recovery while maintaining safety. Specifically, we propose cutting slits into the explosion-proof enclosure and installing tempered glass to prevent explosive gas ingress. This approach effectively suppresses eddy currents on the enclosure surface after its integration into the MCR-WPT system. The core innovation of this work lies in the development of a precise equivalent circuit model that fully incorporates all mutual inductance components to represent eddy current effects in non-ferromagnetic explosion-proof enclosures, and its comprehensive integration into the overall MCR-WPT system circuit. This work provides important theoretical and practical insights that can facilitate the broader application and dissemination of MCR-WPT technology in mining.

2. Model of Mining MCR-WPT System

Traditional WPT systems typically consist of an energy supply coil, transmitting coil, receiving coil, and electrical load [32,33,34,35]. The commonly used four-coil WPT system model is shown in Figure 2. The mining MCR-WPT system differs in that, in underground coal mines, all components involved in the MCR-WPT system need to be completed within metal explosion-proof enclosures. The presence of these enclosures complicates the near-field resonant and inductive coupling between the transmitting coil and receiving coil.

2.1. Introduction of Mining MCR-WPT System

To address the difficulties and challenges faced by MCR-WPT in underground coal mines, this section proposes a low eddy current mining MCR-WPT system, as shown in Figure 3. The system includes a transmitting coil, receiving coil, metal explosion-proof enclosure, insulating partition, power supply module, inverter circuit module, rectifier circuit module, reactive power compensation module, and DC-DC conversion module. And all the above processes need to be carried out within separate explosion-proof enclosures on the transmitting side and receiving side, respectively.

The surface of the metal explosion-proof enclosure that is adjacent to and parallel to both the transmitting coil and the receiving coil is defined as the working surface for mining MCR-WPT. To mitigate eddy currents and enhance coupling, laser technology is used to cut slits on the working surface, and ferrite bars are placed on the transmitter and receiver coils to achieve magnetic focusing and strong coupling. After slitting, a layer of tempered glass is placed inside the metal explosion-proof enclosure to prevent explosive gases from dispersing into the enclosure.

The explosion-proof enclosures employed in underground coal mines are mainly constructed from stainless steel. This material presents a relative magnetic permeability μ_r of approximately 1. It is characterized by a relatively large electrical conductivity, conferring strong electrical conduction capabilities while having weak magnetic conduction properties, thus qualifying as a typical non-ferromagnetic material. For the explosion-proof enclosure in this study, stainless steel has been selected.

2.2. Influence of Metal Explosion-Proof Enclosure on the System

Taking the most classic two-coil WPT system as an example, its equivalent circuit diagram is shown in Figure 4. According to Kirchhoff’s voltage law, the output power and transfer efficiency of the system can be derived, as shown in Equations (3) and (4). It can be seen that the transfer efficiency of the system is affected by factors such as the mutual inductance between the transmitting and receiving coils, load resistance, and operating frequency. When the load and operating frequency are determined, the transfer efficiency is primarily determined by the mutual inductance.

$$\left\{\begin{array}{l}\stackrel{•}{U_S}=(R_S+R_T+j\omega L_T+{\displaystyle \frac{1}{j\omega C_T}})\stackrel{•}{I_1}+j\omega M\stackrel{•}{I_2}\\ 0=j\omega M\stackrel{•}{I_1}+(R_L+R_R+j\omega L_R+{\displaystyle \frac{1}{j\omega C_R}})\stackrel{•}{I_2}\end{array}\right.$$

(1)

where U_S is a high-frequency alternating current (AC) voltage source, R_S is the internal resistance of the voltage source, R_T is the metallic resistance of the transmitting coil, L_T is the inductance of the transmitting coil, and C_T is the compensation capacitor on the transmitting side, R_L is the load resistance, R_R is the metallic resistance of the receiving coil, L_R is the inductance of the receiving coil, and C_R is the compensation capacitor on the receiving side. The field-circuit coupling simulation parameters of the mining MCR-WPT system are shown in

Table 1. The field-circuit coupling simulation Parameters of the Mining MCR-WPT System.

Parameters	The Field-Circuit Coupling Simulation Parameters of the Mining MCR-WPT System
US	220 sin(2π × 105 t + 90°) V
RS	0.1 Ω
RT	0.5 Ω
RR	0.5 Ω
RL	99.5 Ω
CT	1.88 nF
CR	1.93 nF

.

Under the condition of series resonance, I₁ and I₂ can be obtained from Equation (1). Equations (3) and (4) show efficiency η depends on mutual inductance M. When a stainless-steel enclosure is inserted (Figure 5), eddy currents reduce M by 98.5% (

Table 2. Mutual inductance of the two coils, transmission efficiency, average input power and average output power for the three sets of simulations.

Model	Mutual Inductance (mH)	Transmission Efficiency (%)	Average Input Power (W)	Average Output Power (W)
I (No enclosure)	0.5155	68.03	30.61	20.81
II (Enclosure)	0.0076	≈0	1.11	≈0
III (Slitted)	0.4017	62	92.55	47.38

), collapsing η to near zero.

$$\left\{\begin{array}{l}\stackrel{•}{I_1}={\displaystyle \frac{\stackrel{•}{U_S}(R_R+R_L)}{(R_S+R_T)(R_R+R_L)+{\omega }^2{M}^2}}\\ \stackrel{•}{I_2}={\displaystyle \frac{-j\omega M\stackrel{•}{U_S}}{(R_S+R_T)(R_R+R_L)+{\omega }^2{M}^2}}\end{array}\right.$$

(2)

$$P_{out}={\left|\stackrel{•}{I_2}\right|}^2{R}_L={\displaystyle \frac{{\omega }^2{M}^2{U}^2{R}_L}{{\left(\left(R_S+R_T\right)\left(R_R+R_L\right)+{\omega }^2{M}^2\right)}^2}}$$

(3)

$$\eta ={\displaystyle \frac{P_{out}}{P_{in}}}={\displaystyle \frac{{\omega }^2{M}^2{R}_L}{{\omega }^2{M}^2(R_R+R_L)+(R_S+R_T){(R_R+R_L)}^2}}\times 100\%$$

(4)

Taking circular coils as an example, assuming the number of turns of both the transmitting and receiving coils is 1, the equivalent model is shown in Figure 6. The radius of the two rings are R₁ and R₂, and the distance between the planes of the rings is h. Suppose the center of the ring with radius R₁ is located at the origin of the coordinate. The mutual inductance between the two coils M can be calculated using the Neumann formula and the Biot–Savart Law. Take dl₁ and dl₂ on the two rings, respectively.

$$d{\overrightarrow{l}}_1\cdot d{\overrightarrow{l}}_2=R_1{R}_2\cos \theta d{\varphi }_{1}d{\varphi }_2$$

(5)

$$\begin{array}{l}{\overrightarrow{r}}_1=\overrightarrow{i}{R}_1\cos {\varphi }_1+\overrightarrow{j}{R}_1\sin {\varphi }_1\\ {\overrightarrow{r}}_2=\overrightarrow{i}{R}_2\cos {\varphi }_2+\overrightarrow{j}{R}_2\sin {\varphi }_2+\overrightarrow{k}h\end{array}$$

(6)

$$M={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle {\oint }_{l_2}{\displaystyle {\oint }_{l_1}\frac{d{\overrightarrow{l}}_1\cdot d{\overrightarrow{l}}_2}{r}}}={\displaystyle \frac{{\mu }_0}{2}}{\displaystyle {\int }_0^{2\pi }\frac{R_1{R}_2\cos \theta d\theta }{\sqrt{h^2+R_1^2+R_2^2-2R_1{R}_2\cos \theta }}}$$

(7)

As a result, the mining MCR-WPT system with an explosion-proof enclosure can be regarded as a non-ferromagnetic metal plate inserted between the transmitting coil and the receiving coil of the classical two-coil WPT system, as shown in Figure 5. The mutual inductance between the two coils decreases, leading to a reduced coupling coefficient and thus a lower transmission efficiency. Additionally, during energy transmission, eddy currents generated in the metal explosion-proof enclosure on the side of the receiver coil hinder energy transfer by storing it.

According to Lenz’s law, when a metal object is in an alternating magnetic field, vortex-like induced currents will be generated inside. I₁ is the alternating current flowing through the transmitting coil. When the alternating magnetic flux Φ₁ generated by it passes through the metal plate, it excites an induced electromotive force in the metal plate, resulting in the eddy current effect I_e. At this time, the metal plate will generate a new magnetic flux Φ_e that is opposite to the direction of the magnetic flux of the transmitting coil. At the same time, a magnetic flux Φ₂ opposite to the direction of the magnetic flux of the transmitting coil is also generated in the receiving coil. Under the action of eddy currents Φ_e, the magnetic flux Φ₁ passing through the receiving coil in the same direction as the transmitting coil is weakened, and the mutual inductance between the transmitting coil and the receiving coil is reduced. From the perspective of energy, due to the existence of resistance in the metal plate, the induced current will continuously generate Joule heat, leading to a rise in the temperature of the explosion-proof enclosure, system energy loss, and a decrease in transmission efficiency.

2.3. Equivalent Circuit Model of the System

As described in the previous section, the introduction of a metal explosion-proof enclosure imposes effects on the WPT system in the form of eddy currents, hindering near-field resonance and inductive coupling. The eddy current effect of non-ferromagnetic metals in the mining MCR-WPT system can be equivalent to a series circuit of resistance and inductance. The equivalent circuit model of the mining MCR-WPT system is shown in Figure 7. U_S is the high-frequency power supply voltage after inversion. R_S is the internal resistance of the voltage source. R_T and R_R are, respectively, the metallic resistance of the transmitting coil and the receiving coil under high-frequency conditions. C_T and C_R are the resonant capacitors of the two coils, respectively. L_T and L_R are the equivalent inductances of the two coils, respectively. R_L is the load resistance. L₀ and R₀ represent the inductance and internal resistance of the equivalent short-circuit loop current of the explosion-proof enclosure, respectively. M₁₄ is the mutual inductance between the transmitting coil and the receiving coil. M₁₂ represents the mutual inductance between the transmitting coil and the explosion-proof housing at the transmitting end. M₁₃ represents the mutual inductance between the transmitting coil and the explosion-proof housing at the receiving end. M₃₄ represents the mutual inductance between the receiver coil and the explosion-proof housing at the receiving end. M₂₄ represents the mutual inductance between the receiving coil and the explosion-proof housing at the transmitting end. M₂₃ represents the mutual inductance between the explosion-proof housing at the transmitting end and the explosion-proof housing at the receiving end.

According to Kirchhoff’s voltage law, Equation (8) can be listed.

$$\left[\begin{array}{cccc}{Z}_{11}& j\omega M_{12}& j\omega M_{13}& j\omega M_{14}\\ j\omega M_{12}& Z_{22}& j\omega M_{23}& j\omega M_{24}\\ j\omega M_{13}& j\omega M_{23}& Z_{33}& j\omega M_{34}\\ j\omega M_{14}& j\omega M_{24}& j\omega M_{34}& Z_{44}\end{array}\right]\left[\begin{array}{c}\stackrel{•}{I_1}\\ \stackrel{•}{I_2}\\ \stackrel{•}{I_3}\\ \stackrel{•}{I_4}\end{array}\right]=\left[\begin{array}{c}\stackrel{•}{U_S}\\ 0\\ 0\\ 0\end{array}\right]$$

(8)

where I₂ and I₃ are the currents of the equivalent circuit of the eddy current effect in the explosion-proof enclosure at the receiving end and the transmitting end. I₁ and I₄ are the currents in the transmitting coil and the receiving coil, respectively. And among

$$\left\{\begin{array}{l}{Z}_{11}=R_S+R_T+j\omega L_T+{\displaystyle \frac{1}{j\omega C_T}}\\ Z_{22}=R_0+j\omega L_0\\ Z_{33}=R_0+j\omega L_0\\ Z_{44}=R_R+R_L+j\omega L_R+{\displaystyle \frac{1}{j\omega C_R}}\end{array}\right.$$

(9)

Through the solution of Equation (8), the transmission efficiency and equivalent impedance of the mining MCR-WPT system under the resonant condition can be derived as:

$$\eta ={\displaystyle \frac{P_{out}}{P_{in}}}={\displaystyle \frac{\frac{{\left(X_{41}\right)}^2}{{\left(\left(Q+R+T\right)+jS\right)}^2}{\left|\stackrel{•}{U_S}\right|}^2{R}_L}{\frac{X_{11}}{\left(Q+R+T\right)+jS}{\left|\stackrel{•}{U_S}\right|}^2}}={\displaystyle \frac{{\left(X_{41}\right)}^2{R}_L}{X_{11}\left(\left(Q+R+T\right)+jS\right)}}$$

(10)

$$Z_{in}=R_S+R_T+{\displaystyle \frac{{\left(\omega M_{14}\right)}^2}{R_R+R_L}}+{\displaystyle \frac{R_0{\omega }^2\left(M_{12}^2+M_{13}^2\right)}{R_0^2+{\omega }^2{L}_0^2}}-\left(j\omega L_0\right)\cdot {\displaystyle \frac{{\omega }^2\left(M_{12}^2+M_{13}^2\right)}{R_0^2+{\omega }^2{L}_0^2}}$$

(11)

$$\left\{\begin{array}{l}{X}_{11}={\omega }^2\left(Z_{22}{M}_{34}^2+Z_{33}{M}_{24}^2+Z_{44}{M}_{23}^2\right)+Z_{22}{Z}_{33}{Z}_{44}+j\left(2{\omega }^3{M}_{23}{M}_{24}{M}_{34}\right)\\ X_{41}=-{\omega }^2\left(Z_{22}{M}_{13}{M}_{34}+Z_{33}{M}_{12}{M}_{24}\right)+j\left({\omega }^3\left(-M_{14}{M}_{23}^2+M_{12}{M}_{23}{M}_{34}+M_{13}{M}_{23}{M}_{24}\right)-\omega \left(Z_{22}{Z}_{33}{M}_{14}\right)\right)\end{array}\right.$$

(12)

$$\left\{\begin{array}{l}Q={\omega }^4\left(M_{12}^2{M}_{34}^2+M_{13}^2{M}_{24}^2+M_{14}^2{M}_{23}^2-2M_{12}{M}_{13}{M}_{24}{M}_{34}-2M_{12}{M}_{14}{M}_{23}{M}_{34}-2M_{13}{M}_{14}{M}_{23}{M}_{24}\right)\\ R={\omega }^2\left(Z_{11}{Z}_{22}{M}_{34}^2+Z_{11}{Z}_{33}{M}_{24}^2+Z_{11}{Z}_{44}{M}_{23}^2+Z_{22}{Z}_{33}{M}_{14}^2+Z_{22}{Z}_{44}{M}_{13}^2+Z_{33}{Z}_{44}{M}_{12}^2\right)\\ S=2{\omega }^3\left(Z_{11}{M}_{23}{M}_{24}{M}_{34}-Z_{22}{M}_{13}{M}_{14}{M}_{34}-Z_{33}{M}_{12}{M}_{14}{M}_{24}-Z_{44}{M}_{12}{M}_{13}{M}_{23}\right)\\ T=Z_{11}{Z}_{22}{Z}_{33}{Z}_{44}\end{array}\right.$$

(13)

Based on the analysis of the influence of the metal explosion-proof enclosure on the system presented in the previous section, and in conjunction with Equations (10) and (11) derived from the equivalent circuit model, it can be concluded that the integration of a non-ferromagnetic metal explosion-proof enclosure into the MCR-WPT system results in an increase in the real part and a decrease in the imaginary part of the equivalent input impedance due to the eddy current effect. Specifically, this is reflected as an increase in the equivalent resistance and a decrease in the equivalent inductance, accompanied by a reduction in the mutual inductance between the transmitting and receiving coils. Since the lumped capacitance is generally unaffected by external disturbances, the decrease in inductance causes the resonant frequency to shift toward higher frequencies. The alteration in input impedance also introduces a phase difference between the input voltage and current. If the imaginary part of the input impedance is negative, the system exhibits capacitive characteristics, resulting in the input current leading the input voltage in phase. The specific phase difference θ can be calculated using the formula θ = arctan (X_in/R_in), where X_in represents the imaginary part of Z_in and R_in represents the real part of Z_in. This explains the phase shift observed between the input current and voltage. Consequently, a deviation occurs between the power supply’s driving frequency and the resonant frequency of the system, leading to a decline in the transmission efficiency.

Although the method of disrupting eddy currents through slit technology is simple, direct, and commonly used, it still holds novelty in the specific application scenario selected in this paper. As revealed by the subsequent research, when a metal explosion-proof enclosure is integrated into the MCR-WPT system, eddy currents generated on the working surface prevent the system from transmitting power normally. In underground coal mines, to meet explosion-proof requirements as much as possible, slits rather than openings are used to disrupt eddy current paths. This approach maximizes the retention of the explosion-proof enclosure’s structural rigidity and effectively enhances the safety and reliability of system operation. Beyond the simplicity of the technology itself, the key lies in clarifying the mechanism by which eddy currents affect system operation. On this basis, slits not only disrupt eddy currents but also eliminate their impacts on the mutual inductance between the transmitting and receiving coils and the system’s coupling coefficient, providing a feasible solution for stable system operation. As illustrated in Figure 7 in the preceding text, the eddy current effect is modeled as a series circuit constituted by an inductor and a resistor, which is visually manifested in the equivalent circuit model. From an overall perspective, the proposal of this system and its application in the special environment of underground coal mines present unique challenges. This paper can be regarded as an innovative integration of simple technical solutions, WPT, and the operational environment of underground coal mines.

3. Modeling and Simulation

Based on the mining MCR-WPT system proposed in Section 2, ANSYS 2022 Maxwell and Simplorer are used as the research tools for field-circuit (transient magnetic field and circuit) coupling simulation studies to analyze the characteristics of this model. In this section, a mining MCR-WPT model, as shown in Figure 3, is established in the transient magnetic field, and it is connected to the circuit model shown in Figure 7 to achieve a coupled simulation. By comparing and analyzing the classical two-coil MCR-WPT, the MCR-WPT with an explosion-proof enclosure, and the MCR-WPT with a slotted explosion-proof enclosure (the mining MCR-WPT system), the influence of the introduction of the explosion-proof enclosure on the system is studied, and the effectiveness of the slit technology in restoring the system’s transmission performance is verified.

3.1. Scenario Description and Model Establishment

In the scenario of mine applications, the dimensions and material selection of the simulation model established in this section all comply with the actual working conditions in underground coal mines and explosion-proof requirements, and the simulation is jointly completed based on ANSYS 2022 Maxwell 3D and Simplorer. The field-circuit coupling simulation models of the MCR-WPT with a slotted explosion-proof enclosure are shown in Figure 8. The dimensions and structures of the explosion-proof enclosures at the transmitting end and the receiving end are consistent. The distance between the transmitting and receiving coils is set to 9 cm, while the distance between the outer surfaces of the transmitting and receiving enclosures is 5 cm. The enclosure material is stainless steel, and the thickness of the enclosure is 5 mm. The surface of the explosion-proof enclosure is periodically slotted using laser technology. The slots are linear, 50 cm in length, 0.2 cm in width, and evenly spaced at 0.5 cm intervals, resulting in a total of 144 slots distributed across the 60 cm × 52 cm working surface. The slotting design aims to maximize the disruption of eddy current paths, thereby suppressing eddy current losses and enhancing transmission efficiency, while preserving structural rigidity and explosion-proof integrity. The present slot configuration was selected based on comparative simulations that balanced eddy current suppression against mechanical strength. Although the specific pattern presented here is linear and uniformly spaced, the underlying principle is universal; alternative geometries such as vertical slots, inclined slots, or window-based designs may also be effective. The slots are oriented perpendicular to the direction of the induced eddy currents to maximize their disruption effect, as determined by the magnetic field distribution around the coils.

The dimensions and structures of the transmitting coil and the receiving coil are identical. Both are multi-turn closely wound rectangular coils with 50 turns, and the material is copper. As shown in Figure 9, the purple part represents the transmitting coil. To improve magnetic coupling and reduce flux leakage, strip-shaped ferrite bars are placed on both coils. The high permeability of the ferrite material provides a low-reluctance path for the magnetic flux, effectively guiding it through the central area of the transmitter coil and facilitating a closed loop through the receiver coil. To ensure adequate coverage of the winding projection area, three ferrite strips are uniformly distributed beneath each coil, as illustrated in Figure 9.

In the external circuit of the model coupling as shown in Figure 8, a sinusoidal alternating current voltage source with an amplitude of 220 V, a frequency of 100 kHz, and an initial phase of 90° is used as the system excitation. R_S is the internal resistance of the voltage source, R_T is the metallic resistance of the transmitting coil, C_T is the reactive power compensation capacitor on the transmitting side, Winding1 is the winding of the transmitting coil, R_R is the metallic resistance of the receiving coil, R_L is the load driven by the system, C_R is the reactive power compensation capacitor on the receiving side, and Winding2 is the winding of the receiving coil. Particularly, during the simulation process, the self-inductances of the transmitting coil and the receiving coil, that is, the values of L_Winding1 and L_Winding2 in the simulation software, need to be calculated through the transient magnetic field first. Then, according to the series resonance condition, the values of the reactive power compensation capacitors on the transmitting side and the receiving side at 100 kHz are calculated. Notably, all circuits are in operation inside the explosion-proof enclosure.

In the simulation context, given that the excitation source operates at a frequency of 100 kHz, the corresponding period of the excitation signal amounts to 10 μs. In accordance with the Nyquist sampling theorem, setting the simulation step size to 0.1 μs suffices to fulfill the theorem’s stipulations. Nevertheless, as the MCR-WPT system resides in a critically damped state during resonance, it exhibits a propensity for non-convergence throughout the time-domain finite element simulation process. Moreover, reaching a steady state demands an extended period. Consequently, the overall simulation time span is configured to span 22 cycles, equating to 220 μs.

Preliminary simulations reveal that, in the model depicted in Figure 9 during the energy transfer procedure, the eddy currents engendered by the explosion-proof enclosure predominantly congregate on the working surface. For this specific section of the enclosure, meshing is implemented based on a length dimension of 5 mm and a depth dimension of 0.004 mm (smaller than the skin depth of 4.26 × 10⁻³ mm at 100 kHz) to accurately capture the eddy current distribution within the skin depth. The skin depth is calculated as δ = 1/(πμσf)^1/2, where μ is the permeability, σ is the conductivity, and f is the frequency. This fine meshing ensures that the eddy current effects in the non-ferromagnetic metal are resolved with sufficient accuracy. Simultaneously, to alleviate the computational burden imposed on the computer by mesh settings, curtail the calculation time, and minimize storage space requirements, the adaptive meshing function provided by the software is adopted for the remaining portions of the model.

3.2. Simulation and Results Analysis

Upon meticulously setting the aforementioned conditions during the modeling phase, the field-circuit coupling simulation can be initiated. Within the transient magnetic field, it becomes possible to observe the energy distribution and eddy current dispersion patterns of the explosion-proof enclosure. By calculating the ohmic losses incurred by the explosion-proof enclosure, an assessment of its heating behavior can be made. Concurrently, in the circuit simulation domain, the input and output voltages as well as currents of the three sets of MCR-WPT systems can be monitored, facilitating an evaluation of their transmission efficiency traits. For the sake of clarity in exposition, in this section, the classical two-coil MCR-WPT system is designated as Model I; the MCR-WPT system equipped with an explosion-proof enclosure is termed Model II; and the MCR-WPT system featuring a slotted explosion-proof enclosure, which is, in fact, the mining MCR-WPT system, is defined as Model III.

As elucidated in the preceding analysis, the mutual inductance between the transmitting coil and the receiving coil within the MCR-WPT system constitutes a pivotal parameter that mirrors the coupling extent and transmission traits of the system. In light of Equations (4) and (10), it is evident that the magnitude of the mutual inductance between the aforesaid coils exerts a direct influence on the transmission efficiency of the system. The transmission efficiency of the system can be ascertained by computing the average input power and average output power over a specific period subsequent to attaining a steady state, with reference to the input-side voltage and current as well as the output-side voltage and current, as depicted in Figure 10. Table 2 presents a comparison of the mutual inductance between the transmitting coil and the receiving coil for each of the three models, along with their respective transmission efficiencies. The data tabulated therein reveals that Model I exhibits the highest coupling coefficient and transmission efficiency; upon the introduction of the explosion-proof enclosure, during the energy transmission process, eddy currents are generated on the working surface of Model II. These eddy currents function as a magnetic shield, confining the energy within the working surface and impeding its propagation, thereby leading to a diminution in both the coupling coefficient and transmission efficiency of Model II; when the slit technique is employed to disrupt the eddy currents on the working surface, the coupling coefficient and transmission efficiency of Model III are recuperated.

Figure 10a–c, respectively, illustrate the voltage and current waveforms of the input-side voltage source and the output-side load for Model I, Model II, and Model III, within the time interval from 150 μs to 180 μs after the simulation reaches steady state. Specifically, the pink curve represents the current of the input-side voltage source, the blue curve denotes its output voltage, the orange curve indicates the current through the receiving-side load, and the green curve corresponds to the voltage across the receiving-side load.

The input current amplitude differs significantly among the three models: approximately 0.3 A in Model I (no enclosure), 0.4 A in Model II (solid enclosure), and 1.5 A in Model III (slotted enclosure). Although the system in Model II approaches a short-circuit condition due to ineffective coupling, the input current does not increase substantially because the eddy currents induced in the metal enclosure introduce additional equivalent impedance—both resistive and inductive components—that collectively limit current rise. In contrast, the slotted enclosure in Model III suppresses eddy current paths, reduces reflected impedance, and restores resonance conditions. As a result, the input current increases significantly, indicating improved energy transfer rather than increased loss. Therefore, current amplitude alone is not a definitive indicator of system performance; a more reasonable assessment should be based on input impedance magnitude or transmission efficiency.

As indicated by Equation (11), the incorporation of the explosion-proof enclosure increases the real part and reduces the imaginary part of the equivalent input impedance of the system, leading to a rightward shift in the resonance point. Consequently, under the 100 kHz excitation, an input-side compensation imbalance occurs in both Model II and Model III. In Figure 10c (slotted enclosure model), a phase difference between U_S and I_S is clearly present. This occurs because the introduction of the slotted enclosure, with its equivalent impedance as expressed in Equation (11), causes slight detuning of the system. As a result, the input impedance Z_in is no longer purely resistive, and the imaginary part X_in becomes non-zero, introducing a genuine phase shift. Regarding the nature of the phase shift—whether lagging or leading—it can be determined from the expression Z_in = R_in + jX_in. The phase angle θ is given by θ = arctan (X_in/R_in), the impedance is inductive, and the current lags the voltage; if the impedance is capacitive, and the current leads the voltage. For Model III, Equation (11) reveals that X_in < 0, indicating that the system exhibits capacitive detuning. Thus, the input current leads the input voltage, which is consistent with the behavior observed in Figure 10c.

For the purpose of observing the field distribution, the time instant of 150 μs is selected. Upon the integration of the explosion-proof enclosure into Model I, eddy currents are predominantly generated and concentrated on the working surfaces of the transmitting-side as well as the receiving-side explosion-proof enclosures within Model II. As is evident from the current density distribution on the surfaces of the transmitting coil and the receiving coil of Model II, as depicted in Figure 11, subsequent to the introduction of the explosion-proof enclosure, the receiving coil scarcely manages to induce any current. This effectively validates the obstructive impact of eddy currents on wireless power transmission. The same conclusion can also be corroborated by the behaviors of I_R and U_R illustrated in Figure 10b.

The current density distribution and energy distribution on the working surfaces of the transmitting-side and receiving-side explosion-proof enclosures of Model II are, respectively, presented in Figure 12 and Figure 13. It can be discerned therefrom that the majority of the energy and current are centered around the transmitting-side explosion-proof enclosure of Model II. Consequently, it becomes arduous for the receiving side to capture energy, and both the near-field imaging and inductive coupling of the system are adversely affected.

Upon making slits on the working surfaces of both the transmitting and receiving sides of Model II and subsequently installing explosion-proof tempered glass to isolate explosive gases, the current density distribution on the surfaces of the transmitting coil and the receiving coil of Model III is presented as shown in Figure 14. Given that this distribution is observed at the time instant of 150 μs, in combination with Figure 10c and upon verification of the exported data, it is ascertained that the induced current on the receiving coil is marginally greater than that on the transmitting coil. Through calculation, the transmission efficiency of the system of Model III can attain 62%, recovering 90% of the transmission efficiency relative to that of Model I. Moreover, by leveraging the field calculator, it can be determined that the ohmic losses of the explosion-proof enclosures on the transmitting and receiving sides of Model III amount to 2.552 J and 2.578 J, respectively, fulfilling the operational prerequisites in underground coal mines.

To further demonstrate thermal safety compliance with the GB 3836.1-2021 standard [36], which specifies a maximum surface temperature limit of 150 °C, a conservative first-order thermal estimation was performed based on the simulation results and fundamental heat transfer principles. The ohmic loss of 2.552 J in the transmitter enclosure was measured over 220 μs (22 cycles at 100 kHz), representing energy dissipated during the system’s start-up transient. The average power loss during this period is calculated as P_{loss_avg} = 2.552 J/220 × 10⁻⁶ s ≈ 11.6 kW. However, for thermal design, the steady-state power loss is of primary concern. In a well-tuned resonant WPT system, the majority of power is delivered to the load. The output power of Model III is P_{out_avg} = 57.38 W, with an efficiency of 62%. Thus, the total system loss is P_{total_loss} = P_{out_avg} × 0.62 − P_{out_avg} ≈ 21.8 W. Assuming conservatively that the enclosure accounts for 20% of the total loss, the steady-state power loss for the enclosure is P_{loss_ss} = 0.20 × 21.8 W ≈ 4.36 W. The steady-state temperature rise is estimated using ΔT = P_{loss_ss} × Rth, where Rth is the thermal resistance. Using a conservative value of Rth = 5.0 °C/W for natural convection, the temperature rise is ΔT = 4.36 W × 5.0 °C/W= 21.8 °C. With more realistic parameters—a thermal resistance of Rth = 1.0 °C/W for a large metal enclosure and an enclosure loss contribution of 5%—the steady-state power loss is reduced to P_{loss_ss} = 0.05 × 21.8 W ≈ 1.1 W, resulting in a temperature rise of ΔT = 1.1 W × 1.0 °C/W = 1.1 °C. Considering a maximum ambient temperature of 40 °C in underground coal mines, the worst-case and realistic surface temperatures are estimated to be 61.8 °C and 41.1 °C, respectively. Both values are significantly below the 150 °C safety limit stipulated by GB 3836.1-2021, confirming that the design maintains safe operating temperatures under all realistic conditions.

Consequently, the mine MCR-WPT system with a slotted explosion-proof enclosure proposed in this study not only complies with the safety production requirements in underground coal mines but also efficaciously addresses the shielding issue caused by the eddy current effect on the energy transmission coupling path, thereby laying a solid theoretical foundation for the further advancement of mine MCR-WPT technology.

The simulation results validate the feasibility of the model. However, only phased progress has been made so far. Furthermore, to promote in-depth research, future efforts will be concentrated on the optimal design of explosion-proof enclosures with diverse slotting forms, the analysis of coupling mechanisms featuring different coil forms, the exploration of the optimal load-efficiency matching, the investigation of the impact of different reactive power compensation methods on transmission efficiency, as well as the enhancement and construction of experimental setups. It should be particularly noted that this paper undertakes research by adopting the simplest model with classical two-coil WPT, and its sole purpose is to verify the feasibility of the technical route. Consequently, the overall efficiency of the system remains to be improved. This also offers extensive space and a clear direction for subsequent research. It is believed that through continuous exploration and innovation, more remarkable achievements can be attained in this field, thereby making greater contributions to the development and application of mining MCR-WPT technology.

4. Conclusions

In summary, this paper has proposed a novel low eddy current mining MCR-WPT system architecture incorporating slotted explosion-proof enclosures. The simulation results comprehensively demonstrate the model’s validity and practical feasibility, confirming its capability to significantly mitigate the adverse effects of eddy currents on magnetic coupling and transmission efficiency. By applying slot-cutting technology to the working surfaces of the stainless-steel enclosures, the system (Model III) achieves a transmission efficiency of 62%, recovering approximately 90% of the performance of the baseline system without enclosures (Model I). Moreover, the ohmic losses remain low, measuring 2.552 J and 2.578 J for the transmitter and receiver enclosures, respectively. Critically, a first-order thermal analysis confirms that the resulting temperature rise in the enclosure remains below 50 °C, which is well within the 150 °C limit prescribed by the Chinese National Standard GB 3836.1-2021 [36] thereby ensuring operational safety under mining conditions.

The core innovation of this work lies not only in the physical slotting technique but, more importantly, in the development of a high-fidelity equivalent circuit model that fully incorporates all mutual inductance components and accurately represents eddy current effects in non-ferromagnetic enclosures. This model provides a foundational theoretical framework for analyzing and optimizing such systems. The comprehensive simulation study, conducted under realistic mine environmental constraints, establishes an essential and reliable foundation prior to any physical experimentation—a necessary step given the high risks and ethical requirements associated with explosive atmospheres. It offers critical insights into system behavior and risk assessment, paving the way for safe future prototyping.

Looking forward, building upon this rigorously validated simulation groundwork, future work will focus on the thermal optimization of slot configurations, experimental validation under methane atmosphere, and robustness testing against misalignment conditions. We are committed to transitioning this simulation-based study into practical deployment, aiming to develop a fully compliant and efficient WPT solution for underground intelligent mines.