The influence mechanism of Halbach array magnetic field control on discharge channels in EDM

Zhiwei Qiu; Jiajing Tang; Zhengkai Li; Mulong Yin; Weiye Peng

doi:10.1088/1402-4896/ad37af

1. Introduction

In Electrical Discharge Machining (EDM), high-frequency pulsed power supplies are used to break the interpolar insulation and form a plasma channel of 8000 °C–12 000 °C between the tool and the workpiece to melt and evaporate the material [1, 2]. During each single pulse discharge, the electrons emitted from the cathode collide with neutral particles in the interpole insulating medium on their way to the anode due to an extremely high-pressure electric field at the discharge gap. At the same time, the charged particles with positive and negative charges moving at high speed collide in the discharge gap, releasing heat, most of the materials melt due to the heat transfer of the high-temperature plasma, a very small number of materials evaporate directly, and the insulating medium between the poles washes away the molten materials. The result is the formation of a discharge crater. After the discharge is over, the plasma disappears. The deionization process takes place, and the temperature in the crater drops, waiting for the next discharge erosion [2, 3]. Due to the non-contact force of EDM, complex geometries can be machined, which has the advantages of high machining accuracy and high stability. Therefore, EDM is widely used in many high-end manufacturing industries, such as molding, automotive, aerospace, instrumentation, etc [4]. However, as the requirements for processing equipment in these high-end manufacturing industries continue to increase, the EDM industry must improve its processing performance. Therefore, how to improve the performance of EDM has always been a hot topic of research. Since the magnetic field can exert the Lorentz force on charged particles moving in the discharge channel, thus affecting the processing performance, and is easy to set and adjust [5, 6], a great deal of research has been done on the active control of the magnetic field in EDM.

In theoretical and simulation studies, Heinz [7] analyzed the influence mechanism of magnetic field on plasma binding and plasma stability improvement when magnetic field acts on non-magnetic materials processed by fine EDM, and proved the mechanism of Lorentz force on material erosion. Beravala [8] used the air and magnetic field composite method to assist EDM to improve the material removal efficiency, and analyzed the effects of plasma expansion, electron free path reduction, and liquid-air mixing media in detail, and the results proved that the prediction model has high accuracy. Manesh [9] established a two-dimensional magnetic field-assisted EDM single-pulse material ablution model, and found that the Lorentz force under the magnetic field is conducive to the erosion of the material through the analysis of simulation and experimental results. Wang Yan [10, 11] found that the electron diffusion decreased by 5% under the action of the magnetic field by establishing a multi-electron motion model. Mansoor et al [12] verified the evolution of the plasma channel using a finite element-based electrothermal model and found that the Lorentz force generated by the electromagnetic field prevented the radial expansion of the discharge plasma channel during processing, thereby reducing electrostatic repulsion and improving energy efficiency.

In an experimental study, Heinz et al [13] applied a magnetic field parallel to the surface of the workpiece to investigate the mechanical effect of the Lorentz force at the melt pool on the non-magnetic material, which was found to increase material removal. Shabgard et al [14] revealed that the MF-EDM process resulted in a decrease in the radius of the discharge crater on the surface of the workpiece, and an increase in depth and volume. In addition, the magnetic field set near the electrode gap improves the plasma flush efficiency (PFE) and reduces the thickness of the recast layer. The magnetic field significantly increases MRR, lowers SR, and reduces surface cracks. Yeo et al [15] found that magnetic field-assisted micro-EDM achieved more effective cleaning of debris in micropores. Govindan et al [16] investigated the effect of magnetic induction intensity on processing depth. The results show that the depth of the processing crater increases with the increase of magnetic induction intensity. Lin and Lee [17] investigated the processing characteristics of magnetic-assisted EDM. They selected peak currents and pulse durations to determine the effects of applied magnetic fields on MRR, TWR, and SR during EDM. They also used scanning electron microscopy (SEM) to study the magnetic-assisted EDM process. Their results show that this method can improve processing performance. Teimouri and Baseri [18] investigated the effects of applied magnetic fields on MRR, TWR, SR, and gouge in a dry EDM process. They also determine the optimal tool material and geometry by considering MRR. Their results showed that with this approach, MRR increased and SR decreased. Gholipoor et al [19] applied a rotating magnetic field to the machining area during near-dry EDM to study the changes in processing performance and quality indexes in the new environment. Studies have shown that the presence of a rotating magnetic field enhances the debris scouring effect between the discharge gaps and reduces the number of abnormal discharge pulses. The discharge waveform is more stable and efficient. Geeta [20] applied a magnetic field to EDM metal mixing, and the results showed that the magnetic field could improve the processing efficiency, increase the wear of the electrode, reduce the surface roughness, and enhance the micro-hardness of the surface of the material at low current. Joshi et al [21] introduced a hybrid EDM process assisted by a pulsating magnetic field and demonstrated that a resultant force is generated due to the action of the magnetic field and the Lorentz force [22], thereby increasing the energy density and narrowing the path of the average degrees of freedom electrons.

The above results confirm that the use of active magnetic field control can effectively improve the process characteristics of EDM. However, there are still some deficiencies in these literatures, including (1) the inductance lines of general magnetic fields are too dispersed, and the experiment is not easy to operate. (2) The mechanism of action of permanent magnets on discharge channels is not clear, and (3) the effect of ion channel shape changes on treatment efficiency has not been studied. In view of the above defects, this paper introduces the Halbach array magnetic field into the field of active magnetic field adjustment of electrical discharge machining for the first time, and the Halbach array is a excellent type of magnet structure, which can gather the magnetic field lines on one side of the magnet and weaken the magnetic field lines on the other side by arranging the permanent magnets in different magnetization directions according to certain rules, so as to obtain a relatively ideal unilateral magnetic field. The underlying theory was proposed by Mallison in 1973 [23],and a design for an array was first described by Halbach in 1980 [24]. In addition, the magnetic inductance lines of the Halbach array magnetic field are more dispersed than those of ordinary magnetic fields, making it easier to operate when actively controlling the magnetic field. Halbach arrays have been used in a wide range of magnet systems, from industrial applications such as magnetic bearings [25] and brushless AC motors [26] to high-tech applications such as nuclear magnetic resonance (NMR) equipment [27] and magnetic levitation (magnetic levitation) systems [28]. In this paper, a Halbach dual-array structure was selected from many Halbach arrays for study.

In this paper, a physical model of magnetic field-actively controlled EDM is established, and the advantages of Halbach array magnetic field in magnetic induction line distribution and magnetic induction intensity are modeled and analyzed. The trajectory model of charged particles in the electrical spark discharge channel under the active control of the Halbach array magnetic field was established, and the motion trajectories of different charged particles in the discharge channel under the action of the Halbach array magnetic field were discussed, and the theoretical analysis was carried out according to the simulation results. Finally, a control group with or without active control of magnetic field was established to control the distance between the workpiece and the working surface of the Halbach array permanent magnet, and the effect of the active control of the Halbach array magnetic field on the EDM efficiency and surface quality was studied through experiments.

2. Physical model of EDM actively controlled by magnetic field

The working fluid inevitably contains some impurities, which makes the working fluid show some electrical conductivity. When the voltage between the electrode and the workpiece increases to about 100V/μm, the electrons will escape from the surface of the cathode, and the electrons will move to the anode at high speed under the action of the electric field, hitting the molecules or neutral atoms in the working fluid, resulting in collision ionization, and the impact will form negatively charged particles and positively charged particles, resulting in an avalanche increase of charged particles, which will decompose the medium and form a discharge channel. The discharge channel is a plasma composed of positively charged particles, negatively charged particles and neutral particles, the charged particles move at high speed and collide violently, the kinetic energy is converted into heat energy through the collision, so a lot of heat will be generated in the discharge gap, so the temperature of the discharge channel is very high, and the temperature in the center of the channel can be as high as 10 000 °C or more. In the discharge channel, the moving charged particles produce an excited electromagnetic field, which is partly the charged particle-induced field, which is closely related to the velocity of the charged particles and the distance from the charged particles. The other part is the charged particle radiation field, which is related to the charged particle velocity, the charged particle acceleration, and the distance from the charged particle. In the process of the formation of the discharge channel, the excitation electromagnetic field produces a centripetal magnetic compression effect on the discharge channel, and the discharge channel is also affected by the inertial dynamic compression effect of the surrounding medium.

Magnetic field control refers to the application of a magnetic field in the discharge gap, which can apply the Lorentz force to the moving charged particles, thus changing the trajectory of the motion.As shown in figure 1(a), in the absence of active control of the magnetic field, the excited electromagnetic field generated by the movement of charged particles uniformly surrounds the discharge channel, and as the discharge channel expands around it, the axial symmetry of this electromagnetic field with respect to the discharge channel results in the same Lorentz force in all directions, and eventually an axisymmetric discharge channel is formed. As shown in figure 1(b), when there is active control of the magnetic field, the actively applied magnetic field will cancel out a part of the excited electromagnetic field, so that the magnetic field around the discharge channel is no longer axisymmetric with respect to the discharge channel, and when the discharge channel expands, the magnitude of the Lorentz force in each direction is different, resulting in a non-axisymmetric discharge channel. According to the calculation formula of Lorentz force, the magnetic induction intensity of the actively controlled magnetic field is proportional to the Lorentz force on the charged particles, so the greater the magnetic induction intensity of the actively controlled magnetic field, the more obvious the influence on the cross-sectional area and shape of the channel, and the change of the cross-section of the discharge channel can not only change the collision frequency between the charged particles in the discharge channel and the particles in the working fluid, thereby changing the energy of a single spark discharge, but also change the energy distribution on the cross-section of the discharge path.

**Figure 1.** (a) Physical model of non-magnetic field-controlled EDM (b) Physical model of magnetic field-controlled EDM.
Download figure:
Standard image High-resolution image

Based on the analysis of the physical model of active controlled EDM, this paper introduces the Halbach array magnetic field into the field of active control EDM for the first time, when the same material and the same volume of permanent magnets are used, the array can obtain greater magnetic induction intensity, and the magnetic induction lines of the array are more dispersed, and it is easier to adjust the discharge channel. Under the active control of the Halbach array magnetic field, the cross-sectional area of the discharge channel becomes larger, which increases the probability of the charged particles in the discharge channel colliding with the molecules or neutral atoms in the working fluid, resulting in more electrons being ionized out of the dielectric, making the discharge channel more energetic and ultimately affecting the efficiency and surface quality of the EDM.

3. Modeling and analysis of halbach array permanent magnets

The Halbach array permanent magnets in this paper are composed of a NdFeB, b NdFeB, stainless steel, and iron in a certain order. Among them, a NdFeB and b NdFeB are named to distinguish different sizes. In order to concentrate the magnetic field more in the middle part of the permanent magnet array, we improved it based on the literature [29] and obtained the magnetic field array used in this paper. Among them, the length ratio of a NdFeB, b NdFeB, stainless steel and iron on the action surface needs to meet 26:15:10:12. In this paper, the Halbach single-array permanent magnet and the Halbach dual-array permanent magnet are modeled and compared with the ordinary single-array permanent magnet and the ordinary dual-array permanent magnet, respectively. Simulations were performed using the Magnetic Fields, No Currents module of COMSOL Multiphysics 6.0.

3.1. Modeling and analysis of Halbach single-array permanent magnets

3.1.1. Modeling of Halbach single-array permanent magnets

As shown in figure 2, the Halbach single-array permanent magnet and the ordinary single-array permanent magnet were modeled respectively, with the modeling dimensions shown in table 1 and the modeling parameters shown in table 2. In the figure, the direction of the arrow is the N-pole direction of NdFeB, and the overall modeling length of the permanent magnet is 114 mm. The line segment 1 in the figure is located on the S pole side of b NdFeB, and the line segment 2 is located on the N pole side of b NdFeB, and the two line segments are used to compare and illustrate the magnetic induction intensity of the upper and lower sides of the two array permanent magnets, and the vertical distance between line segment 1 and line segment 2 from the surface of the permanent magnet is equal, both of which are 10mm. The magnetization of NdFeB is defined as 400 kA m⁻¹ in the N-pole direction.

Table 1. Permanent magnet modeling parameters.

Name	Dimensions (L × W)
a NdFeB	26 × 15 mm
b NdFeB	15 × 15 mm
Stainless stell	12 × 15 mm
Iron	10 × 15 mm

Table 2. Material properties.

Material	Relative permeability	Electrical conductivity	Relative permittivity
Iron	4000	10	14.2
Stainless stell	1.01	5	300
Deionized water	1	1 × 10⁻⁴	81.5

3.1.2. Inductance line distribution of Halbach single-array permanent magnets

Figure 3 shows the magnetic induction line density of a Halbach single-array permanent magnet versus a common single-array permanent magnet. The direction of the blue arrow in the figure indicates the direction of the magnetic field, and the magnitude of the blue arrow indicates the magnitude of the magnetic induction. As can be seen from the simulation results in figure 3, the magnetic inductance lines of the Halbach single array permanent magnet are more dispersed, so it is easier to operate during active control, and the magnetic inductance lines of the Halbach single array permanent magnet are no longer symmetrical as those of ordinary single array permanent magnets, because the axes between the two adjacent permanent magnets are perpendicular and the magnetic inductance lines are opposed to each other.

3.1.3. Magnetic induction intensity analysis of Halbach single-array permanent magnets

Figure 4 is obtained by selecting line segments 1 and line 2 in figure 2 using the one-dimensional drawing group in the software results post-processing, respectively. Figure 4 shows the magnetic induction intensity of a Halbach single-array permanent magnet versus a common single-array permanent magnet. As can be seen from the simulation results in figure 4, the peak magnetic induction intensity of the ordinary single-array permanent magnet is 0.06827T near the left and right ends of the entire permanent magnet. The simulation results also show that the magnetic induction intensity on the upper and lower sides of the Halbach single array permanent magnet presents an obvious asymmetrical distribution, with one side of line segment 1 being the obvious magnetic field enhancement side and one side of line segment 2 being the magnetic field weakening side. The peak value of the magnetic field enhancement side is 0.1265T, which is much larger than the maximum magnetic induction intensity of the ordinary single-array permanent magnet of 0.06827T, an increase of 85.29%, and the peak value of the magnetic field weakening side is also as high as 0.0872T, which is still 25.56% larger than the peak of the magnetic induction intensity of the ordinary single-array permanent magnet. Whether it is measured by the magnetic field enhancement side or the magnetic field weakening side, the peak magnetic induction intensity of the Halbach single array permanent magnet is located in the middle of the whole permanent magnet, and the part of 95mm on the magnetic induction intensity enhancement side is larger than the peak value of the ordinary single array permanent magnet, accounting for about 83.33% of the whole Halbach single array permanent magnet.

3.2. Modeling and analysis of halbach dual-array permanent magnets

3.2.1. Modeling of Halbach dual-array permanent magnets

As shown in figure 5, the Halbach dual-array permanent magnet and the ordinary dual-array permanent magnet were modeled respectively, the modeling dimensions are shown in table 1, and the modeling parameters are shown in table 2. The arrow direction is the N-pole direction of NdFeB, and the overall modeling length of the permanent magnet is 193 mm, and other definition conditions are the same as those of the Halbach single-array permanent magnet modeling.

3.2.2. Inductance line distribution of Halbach dual-array permanent magnets

Figure 6 shows the magnetic inductance line density of a Halbach dual-array permanent magnet versus a common dual-array permanent magnet. The direction of the blue arrow in the figure indicates the direction of the magnetic field, and the magnitude of the blue arrow indicates the magnitude of the magnetic induction.As can be seen from figure 6, the magnetic inductance lines of the ordinary dual-array permanent magnets are symmetric with respect to the x-axis and y-axis, while the magnetic inductance distribution of the Halbach dual-array permanent magnets is only symmetrical with respect to the y-axis, and the magnetic inductance lines of the Halbach dual-array permanent magnets are more dispersed than those of the ordinary dual-array permanent magnets.

3.2.3. Magnetic induction intensity analysis of Halbach dual-array permanent magnets

Figure 7 is obtained by selecting line segments 1 and line 2 in figure 5 using the one-dimensional drawing group in the software results post-processing, respectively. Figure 7 shows the magnetic induction intensity of Halbach dual-array permanent magnets versus ordinary dual-array permanent magnets. As can be seen from the simulation results in figure 7, the peak value of the magnetic induction intensity of the ordinary dual-array permanent magnet is relatively dispersed, and the peak value of the magnetic induction intensity is 0.0814T. The simulation results also show that the magnetic induction intensity on the upper and lower sides of the Halbach dual-array permanent magnet presents an obvious asymmetrical distribution, with one side of line segment 1 being the obvious magnetic field enhancing side and one side of line segment 2 being the magnetic field weakening side. The peak magnetic induction intensity of the magnetic field enhancement side of the Halbach dual-array permanent magnet is 0.1356T, which is much larger than the peak magnetic induction intensity of the ordinary dual-array permanent magnet 0.0814T, which is increased by 66.58%. On the enhanced magnetic induction intensity side of the Halbach dual-array permanent magnet, the 160mm part is larger than the peak value of the ordinary dual-array permanent magnet, accounting for about 82.9% of the entire permanent magnet model.

Through the above analysis, it can be found that about 83.33% of the enhanced magnetic induction intensity of Halbach single-array permanent magnets is larger than the peak value of ordinary single-array permanent magnets. About 82.9% of the magnetic induction intensity enhancement side of the Halbach dual-array permanent magnet is larger than the peak value of the ordinary dual-array permanent magnet. The peak value of the enhanced magnetic induction intensity of the Halbach single-array permanent magnet is 0.1265T, which is 85.29% greater than that of the ordinary single-array permanent magnet, and the peak value of the enhanced magnetic induction intensity of the Halbach dual-array permanent magnet is 0.1356T, which is 66.58% larger than that of the ordinary single-array permanent magnet. Although the space utilization rate of Halbach dual-array permanent magnets is about 0.43% lower than that of Halbach single-array permanent magnets, the peak magnetic induction intensity of the Halbach dual-array permanent magnet field enhancement side is 0.1356T, and the peak magnetic induction intensity of the Halbach single-array permanent magnet magnetic field enhancement side is 0.1265T. According to the analysis of the physical model of actively-controlled magnetic field EDM in section 2, the greater the magnetic induction intensity, the more obvious the effect of the magnetic field on the discharge channel, therefore, the Halbach dual-array permanent magnet was selected for the study, and the Halbach dual-array permanent magnet was used for modeling and experiments in the subsequent chapters.

4. Modeling and simulation of the formation process of the single-pulse discharge channel with active control of the magnetic field

4.1. Force analysis of a single charged particle in a discharge channel

The spark single pulse discharge erosion mainly includes four processes: the ionization breakdown of the working liquid (medium) between the poles to form a discharge channel; Thermal decomposition of working fluid, melting of electrode materials, thermal expansion of gasification; electrode material throwing; Deionization of interpolar working fluids [30]. The magnetic field mainly affects the charged particles in the discharge channel, thus affecting the size of the discharge channel, which in turn affects the thermal erosion process. Therefore, it is necessary to analyze the effect of the magnetic field on the discharge channel of EDM. To further analyze the trajectories of charged particles, this section analyzes the force diagram of a single charged particle under the action of electric and magnetic fields, as shown in figure 8. The direction of the electric field is shown by the black arrows in the diagram, and the external electric field force remains constant and its direction is perpendicular to the surface of the workpiece. The direction of the actively controlled magnetic field is parallel to the surface of the workpiece, and the perpendicular paper side is outward. In order to simplify the analysis, the following assumptions must be made before the analysis: (1) The electric and magnetic fields are uniform and constant. (2) The initial velocity of the electron is zero, and the injection direction is perpendicular to the electrode surface. (3) A single spark discharge escapes N electrons from the cathode and M monovalent iron ions from the anode.

**Figure 8.** Force analysis of individual electrons and individual monovalent iron ions in the discharge gap.
Download figure:
Standard image High-resolution image

Then the formula for the force on a single charged particle is expressed as:

$\begin{eqnarray}&&{\bf{F}}={{\bf{F}}}_{{\bf{E}}}+{{\bf{F}}}_{B}+\displaystyle \sum _{{\rm{n}}=1}^{N}{{\bf{F}}}_{{\bf{e}}{\bf{n}}}+\displaystyle \sum _{{\rm{n}}=1}^{N}{{\bf{F}}}_{{\bf{F}}{{\bf{e}}}_{{\bf{n}}}^{{\boldsymbol{+}}}}\end{eqnarray} \tag{ 1 }$

$\begin{eqnarray}&&{{\bf{F}}}_{{\bf{E}}}=q{\bf{E}}\end{eqnarray} \tag{ 2 }$

$\begin{eqnarray}&&{{\bf{F}}}_{B}=q{\bf{v}}\cdot {\bf{B}}\end{eqnarray} \tag{ 3 }$

$\begin{eqnarray}&&{{\bf{F}}}_{{\bf{e}}}=\displaystyle \frac{k{Q}_{e}{Q}_{e}}{{r}_{e}^{2}}\end{eqnarray} \tag{ 4 }$

$\begin{eqnarray}&&{{\bf{F}}}_{{\bf{F}}{{\bf{e}}}^{{\boldsymbol{+}}}}=\displaystyle \frac{k{Q}_{e}{Q}_{F{e}^{+}}}{{r}_{F{e}^{+}}^{2}}\end{eqnarray} \tag{ 5 }$

In above conservations equations, F is net force experienced by the charged particles, F _E is electric field force experienced by the charged particle, F _B is lorentz force experienced by charged particles, F _e coulomb force exerted by a single electron on a charged particle, F _{F

e} ⁺ is Coulomb force exerted by a single electron on a charged particle; E is electric field, v is velocity of the charged particle, B is magnetic induction, q is the amount of charge of a charged particle, r_e is the distance of the charged particle from the electron, r_Fe ⁺ is the distance of the charged particle from the monovalent iron ion, Q_e is the amount of charge carried by an electron(−1.6 × 10⁻¹⁹ C); Q_Fe ⁺ is the amount of charge carried by monovalent iron ions(1.6 × 10⁻¹⁹ C);k is electrostatic force constant(8.987551 × 10⁹ N·m²/C²).

4.2. Motion model of a charged particle beam in a single-pulse discharge channel under active control of a magnetic field

Figure 9 shows a trajectory model of a charged particle beam under the active control of a magnetic field, with the inlets of electrons and monovalent iron ions located on the top and bottom surfaces of the discharge gap, respectively. The working fluid is deionized water. The surface potential of the workpiece is 30 V, and the other surface potentials are 0 V. Suppose electrons are injected perpendicular to the electrode surface at an initial velocity of 0 m s⁻¹, and monovalent iron ions are injected perpendicular to the surface of the workpiece at an initial velocity of 300,000 m s⁻¹. The discharge gap is 0.03mm. In an actual EDM single pulse discharge, the electron inlet is emitted from a single point, but considering the convergence requirements of the simulation model, the electron inlet is defined as a circle with a diameter of 2 μm. The monovalent iron ion inlet is a circular shape with a diameter of 10 μm. Since the magnetic field formed by the Halbach dual-array permanent magnets is a non-uniform magnetic field, the closer the permanent magnets, the greater the magnetic induction intensity, and the magnetic induction intensity near the inlet of electron and monovalent iron ions is 0.6T.

The electric field distribution of the static electrons is spherically symmetrical and can be expressed as:

$\begin{eqnarray}&&{\bf{E}}=\displaystyle \frac{e{{\bf{r}}}_{{\bf{0}}}}{4\pi {\varepsilon }_{0}{r}_{0}^{2}}\end{eqnarray} \tag{ 6 }$

where ε₀ is the vacuum permittivity and r₀ is the distance from the electron.

The Lienard–Wiechert potential based on motion electronics:

$\begin{eqnarray}&&{\bf{A}}=\displaystyle \frac{{\mu }_{0}e{{\bf{v}}}_{{\bf{e}}}}{4\pi \left({r}_{0}-\displaystyle \frac{{{\bf{v}}}_{{\bf{e}}}\cdot {{\bf{r}}}_{{\bf{0}}}}{c}\right)}\end{eqnarray} \tag{ 7 }$

where c is the speed of light and μ₀ is the vacuum permeability.

The electromagnetic field excited by the moving electrons at the same time can be expressed as:

$\begin{eqnarray}&&{\bf{E}}=\displaystyle \frac{e}{4\pi {\varepsilon }_{0}{r}_{0}^{2}}\displaystyle \frac{\left(1-\displaystyle \frac{{v}_{e}^{2}}{{c}^{2}}\right)\left({{\bf{r}}}_{{\bf{0}}}-\displaystyle \frac{{{\bf{v}}}_{{\bf{e}}}}{c}\right)}{{\left(1-\displaystyle \frac{{{\bf{v}}}_{{\bf{e}}}\cdot {{\bf{r}}}_{{\bf{0}}}}{c}\right)}^{3}}+\displaystyle \frac{e}{4\pi {\varepsilon }_{0}{r}_{0}}\displaystyle \frac{{{\bf{r}}}_{{\bf{0}}}\times \left[\left({{\bf{r}}}_{{\bf{0}}}-\displaystyle \frac{{{\bf{v}}}_{{\bf{e}}}}{c}\right)\times \displaystyle \frac{d{{\bf{v}}}_{{\bf{e}}}}{dt}\right]}{{\left(1-\displaystyle \frac{{{\bf{v}}}_{{\bf{e}}}\cdot {{\bf{r}}}_{{\bf{0}}}}{c}\right)}^{3}}\end{eqnarray} \tag{ 8 }$

$\begin{eqnarray}&&{\bf{B}}=\displaystyle \frac{1}{c}{{\bf{r}}}_{{\bf{0}}}\times {\bf{E}}\end{eqnarray} \tag{ 9 }$

According to equations (8) and (9), the electromagnetic field excited by the electron beam is composed of two parts: one is the radiation field, which depends on the velocity of the electron, the distance of the electron, and the acceleration of the electron, and the other is the electron-induced field, which depends on the velocity and distance of the electron. In the actual machining process, the electron has a high velocity and acceleration, the excited electric field is not spherically symmetrical, the electric field lines along the direction of the electron motion are sparse, and the electric field lines perpendicular to the direction of the electron motion are dense.

The electric field in the simulation model consists of two parts: an external electric field and an electric field excited by an electron beam moving at high speed. The entire electric field obeys Gauss's law:

$\begin{eqnarray}&&-{\rm{\nabla }}\cdot \left({\varepsilon }_{0}{\rm{\nabla }}{\rm{V}}-{\rm P}\right)={\rho }_{v}\end{eqnarray} \tag{ 10 }$

where P is the polarization vector and ρ_v is the space charge density.

Since the velocity of an electron is much smaller than the speed of light, a simplified Lagrangian equation can be obtained:

$\begin{eqnarray}&&L=\displaystyle \frac{{m}_{p}{v}^{2}}{2}+qA\cdot v-qV\end{eqnarray} \tag{ 11 }$

where m_p is the mass of the particle, v is the velocity matrix of the particle, q is the charge of the particle, A is the magnetic vector potential matrix, and V is the potential.

Differentially transform the above equation to obtain:

$\begin{eqnarray}&&\displaystyle \frac{d}{dt}\left({m}_{p}v\right)=-q\displaystyle \frac{\partial A}{\partial t}-q{\rm{\nabla }}V+q\left(v\times {\rm{\nabla }}\times A\right)\end{eqnarray} \tag{ 12 }$

When the processing medium used in EDM is deionized water, the collision between electrons, the collision between electrons and water molecules, and the collision between electrons and monovalent iron ions will change the trajectory of electrons and monovalent iron ions, and these collisions follow Monte Carlo law, and the types of collisions include elastic collisions, neutralization collisions and ionization collisions, the expressions are as follows:

$\begin{eqnarray*}&&e+e\to e+e\,\left({\rm{Elastic}}\,{\rm{collisions}}\right)\end{eqnarray*}$

$\begin{eqnarray*}&&F{e}^{+}+F{e}^{+}\to F{e}^{+}+F{e}^{+}\,\left({\rm{Elastic}}\,{\rm{collisions}}\right)\end{eqnarray*}$

$\begin{eqnarray*}&&e+{H}_{2}O\to ne+{H}_{2}{O}^{+n-1}\,\left({\rm{Ionization}}\,{\rm{collisions}}\right)\end{eqnarray*}$

$\begin{eqnarray*}&&e+F{e}^{+}\to Fe\,\left({\rm{Neutralize}}\,{\rm{collisions}}\right)\end{eqnarray*}$

In an elastic collision, the elastic collision obeys the law of conservation of momentum, so the kinetic energy of the electrons remains almost unchanged after the collision, and the scattering angle of the electrons after the collision can be expressed as:

$\begin{eqnarray}&&\cos \left(\chi \right)=\displaystyle \frac{2+\varepsilon -2{\left(1+\varepsilon \right)}^{R}}{\varepsilon }\end{eqnarray} \tag{ 13 }$

where R is a random number between 0~1.

In the ionization collision, the electrons escaping from the electrode surface move to the workpiece at high speed under the action of the electric field and hit the water molecules, resulting in collision ionization, and the formation of more electrons and positively charged particles, resulting in an avalanche of electrons, so that the deionized water is broken down and a discharge channel is formed. In the neutralization collision, the electrons are absorbed by the monovalent iron ions, the monovalent iron ions become iron atoms, and the kinetic energy is also absorbed by the monovalent iron ions.

As shown in figure 10, the force analysis of charged particles in the discharge gap under the action of the Halbach array magnetic field, the direction of the Halbach array magnetic field is parallel to the working surface, the external electric field remains unchanged, and its direction is perpendicular to the working surface, the charged particles will move in the z-axis direction due to the electric field force, according to the left-hand rule, it can be clearly seen in the top view that the charged particles are subjected to the Lorentz force in the y-axis direction under the action of the Halbach array magnetic field. The Lorentz forces experienced by charged particles are:

$\begin{eqnarray}&&{{\bf{F}}}_{{{\bf{B}}}_{{\bf{e}}}}={q}_{e}\cdot {{\bf{v}}}_{{\bf{e}}}\cdot {\bf{B}}\end{eqnarray} \tag{ 14 }$

$\begin{eqnarray}&&{{\bf{F}}}_{{{\bf{B}}}_{{{\bf{F}}}_{{\bf{e}}}^{{\boldsymbol{+}}}}}={q}_{{F}_{e}^{+}}\cdot {{\bf{v}}}_{{{\bf{F}}}_{{\bf{e}}}^{{\boldsymbol{+}}}}\cdot {\bf{B}}\end{eqnarray} \tag{ 15 }$

Where q_e is the amount of charge carried by an electron(−1.6 × 10¹⁹ C); q_Fe+ is the amount of charge carried by monovalent iron ions(1.6 × 10⁻¹⁹ C);v_e is the speed of motion of electrons; v_Fe ⁺ is the velocity of movement of monovalent iron ions; B is the strength of the magnetic field.

**Figure 10.** Force analysis of charged particles in the discharge gap under the action of a Halbach array magnetic field.
Download figure:
Standard image High-resolution image

4.3. Simulation of the motion of charged particle beams in the discharge gap under the active control of the magnetic field

Due to the limited computing power, the amount of electrons and monovalent iron ions simulated was 7000, and the simulation period was 320ps. The distribution of water molecules is defined in terms of material density, and meshing selects physics-controlled conventional partitioning. This simple model of motion trajectory is qualitatively described from the point of view of electron bombardment. Table 3 shows the simulation parameter settings. In the following simulation, the direction of observation of the resulting plot is shown in figure 11, figure 12 shows the simulation results of the electron beam motion with or without active control by a magnetic field.

Table 3. Charged particle beam motion simulation parameter settings.

Parameter	Symbol	Unit	Numeric value
Discharge voltage	U	V	30
Electronic inlet diameter	D_e	$\mu m$	2
Electron initial velocity	v	$\mu m$	0
Monovalent iron ion inlet diameter	D_Fe	m/s	10

**Figure 11.** Schematic diagram of the direction of observation of charged particle beams.
Download figure:
Standard image High-resolution image

As can be seen from figure 12, when there is no magnetic field assistance, the electron beam reaches the workpiece and eventually forms a circle that does not deviate from the emission circle, while when the magnetic field assistance is applied, the electron beam becomes an elliptical shape that deviates from the emission circle. This happens for the following reasons: In the absence of magnetic field assistance, due to the continuous flow of electrons from the emission circle into the discharge gap, there is a Coulomb repulsion force between the electrons and electrons, and there is also an elastic collision between the electrons, which makes the electron beam expand uniformly outward, and finally form a regular circle.

Figure 13 shows a schematic diagram of the Lorentz force applied to an ion channel by a Halbach dual-array permanent magnet, with the green dot being the electron beam inlet, the direction of incidence perpendicular to the paper downward, and the red ellipse being the elliptical electron beam. The black arrow is the Lorentz force exerted on the discharge channel in region 1, the red arrow is the Lorentz force exerted on the discharge channel in region 2, and the orange arrow is the net force exerted on the discharge channel. The direction of the blue arrow in the figure indicates the direction of the magnetic field, and the magnitude of the blue arrow indicates the magnitude of the magnetic induction.

When there is active control of the magnetic field, two regions are delineated and named 1 and 2 with the positions shown in the figure, and the direction of the Lorentz force to the electron can be judged according to the direction of the magnetic induction line shown in the simulation results and the left-hand rule, so as to obtain the two components shown by the arrows in the figure. The Lorentz force generated in region 1 causes the electrons to move in the direction shown in the diagram, but as the electrons move farther away from the permanent magnet, the Lorentz force exerted on the electrons within region 1 becomes smaller and smaller. The Lorentz force generated in region 2 causes the electrons to move in the direction shown in the diagram and the electrons get closer and closer to the permanent magnet, so the Lorentz force exerted on the electrons in region 2 is increasing. Due to the inhomogeneity of the force, the final shape of the electron beam is elliptical. As can be seen from the diagram, the horizontal resultant force of the electron beam is directed downward to the right, so the centroid of the resulting electron beam deviates downward from the emission circle.

Figure 14 shows a plot of the electron beam cross-sectional area with and without active magnetic field control. As can be seen from the graph in figure 14, the cross-sectional area of the electron beam expands dramatically within 240 ps and tends to stabilize between 240 ps and 320 ps, regardless of whether the magnetic field is actively regulated. The results at the end of the simulation show that the cross-sectional area of the electron beam under the active control of the magnetic field is ultimately larger than that of the electron beam without the active control of the magnetic field. At the end of the simulation, the cross-sectional area of the electron beam without active control of the magnetic field is 143.89 μm², and the cross-sectional area of the electron beam under the active control of the magnetic field is 153.81 μm², and the simulation results show that the cross-sectional area of the electron beam increases by about 6.89% under the active control of the magnetic field of the 0.6T Halbach array.

Figure 15 shows the simulation results of the monovalent iron ion beam motion with or without active control of the magnetic field. As can be seen from figure 15, the active control of the magnetic field of the Halbach array has no significant effect on the shape of the monovalent iron ion beam, which is different from the effect of the magnetic field on the electron beam, mainly because the mass of the monovalent iron ion is much greater than the mass of the electron, and the same electric field acting on the monovalent iron ion can only produce a small velocity, and according to the previous equation (3), it can be concluded that the Lorentz force on the monovalent iron ion is much smaller than the Lorentz force on the electron. As a result, there was no significant change in the shape of the monovalent iron ion beam.

Figure 16 shows the cross-sectional area of the monovalent iron ion beam with and without active magnetic field control. As can be seen in figure 16, the cross-sectional area of the monovalent iron ion beam increases significantly with or without active magnetic field control. Due to the small Lorentz force on the monovalent iron ions, at the end of the simulation, there was no significant change in the cross-sectional area of the monovalent iron ion beam in the absence of active magnetic field control and in the presence of active magnetic field control.

According to the simulation results of the charged particles in the discharge channel using the Halbach array magnetic field, the cross-sectional shape of the electron beam changes greatly, from a circle to an eccentric ellipse, and the cross-sectional area of the electron beam increases by 6.89% under the active control of the 0.6T Halbach array magnetic field. However, there was no significant change in the cross-sectional shape and size of the monovalent iron beam. Therefore, the magnetic field mainly affects the EDM efficiency by controlling the electrons in the discharge channel.

5. Magnetic field active control processing experiment

5.1. Test method and design

5.1.1. Experimental methods.

Based on the DM71 (DM71 China Taizhou Changde Co., Ltd) precision EDM machine tool, the permanent magnet bracket was designed to place the Halbach dual-array permanent magnet and build the test platform. The experiment was processed by immersion liquid positive polarity. A schematic diagram of the test setup is shown in figure 17. The Halbach dual-array permanent magnets consist of multiple small NdFeB permanent magnets, stainless steel, and iron combined, and the adhesion sequence is shown in figure 5(a), and the Halbach dual-array permanent magnet part dimensions are shown in table 4. The grade of NdFeB is N35 and the surface magnetization is 0.5T.The grades of other experimental materials are shown in table 4.

**Figure 17.** Schematic diagram of the experimental setup.
Download figure:
Standard image High-resolution image

Table 4. Dimensions of each part of the Halbach dual-array permanent magnet.

Name	Dimensions (L×W×H)	Grade
a NdFeB	6.5 × 40 × 20 mm	N35
b NdFeB	3.75 × 40 × 20 mm	N35
Stainless steel	3 × 40 × 20 mm	0Cr18Ni9
Iron	2.5 × 40 × 20 mm	#45

The physical diagram of the Halbach dual-array permanent magnet is shown in figure 18(a). The green plane marked by the yellow arrow in the figure is the working surface of the Halbach dual-array permanent magnet, which is also the enhanced side of the magnetic induction intensity analyzed in the third part of this paper. For the sake of simplification, the following are collectively referred to as working surfaces. The processing photos are shown in figure 18(b) and figure 18(c).The tool electrode material is made of copper tungsten alloy electrode with low loss, with a diameter of Φ 5mm with a density of 13.8 × 10⁻³g/mm2. The workpiece material used in the experiment is 45 # steel with a cubic shape and a size of 10mm × 10 mm × 10 mm, with a density of 7.8 × 10⁻³ g mm⁻². Deionized water was chosen as the insulating working liquid. The working materials are shown in table 5.

Table 5. Working materials.

Working materials	Grade
Tool electrode	Copper-tungsten alloy
Workpiece material	#45
Insulating working liquid	Deionized water

5.1.2. Design of experiments

5.1.2.1. Determination of the range of variables

Direct measurement of magnetic field strength using the TD8620 digital Tesla meter. When the distance from the working face is far away, the magnetic induction intensity is measured with a Tesla meter, and the magnetic induction intensity is almost zero. However, the magnetic induction intensity of ordinary dual array permanent magnets still remains at a relatively large value. For this abnormal phenomenon, a 1:1 modeling was conducted based on the actual Halbach dual array permanent magnet to determine the optimal operating distance for the magnetic induction strength of the Halbach dual array permanent magnet. The modeling results are shown in figure 19(a). The black plane marked by the black arrow in the figure represents the working surface. The direction of the red arrow in the figure represents the magnetic induction intensity detection line. Figure 19(b) shows the variation curve of the magnetic induction intensity from the working surface in this detection direction. From figure 19(b), it can be seen that the magnetic induction intensity is high when the distance from the working face is relatively close, and the maximum magnetic induction intensity can reach 0.43T. As it gradually moves away from the working face by 10mm, the magnetic induction intensity drops sharply. Then, there was a slight increase in magnetic induction intensity within a distance of 10–15 mm, and after a distance of 15mm, the magnetic induction intensity gradually decreased, eventually decaying to 0T at approximately 40 mm.

5.1.2.2. Calculation method

According to the simulation results in figure 19 and the actual measurements, the magnetic induction intensity of the array magnetic field changes significantly within 10mm from the surface of the permanent magnet, and then the magnetic induction intensity is slightly improved in a short distance, so the distance between the workpiece and the surface of the permanent magnet is 0mm, 5mm, 10mm, and 15mm. The magnetic induction intensity is changed and controlled by changing the position of the permanent magnet bracket from the surface of the workpiece. Since the parameter level 1 condition is no magnetic field, only four positions need to be measured for magnetic field strength. The measurement position of the magnetic induction intensity is shown in figure 20. The measurement results are shown in table 6.

**Figure 20.** The measurement position of the magnetic induction intensity.
Download figure:
Standard image High-resolution image

Table 6. Experimental conditions for processing.

Parameter level	0	1	2	3	4
The distance between the workpiece and the working face	No magnetic field	0mm	5mm	10mm	15mm
Measured magnetic induction intensity	0T	0.43	0.1T	0.01T	0.02T

The single-factor test method was used to control the distance between the workpiece and the working surface of the Halbach dual-array permanent magnet, and the machining effect before and after the magnetic field was actively controlled. Calculating the material removal rate and electrode loss rate. The processing test conditions are shown in table 6. Other processing conditions: discharge voltage 100V, peak current 9A, pulse width 400 μs, pulse interval 30 μs.

The formula for calculating the material removal rate (MRR) is shown in equation (16):

$\begin{eqnarray}&&MRR=\displaystyle \frac{{m}_{1}-{m}_{2}}{{\rho }_{1}{t}_{1}}\end{eqnarray} \tag{ 16 }$

Where m₁ is the quality of the workpiece before machining (g),m₂ is the quality of the workpiece after machining (g),ρ₁ is the density of the workpiece (g/mm³),t₁ is processing time (min);

The formula for calculating the tool wear ratio(TWR) is shown in equation (17):

$\begin{eqnarray}&&TWR=\displaystyle \frac{{m}_{3}-{m}_{4}}{{\rho }_{2}{t}_{2}}\end{eqnarray} \tag{ 17 }$

Where m₃ is the quality of the workpiece before machining (g),M₄ is the quality of the workpiece after machining (g),ρ₂ is the density of the workpiece (g/mm³),t₂ is processing time (min);

To avoid chances, 3 workpieces were machined for each parameter, and the material removal rate (MRR) and tool wear ratio(TWR) were calculated and averaged. The TR200 roughness gauge is used to measure the surface roughness value of the machined workpiece, measure the 3 positions of the surface, and take the average value.

5.2. Results and discussion

5.2.1. The effect of the distance between the workpiece and the working face on the material removal rate

As shown in figure 21, when the distance between the workpiece and the working surface is 0mm, the material removal rate increases the most, increasing by about 18.18%. At 10mm and later, there was no significant change in the material rejection rate. When there is no discharge channel formed between the electrode and the workpiece, under the active control of an external magnetic field, the excitation temperature of electrons decreases with the increase of magnetic induction intensity [31]. As the magnetic induction intensity increases, the energy required for electrons to escape from the electrode surface decreases, allowing more single discharges to occur in a multi pulse discharge. During the formation of the discharge channel between the tool electrode and the workpiece, due to the external magnetic field, the ion channel is subjected to the Lorentz force. As shown in the analysis results in figure 13, the ion channel is dragged by the Lorentz force, and the cross-sectional area of the final discharge channel is larger than that without the application of a magnetic field. This increases the number of ionization collisions between electrons and molecules and neutral atoms in the insulating working fluid, leading to an increase in the degree of ionization of the insulating working fluid and thus an increase in the current in the discharge channel [32]. Ultimately, it increases the energy of the discharge channel, leading to an increase in the material's erosion rate. According to the research results of ZAIDI et al [33], under different magnetic fields, the Lorentz force increases the velocity of electrons and their degrees of freedom of motion. Under a 0.3T magnetic field, the velocity of electrons can increase by nearly twice, indicating that active control of the magnetic field enables electrons to have a greater velocity of motion. According to the kinetic energy equation, a higher electron velocity will give electrons greater energy. The above reasons will lead to an increase in material erosion rate after applying magnetic field active control.

5.2.2. The effect of the distance between the workpiece and the working surface on the tool wear ratio

As shown in figure 22, the tool wear ratio is significantly reduced when the distance between the workpiece and the working surface is 0 mm, and the tool wear ratio is reduced by about 46.87%. When the distance between the workpiece and the working surface is 10mm or more, the magnetic induction intensity is too small, and the tool wear ratio is not significantly reduced. Under the active control of the magnetic field, the ablation debris can be magnetized by the magnetic field, and the ablation debris is subjected to a magnetizing force parallel to the processing surface under the action of the magnetic field, and the ablation debris can better discharge the discharge gap. The discharge environment between the two poles is improved, and the number of abnormal discharges is reduced, thereby reducing the electrode loss rate. After the distance between the workpiece and the working surface is 10mm, the magnetic induction intensity is too small, and the magnetizing force of the etched debris is too small, so the reduction of the tool wear ratio is not obvious.

5.2.3. The effect of the distance between the workpiece and the working surface on the surface roughness

Figure 23 shows the trend of the influence of the distance between the workpiece and the working surface on surface roughness, as well as the corresponding electron microscope photos at the corresponding positions. From the results, it can be observed that the machining surface is very rough without active magnetic field control, with a surface roughness of up to 4.186 μm. As shown in figure 23(a), a clear phenomenon of melting and re solidification can be observed in the electron microscope image, and the depth of the discharge craters is very deep. When the distance between the workpiece and the working surface is 0mm, the surface roughness rapidly decreases to 2.53 μm, as shown in figure 23(b). The discharge craters are large and shallow, the machining surface is flat, and the phenomenon of melting and re solidification is significantly reduced. When the distance between the workpiece and the working surface is greater than or equal to 5mm, there is no significant change in the surface roughness value. The surface roughness values in figure 23(c), figure 23(d), and figure 23(e) are 4.025μm, 4.123 μm and 4.061 μm. This indicates that when the distance between the workpiece and the working surface is greater than 5mm, the influence of the Halbach array magnetic field on surface roughness is very small. At the same time, no shallow depth of the discharge crater and a reduction in melt re-curing phenomena were observed from figure 23(c), figure 23(d) and figure 23(c). When the distance between the workpiece and the working surface is 0mm, the Halbach array magnetic field will affect the surface roughness from two aspects. Firstly, the Halbach array magnetic field will affect the cross-sectional area of the discharge channel, as shown in figure 13. After the magnetic field controls the electrical discharge machining, the cross-sectional area of the discharge channel is larger, indicating a more uniform energy distribution and less energy per unit area. Therefore, a single discharge will generate shallower discharge craters. Electrical discharge machining is the result of multiple single spark craters stacked together. When the depth of the craters becomes shallower during each spark discharge, the final surface roughness of the machined surface is lower. Secondly, the Halbach array magnetic field can affect the movement of debris. Without active control of the magnetic field, during the electric spark discharge stage, eroded electrode fragments and workpiece material fragments will be thrown out due to explosive forces. The two types of fragments collide in the discharge gap, and under the combined effect of collision and explosion forces, the fragments will eventually be discharged in a disordered trajectory from the discharge gap. Due to the fact that the workpiece material used in this paper is a magnetic conductive material, when the Halbach array magnetic field actively controls electrical discharge machining, the debris generated by the workpiece is magnetized, and the debris is subjected to additional magnetic force parallel to the machining surface, which accelerates the movement speed of the debris parallel to the workpiece surface. At the same time, the movement direction of the debris is relatively fixed, and a large amount of debris will move in a fixed direction. The directional movement of the debris plays a role in magnetic grinding and precision machining, and the surface of the workpiece can be polished. When the distance between the workpiece and the working surface is greater than or equal to 5mm, the magnetic induction intensity of the Halbach array magnetic field is too small, making the influence of the Halbach array magnetic field on the discharge channel too small, and the cross-sectional area does not increase significantly. Moreover, a too small magnetic induction intensity results in a low degree of magnetization of the debris, a low degree of directional movement, a slow rate of debris increase, poor magnetic grinding and precision machining effects, and no significant reduction in surface roughness.

From figure 24, it can be observed that when the distance between the workpiece and the working surface is 0mm, the Halbach array magnetic field actively controls the reduction of surface roughness in electrical discharge machining, which is 39.56%. When the distance between the workpiece and the working surface is greater than or equal to 5mm, there is no significant decrease in surface roughness. As analyzed in figure 19 in this paper, the magnetic induction intensity of the Halbach array magnetic field drops sharply from 0.43 T to 0.01 T within 10 mm of the working surface, and to 0.1 T at 5 mm. The magnetic induction intensity is too small at a distance of 5mm or more from the working surface, resulting in the little influence of the Halbach array magnetic field on the discharge channel and debris, and finally the surface roughness is not significantly reduced.

Through the experimental study of Halbach dual-array permanent magnet actively controlled EDM, it is found that when the working surface of Halbach dual-array permanent magnet is less than 5mm away from the workpiece, it has obvious effects on the material abolition rate, electrode loss rate and surface roughness. When the distance between the workpiece and the working surface is 5–10mm, it only has a great impact on the material abolition rate and electrode loss rate. When the distance between the workpiece and the working surface is more than 10mm, there is no obvious effect on the material abolition rate, electrode loss rate and surface roughness. When the workpiece is 0mm away from the working surface, the material removal rate is increased by 18.18%, the tool wear rate is reduced by 46.87%, and the surface roughness is reduced by 39.56%.

6. Conclusions

Through the above theoretical, simulation and experimental studies on the active control of EDM by the Halbach array magnetic field, the following conclusions can be obtained.

(1)
Through the modeling and analysis of single and dual Halbach array permanent magnets and ordinary array permanent magnets, it is found that the magnetic inductance lines of Halbach array permanent magnets are more dispersed than those of ordinary array permanent magnets, and the magnetic inductance intensity of the upper and lower sides of Halbach array permanent magnets is not equal, and the peak magnetic inductance intensity of the magnetic field enhancement side of Halbach single array permanent magnet is 85.29% larger than that of ordinary single array permanent magnets. The peak value of magnetic induction intensity on the enhanced side of Halbach dual-array permanent magnet is 66.58% larger than that of ordinary dual-array permanent magnet.
(2)
By modeling and analyzing the Halbach dual-array permanent magnets and Halbach single-array permanent magnets, it is found that the space utilization rate of the Halbach dual-array permanent magnets is reduced by 0.43%, and the peak magnetic induction intensity of the magnetic field enhancement side is increased by 7.19%.
(3)
According to the simulation results of the effect of Halbach dual-array permanent magnets on the charged particles in the discharge channel, the cross-section of the electron beam changes from a circle to an off-center oval, and the cross-sectional area of the electron beam increases by 6.89% under the active control of the Halbach array magnetic field of 0.6T. However, there was no significant change in the cross-sectional shape of the monovalent iron ion beam. Therefore, the active control of the magnetic field mainly affects the processing efficiency by affecting the electrons in the discharge channel.
(4)
When the distance between the workpiece and the working surface of the Halbach dual-array permanent magnet is less than 10mm, the material ablution rate of the workpiece increases significantly, and the electrode loss rate and surface roughness are significantly reduced. When the distance between the workpiece and the working surface is 0mm, the material removal rate increases by up to 18.18%, the electrode loss rate decreases by up to 46.87%, and the surface roughness decreases by up to 39.56%.
(5)
Magnet active control EDM can increase the energy of the discharge channel.
(6)
The magnetic field actively controlled EDM can promote the discharge of the erosion debris in the discharge gap, play the role of magnetic grinding and finishing, reduce the tool wear ratio and surface roughness, and improve the processing quality.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 52005308 and 52105464), China Postdoctoral Science Foundation (No. 2022M711091), Postdoctoral Innovation Project of Shandong Province (No. SDCX-ZG-202202001), Science and Technology Support Plan for Youth Innovation of Colleges and Universities of Shandong Province of China (No. 2023KJ332).

Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

The influence mechanism of Halbach array magnetic field control on discharge channels in EDM

Article metrics

Submit

Permissions

Author e-mails

Author affiliations

Author notes

ORCID iDs

Dates

Abstract