Title: The importance of coil conductivity and eddy current effects in the analysis of electromagnetic forming process
Abstract: High VoltageVolume 7, Issue 2 p. 390-404 ORIGINAL RESEARCH PAPEROpen Access The importance of coil conductivity and eddy current effects in the analysis of electromagnetic forming process Quanliang Cao, Quanliang Cao orcid.org/0000-0003-3691-2311 Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorLiangyu Xia, Liangyu Xia Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorXian Li, Xian Li Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorLimeng Du, Limeng Du Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorZhipeng Lai, Zhipeng Lai Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorXiaotao Han, Corresponding Author Xiaotao Han [email protected] orcid.org/0000-0002-7089-9598 Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, China Correspondence Xiaotao Han and Liang Li, Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, 430074, China. Email: [email protected] and [email protected] for more papers by this authorLiang Li, Corresponding Author Liang Li [email protected] Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, China Correspondence Xiaotao Han and Liang Li, Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, 430074, China. Email: [email protected] and [email protected] for more papers by this author Quanliang Cao, Quanliang Cao orcid.org/0000-0003-3691-2311 Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorLiangyu Xia, Liangyu Xia Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorXian Li, Xian Li Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorLimeng Du, Limeng Du Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorZhipeng Lai, Zhipeng Lai Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, ChinaSearch for more papers by this authorXiaotao Han, Corresponding Author Xiaotao Han [email protected] orcid.org/0000-0002-7089-9598 Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, China Correspondence Xiaotao Han and Liang Li, Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, 430074, China. Email: [email protected] and [email protected] for more papers by this authorLiang Li, Corresponding Author Liang Li [email protected] Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, China State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan, China Correspondence Xiaotao Han and Liang Li, Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, 430074, China. Email: [email protected] and [email protected] for more papers by this author First published: 26 May 2021 https://doi.org/10.1049/hve2.12109Citations: 6 Associate Editor: Jian Wu AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Coil design and performance analysis are critical for the electromagnetic forming (EMF) process. However, previous studies have paid little attention to the coil conductivity, and the influence of eddy current effects in coil conductors on the forming process has not been well understood. This work aims to address these problems through numerical and experimental investigations. It is found that, due to the eddy current effects, the uneven current distribution appears in the coil conductors that should be considered when analysing the forming process, especially for the cases with large heights of coil conductors. Meanwhile, an important and interesting discovery is that the eddy current effects can greatly reduce the sensitivity of the coil conductivity with respect to the workpiece deformation, the Joule heating and the temperature rise in the coil. For instance, compared to a forming system with a copper coil, the forming height of the workpiece in the forming system with an AA5083-O coil is reduced by less than 5 % under the same discharge energy, while the copper conductivity is 3.3 times that of the latter. These results are helpful to understand the EMF process and achieve the optimal design of coils. 1 INTRODUCTION Electromagnetic forming (EMF) is a high-speed forming technology that uses Lorentz forces to accomplish plastic forming of metallic workpieces. Compared with quasi-static forming processes, the EMF process has great potential in improving the forming limits, suppressing springback and wrinkling [1-3]. Therefore, EMF has been widely applied for the forming of lightweight and highly conductive workpieces, such as tube forming [4-7] and sheet metal forming [8-11]. Furthermore, the EMF has inspired other Lorentz-force-based forming methods such as electromagnetically assisted forming and driven forming [12]. For EMF and related forming processes, the coils play a vital role because they generate the pulsed magnetic fields for the required Lorentz forces. They also determine the distribution characteristics of the Lorentz forces. Therefore, the coil design is always an important topic in the field of EMF [1, 13, 14]. Until now, various coil configurations have been developed that are tailored for different forming requirements. For instance, to achieve sheet metal forming with uniform Lorentz forces, Oliveira et al. [15] developed a double-spiral coil with the same direction of current flowing. The coil structure solves the problem that the Lorentz forces acting on the centre region of the workpiece are much smaller when using conventional single spiral coil. Kamal and Daehn [16] developed a uniform pressure actuator, which consists of an inner coil and an outer conductive channel. Wu et al. [17] further developed a new type of uniform pressure actuator by changing the position of the coil and the conductive channel. They demonstrated that the new actuator can improve both forming efficiency and operating life. To achieve the special shape of tubes, Li et al. [18] developed a coil structure with variable wire spacing. They showed that this approach can be used to improve the forming accuracy for EMF of variable-diameter tubes. Qiu et al. [19] developed a multi-layer concave coil and showed that it can be used to improve the homogeneity of tube bulging. Besides, it has been demonstrated that the introduction of a field shaper in electromagnetic tube forming systems can be used to alter Lorentz force distributions in the tube, which provides a way to concentrate the Lorentz forces in a specific region of the tube [20], and to achieve a uniform-pressure tube forming [21]. To achieve better control of the Lorentz force distribution in the workpiece, multi-coil and multi-power-supply forming systems were used. Lai et al. [22] developed a dual-coil EMF system for deep drawing of sheet metal with a drawing ratio of 3.25. Two coils were used to generate the axial and radial Lorentz forces, respectively, which were needed for both axial deformation and radial material flow. Lai et al. [23] further clarified the plastic deformation and flow behaviour of sheet metal during the double-coil deep drawing process. Li et al. [24] developed a dual-coil EMF system and showed that it can be used to improve the tube forming capability due to a background magnetic field. Ouyang et al. [25] showed that the dual-coil EMF system can be further used to generate considerable attractive Lorentz forces and achieve attractive forming of tubes. In addition to the above mentioned topics, the mechanical and thermal design of the coils is also important in terms of service life and efficiency. Typically, high electromechanical stresses caused by large currents will lead to the occurrence of mechanical failure in the coils [26]. To prevent coil damage during the EMF process, additional high-strength materials are required to improve the mechanical strength of the coil. For instance, Seth et al. [2] embedded a flat spiral coil within an epoxy resin with a Kevlar face sheet. Similarly, Golovashchenko et al. [27] embedded a flat spiral coil within an insulating block made of Micarta material, and added a fitted steel bandage around the coil to prevent its expansion. Later, Qiu et al. [28] introduced the pulse magnet reinforcement method to greatly enhance the strength of a multi-layer multi-turn coil, where each turn of the coil was reinforced by Zylon fibres. In terms of the coil thermal design, Gies et al. [29] pointed out the importance of exploring the Joule heating and temperature characteristics of the forming coil in the EMF process. They also studied the effect of multiple factors such as discharge energy, cycle time and workpiece material on the coil temperature. To improve the coil’s temperature-increase problem during the EMF process, Cao et al. [30] developed a discharge circuit configuration with a crowbar circuit and showed that it can be used to control the current waveform. It was demonstrated that the circuit configuration can also be used to reduce Joule heating in the coil without affecting the forming efficiency. Qiu et al. [31] proposed a half-wave-current EMF method, which cuts off the current after the first half-wave to reduce Joule heating and the temperature rise in the coil. Du et al. [32] proposed a technically simple and cost-effective method to control the current waveform by adjusting the damping coefficient of the discharge circuit, and demonstrated that it can greatly decrease the Joule heating in the coil. In short, there have been many studies focussing on both coil function and service life. Currently, copper is used by default as the conductor material for coils because it has good electric conductivity that is beneficial to improving the forming efficiency. In fact, from the perspective of enhancing the strength of the coil, it is beneficial to use a higher strength conductor material. However, as stated by Gies and Tekkaya [33], an increased mechanical strength of the conductor material often leads to reduced electric conductivity. Then the question arises whether metallic materials with low conductivity are suitable for coils in EMF. To the best of our knowledge, there is no report discussing the influence of the conductivity of the conductor material on the forming process—especially the workpiece deformation and thermal characteristics of the coil. Before this can be clarified, the influence of the eddy current effects in the coils on the forming process should be explored. This is important because it could affect the coil impedance and the current distribution [34], and it is related to the coil material’s conductivity. In this work, both simulations and experiments are carried out to solve these problems. In the following sections, the numerical model, the experimental setup, the corresponding numerical analysis and experiments are described in detail. 2 NUMERICAL MODEL AND EXPERIMENTAL SETUP 2.1 Multi-physics numerical modelling EMF involves a complex multi-physics coupling process. Previous studies mainly focussed on the effect of workpiece deformation on the coupling between the electromagnetic field and mechanical field. However, the influence of eddy current effects in the coils is also an important issue that is related to multiple physical quantities and has not attracted much attention. Eddy current effects could change the current distribution in the coil conductors and alter the impedance of the coil, which in turn affects the characteristics of the discharge current. In addition, the uneven current distribution in the coil conductors could inevitably affect the eddy currents induced in the workpiece and magnetic field distribution, which will further affect the generated Lorentz forces and workpiece deformation. A circuit-electromagnetic-mechanical-thermal coupled numerical model of EMF with considerable eddy current effects is shown in Figure 1a. Five sub-models were adopted and solved using the COMSOL Multiphysics Software package (V5.4): (1) The ‘global ordinary differential equations and differential algebraic equations’ model. (2) The ‘magnetic fields’ sub-model. (3) The ‘solid mechanics’ sub-model. (4) The ‘heat transfer in solids’ sub-model. (5) The ‘moving mesh’ sub-model. The first four sub-models are used for circuit analysis, electromagnetic analysis, workpiece deformation analysis and thermal analysis, respectively. The fifth sub-model is based on an arbitrary Lagrangian-Eulerian (ALE) method, and it is used to update the air element mesh shape in the area around the workpiece as the workpiece deforms. Based on the coupled model, once the structural parameters and discharge parameters of the forming system are determined, the discharge current in the coil, the Lorentz force acting on the workpiece, the plastic deformation of the workpiece and the temperature of the coil can be obtained. In addition, a numerical procedure, which does not consider the eddy current effects in the coil, was also developed—see Figure 1b. In this procedure, a uniform current density was loaded for electromagnetic analysis. Because the differences in electromagnetic and mechanical calculations between the two models in Figure 1 are mainly investigated in the subsequent sections, the thermal sub-model is not included in Figure 1b. The used discharge current in the coil in Figure 1b is consistent with that in Figure 1a, which can be realized by reading the current data from Figure 1a and importing them into Figure 1b for processing. Note that, the conductivity of the coil conductors in Figure 1b needs to be set to zero to completely avoid the influence of the eddy currents in the coil, while the actual conductivity of these materials should be provided in Figure 1a to consider the eddy current effects. FIGURE 1Open in figure viewer Schematic flowchart of two types of numerical models used in the simulations: (a) The coupled circuit-electromagnetic-mechanical-thermal numerical model; (b) The coupled electromagnetic-mechanical model with an initial loading of uniform current density The schematic cross-section of the EMF system used in this work is shown in Figure 2. Because the coil is concentric and coaxial with the sheet metal and die, a 2D axisymmetric model is used in the simulations—see Figure 3. It mainly consists of the domains of coil conductors, air and sheet metal. Because the sheet metal is free bulging, and the material flow in the flange region is negligible, the interaction between the sheet metal and the die is not considered in the simulations. Therefore, there is no need to build a model of the die, and only fixed constraints are used for the upper and lower surfaces of the sheet metal in the flange region. Details of the parameters in the simulations are shown in Table 1. The corresponding equations and descriptions of these physical models are as follows. FIGURE 2Open in figure viewer Schematic cross-section of the electromagnetic sheet metal forming system FIGURE 3Open in figure viewer Modelling and meshing of the electromagnetic forming system TABLE 1. System parameters used in the simulations Components Parameters Value Power supply and circuit line Capacitance ( C ) 150 μF Line resistance ( R 1 ) 17 mΩ Line inductance ( L 1 ) 2.3 μH Coil (copper) Inner radius of coil 15 mm Outer radius of coil 60 mm Height of coil (conductor region) 20 mm Coil turns 8 Turn-to-turn distance 3 mm Specific heat capacity 390 J/(kg°C) Mass density 8960 kg/m3 Electrical conductivity 5.56 × 107 S/m Coil (AA5083-O) Inner radius of coil 15 mm Outer radius of coil 60 mm Height of coil (conductor region) 20 mm Coil turns 8 Turn-to-turn distance 3 mm Specific heat capacity 880 J/(kg°C) Mass density 2700 kg/m3 Electrical conductivity 1.68 × 107 S/m Workpiece (AA5052-O) Radius of workpiece 100 mm Thickness of workpiece 1 mm Distance between coil and workpiece 2.5 mm Electrical conductivity 2.03 × 107 S/m Young modulus 68 GPa Poisson ratio 0.33 Mass density 2740 kg/m3 2.1.1 Electric circuit equations The used EMF system mainly consists of a capacitor bank, an air-pressure switch, a coil, a workpiece and a die. Its equivalent circuit model is shown in Figure 4. When the switch is closed, the capacitor bank begins to energise the coil, and a pulsed current is generated. The pulsed current flowing through the coil can cause electromagnetic coupling between the coil and the nearby workpiece according to the principle of electromagnetic induction. This also produces thermal and mechanical effects in these two components. FIGURE 4Open in figure viewer The equivalent circuit model of the electromagnetic forming system The equivalent circuit equations are used to calculate the discharge current, which can be written as [30, 35]. ( R 1 I L + L 1 d I L d t ) + V c o i l = U c (1) U c + 1 C ∫ 0 t I L d t = U 0 (2) V c o i l = R c I L + L c d I L d t + M c − w d I w d t (3)where R 1 is the line resistance, I L is the discharge current, L 1 is the line inductance, V c o i l is the voltage of the coil, U c is the voltage of the capacitor bank, C is the capacitance of the capacitor bank, U 0 is the initial voltage of the capacitor bank, R c is the coil resistance, L c is the self-inductance of the coil, M c − w is the mutual inductance between the coil and the workpiece, I w is the eddy current in the workpiece. 2.1.2 Electromagnetic field equations The electromagnetic field equations are mainly used to predict the coil current distribution, the eddy current distribution in the workpiece and the Lorentz force acting on the workpiece. They can be expressed as follows: ∇ × ( 1 μ ∇ × A ) = J − σ c ∂ A ∂ t + σ c v × ( ∇ × A ) (4) B = ∇ × A (5)where A is the magnetic vector potential, μ is the medium permeability, J is the used external current density, σ c is the medium conductivity, and v is the medium speed. The third term on the right side of Equation (4) is used to reflect the motional electromotive force caused by the workpiece motion. The prerequisite for the above electromagnetic calculation is that the coil current is known, while the current calculation in Equations (1) and (2) needs a known coil voltage. Therefore, the following equations are required: I L = ∫ S J k d S (6) V coil _ k = 2 π r ( J k σ c + ∂ A ∂ t ) (7) V coil = ∑ k = 1 n V coil _ k (8)where J k is the current density of the k-turn conductor of the coil, S is the cross-section of each coil turn and V coil _ k is the voltage of the k-turn coil. Note that, when performing simulations without considering the eddy current effects in the coil conductors, the current density in the coil conductors is described by I L / S and it remains the same in all conductors, where S is the cross-section area of each coil turn. In the ‘magnetic fields’ model, a single-conductor model using the current I L as input is adopted, and a boundary condition of magnetic insulation is applied along the outer boundary of the air domain in Figure 3. 2.1.3 Mechanical field equations The mechanical field equations are used to predict the workpiece deformation under the action of Lorentz forces. The main equation is described by ρ d ∂ 2 u ∂ t 2 − ∇ × σ = f m (9)where ρ d is the density of workpiece, u is the displacement vector of workpiece, σ is the dynamic flow stress and f m is the Lorentz force density. The Lorentz force density can be obtained using the following equations: J e = − σ c ∂ A ∂ t + σ c v × B (10) f m = J e × B (11)where J e is the eddy current in the workpiece, A and B are obtained by solving the above electromagnetic field equations. The v in Equations (4) and (10) can be obtained by the derivative of u in Equation (9) with respect to time. In the ‘solid mechanics’ model, a fixed constraint is applied along the surfaces of the workpiece between the blank holder and the die, and other regions of the workpiece can be deformed freely according to the calculated workpiece displacement. 2.1.4 Thermal field equations The thermal field equations are used to predict the temperature rise of coil conductors during the EMF process. They are: Q k = ρ d C c ∂ T ∂ t + ∇ × ( − k T ∇ T ) (12) Q k = 1 σ c J k ⋅ J k (13)where Q k is the heat source that refers to the Joule heating of the k-turn conductor, C c is the heat capacity, T is the temperature of conductor, k T is the thermal conductivity. Note that in the subsequent simulations, because we mainly focus on the temperature rise in the coil conductors in a short period of time (within a few milliseconds), the adiabatic boundary condition is loaded around the coil conductors to simplify the calculations. In addition, the adiabatic boundary condition can eliminate the influence of the difference in heat conduction of materials on the temperature rise, so that the effect of Joule heating on the temperature rise can be better reflected in the simulations. 2.2 Experimental setup Two coils made of copper and 5083-O aluminium alloy, were used for comparative analysis. Their manufacturing process is shown in Figure 5. A copper block is processed into a spiral 8-turn coil with a cross-section of 3 × 20 mm and an inner diameter of 15 mm. The coil is nested within an epoxy groove to anchor it. A capacitor bank with a capacitance of 150 μF was used for the discharge. FIGURE 5Open in figure viewer The manufacturing process of the coil: (a) Exploded view of the coil assembly; (b) Three-fourths structure diagram of the coil; (c) Photograph of the fabricated copper coil; (d) Photograph of the fabricated AA5083-O coil The tooling diagram of the experimental setup is shown in Figure 6, including the coil, hydraulic press, Rogowski coil, oscilloscope, coaxial cable, and epoxy cushion blocks. The hydraulic press provides a pressure of about 2 MPa to constrain the material flow in the flange region of the sheet metal and to suppress wrinkles. An LCR metre (Hioki IM3536) and thermal infrared imager (Fluke Ti 25) were used to measure the coil impedance parameters and the surface temperature of the coil, respectively. According to our previous work [36], the conductive materials around the workpiece could affect electromagnetic calculations. Therefore, to avoid unnecessary complications in the simulations, the components including the die and the blank holder around the workpiece, were made of non-conductive materials. In addition, the coil and the press, as well as the workpiece and the press, were separated by non-conductive epoxy-resin blocks to eliminate the effect of metal parts within the press on the calculations. FIGURE 6Open in figure viewer Photograph of the experimental setup 3 RESULTS AND DISCUSSION 3.1 Influence of eddy current effects on current distribution Typically, the eddy current effects can be divided into skin effect and proximity effect [37, 38]. The skin effect means that the current tends to be distributed along the outer surface of the conductor when it is time varying. The proximity effect means that the current tends to be deflected to one side, when a time-varying current flows through adjacent conductors. During the EMF process, the eddy current effects of coils are unavoidable due to the presence of pulsed currents in the coil conductors and the surrounding conductive workpiece. It is to be noted that the eddy current effects are complex because they involve the skin effect within conductors and the proximity effect between nearby conductors as well as the proximity effect between the conductor and workpiece, and therefore the skin effect and proximity effect are not analysed separately in this work. To reveal the influence of eddy current effects on the current distribution in the coil conductors, the following two cases are analysed in this section: case A is performed with a non-deformed workpiece, which is done to represent the initial state of the forming process. Case B is performed without a workpiece to approximately represent the case that the workpiece is far away from the coil in the simulations. The copper coil is used and the discharge voltage is 6 kV. The other parameters are shown in Table 1. Figures 7 and 8 show the calculated characteristics of the current density in the coils for case A and case B, respectively. It can be clearly seen that the current in the coil conductors is unevenly distributed regardless of whether there is a workpiece. However, there is a clear difference in the current distribution between the two cases. In case A (see Figure 7a), the coil current tends to be concentrated at the bottom of the coil. This is caused mainly by the proximity effect between coil conductors and workpiece. During the discharge process, the generated pulsed magnetic field by the coil induces an eddy current in the workpiece with an opposite direction to the coil current. As a result, the current in the coil is mainly distributed near the workpiece under the action of the proximity effect. In addition, due to the presence of the other two eddy current effects, the current distribution in the centre plane of the coil is not uniform—see Figure 7b. In case B, due to the eddy current effects, the current density at the edge of the conductor is higher than that at the centre of the conductor—see Figure 8a and b. Unlike case A, the current density in the coil conductors in case B is symmetrically distributed (up and down) due to the absence of the proximity effect between the coil and workpiece—see Figure 8a and c. On the other hand, it is asymmetric in Figure 7a and c. A similar current density distribution along the conductor height in Figure 7c was also previously reported by Gies and Tekkaya [39], when they performed an electromagnetic analysis of the EMF coils. Note that, since the equivalent inductance of the EMF system in case A decreases due to the presence of the workpiece [13], the calculated current is larger than that in case B,—see the inserted maps in Figures 7 and 8. FIGURE 7Open in figure viewer Current distribution in case A (in the presence of a workpiece): (a) The surface current distribution diagram of the coils at the peak time of the fir
Publication Year: 2021
Publication Date: 2021-05-26
Language: en
Type: article
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