Investigation of the Effect of Coil Current Waveform on Electromagnetic Tube Forming

Abstract

The coil current frequency and waveform have a great impact on the forming performance of the workpiece in electromagnetic forming. However, existing research is mostly limited to analyzing the influence of either frequency or waveform on the forming outcome independently, which makes it challenging to fully reveal the intrinsic relationship between current parameters and forming results. In this work, three discharge circuit structures are developed to generate different coil currents composed of various frequencies and waveforms, and their effects on deformation of AA6061 Aluminium alloy tube are systematically investigated through numerical and experimental approaches. Results show that a conventional circuit can generate an attenuated oscillating sinusoidal waveform consisting of several pulse half-waves, while a circuit composed of a thyristor switch can generate a half-wave current, and a circuit consisting of a crowbar circuit can generate a current with a slow decay rate. Further, it is found that at a high-frequency discharge, a current having a slow decay rate is favorable for forming efficiency, as well as reducing coil temperature, while at a low-frequency discharge, the current waveform has almost no effect on the forming efficiency; thus, a half-wave current is highly recommended to significantly reduce the coil temperature. The obtained results are of great significance in guiding the design of coil currents and optimizing electromagnetic forming technology.

1. Introduction

Electromagnetic forming (EMF), as an advanced high-speed forming technology, has been widely applied in lightweight alloys and composite materials thanks to its high strain-rate effect and non-contact loading [1,2,3,4]. Research indicates that electromagnetic forming enhances the forming limit of lightweight alloys and reduces issues such as wrinkling and springback, especially when compared to traditional quasi-static forming techniques [5,6,7]. For example, Lin et al. [8] demonstrated that the forming limit of 2024-O aluminum alloy material treated by EMF can be increased by 36.9% compared to quasi-static stamping. Cui et al. [9] indicated that the EMF can be applied to springback calibration of U-shaped bending samples, and that higher discharge voltages result in lower springback.

In electromagnetic forming, the primary driving force for plastic deformation of the workpiece is the Lorentz force generated by the coil current. Thus, the size, frequency, and waveform of the coil current directly influence the distribution of the Lorentz force, which subsequently impacts the deformation behavior of the workpiece. Normally, the current generated in the coil is an attenuated oscillating sinusoidal waveform consisting of several pulse half-waves [10]. To alter the current in the coil, three methods are mainly applied as follows:

(1)

By adjusting the discharge voltage. This is a common method to change the current size. Usually, the greater the discharge voltage, the greater the amplitude of the current and the greater the Lorentz force produced. Paese et al. [11] discovered that the forming depth of sheet metal increases as discharge voltage rises, and similar results can be found in tube metal forming reported by Ouyang et al. [12]. It should be noted that changing the voltage can only change the current amplitude, and cannot change the current frequency or waveform.

(2)

By adjusting the discharge capacitance. This is the most widely used method to adjust the current frequency. Typically, the larger the capacitance, the smaller the frequency of the generated current. Based on this, Yu et al. [13] investigated the impact of current frequency on tube compression and found that a decrease in frequency results in a longer current period and reduced amplitude, leading to extended delay time and overall forming process. Further, there exists an optimum frequency to increase forming efficiency, which is located at a frequency where the relative skin depth is between 0.61 and 0.70. Dong et al. [14] explored how current frequency impacts the tube expansion, showing that the maximum expansion occurs at a frequency where the relative skin depth is below 1. They also found that the optimal frequency varies with the system’s resistance and inductance. Cao et al. [15] investigated the impact of current frequency on electromagnetic sheet forming and found that maximum deformation occurs at two optimal frequencies, corresponding to relative skin depths of 1 and 1.6. However, these works are mainly accomplished through finite element simulations.

(3)

By developing new discharge circuit structure. The above ways can only adjust the amplitude and frequency of the current, but cannot change the current waveform, where the current waveform is always distributed as an attenuated oscillating sinusoidal waveform consisting of several pulse half-waves. Cui et al. [16] demonstrated that the forming process of sheet is mainly done in the first current pulse, and Qiu et al. [17] indicated that the current after the first current pulse increases the thermal load on the coil. To solve this, several novel discharge circuit structures have been proposed in recent years. For instance, Cao et al. [18] proposed a crowbar circuit with a diode and resistor placed across the capacitor, allowing control of the current decay rate through adjustment of the crowbar resistance. This circuit has been proven to significantly reduce temperature rise without sacrificing forming efficiency. Deng et al. [19] discovered that a slow-rising and fast-falling current waveform could change the direction of the Lorentz force, enabling an attractive electromagnetic forming. Based on this, Xiong et al. [20] developed a double-circuit system to produce an approximate current waveform, thus realizing an attractive tube forming. Ouyang et al. [21] proposed a new circuit consisting of an additional inductive load and a crowbar circuit for attractive tube forming, where the inductive load can slow down the rise of the current, while the crowbar circuit can accelerate the decay of the current.

In summary, the coil current plays a pivotal role in determining the outcomes of the EMF process. However, existing studies often examine the effects of current frequency and waveform independently, making it challenging to fully reveal the complex physical mechanisms involved in the electromagnetic forming process. In this paper, tube expansion is taken as an example, and the influence of current frequency and waveform on the tube forming behavior and coil performance are systematically investigated through numerical and experimental methods. The new findings not only deepen the understanding of the mechanism of electromagnetic forming, but also provide new ideas for the optimization of process parameters.

2. Principle and Method

2.1. Principle

Figure 1 illustrates the schematic of electromagnetic tube bugling, where a coil is placed inside the tube. Upon discharging the capacitor, a pulsed current is generated in the coil, producing a magnetic field and inducing eddy current in the tube. The interaction between the magnetic field and the eddy current generates a Lorentz force, causing the tube to undergo rapid plastic deformation at high speed.

In this paper, the effect of current waveform (oscillation mode and frequency) on tube deformation behavior is discussed. To obtain different oscillation modes of currents, three types of circuit structures are used, as shown in Figure 2. Figure 2a presents the most widely used circuit structure in electromagnetic forming, where the coil and tube can be treated as a series connection of a resistance (Rc) and an inductance (Lc). According to circuit theory, the current i following through the coil can be determined as [22]:

$$i={\displaystyle \frac{U_0}{{\omega }_{\mathrm{d}}{L}_{\mathrm{c}}}}{e}^{-\alpha t}\sin ({\omega }_{\mathrm{d}}t)$$

(1)

where C is the capacitance, U₀ is the discharge voltage, attenuation coefficient α = R_c/2L_c, oscillation angle frequency ${\omega }_{\mathrm{d}}=\sqrt{{\displaystyle \frac{1}{L_{\mathrm{c}}c}}-\left({\displaystyle \frac{R_{\mathrm{c}}}{{2L}_{\mathrm{c}}}}\right)}$. It can be seen that the current presents as an attenuated oscillating sine wave.

Usually, the first current waveform is critical for the forming process, but this has not been verified by experiment. To solve this, a thyristor switch (T) is applied to replace the traditional mechanical switch (S), as shown in Figure 2b. Due to the unidirectional conductivity of the thyristor switch, the current is cut into a half-wave. The current i following through the coil can be determined as follows:

$$i={\displaystyle \frac{U_0}{{\omega }_{\mathrm{d}}{L}_{\mathrm{c}}}}{e}^{-\alpha t}\sin ({\omega }_{\mathrm{d}}t)(t<{\displaystyle \frac{\pi }{{\omega }_{\mathrm{d}}}})$$

(2)

In addition, the effect of the current decay rate on tube forming is also discussed, which can be achieved by using a crowbar circuit with a resistor (R_d) and diode (D) across the capacitor, as shown in Figure 2c. By adjusting the value of the resistance R_d, the current decay rate after the first peak can be altered. The current i following through the coil can be determined by solving the following equation group:

$$\left\{\begin{array}{l}i{R}_c+L_c{\displaystyle \frac{di}{dt}}-u_c=0\\ u_c=U_0-{\displaystyle \frac{1}{C}}{\displaystyle {\int }_0^t(i+i_d)}\\ i_d=0u_c>0\\ i_d={\displaystyle \frac{u_c}{R_{\mathrm{d}}}}{u}_c<0\end{array}\right.$$

(3)

where i_d is the current following through the crowbar circuit, and u_c is the voltage on the capacitor.

Further, to alter the frequency of current, it is well known that the frequency f of current can be calculated as follows:

$$f={\displaystyle \frac{{\omega }_{\mathrm{d}}}{2\pi }}={\displaystyle \frac{\sqrt{\frac{1}{L_{\mathrm{c}}C}-{(\frac{R_{\mathrm{c}}}{2L_{\mathrm{c}}})}^2}}{2\pi }}$$

(4)

It can be seen that the frequency f is related to coil parameters and capacitance. Since the parameters of the coil are fixed, the current frequency can be varied by changing the capacitance parameters. Usually, the higher the capacitance, the lower the current frequency. In summary, different current waveforms can be obtained by switching the above three circuit configurations and the value of capacitance, which provides a technical way to study the effect of current waveforms on tube forming.

2.2. Numerical Method

To investigate the deformation of a tube under different current waveforms, a numerical model implemented with LS-DYNA R13.0 software is presented. The geometric structure of the EMF system is shown in Figure 3a. A coil with 14 turns of wire is arranged in a single layer, reinforced with a zylon layer to prevent coil failure. The tube material is 6061-O aluminum alloy, with an inner diameter of 75 mm, a thickness of 2 mm, and a height of 180 mm. The stress–strain behavior of the AA6061-O aluminum alloy is defined as follows:

$${\sigma }_{qs}=120.14{\epsilon }^{0.157}$$

(5)

To incorporate the strain rate on material flow stress, the Cowper–Symonds constitutive model was used, given by the following equation:

$$\sigma ={\sigma }_{\mathrm{q}s}\left[1+{\left({\displaystyle \frac{{\epsilon }_P}{P}}\right)}^m\right]$$

(6)

where εp represents the plastic strain rate, with specific values for P and m set at 6500 s⁻¹ and 0.25 for aluminum [23], respectively. Additional details on the key physical parameters of the materials used are provided in

Table 1. Main physical parameters of the used materials.

Parameters	Materials
	Tube (AA6061-O)	Coil (Copper)
Density (kg/m3)	2.7 × 103	8.9 × 103
Elastic modulus (GPa)	68.8	126
Poisson ratio	0.33	0.34
Yield strength (MPa)	55.2	100
Electrical resistivity (Ω × m)	3.66 × 10−8	1.66 × 10−8

.

Figure 3b gives the numerical model for the EMF system, where only the coil and tube are meshed. To reduce the calculation burden, only a 1/4 model was built. Meanwhile, to improve the calculation accuracy, the maximum size of the mesh is not more than 1 mm. The currents following through the coil are obtained from the experiment. In the calculation process, the electromagnetic field and the structural field are mainly solved, where the former is used to calculate the Lorentz force, while the latter is used to calculate the deformation of the tube. The timestep for electromagnetic field is set as 1 μs, and after every 10 μs, the deformation information of the tube is reacquired, thus guaranteeing the accuracy of the calculated Lorentz force. More information on building the simulation model can be found in the reference reported by Zhu et al. [3].

2.3. Experimental Setup

Figure 4 shows the built EMF system, including the circuit and the coil. In this case, the capacitor bank consists of eight 80 μF capacitors, which can achieve a maximum capacity of 640 μf. The charger can charge the capacitor up to a maximum voltage of 30 kV. The MKPD-200 thyristor switch can withstand a maximum current of 200 kA and a voltage of 25 kV, while those for ZPDZ-120 diode are 120 kA and 30 kV, respectively. By utilizing these devices, different discharge circuit structures shown in Figure 2 can be realized. Further details on the discharge parameters are provided in

Table 2. Main discharge parameters.

Discharge Parameters	Symbol/Unit	Value
Discharge voltage	U0kV	0~30
Capacitance	C/μF	160~640
Coil resistance	Rc/mΩ	9
Coil inductance	Lc/μH	10
Crowbar resistance	Rd/mΩ	0~1000

. The tube is sleeved around the outside of the coil, and the coil is held down by a press to ensure no movement during discharge.

3. Result and Discussion

3.1. Current Realization and Model Verification

Figure 5a presents the current waveforms under different circuit types, where a discharge of 8 kV and a capacitance of 160 μF are applied. Note that circuit I represents a conventional circuit (see Figure 2a), circuit II represents a half-wave circuit (see Figure 2b), and circuit III represents a circuit containing a crowbar circuit with an extremely small resistance (Rd = 10 mΩ, see Figure 2c). It can be seen that the peak moment of the current is the same under all three types of circuits, which is measured to be 63 μs. In circuit I, the current decays rapidly, reverses past zero, and then continues to oscillate and decay. In circuit II, the current decays to 0 and is cut into a half waveform, and the duration is determined to be 130 μs. In circuit 3, the current slowly decays to zero. To alter the current frequency, a larger capacitance was applied. Figure 5b gives the current waveforms at a discharge of 5 kV and a capacitance of 640 μF, where the peak moment of the currents is measured to be 125 μs, while the attenuation characteristics are similar to those in Figure 5a.

To validate the developed simulation model, Figure 6a shows the simulation and experimental results of the deformed tube, where a discharge of 10 kV and a capacitance of 160 μF were applied under circuit I. Results show that the forming height and length of the tube are 95.2 mm and 174.8 mm in the experiment, respectively, while those are 94.6 mm and 175.5 mm in the simulation. This demonstrates a strong agreement between the simulation and experimental results. Figure 6b shows the dynamic deformation process of the tube, and it can be observed that the center of the tube deforms first due to the Lorentz force, and the tube is finally shaped as a convex profile. The deformation process of tubes under other discharge parameters is similar, and no more discussion is presented here.

3.2. Deformation Analysis Under a High-Frequency Current

In this section, the deformation of the tube under three circuits at a capacitance of 160 μF is discussed. Figure 7 shows the forming height of tubes under different discharge voltages. Note that only the discharge of 8 kV in circuit II is performed in the experiment since a strong current with high frequency would destroy the thyristor switches. It can be seen that circuit III has the largest amount of forming height, followed by circuit I, and circuit II has the least, no matter what the voltage is. For example, at a voltage of 10 kV, the forming heights for circuits I, II, and III are 7.8 mm, 7 mm, and 11.3 mm, respectively. Compared to circuit I, the forming height in circuit III can be significantly increased, by 45%.

To explain this phenomenon, Figure 8 shows the Lorentz force, displacement, and velocity of the tube central point at a discharge of 10 kV in circuit I. Results show that the Lorentz force gets its maximum value of about 13.6 GN/m³ at 54 µs, which is similar to the peak moment of the coil current (63 μs, see Figure 5a). At this moment, there is almost no displacement occurring at the central point. As the current decays, the Lorentz force begins to decay, while the velocity of the tube increases dramatically, reaching a peak value of 80 m/s at 90 µs. Meanwhile, the displacement is measured to be 3 mm. The force decays to 0 at 110 µs and begins to reverse slightly. This reversal is common in electromagnetic forming, which is caused by the phase difference between the eddy current and the coil current. Therefore, in the first current half-wave, the deformation of tube is observed to be 5.3 mm, accounting for 68% of the total deformation. When the second half-wave of the current arrives, the tube experiences force again. Due to the attenuation of the current, the maximum force generated by the second current half-wave is only 4.7 GN/m³, but it could also prevent the velocity attenuation. Under the effect of Lorentz force and inertia, the deformation of the tube continues to increase. Based on the displacement curve, the end time of deformation is determined to be 230 µs. Therefore, the Lorentz force generated by the third current half-wave has no gain effect on the deformation. In the end, the deformation of the tube is measured to be 7.8 mm.

Figure 9 shows the Lorentz force, displacement, and velocity of the tube central point in circuit II. Results show that the deformation process of the tube in the first current half-wave (before 130 µs) is the same as that in circuit I, and no more discussion is presented here. When the first current half-wave is cut off, the current disappears. Therefore, the tube is no longer subjected to Lorentz force, and the deformation of the tube relies on an inertial effect. It can be observed that the forming speed of the tube decreases sharply, and the deformation end time is advanced to 190 µs, with a deformation height of 7 mm. Combining the results of Figure 5 and Figure 6, it is found that the Lorentz force generated by the second current half-wave also contributes to the increase in deformation, but it is very small, with only a 14% enhancement in forming height.

Figure 10 shows the Lorentz force, displacement, and velocity of the tube central point in circuit III. Results show that the deformation process of the tube before the first current peak value (before 63 µs) is the same as that in circuit I and II. Due to the fact that the current in circuit III decays slowly, the Lorentz force also decays slowly. The maximum velocity of the tube is observed at 100 µs, measuring 88 m/s, which is a 10% increase compared to that produced by the previous two circuits. In addition, throughout the deformation process, the force is constantly impeding the velocity decay. Therefore, the velocity decay is slower, and the forming time is longer. In the end, the deformation height of the tube can reach 11 mm.

Further, the effect of the current decay rate on tube forming in circuit III is investigated. Figure 11a shows the current waveforms at different crowbar resistances. Results show that the current decay rate increases with resistance, and the current starts to oscillate when the resistance reaches 200 mΩ. When the resistance is larger, the amplitude of the oscillation increases. It can be predicted that when the resistance is large enough, the generated current is equivalent to that in circuit I. Figure 11b gives the forming height of the tube under different crowbar resistances. It can be found that the amount of forming height first decreases sharply, and this is because the decay of the Lorentz force increases with the current decay rate. Then, the forming height increases slightly, which can be attributed to the appearance of the second current half-wave. In summary, a current having a slow decay is more favorable for the forming efficiency under a high-frequency discharge condition.

3.3. Deformation Analysis Under a Low-Frequency Current

In this section, the deformation of the tube under three circuits at a capacitance of 640 μF is discussed. Figure 12 shows the forming height of tubes under different discharge voltages. Results show that the amount of forming height in three circuits is similar, no matter at what voltage, which is distinctly different from that in a high-frequency discharge condition.

To explain this, Figure 13 shows the Lorentz force, displacement, and velocity of the tube central point at a discharge of 5 kV in circuit I. It can be observed that the Lorentz force gets its maximum value of about 10 GN/m³ at 100 µs, which is ahead of the peak moment of the coil current (125 µs, see Figure 5b). At this moment, the tube has begun to deform, and a displacement of 1 mm is observed. When t = 125 μs (the time for current peak value), the displacement of the central point reaches 2.8 mm, As the deformation proceeds, the current decays, the Lorentz force decays, the velocity decreases, and the displacement increases. The final displacement is measured to be 9.3 mm, and the whole forming process lasts 270 μs, which is close to the duration for the first half-wave in circuit I (260 μs). This indicates that the deformation of the tube is mostly completed in the first current half-wave, and the force induced in the second half-wave is too weak to cause deformation of the tube.

Figure 14 illustrated the Lorentz force, displacement, and velocity of the tube central point in circuit II. The deformation process is similar to that of circuit 1 since the forming process is almost complete in the first current half-wave, and the final displacement is observed as 9.3 mm.

Figure 15 shows the Lorentz force, displacement, and velocity of the tube central point in circuit III. The difference between the three currents occurs after the peak current (t = 125 μs); therefore, its deformation process is the same as that described in circuit I before t = 125 μs. After t = 125 μs, the current decays slowly due to the crowbar circuit. Although the Lorentz force also decays slowly, the deformation that happened before t = 125 μs would weaken the Lorentz force on the tube since the force is associated with the distance between the coil and the tube. This is quite different from that in a high-frequency discharge, where the tube has not been deformed before the current starts to decay. As a result, only minor forming height improvement is observed, measuring 10.3 mm. It can be predicted that the displacements produced by the three circuits would be consistent when the frequency of the current is further reduced. In summary, the rate of current decay has little effect on forming efficiency under a low-frequency discharge condition.

3.4. Effect of Current Waveform on Coil Performance

The maximum stress load and temperature rise are the two key factors that influence the service life of a coil, where the former can lead to coil structural damage while the latter can cause insulation damage. Thus, the effect of coil current waveform on these two factors is further discussed in this section.

Figure 16 shows the maximum stresses generated in the Zylon layer of the coil for three currents (see Figure 5b), where the discharge voltage is 5 kV and the capacitance is 640 μf. Results show that the current waveform has no effect on the maximum stresses, which is determined to be 410 MPa. This is because the maximum stress occurs at the moment of peak current, which is the same under the three currents.

Figure 17 illustrates the Joule heat produced by the three currents, where the heat is primarily caused by the coil’s resistance, with higher Joule heat leading to a greater temperature rise. It can be seen that circuit I produces the greatest amount of Joule heat since the energy of the capacitor is mainly dissipated in the coil. Conversely, both circuit II and circuit III can reduce Joule heat by a factor of 25% and 64%, respectively. This is due to the fact that in circuit II the current is interrupted in time, whereas in circuit III the resistance on the crowbar circuit can consume some of the energy. Therefore, for a high-frequency discharge condition, circuit III is highly recommended because it can reduce coil temperature rise while increasing forming efficiency. For a low-frequency discharge condition, circuit II is recommended since it can significantly reduce temperature rise without affecting forming efficiency.

4. Conclusions

This work explores the influence of coil current frequency and waveform on electromagnetic tube forming through numerical and experimental methods, where three circuits are used to produce various current waveforms. The main findings are summarized as follows:

(1)

It is demonstrated that at high-frequency discharge, a current having a slow decay rate can improve the forming efficiency. This is because the tube is not deformed until the current has decayed, which allows the subsequent Lorentz force to be strong to promote forming.

(2)

It is demonstrated that at low-frequency discharge, the current waveform has little effect on forming efficiency. This is because the tube has already been deformed before the current decays, which weakens the following induced Lorentz force.

(3)

It is found that a current having a slow decay rate is highly recommended in high-frequency discharge since it can increase the forming efficiency as well as reduce the temperature rise of the coil, while a current having a half-wave is favorable for low-frequency discharge since it can significantly reduce the coil temperature.

The proposed circuits (circuits II and III) have some advantages in improving the forming efficiency or reducing the coil temperature rise compared to the traditional one (circuit I), but the additional circuit devices also increase the equipment cost. Future work will further explore the effect of current waveform on other metal materials and its influence on the microstructural changes of materials.