Original Article
Current frequency effect on
electromagnetic tube expansion
Proc IMechE Part C:
J Mechanical Engineering Science
2020, Vol. 234(23) 4636–4644
! IMechE 2020
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DOI: 10.1177/0954406220927059
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SK Dond1 , Hitesh Choudhary2, Biswaranjan Dikshit2,
Tanmay Kolge2 and Archana Sharma2
Abstract
The frequency of capacitor discharge current is an important parameter in the electromagnetic forming process. In the
present work, simulation and experimental study of electromagnetic expansion is carried out on aluminium tubes
(Al 5052). The aim is to obtain the optimum frequency of discharge current that gives maximum tube expansion at
constant discharge energy. A two-dimensional axisymmetric numerical simulation model is developed that couples
electromagnetic and structural phenomenon sequentially. After validating experimental tube expansion results with
numerical results, the simulation model is used to analyse the effect of variation in current frequency on the tube
expansion. It is observed that maximum tube expansion and higher process efficiency occurred at 5 kHz, which is
considered as the optimum frequency for the used experimental set-up. In contrast to sheet forming case observed
in the literature, results of tube expansion show that maximum forming is obtained at a frequency where the ratio of skin
depth to tube thickness is less than 1. It is also noticed that the optimum frequency depends on the inductance and
resistance of the system.
Keywords
Electromagnetic tube expansion, optimum frequency, sequentially coupled simulation, plastic deformation
Date received: 23 December 2018; accepted: 21 April 2020
Introduction
Electromagnetic forming (EMF) is an impulse forming process and finds the applications in automobile
and aerospace industries. The process has advantages
like forming, crimping and welding of lightweight
materials such as aluminium.1 In EMF process, a
damped sinusoidal current in the range of kiloamperes is passed through the electromagnetic coil and
coil current generates an eddy current in the magnetically coupled workpiece (WP). Lorentz force responsible for forming is generated by the interaction of
current density induced in WP and the magnetic flux
density present between the coil and WP. A comprehensive review and assessment of the EMF process is
presented.2 Several analytical and numerical studies
have been done for EMF circuit analysis of tube
and sheet forming. To investigate the EMF process,
numerical simulation is the best choice as it helps to
get a realistic insight of the procedure which is difficult
in practice. The numerical investigation is performed
by various software using loosely coupled and sequentially coupled techniques.3–5 The advantage of the
sequential approach is that the effect of WP deformation on electromagnetic force distribution is considered and hence it has better accuracy over
a loosely coupled approach.4,6 Electrical and mechanical parameters involved in the EMF process decide
the resultant forming of WP. The effect of coil–tube
relative position and inertia on the tube forming
shapes is reported by the author.7 Reducing the coil
length reduces the inductance and resistance of the
coil and that causes an increase in peak current and
frequency. It is observed that magnetic pressure acting
on tubular WP varies with length and resistivity of
tubular material.8 Forming behaviour also strongly
depends on the characteristic of the current passing
through the coil. Not only peak current, but the duration of the current pulse also affects the resultant
sheet forming.9 Optimum selection of capacitance
value and discharging energy play a key role in controlled forming of WP. Variation in capacitance varies
the frequency of the current flowing through the
1
Homi Bhabha National Institute, Mumbai, India
Bhabha Atomic Research Centre, Mumbai, India
2
Corresponding author:
SK Dond, Homi Bhabha National Institute, Mumbai 400094,
Maharashtra, India.
Email: shandond12@gmail.com
Dond et al.
4637
electromagnetic coil. The EMF process efficiency is
less2 and hence, it is required that the system parameters be set so as to get maximum efficiency. For given
discharge energy, there exists a frequency of the current in the coil that gives maximum deformation. Very
few documents are available that have studied the
effect of frequency on WP forming. The effect of frequency on tube compression is studied10 and relates
the optimum frequency to high plastic strain energy.
In the frequency domain approach,11 electromagnetic
fields obtained from FEM modelling are used to estimate the optimum frequency but WP deformation is
not considered during the analysis. For sheet forming
case,12,13 the ratio of skin depth to tube thickness is
found to be near or more than 1 when optimum frequency achieved. In the present work, the electromagnetic expansion study is performed on tubular WP
made up of aluminium. A 2D axis-symmetric, sequentially coupled simulation is carried out using
COMSOL FEM software and obtained results are
compared with experimental ones. Numerical simulation model is further used to find the optimum frequency of discharge current and to study the effect of
process parameters like the tube thickness, inductance
and resistance of the system on optimum frequency.
This study can be useful in deciding the system and
coil design parameters to get more efficiency.
Figure 1. Schematic of the system.
EMF system
EMF uses a high energy magnetic field to deform
materials. This field is produced by the transient
under-damped current that flows through the coil.
The time varying, pulsed magnetic field is generated
by the coil and is linked with surrounding metallic
WP. Eddy current gets induced in a WP with direction
opposite to that of electromagnet current direction.
Figure 1 shows the schematic of the EMF process
for tubular WP. Here, dot and cross show the direction of the current in the coil and WP. Initially, energy
is stored in the electrostatic form in the capacitor
bank by charging the capacitors. The energy stored
in the capacitor is given by equation (1). Here W is the
energy stored in the capacitor bank, V is the charging
voltage of the capacitor bank and C is the capacitance
value. High power rating switch is used to discharge
the capacitive energy into the coil
1
W ¼ CV2
2
ð1Þ
The electrical equivalent circuit of EMF is presented in Figure 2. Coil and WP are mutually coupled.
RC and LC are coil resistance and inductance whereas
RW and LW are tubular WP resistance and inductance. System resistance RS and inductance LS can
be obtained from the current waveform of the
system short circuit test. R and L are total resistance
Figure 2. Electrical equivalent circuit of the EMF system.
and inductance of the system, respectively. As the WP
moves away from the coil, the magnetic coupling
between the coil and WP decreases and resultant R
and L also vary. However, in EMF, the most important part of the electric current that causes the displacement is very short14 and hence R and L can be
assumed constant during forming. The damped sinusoidal current flowing through the coil is given by
equation (2) and it is determined by the resonant circuit formed by equivalent resistance, inductance and
capacitance of the system. Here x is the angular frequency of the current and b is the damping coefficient.
x and b depends on the system parameters as given by
equations (3) to (5)
V t
e sin !t
!L
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
! ¼ !0 2
I¼
ð2Þ
ð3Þ
R
2L
ð4Þ
1
!0 ¼ pffiffiffiffiffiffiffi
LC
ð5Þ
¼
4638
Proc IMechE Part C: J Mechanical Engineering Science 234(23)
1
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f0 r
r ¼
h
ð6Þ
ð7Þ
placed over the coil. The supporting structure is
used to maintain the same axial position of the coil
and tube. Figure 3 shows the coil and WP dimensions.
The material properties are given in Table 1.
The magnetic field diffusion in the WP depends on
the skin depth. The skin depth d is given by equation
(6) where f is the coil damping current frequency, is
the WP electrical conductivity and l0 and lr are permeability of free space and the relative permeability,
respectively. dr is the relative skin depth as given
by equation (7). It is the ratio of skin depth to WP
thickness h.
Experiment detail
EMF set-up as shown in Figure 4 consists of the
power supply, capacitor bank and coil–tube arrangement. The power supply has an energy capacity of 40
kJ. The capacitors are rated for 20 kV charging voltage and 224 mF capacitance. A solenoid coil (Figure 5)
is used to generate the magnetic field. The tube is
Figure 5. The coil with reinforcement and used Al 5052 tube.
Table 1. Material properties of the EMF system.
Material
Parameter
Value
Coil (copper)
Number of turns
Electrical conductivity
Yield stress
Electrical conductivity
Density
Poisson’s ratio
7
58 MS/m
89.6 MPa
20 MS/m
2680 kg/m3
0.33
Workpiece (Al 5052)
Figure 3. Dimensions of the experimental system.
Figure 4. Experimental setup and arrangement of coil and tube.
Dond et al.
4639
Tube forming experiments are carried out at different energies by varying the operating voltage and at a
constant capacitance of 112 mF. Resistance R and
inductance L of the EMF system are 18 mX and
1.7 mH, respectively. Discharging current through
the coil is measured by Rogowski coil and recorded
on an oscilloscope.
Numerical simulation
An axisymmetric, 2D sequentially coupled simulation
is performed using COMSOL. Following assumptions
are made in electromagnetic field analysis:
1. The cylindrical coordinate system is used for the
developed 2D model that consists of axial and
radial components.
2. The electrical conductivity of material and permeability is constant and isotropic.
3. The effect of temperature on material properties is
not considered.
In the electromagnetic field model, magnetic vector
potential method is used as given below to solve the
Maxwell equations
r B ¼ 0
r E ¼
ð8Þ
@B
@t
r B ¼ J þ "
ð9Þ
@E
@t
ð10Þ
where E is the electrical intensity (V/m), J is the current density in the coil (A/m2), B represents magnetic
flux density (T) and 2 is permittivity of the medium
(F/m). Representing magnetic and electric field using
magnetic vector potential A such that
B ¼ r A
ð11Þ
@A
E ¼ @t
ð12Þ
The obtained force is used as an input in the mechanical model. In the structural mechanics model, tube
displacement is calculated using the following
equation
@2 u
r s ¼ fm
@t2
Here, q is the mass density of WP, u is displacement
vector, s is stress tensor and fm is magnetic force
density. Figure 6 shows the flowchart of the sequential
coupled model used for simulation. The coil current
obtained by the Rogowski coil is used as input to the
simulation model. Time step is kept at 0.5 ms and at
each time step tube geometry is updated based on
Lorentz force. Numerical simulation model consists
of a coil, tube and air. The coil and tube regions are
meshed through mapped meshing. Figure 7 shows the
meshing used in the numerical model. The electromagnetic and structural models are run till the tube
gets final displacement. Tubular WP flow stress
approximated by Johnson–Cook material model is
as given by equation (16). The temperature effect is
ignored here as any small rise in temperature has little
influence on electrical and mechanical properties of
WP material15–17
T T0 m
¼ ½Aw þ Bw ðp Þn ½1 þ Cw lnð_p Þ 1 Tm T0
ð16Þ
Here, 2p is an effective plastic strain; _p is strain
rate; and Aw, Bw, Cw, n and m are material
Substitution of equations (11) and (12) in equations (8) to (10) can give
r
1
@A
r A ¼ J @t
ð13Þ
where is the conductance of medium (S/m) and @@tA
is the induced current density in the tube (A/m2).
According to the Lorentz formula, electromagnetic
force acting on WP is given as
1
B
F ¼ J B ¼ ðr BÞ
ð15Þ
ð14Þ
Figure 6. Flowchart of sequential coupled simulation.
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Proc IMechE Part C: J Mechanical Engineering Science 234(23)
Figure 8. Expanded tubes (a) 26.8% and (b) 40.2% change in
diameter.
Figure 7. The 2D axis symmetric arrangement and meshing.
constants.18 First bracket represents stress as a function of strain. Second and third bracket represents
stress as a function of strain rate and temperature,
respectively.
Results and discussion
Comparison of experimental and simulation results
Experimentally deformed tubes are shown in Figure 8.
It is observed that the increase in discharge energy
increases the tube expansion as expected. For the
operating voltage of 16 kV and capacitance of
112 mF, 26.8% increase in tube central diameter is
obtained whereas an increase in voltage to 18 kV
showed a 40% increase in tube diameter. Maximum
deformation of the tube occurred at the central region
and less towards the tube ends. This happened
because of the tube length being higher than the coil
length. Maximum magnetic flux concentration
occurred at the tube surface facing coil central turn
and that causes higher magnetic pressure at the
middle part of the tube. Comparison of experimental
and simulation results is shown in Figure 9.
Numerical results of tube final displacement and
experimental results have a maximum error of 8%
thereby validating our simulation model. The reason
for error can be attributed to the following reasons.
(a) The difference in experimental and literature
values of material properties and plasticity model
constants. (b) Assumptions taken in the numerical
model. Figures 10 to 12 indicate the simulation
results obtained at 16 kV, 112 mF condition.
Figure 9. Radial displacement of the tube by experiment and
by simulation.
The displacement and velocity variation with time at
the maximum displacement point of the tube is shown
in Figure 10. Peak velocity of the tube is found to be
delayed by 15 ms from peak coil current which might
be due to tube inertia. Figure 11 shows the surface
plot of the tube deformation. The formed shape of
the tube is similar to the experimental one.
Displacement of the tube along the surface is indicated by different colours. Forming efficiency is
obtained from the ratio of plastic strain energy of
the tube to energy stored in the capacitor. Figure 12
shows these energy plots where 10.5% forming efficiency is obtained. This low efficiency can be understood in the following way: As observed in Figure 10,
tube deformation completes well before the total current dies out. Only first and second current pulses are
utilized for WP deformation while the consequent
pulses result in just heating. This affects the efficiency
of the process.
Dond et al.
4641
Figure 10. Coil current, tube velocity and displacement at
16 kV, 112 mF.
Figure 13. Radial displacement of the tube with a thickness
of 2.4 mm.
Figure 11. Radial displacement of tube obtained at 16 kV,
112 mF.
Figure 12. Energy plots at 16 kV, 112 mF.
Optimum frequency estimation by the
numerical model
The simulation model is further used to observe the
effect of frequency on the forming of the tube at
constant discharge energy condition. The operating
voltage and capacitance are varied in such a way to
keep total discharge energy given by equation (1) as a
constant at 16 kJ. Change in capacitance causes
changes in damping frequency of coil current.
Displacement of the tube is measured at the maximum
forming point. Figures 13 and 14 show displacement
obtained at different frequencies for different tube
thickness. The maximum displacement of the tube is
found to occur at 5 kHz and we called it optimum
frequency. The relative skin depth dr at the optimum
frequency in Figures 13 and 14 is less than 1. This
observation is in contrast to the sheet forming
case13,14 where dr is equal to and more than 1 when
maximum forming occurs in the sheet.
The optimum frequency phenomenon can be
explained with the magnetic pressure profile acting
on tube and skin depth. Magnetic pressure is proportional to the square of magnetic flux density and the
magnetic field is proportional to the coil current.
Higher the current, higher the magnetic field. Skin
depth d increases with the decrease in frequency
value (equation (6)). When d is more than the thickness of the tube, in other words, dr is more than 1, the
magnetic field penetrates through tube thickness. The
effective magnetic pressure acting on the tube gets
reduced. That is why the tube forming reduces when
the dr is more than 1. Increase in frequency increases
the current peak value with a decrease in current pulse
duration as shown in Figure 15. At high frequency,
though the magnetic pressure is high because of
higher peak current, the duration of this pressure
pulse is very short due to rapid damping of the current
pulse. As mentioned in Cui et al.,9 resultant WP forming is not only affected by peak current but also by the
duration of the current waveform. That is why the
tube forming slowly reduces when the frequency is
increased beyond an optimum value. It means that
not only the peak current but the optimum combination of peak current and the duration of the current
4642
Figure 14. Radial displacement of the tube with thickness
3 mm.
Figure 15. Coil current waveform at various frequencies.
pulse resulted in maximum tube forming. For the used
experimental set-up, the optimum combination
occurred at 5 kHz. Further, the analysis is carried
out to study the effect of different parameters on the
optimum frequency and corresponding relative skin
depth.
Effect of tube thickness
Figures 12 and 13 show the effect of tube thickness
variation at constant discharge energy. It is observed
that tube forming is reduced with an increase in
tube thickness. With an increase in tube thickness,
the optimum frequency is unaffected but the relative skin depth dr at the optimum frequency is
decreased. This is because the optimum combination
of the current peak and duration of the current pulse
(current frequency) have resulted in a maximum
push to the material. The tube thickness has no
effect on the peak current or on the current frequency.
The process efficiency of EMF is shown in Figure 16
for different thickness of tubes. Maximum efficiency
Proc IMechE Part C: J Mechanical Engineering Science 234(23)
Figure 16. Process efficiency variation with frequency.
Figure 17. Effect of frequency variation with higher
inductance on tube expansion.
occurred at same optimum frequency and it is more
for thin tubes as these are more susceptible to
deformation.
Effect of system inductance and resistance
Variation in WP resistivity, WP length, the coupling
between the coil and the WP and coil material affect
the magnetic pressure acting on the WP. From the
electrical circuit point of view, all these variations
affect the resultant inductance and resistance of the
EMF system. To take into account this parametric
variation, the effect of resistance and inductance variation on optimum frequency is studied. The simulation model is used to study the effect of inductance
and resistance value on optimum frequency.
Figures 17 and 18 depict the tube expansion performed at reduced damping coefficient b value for
2.4 mm thickness tube. b depends on resistance and
inductance of the system as given by equation (4). In
the first case, b is reduced from 5294 to 3500 S1 by
increasing the inductance and in the second case b is
Dond et al.
4643
Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
ORCID iD
SK Dond
https://orcid.org/0000-0002-1793-792X
References
Figure 18. Effect of frequency variation with lower resistance
on tube expansion.
reduced to 3500 S1 by reducing the resistance value.
Compared to previous results, it is observed that optimum frequency increased with an increase in inductance, whereas it decreased with a decrease in
resistance. Thus, the optimum frequency does not
remain constant. It depends on system parameters
and this fact should be taken into consideration
during the design of EMF system configuration.
System parameters can be set in such a way that
the coil current oscillates with optimum frequency.
This gives higher process efficiency along with
increase coil life.
In this paper, a 2D axisymmetric numerical model
is developed for the analysis. However, a close observation of formed tube shows that the tube forming
is not exactly axisymmetric due to the helical nature
of the coil. The future aim is to develop a 3D numerical model to improve the numerical simulation
accuracy.
Conclusions
Electromagnetic expansion of aluminium tubes is carried out experimentally and the tube displacement
results are compared with results of the 2D numerical
simulation model. Effects of system parameters on
the optimum frequency and relative skin depth are
further analysed using a simulation model. From
the observed results, the following conclusions are
drawn: (1) Maximum expansion of tubular WP
occurred at a frequency where the relative skin
depth is less than 1 in contrast to sheet forming
case observed in the literature. (2) The frequency of
current and the relative skin depth at which maximum
forming occurs are not constant. The optimum frequency changes with resistance and inductance of
the system.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of
this article.
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