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Metodologia para el Diseño de Bobinas de Bitter

Journal of Manufacturing Processes 16 (2014) 551–562
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Journal of Manufacturing Processes
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Technical Paper
Bitter coil design methodology for electromagnetic pulse metal
processing techniques
Oleg Zaitov a,∗ , Vladimir A. Kolchuzhin b
Belgian Welding Institute, Technologiepark 935, B-9025 Zwijnaarde, Belgium
Chemnitz University of Technology, Department of Microsystems and Precision Engineering, Reichenhainerstrasse 70, D-09126 Chemnitz, Germany
a r t i c l e
i n f o
Article history:
Received 2 March 2014
Received in revised form 2 June 2014
Accepted 15 July 2014
Available online 22 August 2014
Bitter coil
Magnetic pulse welding
Design methodology
a b s t r a c t
Electromagnetic pulse metal processing techniques (EPMPT) such as welding, forming and cutting have
proven to be an effective solution to specific manufacturing problems. A high pulse magnetic field coil
is a critical part of these technologies and its design is a challenging task. This paper describes a Bitter
coil design using a newly developed methodology for a simplified analytical calculation of the coil and
complementary finite element models (FE) of different complexity. Based on the methodology a Belgian
Welding Institute (BWI) Bitter coil has been designed and tested by means of short circuit experiments,
impedance and B-field measurements. A good agreement between the calculated and the experimental
design parameters was found.
© 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction
In a certain range of thicknesses and materials combinations
EPMPT are more competitive than of the same name conventional
manufacturing methods. However a wide industrial use of the technologies is limited, partly due to a lack of compact engineering
guidelines for a coil design. The main purpose of the present work
is to develop such guidelines for a pulsed Bitter coil.
Different types of coils categorized by Furth et al. [2] can be used
for EPMPT. On the basis of an analysis of manufacturing techniques,
principal design solutions and performance characteristics of the
above-named coils described by Lagutin and Ozhogin [10] one can
conclude that within tubular applications the Bitter coils have high
reliability, manufacturability and maintainability. These are the
key characteristics for an industrial implementation of the coils
and factors which defined our choice to develop the calculation
methodology for them. The coil is an assembly of the alternating
conducting and insulating discs, each with a radial slit as shown in
Fig. 1. The contact between the disks is realized due to their overlap.
The Bitter coils can be used with fieldshapers (FS). Unfortunately a joint analytical treatment of the coil and a FS is hugely
limited. However the FS can be partially taken into account, but in
∗ Corresponding author. Tel.: +33 630775462.
E-mail addresses: [email protected] (O. Zaitov),
[email protected] (V.A. Kolchuzhin).
order to be brief in this article we are focused on the calculation
methodology for the direct acting Bitter coil.
The coil design is a complex task and mainly includes the determination of appropriate coil materials, sizes, the electromagnetic
parameters such as an inductance, a resistance and the B-field as
well as thermal and stress loadings. Most publications dedicated
to the high magnetic coil design deal with pulsed coils having
constant current density distribution which is according to Kratz
and Wyder [9] approximately realized in multi-layer multi-turn
coils. Some of the relevant publications within the constant current
density coils design are represented below. Wood et al. [18] proposed an approach to a material selection for such a coil. Knoepfel
[8] suggested a methodology to calculate main electromagnetic
parameters of the coil: the inductance, the resistance and the central field. Similar methodology to find the main design parameters
of the coil and its strength was proposed by Dransfeld et al. [1]. A
relatively precise and complete design of the coil can be fulfilled in
software developed by Vanacken et al. [16].
The pulsed Bitter coils have the current density distribution
which is approximately in inverse proportion to the inner radius
and therefore the above-mentioned publications become irrelevant
in the present case. Nevertheless methods of finding single design
parameters of the pulsed Bitter coils are found in specialized literature. For example, the inductance of the Bitter coil can be calculated
using a method proposed by Grover [5]. Knoepfel [8] suggested a
formula to find the central field of the coil. Moreover, the most
comprehensive physical and mathematical interpretations of the
general design principles and different calculation techniques of
1526-6125/© 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
known. Ideally the field distribution law in the gap coil-WP must
be specified based on demands of an application. A step-by-step
explanation of the scheme is represented below.
Initial data for calculation:
1. Demanded parameters of the field: an amplitude magnetic field
in the centre of the gap coil-WP Bmax and the rise time of the
field are given.
2. A WP geometry characterized by an outer radius r0 , a wall thickness r and a work area length l as well as WP material
properties represented by the conductivity , the heat conductivity , the specific heat capacity c, the yield strength y and the
mass density m are known.
3. Pulse generator data such as the storable energy W, the maximum current amplitude I0 , the short circuit frequency f0 , the
inductance Li and the resistance Ri are convenient to know for a
simulation of a current pulse but this information is not obligatory and can be specified later during the design process.
Fig. 1. A principal construction of the Bitter coil: 1 – Bitter plate, 2 – connecting
lead, 3 – contact, 4 – current path, 5 – flange, 6 – insulator.
the design parameters of the pulsed Bitter coils are given by Kratz
and Wyder [9].
Despite a sufficient, mainly academically orientated theoretical
knowledge on the pulsed Bitter coils design, a simplified but complete industrial design methodology does not exist. The fact has
prompted us to rework a thorough academic approach into a compact, industry-friendly methodology of the analytical calculation of
the pulsed Bitter coil. This has been done by analysing an applicability of theoretical models describing electromagnetic, strength and
thermal parameters of the coil and adjusting them to the present
coil embodiment. A principal novelty of the methodology is that
every design parameter is modified by asymmetry factors reflecting real geometry of the coil. An implementation of the asymmetry
factors and a frequency-dependent resistance has improved precision of the methodology. Moreover several supporting FE models
have been developed aiming to partly verify the analytical approach
and to get a deeper insight into the design parameters. As it will be
shown further the methodology of the analytical calculation is an
effective tool for defining the main design parameters. Furthermore
each step of the methodology can be fulfilled on a paper. Finally
short circuit experiments, impedance and B-field measurements
have been used for a verification purpose. A list of the symbols
used in this article and the corresponding meanings is represented
in Table 1.
2. Methodology of the analytical calculation
The analytical approach can only be applied to the coil having cylindrical symmetry, which means that there is no change
in geometry when rotating about one axis, and when magnetoresistance phenomenon, eddy currents, plastic deformations and
thermal stresses are neglected. Additionally the field at each instant
of time is calculated as the static field of the coil with a certain current density. With the limitations stated above the methodology
can be schematically represented in Fig. 2.
The present scheme assumes an approach to the coil design provided that the demanded magnetic field in the gap coil-WP, its
rise time, WP geometry and parameters of the pulse generator are
Material assignment. An insulating material represented by
allowable working temperatures, dielectric and ultimate compression strengths, and a coil material described by the conductivity ,
the heat conductivity , the specific heat capacity c, the ultimate
tensile strength UTS and allowable working temperatures initial Ti
and final Tf must be defined.
Maximum field in the gap coil-WP. It is known that the maximum
achievable field in the gap coil-WP (FS-WP) must be at least 40 Tesla
and the rise time must not exceed 25 ␮s for the most of welding
applications. Using an efficiency coefficient introduced by Wilson
and Srivastava [17] one can connect the fields in the gaps coil-FS
and FS-WP.
Coil geometry value assignment. An inner radius r1 is defined by
the WP outer radius r0 and an insulation gap g which is typically
0.75–1.5 mm, a nominal length of the coil l0 is determined by the
work area length of the WP l, while an outer radius r2 , thicknesses
of a turn and the insulation between the turns h, as well as the
asymmetry parameters ϕ, , of the turns can be defined using
the parameters of the existing prototypes or arbitrarily. Finally a
nominal number of turns N can be estimated.
Auxiliary calculations. These are calculations of two form-factors
of the coil ˛ and ˇ reflecting relations between the sizes of the
coil, an “effective” number of turns allowing to transform the
asymmetrical real to the ideally symmetrical coil, the skin depth
characterizing an attenuation of electromagnetic waves in a conductor, the demanded current in the coil, a so-called filling factor
describing a structure of conducting and insulating regions in the
total cross-section and the material integral connecting physical
properties of the coil material with an amplitude field and a pulse
Design limitations. The maximum achievable field in the coil is
mainly limited by two factors. The first is the mechanical strength
of the coil depending on the ultimate tensile strength UTS of the
coil material, its geometry and a distribution of the current density
in it. The second factor is the thermal one and is determined by the
thermal physical properties of the coil material, its geometry, the
distribution of the current density in the coil, the allowable temperature range and the demanded pulse length. Both factors have to
be considered in conjunction and the strongest factor defining the
maximum achievable field must be selected. Finally the demanded
field in the gap coil-WP Bmax and the maximum achievable field B0
are compared and a decision is made according to the scheme.
Inductance of the coil. Geometrical parameters of the coil such as
the inner r1 and the outer r2 radiuses, the length lcoil , the effective
number of turns and a self-inductance factor (˛,ˇ) depending
on the coil geometry and the current density distribution in the coil
determine the inductance of the coil Lcoil .
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
Table 1
List of symbols.
Maximum demanded field in the gap coil-WP
Maximum achievable field in the gap coil-WP
Rise time of the field
WP outer radius
WP wall thickness
WP work area length
Electrical conductivity
Heat conductivity
Heat capacitance
Yield strength
Discharge energy
Maximum current amplitude of a generator
Short circuit frequency of the generator
Inner inductance of the generator
Inner resistance of the generator
Ultimate tensile strength
Allowable initial temperature
Allowable final temperature
Inner radius of the coil
Outer radius of the coil
Thickness of a turn
Skin depth
Stress-determined current density
Thermally-determined current density
Equivalent inductance of the coil and the WP
Equivalent resistance of the coil and the WP
Discharge voltage
Electric field strength
Insulation thickness
Electric displacement
Nominal number of turns
Nominal number of intermediate turns
Nominal number of end turns
Cut angle
Contact angle
Connecting angle
Filling factor
Resistance of the coil. There are two approaches to the resistance
calculation in the methodology. First approach is made based on
the equation of Ohmic power in the coil and operates with the
resistivity of the coil material , its geometry and the filling factor describing a structure of conducting and insulating regions
in the total cross-section. This method does not take into account
an increase of the resistance with frequency caused by skin and
proximity effects. The second approach enables calculating the
frequency dependent resistance and was adopted from induction
heating technique.
Simulation of an equivalent circuit. Having defined the resistance
and the inductance of the coil and knowing the parameters of the
generator, a current pulse can be easily simulated using a differential equation of damped current oscillations at the given initial
conditions. The rise time T/4 and the amplitude of the obtained
current pulse Im must be compared with the demanded rise time of
the magnetic field specified in the performance requirements and
the maximum current in the coil Imax found earlier. The calculated
values must approximately fit the demanded values. Otherwise
the previous calculation steps are repeated using a new geometry and a material until the aforementioned correspondence is
2.1. Auxiliary calculations
After the value assignment of the coil geometry, the inner r1
and the outer r2 radiuses, the thickness of the turn , the insulation thickness h, the nominal number of turns N and the asymmetry
parameters ϕ, , are approximately defined. Practically the asymmetry parameters represent cuts and contact surfaces in the real
coil (Fig. 3).
Therefore the real coil to be manufactured with N nominal turns
is obtained by adding the asymmetry parameters to the ideally
symmetrical coil with turns.
Effective number of turns
Self-inductance factor
Calculated amplitude current through the coil
Maximum demanded current through the coil
Relative magnetic permeability
Permeability of vacuum
Angular frequency
Stress-determined field
Thermally-determined field
Pulse shape factor
Damped angular frequency
Inductance of the coil
Resistance of the coil to the constant current
Active resistance to alternating current
Gap between the coil and the WP
Material integral
Permittivity of space
(˛, ˇ)
FMat (Ti ,Tf )
A2 s/m4
1. Similar to Izhar and Livshiz [6] method, “effective” intermediate
M and end X turns can be found from the following expressions:
360◦ − (ϕ +
360◦ − (ϕ/2 +
+ )
Then total number of efficient turns consists of “effective” intermediate and endplates is found from expression (3):
Therefore the active length of the coil is found from (4):
lcoil =
· + ( − 1) · h
Now a simplification of cylindrical symmetry of the coil can
be applied: the real Bitter coil with N nominal turns but with a
symmetry breakdown is reduced to the ideal symmetrical coil
having turns. The obtained value must be rounded up to an
integer number.
2. According to Kratz and Wyder [9] the form-factors of the ideal
coil are found from (5) and (6):
2 · r1
3. If the current density distribution in the coil is described by a
function f(r,z) = r1 /r, which is typical for the Bitter coils, than the
filling factor is defined from expression:
f (r, z)dz
f (r, z)dz
The numerator describes the current density distribution in
the conductor and the denominator describes the distribution
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
Fig. 2. Graphical interpretation of the analytical approach.
in the whole cross-section, taking into account the insulation.
After integration of (7) a convenient analytical form is obtained:
5. The skin depth is a well-known characteristic describing an
attenuation of electromagnetic waves in a conductor is found
from (10):
4. The maximum current in the coil can be approximated as (9):
Bmax · lcoil
· 0
· 0·ω
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
Fig. 3. Intermediate and end Bitter plates.
6. The physical properties of the conductor material and the initial and the final temperatures define the material integral
FMat (Ti , Tf ):
FMat (Ti , Tf ) =
m · c(T )
(T )
Higher values of FMat (Ti , Tf ) allows to generate higher fields and
longer pulses.
jth =
2.2.1. Strength limitation
Various models allow calculating the strength of different types
of coils. Gersdorf et al. [3] developed the spatial-averaged model
which averages properties of a conductor material and an insulator
material and operates with a new material having these averaged
properties. Mechanical stresses in a turn are found from the equilibrium equation including a volume Lorentz force, a tangential
force and a force which is caused by a pressure difference on its
inner and outer surfaces. Liedl et al. [11] proposed the layer model
considering mechanical properties of the insulator material and
the conductor materials separately. This model does not take into
account an axial force caused by the radial component of the magnetic field. In the present paper the “free-standing wire” model and
its mathematical description suggested by Kratz and Wyder [9] are
used. The model assumes that there are no mechanical interactions
between the turns. This means that mechanical loads are not transferred from one turn to another and only circumferential stresses
occur in the turns as a response to the radial Lorentz forces trying
to expand the coil. Assuming an infinite length of the coil analytical expressions for a calculation of the stress-determined current
density and the field can be written as (12) and (13):
B =
· r2 ·
for the most of cold deformed aluminium alloys in order to prevent recrystallization and a loss of strength. Insulating materials
can have even lower temperature limit which must be taken into
account. Then a maximum achievable thermally-determined current density and corresponding field are found from (14) and (15):
2.2. Design limitations
j =
Fig. 4. Self-inductance factor (˛,ˇ) for an ideal Bitter coil with current density
proportional to 1/r as a function of the shape parameters ˛ and ˇ [9].
2.2.2. Thermal limitation
According to Lagutin and Ozhogin [10] the thermal limitation
can be represented as an upper bound of a temperature rise in
a skin layer during a pulse. Knoepfel [8] considered an extreme
limiting condition when the upper bound corresponds to the melting temperature of the coil material. In the present methodology
the temperature rise during the pulse is not calculated directly,
instead this value is assigned based on the coil material properties
as it was proposed by Kratz and Wyder [9]. For example, according to Mathers [12] the upper temperature must not exceed 200 ◦ C
Bth =
FMat (Ti , Tf )
tpulse · ς
· · r2 · jth
Finally according to Kratz and Wyder [9] the maximum achievable field for the coil is determined by the stronger of the two
above-mentioned limiting conditions and has a mathematical
interpretation in a form of (16):
B0 = Min(Bth , B ) · ln(˛)
When Bth < B the maximum achievable field is fully determined
by the thermal limiting factor, in the opposite case of B < Bth the
capabilities of the coil are limited by the strength factor. In accordance with the scheme (Fig. 2) next step of the methodology will be
available if the demanded field Bmax in the gap FS-coil is less than
the maximum achievable field found above.
2.3. Calculation of the inductance and active resistance of the coil
Several methods for the inductance calculation were found in
literature. Each method uses a set of similar fixed parameters
but distinguishes itself from another by a specific term. Izhar and
Livshiz [6] suggested a formula which takes into account the skin
effect for a coil and therefore reflects frequency-dependent inductance. An approach developed by Kalantorov et al. [7] operates by
a parameter depending on a ratio of the height of the turn and
the mean diameter of the coil and more suitable for the inductance calculation at relatively low frequencies. Finally a comparison
of the calculated and the experimental inductances showed that
a method including a self-inductance factor (˛,ˇ) (Fig. 4) as the
specific term proposed by Kratz and Wyder [9] resulted in the best
precision (17)
Lcoil =
· r1 · (˛, ˇ)
Practically the equivalent inductance of the coil and the WP
is of particular interest and on the basis of a formula for a oneturn coil and a coaxial cylinder equivalent inductance proposed by
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
2.4. Simulation of an equivalent circuit
As is well known damped current oscillations caused by a discharge of a capacitor bank in a circuit with an inductance and an
active resistance is described by Eq. (24):
d2 I(t)
+ ω2 I(t) = 0
+ 2ˇ
dt 2
On the basis of initial conditions (25) particular solution of the
equation (24) is found from (26)
q(t = 0) = CU and I(t = 0) = 0
U −ˇt
sin(ωd t)
I(t) = −
In accordance with the scheme the calculation is completed
when the amplitude current in the coil Im and its rise time T/4 satisfy
the corresponding demanded values Imax and .
2.5. Calculation of central magnetic field using Fabry formula
Fig. 5. Contact
resistance versus compressing forces:
1 – Cu-Cu contact, oxidized
clean surface Ra 6.3; 3 – Contact between Cusurface Ra 6.3; 2 – Cu-Cu contact,
Cr-Zn plates, oiled surfaces Ra
√3.2; 4 – Cu-Cu, clean surface; 5 – Contact between
Cu-Cr-Zn plates, clean surface Ra 6.3 [4].
Shneerson [14], one can calculate the equivalent inductance of the
Bitter coil as (18), when the conditions (19) are met:
L˙ ≈
· 1+
· r0 · g
· a1 + ln
r0 g ·r0 (18)
In applications different than EPMPT one may be interested in
the central field generated by the coil. A relation between the central magnetic field B01 , the Fabry factor G(˛,ˇ), the magnetic energy
Wm and the inner radius of the coil r1 is reflected in the Fabry formula. According to Kratz and Wyder [9] the Fabry formula for the
Biter coils is given by expression (27). In turn the Fabry factor G(˛,ˇ)
(28) represents the shape of the coil, the type of current density distribution given by the function f(r,z) = r1 /r and the self-inductance
factor (˛,ˇ).
The first resistance calculation was made according to Kratz and
Wyder [9] based on the equation of Ohmic power (Joule heat) in the
P = I 2 Rcoil
B01 =
G(˛, ˇ) =
· Wm
· G(˛, ˇ)
(˛, ˇ)
r 2 f (r, z)
(r 2 + z 2 )
f (r, z)drdz
This is the resistance to a constant current neglecting the contact
resistances between the plates:
Rcoil = N 2 ·
· r1 · ˇ · ln(˛)
There are contacts in the coil and their resistance can also be
taken into account. As shown in Glebov et al. [4] for copper contacts
and different compressing forces between them this resistance is
found from Fig. 5.
Formula (21) does not reflect an increase of the resistance with
an increase of the current frequency which results in an understated
resistance while considering frequencies from the common magnetic pulse technology range (10–25 kHz). Nevertheless Slukhotsky
and Ryskin [15] proposed an analytical form to calculate the frequency dependent resistance:
Rac =
N · coil · 2 · · r1
ıcoil · (22)
As it can be easily seen formula (22) defines the resistance of the
conductor with a length 2··r1 and a cross-section ·ı. It should
be noted here that the nominal number of turns N is used as a
total current path length is counted and there is no need to keep
symmetry as in the case with the inductance. Having the inductance
and the resistance of the coil determined the parameters of any
current course can be found.
By analogy of the equivalent inductance the equivalent resistance of the coil and the WP can be calculated from the following:
R˙ ≈ Rac +
WP · 2 · · r0
ıWP · l
Using designations for the geometrical parameters of the coil
and integrating right part of (28) along the cross-section of the coil
a convenient form of the Fabry factor is obtained:
G(˛, ˇ) =
r ln ⎜
(˛, ˇ) lcoil · ln
The higher the values of G(˛,ˇ) and the magnetic energy Wm ,
and the smaller the inner radius of the coil r1 the stronger the field
is obtained. Having the current course and the inductance of the
coil defined its magnetic energy can be calculated:
Wm (t) = Lcoil ·
Finally, formula (27) is to be applied and the central field can be
3. FE modelling of the coil
The classical theory of electromagnetism is fully represented
by the Maxwell’s equations and complementary constitutive
laws (Table 2).The present time-harmonic magnetic problem is
described by Eqs. (32)–(34), (36) and (37). Time-harmonic magnetic
problem means that H-field can be represented as the following:
H(t) = H 0 ejωt
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
Table 2
Differential form of Maxwell’s equations and complementary constitutive laws.
Maxwell’s equations
divD = divB = 0
rotE = − ∂B
rotH = j + ∂D
Constitutive laws
D = ε0 εE
B= 0 H
j = E
Omitting intermediate computations the diffusion equation for the
H-field can be written in the form of:
∇ 2 H = jω H(t)
Solutions of Eq. (39) for a semi-infinite space, different boundary conditions and the initial condition of Hz (r,z) = 0 for 0 < r < ∞
can be found in Knoepfel [8]. As shown in Meeker [13] the problem is numerically solved in vector potential formulation and with
the use of the Neuman boundary condition, i.e. when flux lines are
perpendicular to the boundary of the problem domain. Input data
of the models, their statuses, hardware configurations and solution
times are given in Table 3.
Fig. 7. Tangential component of B-field along the radius.
one of the verification steps of the analytical model. Results of the
2D modelling are represented below.
3.1. 2D modelling of the coil in FEMM
The analytical approach cannot describe a field distribution
pattern in the coil. For example in welding this means that an estimation of a range of impact velocities is impossible. This task can
be done by modelling of the coil in user-friendly FEMM software
developed by Meeker [13] and distributed under the Aladdin Free
Public License. Moreover the FEMM model can be considered as a
3.1.1. Central B-field
A visualization of the obtained magnetic field is represented in
Fig. 6. The absolute values of the fields along the central radius and
in the gap coil-WP are shown in Figs. 7 and 8 respectively.
The simulation allows defining the magnetic field strength in
every point within and outside of the working volume showing a
significant advantage over the analytical approach.
Table 3
Summary data of the FE-models.
Input data
f, [kHz]
Im , [kA]
, [MS/m]
Number of
Number of
time [h]
257 670
513 324
5 775 365
1 438 560
Intel® CoreTM i5-460 M CPU @ 2.53 GHz, 4.0 GB RAM,
Microsoft® Windows® 7
Intel® CoreTM i7-3930 K CPU @ 3.20 GHz, 64.0 GB RAM,
Microsoft® Windows® 7
Fig. 6. Contour plot of |B|-field in the coil.
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
Fig. 10. Current density distribution along the turns.
Fig. 8. B-field in the gap coil-WP.
3.1.2. Current density distribution in a turn
A visualization of the obtained current density distribution in
the coil is plotted in Figs. 9–11.
The current density distribution along the central parts of the
turns is quite similar whereas across the turns it drastically changes
at the corners. This is explained by the following effect: elementary currents in the corners are coupled with a smaller magnetic
flux than in the central part of the turn and as a consequence
the induced counter electromotive force in the corners is smaller
than in the centre. Practically the simulation can be used for an
optimization of the radiuses of the corners aiming a reduction
of an excessive current density and therefore preventing a local
The current density distribution in the lateral surfaces of the outermost turns is mainly influenced by the proximity effect which is a
result of an induction of the eddy currents in the conductor by magnetic fields generated by the neighbouring conductors. The effect
increases the resistance of the coil and intensifies with a decrease of
the distance between the conductors and an increase of an amount
of the turns.
3.1.3. Inductance and resistance of the coil
The magnetic energy of the coil is computed by FEMM in accordance with formula (40):
Wm =
where the integral is taken over the problem domain. As the current
flowing in the coil is known the inductance is found from (41):
Another method to derive the inductance is to use the “Circuit
Properties” button. For the present case the “Circuit Properties”
data is listed in Table 4.
The inductance is defined by the Flux/Current ratio found for
3 “effective” turns while the resistance is given by the Voltage/Current ratio obtained for 5 “effective” turns as the total current
path length is counted and there is no need to keep symmetry as in
the case with the inductance.
3.2. Parametric study of the coil by 3D FEM analysis
In the present study, we employed the commercial product
ANSYS Emag (part of ANSYS Academic Associate license) as the
most advanced tool of a complex 3D modelling of the coil in the
Fig. 9. Contour plot of current density distribution in the coil.
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
Table 5
List of equipment used in the experiments.
Parameter determined
Short circuit
Field probe
Rocoil FH-4015, Integrator IJ-1729,
Oscilloscope TiePie, Handyscope HS3
EELAB measuring coil
Resistance R
Inductance L
Resistance R = R(f)
Inductance L = L(f)
observed that the coil resistance decreases if the cross-section (the
thickness and the contact angle) increases and if a current path
length (the inner radius and the cut angle) shortens. In order to get a
higher B-field, it is highly important to reduce Joule heat generation.
3.3. Sensitivity analysis
Fig. 11. Current density distribution across the turns at a distance of 0.5 mm from
the inner surface of the turns.
linear harmonic regime [19]. To accurately calculate the current
density distribution, a FE model must have a mesh size at the conductor outer boundary smaller or roughly equal to the skin depth.
This will lead to a very large number of elements, and therefore the
model will become too demanding for a desktop PC. For instance,
at the operating frequency of 10 kHz and considering pure copper
with the resistivity of 2.7 × 10−8 Ohm·m, the skin depth is about
0.8 mm. To avoid a very large number of elements one need to use a
boundary layered mesh which consist of thin, elongated elements
at the boundaries of conductors. The boundary layered 3D mesh
was generated using commercial pre-processing software ANSA
[20]. The mesh was created with the SOLID236 element using 3D
edge-flux formulation. The computational air domain was truncated with a cylindrical volume with the outer radius 2r2 and the
open boundary was modelled with a flux-parallel boundary condition. The contact resistance between the plates is neglected in
the model. The electrical resistivity of aluminium and copper are
4.0 × 10−8 Ohm·m and 2.7 × 10−8 Ohm·m, respectively.
The main objective of the 3D FE-simulation is to perform a parametric study taking into account six input geometrical parameters:
inner radius r1 , outer radius r2 , turn thickness , connection thickness h, contact angle
and cut angle ϕ. Goal functions are the
coil resistance, the inductance, and the B-field. A mesh morphing
is used partially for geometrical modifications and a reuse of the
initial FE-model. The inductance of the coil was calculated from
the magnetic energy using formula (30). The power Prms dissipated
in a conducting coil body under the harmonic excitation can be
calculated as:
· j(x, y, z)2 dV
A local approach is applied to perform a sensitivity analysis by
taking a partial derivative of each output parameter Gj with respect
to an input parameter pi . This method examines small perturbations and a one parameter at a time. The obtained sensitivities are
depicted in Fig. 13.
These sensitivities reflect the calculated parametric dependences. All sensitivities except for the inner radius have negative
value. The outer radius and the thicknesses have minimum and
maximum sensitivities correspondingly.
3.4. Conclusions to 2D and 3D FEM simulation
The 2D numerical model (FEMM) has shown higher capabilities for description of the electro-magnetic design parameters than
the analytical approach as it takes into account the eddy currents
induced in the coil. Another important advantage of the numerical
model over the analytical one is an ability to compute the field at
any point of the space. Nevertheless the method doesn’t include
any strength or temperature estimation of the coil and therefore
cannot be used independently. Finally it can be concluded that the
numerical 2D model may complement the analytical one provided
that results of the calculation of each model are close.
The 3D numerical model using ANSYS Emag is the most
advanced tool of a complex 3D analysis of the coil as it can take
into account the asymmetry parameters of the coil. Moreover the
parametric and the sensitivity analyses are convenient way of the
design optimization.
On this step the analytical, 2D and 3D numerical inductances,
resistances and central magnetic fields and frequencies are defined.
In order to find out an accuracy of each of the models and to verify
the analytical approach experiments are needed.
4. Experimental verification of the methodology
Fig. 12 shows variations of the resistance and the inductance
with regard to a respective geometrical parameter. Green points
correspond to the initial values of the input parameters. It is clearly
For the experimental verification four complementary methods
were used (Table 5). Current curves obtained during short circuit
Table 4
Circuit properties.
3 “effective” turns
Total current [A]
Voltage drop [V]
Flux linkage [Wb]
Flux/current [H]
Voltage/current [Ohm]
Real power [W]
Reactive power [VAr]
Apparent power [VA]
5 “effective” turns
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
Fig. 12. Results of the parametric study of the Bitter coil.
Fig. 13. Results of sensitivity analysis of the Bitter coil.
experiments were the basis for a determination of the average resistance R and the inductance L of the coil. B-field was measured by
the field probe developed by Electrical Energy Laboratory (EELAB),
University of Gent, and finally the resistance and inductance of the
coil were measured in a range of frequencies 20 Hz–150 kHz using
an RLC-bridge.
5. Discussion
The results of the analytical and the numerical calculations as
well as the experimental parameters of the coil are summarized in
Table 6.
As it can be seen from the Table 6, the short circuit
experiment and the RLC-bridge measurement resulted in different inductances and resistances. A possible explanation is
that the RLC-bridge measurement describes a steady-state process with the constant resistance and the inductance while the
corresponding short circuit values describe a transient process
with the instantenious resistance and the inductance which are
significantly influenced by characteristics of switches of a generator. Based on the above-mentioned facts one can conclude that
the RLC-bridge measurement is more suitable for the verification
of the FE models with the time-harmonic approximation of the
coil behaviour. In reality the coil works in the transient regime
which has not been modelled in the present work. Each generator is characterized by a unique transient behaviour and therefore
it is challenging to build one model which can describe different
transient processes.
5.1. Active resistance
Active resistances obtained analytically and experimentally
during the short circuit measurement are relatively alike which
shows a good capability of the analytical approach to describe
behaviour of the coil attached to the generator (Table 7). At the same
time the resistance obtained during the RLC-bridge measurement
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
Table 6
Summary of calculated and measured values.
Calculation technique
Numerical 2D FEM (FEMM)
Numerical 3D FEM (ANSYS Emag)
Verification technique
Short circuit
Field probe
Active resistance to alternating
current Rac , [mOhm]
Inductance L [nH]
Table 7
Parameters of the generator.
B-field [T]
Frequency of
the field f [kHz]
Table 8
Discharge currents and relative errors.
C [␮F]
Umax , [kV]
Ri [mOhm]
Li [nH]
f0 [kHz]
is close to both numerical 2D and 3D resistances which in turn
verifies the numerical models.
5.2. Inductance
Inductances obtained analytically, by both numerical methods
and by the short circuit experiment are relatively close but smaller
than the value from the RLC-bridge measurements. One of the possible explanations of the differences has been mentioned above.
5.3. Central magnetic field
The analytical, both numerical and the experimental fields are
close to each other, which is a positive verification fact.
In order to find out if the differences between the calculated and
the experimental parameters are appropriate (Table 6) behaviour
of the coil connected to the generator is simulated by solving the
differential equation of damped current oscillations (26) for the
measured RLC-bridge and the analytically calculated parameters
and a comparison of the obtained current curves with the short
circuit experiment is made (Fig. 14).
First quarters of periods of the experimental and each of
the simulated damped current oscillations, practically defining a
Rise time [␮s]
current Im [A]
Short circuit
Error relative to short
experiments [%]
Error relative to B-field
measurement [%]
technological effect on the WP, and errors of the analytical calculation relative to the short circuit current and the solution for the
measured RLC-bridge values are represented in Table 8.
The analytically calculated current shows the smallest errors
relative to the solution for the measured RLC-bridge values,
therefore this experimental technique is preferable in the present
case. Based on this fact it can be concluded that the simplified
methodology of the Bitter coil analytical calculation is verified on
the basis of the RLC-bridge measurement with relative errors of 9%
in the central field, 10% in the rise time and 15% in the amplitude
current (Table 8).
In the present work the equivalent inductance (18), resistance
(22) and the frequency of the electromagnetically coupled coilWP system were not checked experimentally. In practice at the
same discharge energy the coil-WP system will generate a higher
frequency and a current than the coil used separately due to a
smaller equivalent inductance and therefore obtaining the maximum allowable rise time less than 25 ␮s only in the coil can
overcome difficulties of the exact coil-WP inductance estimation
and be satisfying. Moreover the maximum current amplitude Im
increases with an increase of the charging energy of the capacitors.
This fact can be used to compensate relatively small differences
between the experimental and calculated current amplitudes.
6. Conclusions
Fig. 14. Current curves obtained with short circuit, RLC-bridge and analytically
defined inductances and resistances.
The industry orientated methodology for the simplified analytical calculation of the pulsed Bitter coil has been developed. Coil
asymmetry characteristics implementation allowed increasing a
precision of the analytical models describing main design parameters of the coil. Additionally the 2D and the 3D FEM model have
been developed aiming to partly verify the analytical approach as
well as to get a deeper insight into the design parameters. Based
on the methodology a Belgian Welding Institute (BWI) Bitter coil
has been designed and tested by means of short circuit experiments, impedance and B-field measurements. Differences between
the calculated and the experimental B-fields, rise times and amplitude currents were found to be 9%, 10% and 15% correspondingly,
which allows drawing a conclusion of a positive verification of
the methodology. Therefore the developed methodology can be
O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562
practically used for the Bitter coils design, including determination
of geometrical, thermal and load-bearing boundaries of the coil and
1. Iterative determination of the geometry and the material of the
coil until they meet the requirements specification.
2. Calculation of the two main electromagnetic parameters of the
coil: the inductance and the resistance which along with the
parameters of the generator form a current pulse.
3. Determination of the maximum allowable magnetic field generated by the coil based on the thermal and the strength properties
of the conductor material, the current density distribution in it
and on the geometry of the coil.
4. Determination of the demanded current pulse satisfying the performance specification.
Additional use of FEMM overwhelms some of the limitations of
the analytical approach mainly due to its ability to calculate the field
at any point of the space. FEMM can also play a role of a fast verification tool for such parameters as the B-field, the resistance and the
inductance of the coil. Finally if time and resources are not limited
ANSYS Emag can be used as an advanced analysing tool taking into
account multiphysical interactions and the asymmetry parameters
of the coil which makes this tool the most comprehensive in the
design optimization and refining.
This work has been done within the ACODEPT (Advanced Coil
Design for Electromagnetically Pulsed Technologies) funded by
the European Commission within the CORNET programme. The
CORNET promotion plan 61 EBR of the Research Community for
European Research Association for Sheet Metal Working has been
funded by the AIF within the programme for sponsorship by
Industrial Joint Research (IGF) of the German Federal Ministry of
Economic Affairs and Energy based on an enactment of the German
Parliament. The authors would like to thank Mitko Bozalakov for
help in conducting the impedance and B-field measurements and
EELAB of University of Gent for providing us with the measuring
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Oleg Zaitov received a bachelor’s degree and a master’s degree in welding engineering from Ufa State Aviation Technical University, Russia, in 2007 and 2009
correspondingly. During 2009–2010 he was working as a researcher in the field of
linear friction welding at the Department of Welding Engineering at Ufa State Aviation Technical University. In 2011 he joined Belgian Welding Institute as a research
engineer with the main focus on magnetic pulse welding (MPW) developments. His
research interests are a development of mathematical models for preliminary weldability assesment in MPW, a coil design and its optimisation for MPW. He has been
writing his PhD within these topics.
Vladimir A. Kolchuzhin received a bachelor’s degree and a master’s degree in
electrical engineering from Novosibirsk State Technical University, Russia, in 1997
and 1999, respectively. During 1999–2003 he was working as a Scientific Assistant
at the Department of Semiconductor devices and Microelectronics at Novosibirsk
State Technical University. In Nov. 2003 he joined the Department of Microsystems
and Precision Engineering at Chemnitz University of Technology, Germany where
in 2010 he received his Doctoral degree. He has been working in the field of the
advanced modelling methods development for MEMS. His research interests are
numerical methods for the nano- and microsystems design.