Journal of Manufacturing Processes 16 (2014) 551–562 Contents lists available at ScienceDirect Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro Technical Paper Bitter coil design methodology for electromagnetic pulse metal processing techniques Oleg Zaitov a,∗ , Vladimir A. Kolchuzhin b a 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 Keywords: 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.  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  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  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.  proposed an approach to a material selection for such a coil. Knoepfel  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. . A relatively precise and complete design of the coil can be fulﬁlled in software developed by Vanacken et al. . 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 . Knoepfel  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 http://dx.doi.org/10.1016/j.jmapro.2014.07.008 1526-6125/© 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. 552 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 . 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  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 length. 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) 551–562 553 Table 1 List of symbols. Symbol Unit Description Symbol Unit Description Bmax B0 r0 r l c y m W I0 f0 Li Ri UTS Ti Tf r1 r2 ı j jth L R U H E T T s mm mm mm MS/m W/m·K J/kg·K MPa kg/m3 J A kHz nH mOhm MPa K K mm mm mm mm A/m2 A/m2 nH mOhm V A/m V/m 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 Density 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 H-ﬁeld Electric ﬁeld strength h D N i b ϕ mm C/m2 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 Form-factors 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 reached. 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. deg deg deg ˛ ˇ Effective number of turns Self-inductance factor Resistivity 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 Capacitance 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) Ohm·m A A Im Imax H/m rad/s T T 0 ω B Bth C ωd Lcoil Rcoil Rac g FMat (Ti ,Tf ) ε0 F rad/s H mOhm mOhm mm A2 s/m4 F/m 1. Similar to Izhar and Livshiz  method, “effective” intermediate M and end X turns can be found from the following expressions: M= 360◦ − (ϕ + 360◦ X= 360◦ − (ϕ/2 + 360◦ ) (1) + ) (2) Then total number of efﬁcient turns consists of “effective” intermediate and endplates is found from expression (3): =i·M+b·X (3) Therefore the active length of the coil is found from (4): lcoil = · + ( − 1) · h (4) 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  the form-factors of the ideal coil are found from (5) and (6): ˛= r2 r1 (5) ˇ= lcoil 2 · r1 (6) 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: r2 = r1 r2 r1 dr dr 0 lcoil 0 f (r, z)dz (7) f (r, z)dz The numerator describes the current density distribution in the conductor and the denominator describes the distribution 554 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: = · lcoil 5. The skin depth is a well-known characteristic describing an attenuation of electromagnetic waves in a conductor is found from (10): (8) 4. The maximum current in the coil can be approximated as (9): Imax Bmax · lcoil = · 0 (9) ı= 2· · 0·ω (10) O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 555 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 ): Tf FMat (Ti , Tf ) = Ti m · c(T ) .dT (T ) (11) 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.  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.  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  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 = 1 · r2 · 1 ln(˛) ln(˛) √ UTS · 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 . UTS / 0 0 (12) (13) 2.2.2. Thermal limitation According to Lagutin and Ozhogin  the thermal limitation can be represented as an upper bound of a temperature rise in a skin layer during a pulse. Knoepfel  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 . For example, according to Mathers  the upper temperature must not exceed 200 ◦ C Bth = FMat (Ti , Tf ) tpulse · ς 0 · · r2 · jth (14) (15) Finally according to Kratz and Wyder  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(˛) (16) 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  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.  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  resulted in the best precision (17) Lcoil = 2 · 0 · r1 · (˛, ˇ) 4· (17) 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 556 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) dI(t) + ω2 I(t) = 0 + 2ˇ dt dt 2 (24) 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 (25) U −ˇt e sin(ωd t) Lωd (26) 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 . Shneerson , one can calculate the equivalent inductance of the Bitter coil as (18), when the conditions (19) are met: L˙ ≈ 2· · 1+ 2 ·· 4·g · 0 · r0 · g · a1 + ln r0 g ·r0 (18) 4·g 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  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). (19) The ﬁrst resistance calculation was made according to Kratz and Wyder  based on the equation of Ohmic power (Joule heat) in the coil: P = I 2 Rcoil (20) B01 = G(˛, ˇ) = 0 · Wm · G(˛, ˇ) r1 2 · (˛, ˇ) (27) r 2 f (r, z) (r 2 + z 2 ) 3/2 drdz/ f (r, z)drdz (28) This is the resistance to a constant current neglecting the contact resistances between the plates: Rcoil = N 2 · · · r1 · ˇ · ln(˛) (21) There are contacts in the coil and their resistance can also be taken into account. As shown in Glebov et al.  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  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 (23) 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) = lcoil r1 ⎜ 2 1 r ln ⎜ ⎝ 2 (˛, ˇ) lcoil · ln l r 1 coil r2 + + l2 coil r2 1 l2 coil r2 2 ⎞ +1 +1 ⎟ ⎟ ⎠ (29) 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 · I(t)2 2 (30) Finally, formula (27) is to be applied and the central ﬁeld can be found. 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 (38) O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 557 Table 2 Differential form of Maxwell’s equations and complementary constitutive laws. Maxwell’s equations divD = divB = 0 rotE = − ∂B ∂t rotH = j + ∂D ∂t Constitutive laws D = ε0 εE B= 0 H j = E (31) (32) (33) (34) (35) (36) (37) Omitting intermediate computations the diffusion equation for the H-ﬁeld can be written in the form of: ∇ 2 H = jω H(t) (39) 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 . As shown in Meeker  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  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. Model Software Input data f, [kHz] Im , [kA] , [MS/m] 2D FEMM 10 75 25 3D ANYS Emag Number of nodes Number of elements Hardware Solution time [h] 257 670 513 324 0.03 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. 6 558 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 overheating. 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 = 1 2 BHdV (40) where the integral is taken over the problem domain. As the current ﬂowing in the coil is known the inductance is found from (41): L= 2Wm I2 (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) 551–562 559 Table 5 List of equipment used in the experiments. Method Equipment Parameter determined Short circuit experiment Field probe RLC-bridge Rocoil FH-4015, Integrator IJ-1729, Oscilloscope TiePie, Handyscope HS3 EELAB measuring coil HAMEG HM 8118 Resistance R Inductance L B-ﬁeld 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 . 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 . 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: Prms 1 = 2 · 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. (42) 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. Parameter Value 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 75,000 138.298+i8474.16 0.133535−i0.000749625 1.78047e−006−i9.995e−009 0.00184397+i0.112989 5.18616e+006 3.17781e+008 3.17823e+008 333.663+i19314.1 0.3045−i0.000877415 4.058e−006−i1.16989e−008 0.00444884+i0.257521 1.25124e+007 7.24278e+008 7.24386e+008 560 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) 551–562 561 Table 6 Summary of calculated and measured values. Method Calculation technique Analytical Numerical 2D FEM (FEMM) Numerical 3D FEM (ANSYS Emag) Veriﬁcation technique Short circuit RLC-bridge Field probe Active resistance to alternating current Rac , [mOhm] Inductance L [nH] 6.84 4.44 3.48 1548 1780 1541 7.7 4.6 1503 2309 Table 7 Parameters of the generator. B-ﬁeld [T] Central Outermost 1.36 1.46 1.44 2.3 2.17 1.5 1.9 Frequency of the ﬁeld f [kHz] 10 10 10 Table 8 Discharge currents and relative errors. Parameter Value C [F] Umax , [kV] Ri [mOhm] Li [nH] f0 [kHz] 160 25 2.95 42.58 60.7 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 Method Rise time [s] Amplitude, current Im [A] Short circuit RLC-bridge Analytical Error relative to short circuit/RLC-bridge experiments [%] Error relative to B-ﬁeld measurement [%] 29.5 28 25 6/10 75,077 78,824 91,018 23/15 9 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 562 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 includes: 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. Acknowledgments 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 equipment. References  Dransfeld K, Haidu J, Herlach F, Landwehr G, Maret G, Miura N, et al. Strong and ultra-strong magnetic ﬁelds and their applications. Springer Verlag; 1985.  Furth HP, Levine MA, Waniek RW. Production and use of high transient magnetic ﬁelds. Rev Sci Instrum 1957;28(11):949–58.  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Release 14.0, Help System, Low-Frequency Electromagnetic Analysis Guide. ANSYS Inc.; 2011.  http://www.beta-cae.gr/ansa.htm 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.