This paper presents a new Finite Element numerical method to analyze the coupling between twisted filaments in a superconducting multifilament composite wire. To avoid the large number of elements required by a 3D code, the proposed method makes use of the energy balance principle in a 2D code. The relationship between superconductor critical current density and local magnetic flux density is implemented in the program for the Bean and modified Kim models. The modeled wire is made up of six filaments twisted together and embedded in a lowresistivity matrix. Computations of magnetization cycle and of the electric field pattern have been performed for various twist pitch values in the case of a pure copper matrix. The results confirm that the maximum magnetization depends on the matrix conductivity, the superconductor critical current density, the applied field frequency, and the filament twist pitch. The simulations also lead to a practical criterion for wire design that can be used to assess whether or not the filaments are coupled

Bouillault, F

Devred, Arnaud

Leroy, Daniel

Satiramatekul, T

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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007 3737Magnetization Modeling of TwistedSuperconducting FilamentsT. Satiramatekul, F. Bouillault, A. Devred, and D. LeroyAbstract—This paper presents a new Finite Element numericalmethod to analyse the coupling between twisted filaments in a su-perconducting multifilament composite wire. To avoid the largenumber of elements required by a 3D code, the proposed methodmakes use of the energy balance principle in a 2D code. The rela-tionship between superconductor critical current density and localmagnetic flux density is implemented in the program for the Beanand modified Kim models. The modeled wire is made up of six fil-aments twisted together and embedded in a low-resistivity matrix.Computations of magnetization cycle and of the electric field pat-tern have been performed for various twist pitch values in the caseof a pure copper matrix. The results confirm that the maximummagnetization depends on the matrix conductivity, the supercon-ductor critical current density, the applied field frequency, and thefilament twist pitch. The simulations also lead to a practical crite-rion for wire design that can be used to assess whether or not thefilaments are coupled.Index Terms—Coupling currents, energy balance principle,magnetization, superconducting multifilament composite wire,twist.I. INTRODUCTIONTHE numerical simulation reported here allows us to de-fine the coupling-decoupling phenomena between super-conducting filaments. We have shown [1] that the magnetizationobtained in the case of uncoupled superconducting filamentsis smaller than the one obtained for coupled superconductingfilaments.In practical multifilament composite wires, the supercon-ducting filaments are embedded in a low-resistivity matrix thatis much less conducting than the superconductor, for example,OFHC copper. Then, there are currents circulating between thefilaments via the conducting matrix, which result in a filamentcoupling. This coupling can be greatly reduced by twistingthe filaments together [2]. The shorter the twist pitch, theless coupled the filaments. An electromagnetic theory of thisproblem is presented in [3].This paper deals with the electromagnetic coupling under ex-ternal field, under self field the coupling is different and thetwisting plays no part. We use a method based on the principleof energy balance to solve the problem of twisted filaments. WeManuscript received August 19, 2006.T. Satiramatekul is with the Faculty of Engineering, Kamphaengsaen, Kaset-sart University, Kamphaengsaen, Nakhon Pathom 73140, Thailand (e-mail: thi-tipong.s@ku.ac.th).F. Bouillault is with LGEP, CNRS UMR 8507, SUPELEC, Plateau deMoulon, 91192 Gif-sur-Yvette Cedex, France.A. Devred is with CEA, DSM/DAPNIA/SACM, 91191 Gif-Sur-YvetteCedex, France, and CERN, AT-MAS, CH-1211, Geneva 23, Switzerland.D. Leroy is with CERN, AT-MAS, CH-1211, Geneva 23, Switzerland.Digital Object Identifier 10.1109/TASC.2007.899414Fig. 1. 2D representation of six-filament wire over one sixth of a twist pitch.Fig. 2. Position of six filaments in the planes z = 0 and z = p=12.also study how the dependence of the current density on themagnetic flux density affects the magnetization of twisted su-perconducting filaments.II. TWISTED WIRE MODELIn order to understand the influence of the twisting on themagnetization, we start by considering a composite wire madeup of six superconducting filaments, twisted together and em-bedded in a low-resistivity matrix.Let us consider the 2D representation given in Fig. 1 of thecurrents and voltages along a sixth of the filament twist pitch, ,as well as the positions of the six filaments in the planesand illustrated in Fig. 2. Given the problem geom-etry, we assume that the electrical potential, , along filament, is of the form(1)where is a reference potential.1051-8223/$25.00 © 2007 IEEE3738 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007In addition, for the currents, we assume that(2)(3)In order to avoid a three-dimensional modeling, we dividethe wire into elementary sections of length. Given theweak slope of the filaments, we assume that, in each section,the problem can be treated as two-dimensional and that thecurrents are independent of . Then, the difference of the cur-rents in the filaments for two successive sections is equal to thecurrent circulating in a section of matrix of length sub-jected to the adequate potentials.By using the principle of energy balance [4] in the planein Fig. 1, we derive(4)where is the power dissipated in a section of conducting ma-trix. In this case, we simply have(5)and(6)where is the electrical conductivity of the matrix and is adimensionless parameter obtained by solving the 2D harmonicLaplace equation, which in our formulation corresponds to:. The equation is solved by the finite element methodwith the boundary conditions(7)By substituting the voltages in (4), we obtain get(8)Equation (8) can be re-written in the following matrix form(9)where(10)(11)where is defined in Fig. 3 and is the current vector in thefilaments at . The currents can be derived from [1](12)with(13)Fig. 3. Definition of the distance d for one sixth of a twist pitch.Fig. 4. Current density distribution in conducting matrix (in A=m ).where is the magnetic permeability of vacuum, is the elec-trical field vector, and and are matrices whose di-mensionless coefficients take into account the wire geometry.The matrix system resulting from the discretization ofMaxwell’s equations in the case of the twisted filaments is [1](14)(15)where is the current density vector, is the source vector,is the mass matrix, and is a dimensionless matrix.On the left side of (15), we find that, if the first term is verysmall compared to the second term, we are in the case of un-coupled filaments. Conversely, if the first term is very largecompared to the second term, we are in the case of coupledfilaments [1].In addition, the current density in the conducting matrix canbe derived from(16)Fig. 4 shows the simulation results in the plane .In order to compute the total magnetization of a multifilamentwire, let us consider the 2D drawing of the currents in Fig. 5. Wefind that there are two contributions: one from the currents in thefilaments, and one from the currents in the conducting matrix.SATIRAMATEKUL et al.: MAGNETIZATION MODELING OF TWISTED SUPERCONDUCTING FILAMENTS 3739Fig. 5. 2D Drawing of currents in wire with twisted filaments.The total magnetization per superconductor unit volume is(17)(18)where is the wire length and is the superconductor cross-sectional area, and and are the magneti-zations due to the currents circulating in the superconductingfilaments and in the conducting matrix.In our case, it can be shown that(19)which yields(20)where depends on the filament twist pitch.III. SIMULATION RESULTS AND DISCUSSIONThe finite elements code applied for superconductors was de-veloped by LGEP. Our work is based on this code. The methodconsists in using a model with two slopes instead of Bean’smodel [5]. But for the superconductors at high critical tempera-ture, the real characteristic differs rather clearly from thatof Bean’s model, a more realistic model can be obtained by aprogressive function [6]. We modeled a wire made up of sixsuperconducting filaments twisted together and embedded inan OFHC copper matrix at 4.2 K). Atime-varying magnetic flux density was applied vertically witha maximum amplitude of 1 T and a frequency of 1 Hz. Initially,the simulations were made according to Bean’s model and as-suming a constant critical current density of .In order to study the influence of twisting on the coupling be-tween filaments and on the magnetization, we varied the fila-ment twist pitch of the filaments, , from 0.03 mm to 300 mm.Fig. 6 illustrates the current density distributions in the six fil-aments in the plane for two extreme twist pitch values.In Fig. (6a) , we can see the case of perfectlycoupled filaments, while in Fig. (6b) , we canFig. 6. Current density distributions at t = T=4 for (a) p = 300 mm and (b)p = 0:03 mm (using Bean’s model).Fig. 7. Current density distributions at t = T for (a) p = 300 mm and (b)p = 0:03 mm (using Kim’s model).see the case of perfectly decoupled filaments. When decoupled,the filaments carry their own go and return currents, whereascoupled filaments behave like one monofilament conductor. Forintermediate values, we can observe cases of partial filamentcoupling.In order to take into account the dependence of the criticalcurrent density on the magnetic flux density, a second set ofsimulations were carried out using Kim’s model, withwith and[1]. The modeled domain, the matrix conductivity valueand the applied magnetic flux densities are the same ones asthose used previously. Fig. 7 shows the current density distribu-tions in the six filaments in the plane for the same twoextreme cases, where again observe perfect coupling and decou-pling [2].Figs. 8 and 9 compare the magnetization cycles obtainedusing Bean’s and Kim’s model for three different twist pitchvalues. We can clearly see the deformation of the cycle in thecase of partially coupled filaments compared to the extremecases of perfect coupling and uncoupling. In the case of Kim’smodel, the deformation is more important at high magnetic fluxdensities than at low magnetic flux densities. This deformationis due to the current density distribution in the filaments [4].From the magnetization cycles, we can deduce the maximummagnetization value. According to our computations, we have3740 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007Fig. 8. Magnetization cycles obtained from Bean’s model.Fig. 9. Magnetization cycles obtained from Kim’s model.confirmation that this value depends on the electrical conduc-tivity of the conducting matrix, on the critical current density,on the frequency of the applied magnetic induction, and on thetwist pitch of the filaments. From (15), it appears that a good in-trinsic parameter to characterize the effect of twisting for a givencritical current density is: or: , where is the skineffect in the conducting matrix. Fig. 10 shows that, for Bean’smodel with , the filaments are cou-pled together when is higher than 5.60. On the other hand,if this value is lower than 0.056, the filaments can be consid-ered as decoupled. Then, in practice, if copper is used for theconducting matrix, it is necessary that the filament twist pitchbe lower than 1 mm to unsure that so that the superconductingfilaments are decoupled. When relying on Kim’s model, the fil-aments are coupled if the value is higher than 1.90, and de-coupled if the value is lower than 0.056. Note that this lowerlimit is the same as the one obtained with Bean’s model, and thatin our computation where , weFig. 10. Maximum magnetization versus p= ratio.find that the maximum magnetization obtained by Kim’s modelis always smaller than that obtained by Bean’s model. The dif-ference between two values is more important in the case ofcoupled filaments than for uncoupled filaments.IV. CONCLUSIONWe studied the effects of superconducting filament twisting incomposite wires. We proposed a method based on the principleof energy balance to solve this problem in 2D. Simulations werecarried out using Bean’s model and Kim’s model. In both cases,we could express the maximum magnetization as a function ofa reduced parameter directly proportional to the filament twistpitch. The value of this parameter enables to assess the where ornot the superconducting filaments are coupled together throughthe low-resistivity matrix of the wire.REFERENCES[1] T. Satiramatekul and F. Bouillault, “Magnetization of coupled and non-coupled superconducting filaments with dependence of current densityon applied field,” IEEE Trans. Magn., vol. 41, pp. 3751–3753, Oct.2005.[2] M. N. Wilson, Superconducting Magnets. Oxford: Oxford UniversityPress, 1983.[3] W. J. Carr, Jr., “Electromagnetic theory for filamentary superconduc-tors,” Phys. Rev. B, vol. 11, pp. 1547–1554, Feb. 1975.[4] T. Satiramatekul, F. Bouillault, and L. Santandrea, “Modeling of mag-netization curves of partially coupled superconducting filaments withdependence of current density according to applied magnetic field,”IEEE Trans. Magn., vol. 42, pp. 887–890, Apr. 2006.[5] M. Maslouh, F. Bouillault, A. Bossavit, and J. C. Verite, “From Bean’smodel to the H-M characteristic of a superconductor: Some numericalexperiments,” IEEE Trans. Appl. Supercond., vol. 7, pp. 3797–3800,Sep. 1997.[6] N. Banno and N. Amemiya, “Numerical analysis of AC loss in highT twisted tape carrying AC transport current in external AC magneticfield. Effect of twisting on loss reduction,” IEEE Trans. Appl. Super-cond., vol. 9, pp. 2561–2564, June 1999.

Magnetization Modeling of Twisted Superconducting Filaments

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