Effect of self-field on the current distribution in Roebel

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Effect of self-field on the current distribution in Roebel-assembled coated conductor cables

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 Supercond. Sci. Technol. 24 095002 (http://iopscience.iop.org/0953-2048/24/9/095002) View the table of contents for this issue, or go to the journal homepage for more

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IOP PUBLISHING

SUPERCONDUCTOR SCIENCE AND TECHNOLOGY

Supercond. Sci. Technol. 24 (2011) 095002 (8pp)

doi:10.1088/0953-2048/24/9/095002

Effect of self-field on the current distribution in Roebel-assembled coated conductor cables M Vojenˇciak1,2, F Grilli1 , S Terzieva1,4 , W Goldacker1 , M Kovácˇ ová3 and A Kling1 1 Karlsruhe Institute of Technology, Institute for Technical Physics, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany 2 Institute of Electrical Engineering, Slovak Academy of sciences, Dubravska cesta 9, 841 04 Bratislava, Slovakia 3 Slovak University of Technology, Faculty of Mechanical Engineering, Nam. Slobody 17, 812 31 Bratislava, Slovakia

E-mail: [email protected]

Received 1 March 2011, in final form 14 June 2011 Published 19 July 2011 Online at stacks.iop.org/SUST/24/095002 Abstract Roebel cables are a promising solution for high current, low AC loss cables made of high-temperature superconductors in the form of coated conductors. High current creates significant self-field, which influences the superconductor’s current-carrying capability. In this paper, we investigate the influence of the self-field on the cable’s critical current and the current repartition among the different strands. In order to investigate the cable’s critical current, we analysed the influence of flux creep on the cable properties. Using the experimental material’s properties derived from measurements on a single conductor as input for our calculations, we were able to predict the critical current of the cable in two limiting situations: good current sharing and complete electrical insulation among the strands. The results of our calculations show good agreement with the measured critical current of three Roebel cable samples. (Some figures in this article are in colour only in the electronic version)

HTS are very sensitive to magnetic field at temperatures close to their critical temperature. Therefore, the self magnetic field significantly influences the critical current density of high current cables [3, 4]. The JC (B) dependence of the critical current density on the magnetic field of the 2G HTS is anisotropic, and its complex shape partly depends on the fact that, besides the intrinsic anisotropy of the pinning force, the superconductor also contains artificial pinning centres [1]. Even though each strand of the cable experiences equal conditions in long length scale, it changes its position with respect to other strands within one transposition length. Therefore the superconductor is exposed to different magnetic fields along one transposition length. The critical current density is influenced by the self magnetic field in this way. The effect of self-field on critical current has been already investigated on cables made of low-temperature superconductors in [5]. In that work, it was pointed out that the calculation of the cable’s self-field is necessary for estimating

1. Introduction High-temperature superconductors (HTS) are promising materials for applications like motors, generators and also fusion magnets. REBCO coated conductors, the so-called second generation HTS (2G HTS), have recently achieved significant improvement in current capability and long length production [1]. However, in many applications, currents in the range of kA are required. Several conductors have to be connected in parallel to achieve such currents. A superconducting Roebel cable seems to be a viable solution for connecting a large number of coated conductors [2]. This cable is flexible and the transposition of the strands provides equal electrical and mechanical conditions for each strand, in long length scale. 4 On leave from: Georgi Nadjakov Institute of Solid State Physics—BAS,

Sofia, Bulgaria. 0953-2048/11/095002+08$33.00

1

© 2011 IOP Publishing Ltd Printed in the UK & the USA

Supercond. Sci. Technol. 24 (2011) 095002

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either to zero or to the critical value JC . This results in the fact that the critical current is reached when the superconductor carries its critical current density in the whole cross-section. For numerical simulations we used the 2D magnetostatic model of the commercial software package Comsol [8]. Infinitely long straight conductors are assumed, with current direction perpendicular to the plane. The state variable is the vector potential A, which in 2D has only one component perpendicular to the plane. Following the procedure described in [9] we found the distribution of the vector potential A, of the magnetic flux density B and of the current density J satisfying a given constraint: the dependence of the current density on the local magnetic flux density. The superconductor was modelled with its real dimensions: a thickness of 1.4 μm, and widths of 4 mm and 1.98 mm for the single tape simulation and the cable simulation, respectively. A mesh with rectangular elements was used for the superconducting area with a discretization of 100 × 10 elements for the conductor’s width and thickness, respectively. We used the same formulae as in [10] for the description of the superconductor’s critical current density JC dependence on the local magnetic field B and its direction θ with respect to the wide face of the superconducting tape, roughly corresponding to the superconductor’s crystallographic ab -planes. For the convenience of the reader we list the formulae below. The following equation describes three kinds of magnetic flux pinning in the superconductor and combines smoothly their influence:

the cable’s critical current. As is shown later in this paper, the distribution of the magnetic field and of the critical current density has consequences for the shape of the overall current– voltage characteristic of the cable. The self-field of Roebel-assembled coated conductors has been recently investigated by various techniques [3, 4]. In those papers the critical current distribution was investigated in a 2D cross-section of the cable under the assumption of equal current distribution among the strands. In [4] the authors found the maximum perpendicular component of the magnetic flux density in the cable and constructed the load line of the cable. Then they compared that load line with the dependence of the critical current on the applied magnetic field measured on the single conductor. In [3] the authors modified a method proposed in [6] for a single tape. The authors of [3] kept the current equal in all the strands by a numerical procedure which leaves the parts with a low magnetic field without current. This is in a certain way similar to ramping up the current in the cable. If the ramping is stopped when a part of the cable reaches the critical current, some parts of the cable are still current free. In this work we investigate the complete transition of the superconductor to normal state. We describe the JC (B) dependence of the superconductor in detail and, in contrast to [3], we investigate the properties of the superconductor until the whole cross-section is filled by the current. A subsequent analysis allows us to construct a smooth current–voltage curve ( E – I curve) of the cable. In this paper we first describe our model of the superconductor used for calculation by the finite element method (FEM). This model describes the dependence of the critical current density on the magnetic field. Then we analyse the electric field distribution along the cable length in section 2. In section 3 we investigate the influence of the magnetic field on a single superconducting tape. The information from this experimental and numerical investigation is used as input for calculating the cable’s critical current in section 4. The main conclusions of this work are summarized in section 5.

1

JC = (JCmab + JCmc + JCmi ) m

(1)

where JCab , JCc and JCi are defined by formulae (2)–(4) and m is constant. J0 p JCab = (2) (1 + BBloc0abfab )β

JCc =

2. Calculation method

JCi =

In this section we present the two models used in our calculations. First we describe the 2D FEM model used for the calculation of the current density distribution in the cable cross-section (section 2.1). In this part we use a formulation based on the critical state model with a vector potential as a state variable. This model is used for simulation of the cable with good current sharing. Then we present the model for computing the electric field along the cable’s length (section 2.2). In this model we assume that the current is evenly distributed in all strands, but electric field appears along the strand. For the calculation of the electric field we use the power-law characteristic. This model is used for the simulation of a cable with completely insulated strands.

J0 p (1 +

Bloc f c β ) B0c

J0i (1 +

Bloc f i α B0i )

(3)

(4)

where the functions f ab , fc , f i define the dependence on direction of magnetic field: if θloc ∈ [−90◦ + δab , 90◦ + δab ] f ab0 (5) f ab = f abπ otherwise

f ab0 = cos2 (θloc − δab ) + u 2ab sin2 (θloc − δab ) f abπ = v 2 cos2 (θloc − δab ) + u 2ab sin2 (θloc − δab ) f c = u 2c cos2 (θloc − δc ) + sin2 (θloc − δc ) f i = cos2 (θloc + δc ) + u 2i sin2 (θloc + δc ).

2.1. The 2D FEM model

(6) (7) (8) (9)

The homogeneous external field is applied by appropriately choosing the vector potential AB on the boundary of the

Our calculation is based on Bean’s model of the critical state [7]—the current density in the superconductor is equal 2


M Vojenˇciak et al

of a single tape and in the case of the tape used in the cable, we propose to use the average pinning force as a quantity directly related to the factor n . From the measurements of the E – I curve on the single tape in various magnetic fields we estimated the n factor. For a given applied magnetic field and its angle, we performed the FEM calculation of the critical current, as is described in section 2.1. Using the results of the FEM model calculation, we estimated the components of the local pinning force density parallel ( f px ) and perpendicular ( f py ) to the wide face of the tape as f px = |J · B y | (12)

Figure 1. Definition of the magnetic field components and angle in the coordinate system used in the calculations.

f py = |J · Bx |

where J is the local current density in the superconductor, Bx and B y are the components of the local magnetic flux density parallel and perpendicular to the wide face of the tape (see figure 1), respectively. We used the absolute value in equations (12) and (13) because the actual orientation of the force is not important for our purpose. We estimated the average pinning force density on the superconductor’s crosssection SSC as f px d S S Fpx = SC (14) Ssc f py d S S Fpy = SC . (15) Ssc For the description of the relation between pinning force and n -factor we used the following fitting function:

simulated domain. In this model, a field with strength Ha and angle θa is obtained by setting

AB = μ0 Ha (y · cos(θa ) − x · sin(θa ))

(13)

(10)

where y and x are the local coordinates of the boundary. The coordinate system and angle definition used is schematically shown in figure 1. 2.2. Electric field calculation For the calculation of the electric field along the cable’s length, the model for describing the superconductor was changed from the critical state model to a smooth nonlinear power-law E – I curve. At liquid nitrogen temperatures (64–77 K) thermal fluctuations cause slow movement of the pinned fluxoids in the direction of the Lorentz force. This effect is known as flux creep [11]. The shape of the E – I curve can be described by equation (11) at currents close to the critical current in a form where the electric field along the conductor E depends on the transport current I : n I E = EC . (11) IC

n = a + b(cFpx + Fpy )d

(16)

with parameters a , b , c and d . This equation form does not have any physical meaning and it is used just to fit the data for the measured n factor, in order to have an analytical expression for n(Fpx , Fpy ) to use in the calculations. Finally, the smooth E – I curve for the superconductor in an arbitrary external magnetic field can be constructed from equation (11). The required parameters IC and n are estimated from FEM calculation and equation (16) respectively.

This equation has two parameters—the critical current IC and a factor of steepness n . The parameter E C is the criterion for the estimation of the critical current and it is usually set to 1 × 10−4 V m−1 (=1 μV cm−1 ). For calculation of the critical current IC we used the model described in section 2.1 and then we continue further with analysis of the electric field. The factor n describes how important the flux creep effect is. A factor n approaching infinity means no flux creep; this corresponds to the critical state model. Lower n means that the flux creep effect is more important and the E – I curve is less steep. Commercial 2G HTS tapes contain artificial pinning centres with anisotropic capability of the magnetic flux pinning. As a consequence, the exact local description of the pinning force and also of the flux creep effect is very difficult. Therefore we used a phenomenological description of the flux creep effect. This effect depends on the Lorentz force which is equal to the product of the electric current and the magnetic field. Since the distribution of the magnetic field and current density in the superconductor can be very different in the case

3. Critical current of the single tape The investigation of the single tape critical current is needed to find the properties of the superconducting material itself. In other words, the single tape investigation is needed to extract the dependence of the material’s critical current density JC on the local magnetic flux density Bloc from results of critical current IC measurements in applied magnetic field Ha . The dependence of the critical current on applied magnetic field is a standard definition of a superconductor’s performance. We measured the electromagnetic characteristics of a commercial 4 mm wide SuperPower 2G HTS tape [12]. The self-field produced during the critical current measurement causes a non-homogeneous distribution of the current density in the tape’s cross-section. Therefore, we had to combine the experimental investigation with a simulation by the FEM [9, 10]. The experiments were focused on a detailed study of anisotropy at various applied magnetic fields. We measured the 3


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Figure 3. Dependence of the n -factor on the components of the pinning force density. Experimental data are shown by lines, while the fitting function is shown by the meshed surface.

Figure 2. Dependence of the critical current on the angle of the applied magnetic field for several values of applied magnetic field. The symbols are experimental data, the lines show the results of the simulation by the FEM.

superconductor and also on the resistance between individual strands, through which the current can be redistributed from one strand to another when it reaches the local critical value. Here we analyse two limiting cases. In the first case we assume that the resistance between the strands is negligible, which means that the current can flow from one strand to another one without dissipation. This means good current sharing within the cable. In the second case we assume that the strands are completely insulated. This means no current sharing within the cable. In the following, we describe the FEM model of the cable. This model is sufficient for the good current sharing case, investigated in section 4.1. In section 4.2, the no current sharing case is investigated. In such a case, we start the analysis with the FEM model, but we continue with analysis of the critical current and electric field distribution along the length of the cable. In section 4.3 we compare our calculated critical currents with experimental results. We performed calculations for three Roebel cables made of commercial coated conductor. The initial 4 mm wide SuperPower tape was punched into meander-shaped strands. The effective widths of the strands were 1.98 mm. The assembled cables consist of 14 ( R 14 × 1), 39 ( R 13 × 3) and 50 ( R 10 × 5) strands. A detailed description of the cables can be found in a previous work of ours [14]. The arrangement of the strands in the cable is shown in figure 4. The cable R 14 × 1 is shown in this figure; to make the figure readable we show only 7 strands, each with a different colour. While cable R 14 × 1 consists of 14 individual strands, cable R 13 × 3 consists of 13 stacks of 3 strands. Strands within one stack are placed parallel face to face to each other. In the figures that follow we use capital letters to identify individual strands within one stack. One colour corresponds to one stack of strands in figure 4. Cable R 10 × 5 consists of 10 stacks, each composed of 5 strands. All the cables have strands arranged in two columns in a cross-sectional view. Although we carried out the same analysis for all three investigated cable samples, we show the figures only for the cable R 14 × 1; the results are qualitatively similar for the other two cable samples.

critical current IC as a function of the angle θa for given field strength Ha of the external field (figure 2). The dependence on the angle shows a good agreement with the data published by the manufacturer [1]. The complex shape of the anisotropy curve is caused by a combination of various pinning mechanisms. The peak close to the crystallographic ab -planes (0◦ and 180◦ ) is caused by the intrinsic pinning of YBCO material, while the peak close to the c crystallographic axis (90◦ and 270◦ ) is related to artificial pinning centres [1, 10]. Different critical currents for orientations of the field at 0◦ and 180◦ were experimentally observed in earlier works [2, 3, 10]. This behaviour can be ascribed to the difference in the surface pinning on the side of the substrate and on the side of the cap layer [13]. We performed a set of calculations for various applied magnetic field strengths Ha and angles of the field θa . Following the procedure described in detail in [10], we found the proper values of the parameters in equations (1)–(9), which for the analysed tape sample were the following: parameter of smoothing m = 8; parameters for the field dependences J0 p = 5.3×1010A m−2 , J0i = 3.212×1010 A m−2 , B0ab = 4.569 mT, B0c = 2.016 mT, B0i = 32 mT, β = 0.4764, α = 0.9; parameters of angular dependences δab = 1.7◦ , δab = −7◦ , u ab = 8.337, u c = 1.794, u i = 1.7, v = 0.9. By comparing the experimentally measured n factor values and the calculated components of the average pinning force, we found the values of the parameters in equation (16) a = 14.9, b = 8.48 × 1021 , c = 2.55 and d = −2.3. Figure 3 shows the dependence of the n factor on the components of the pinning force; in the figure both the experimental data and the fitting function are shown.

4. Critical current of the Roebel cable The shape of the E – I curve and thus the overall critical current of the cable depends on the JC (B) characteristic of the 4


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Figure 5. The distribution of the magnetic flux density (surface) and the magnetic vector potential (isolevels) in the simulated cross-section of the Roebel cable. The total transport current is 496 A.

In the case of good current sharing, the critical current of the whole cable is equal to the sum of the critical currents of the individual strands; when the critical current of one strand is reached, the current is redistributed to the other strands. Since we assume a negligible resistance between the strands, the redistribution of the current does not require additional energy. The critical current of the whole cable is reached when the complete cross-section of the cable is filled with the critical current density. This is exactly the situation simulated by our FEM model. The electric field is constant in the whole crosssection of the cable and also along the length of the cable. For the considered cable samples, the calculated critical current is 496.3 A for the cable R 14 × 1, 1066.8 A for the cable R 13 × 3 and 1325.4 A for the cable R 10 × 5, respectively.

Figure 4. Arrangement of the strands in the 2G HTS Roebel cable. Part (a) shows the 3D view, with the four marked transverse planes; the cross-sectional views (from a direction shown by the arrow in figure (a)) in these planes are shown in part (b).

4.1. Good current sharing For the simulation of a Roebel cable, we used the same FEM model as for the single tape, only the geometry of the superconducting domains was changed to represent the Roebel cable’s cross-section. The simulated geometry consists of rectangular domains with a width of one strand (1.98 mm) and the thickness of the superconducting layer (1.4 μm) arranged in two columns with a vertical separation between the superconducting layers of 0.147 mm. The distribution of the magnetic field and the lines of the vector potential resulting from the simulation are shown in figure 5. The total transport current of the cable is 496 A in this case. The calculated distribution of the current density inside the superconductor in the same cable as in figure 5 is shown in figure 6(a). In the figure, the thickness of the superconducting layer is expanded 100 times for better readability. It can be noted that the distribution of the current density is not symmetrical horizontally or vertically. This is the consequence of the non-symmetrical IC (B) dependence shown in figure 2. We estimated the critical currents of the individual strands in the cable cross-section by integrating the current density over the superconducting domain’s cross-section, see figure 6(b). The bar graph in figure 6(b) shows that the strand’s lowest critical currents are in the top and bottom parts of the cable, where the magnetic field is the highest. The variation of the strand’s critical current in the cable cross-section is around 25% in the case of the cable R 14 × 1 shown in the figure. It is 33% in the cable R 13 × 3 and 29% in the cable R 10 × 5. Note that the width of the strand is 1.98 mm and the self-field critical current is around 63 A.

4.2. No current sharing If we assume that the strands are completely insulated from each other, the current cannot be redistributed from one strand to another one when the current in that strand reaches the critical value. Further increase of the current in a strand results in an increase of the electric field along that strand. In the following, we investigate the electric field variation along the strand length. The strand changes position with respect to the other strands in the cable, see figure 4. This means that along one transposition length, each strand assumes all the positions in the columnar arrangement shown in figure 6(a). Therefore the critical current varies along the strand length; this variation is shown in figure 7. The highest critical current is close to the centre of the straight part of the strand. However, it is not necessarily exact at the centre, because the IC (B) dependence has very complex shape, see figure 2. We did not model the diagonal parts of the strand, therefore the curves in figure 7 are interrupted in these parts. The cables R 13 × 3 and R 10 × 5 are made of stacked strands, see [14] for details. Individual strands within one stack are always parallel, but they experience different magnetic fields, because part of the field is generated by other strands of the same stack. Therefore the variation of the critical current along the superconducting strand is slightly different for each of the stacked strands—see figure 7. 5


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Figure 6. (a) Calculated distribution of the current density superconducting areas in the Roebel cable. (b) Critical currents of individual strands in the cable cross-section.

Consequently, we were able to calculate the n factor variation along the strand length, see figure 9(a). Finally we calculated the variation of the electric field along the strand length for different transport currents, figure 9(b). The highest electric field is in the crossing parts, where the critical current is the lowest. The electric field is lowest in the centres of the straight parts. This part of the strand is always in the middle of the cable, where the critical current is the highest. In the experiment, the electric field is usually measured by voltage taps placed at a distance of an integer number of transposition lengths [4]. In such a case, the measured electric field per unit length is an average of the local electric field on the measured strand. The calculated average electric field is shown by dashed lines in the figure. This value of the electric field is equal to the experimental values measured on the length of one transposition length. By the same calculation, we calculated the electric field for several transport currents and thus we calculated the shape of the E – I curve for the strand. For this calculation we made the assumption that the current is evenly distributed in all strands. The total current of the cable is equal to the current in one strand multiplied by the number of strands in such a case. The average electric field for every strand is the same; therefore the electric field along the whole cable is the same as the electric field along one strand. This is exactly true for the cable R 14 × 1. In the cables with stacked strands, the individual tapes in the same stack have slightly different E – I curves, see figure 10. However, the differences are small, so we assume that the change of the current distribution does not change the magnetic field distribution. We performed the calculation of the electric field as nonlinear resistances connected in parallel in the case of the cables with stacked strands. The critical current of the cable without current sharing was estimated with the standard criterion of 1 μV cm−1 . In our particular case, the critical current is 463.3 A for the cable R 14 × 1, 980.2 A for the cable R 13 × 3 and 1232.1 A for the cable R 10 × 5.

Figure 7. Variation of the strand’s critical current over one transposition length of the cable. Note that the y -scale is individual for each cable, while the x -scale is common for all cables. On the bottom, the meander-like shape of a strand along one transposition length is shown.

An electric field appears along the strand length at currents close to the critical current as a consequence of the flux creep effect. The variation of the magnetic field along the strand length causes a variation of the flux creep effect, resulting in a variation of the electric field along the strand length. The change of the local critical current means that also the distribution of the electrical field is non-homogeneous. For a calculation of the electric field E by equation (11), we had to calculate the n factor first. Therefore, we calculated the distribution of the pinning force in the superconductor. The distribution of the parallel and the perpendicular components of the pinning force density, calculated by equations (12) and (13), is shown in figures 8(a) and (b), respectively.

4.3. Comparison with experiment In table 1, we summarize properties of the investigated cables: their measured critical currents and results of our calculations for both cases, good current sharing and no current sharing. 6


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Figure 8. Distribution of the pinning force density in the superconductor. The pinning force was investigated in terms of two components: the component parallel to the strand’s wide face (a), and the component perpendicular to the strand’s wide face (b).

Figure 9. Variation of the n factor value (a), and electric field distribution (b) along the strand on one transposition length.

Table 1. The measured and calculated critical currents of the cables.

Sample

Measured IC (A)

Calculated IC , good sharing (A)

Calculated IC , no sharing (A)

R 14 × 1 R 13 × 3 R 10 × 5

465 1060 1195

496.3 1066.8 1325.4

463.3 980.2 1232.1

all the strands act as a single conductor and no electric field can be generated until the whole superconductor carries its critical current density. In the case of no current sharing, the electric field appears when the first part of the strand reaches its local critical current. Note that we assume completely uniform properties of the superconductor in the calculations; differences in the critical current in the cable cross-section are caused only by the self-field in our calculations. In real 2G HTS, random defects [1] and the manufacturing process [15] decrease the local critical current; therefore the critical current

Figure 10. The calculated E – I curves of cables with no current sharing assumption.

In the case of good current sharing, the critical current is always higher than in the case of no current sharing. This is caused by the fact that in the case of good current sharing, 7


M Vojenˇciak et al

varies along the tape length, typically by around 10%. The properties of the superconductor were measured on a single piece of tape. The distribution of the mentioned defects is unknown to us and therefore the calculation does not precisely describe the local properties of the tape used for the experiment. We assess the uncertainty to be in the range of 10%. This effect decreases the critical current of the cable, especially in the case of no current sharing. The experimentally measured critical currents of the cable samples R 14 × 1 and R 13 × 3 fall between our limit cases, which suggests that the strands are connected by random contacts of the stabilizing layer of copper. The cable sample R 10 × 5 has a slightly lower critical current than was estimated, and the reason for this different behaviour has still to be understood. The difference between the lowest and the highest possible critical currents is 7.1% for the cable R 14 × 1, 8.8% for the cable R 13 × 3 and 7.6% for the cable R 10 × 5.

Acknowledgments This work was supported partly by the European Commission under ‘NESPA’, contract number MRTN-CT-2006-035619, by BMWi-Germany project ‘Highway’ under contract number FZK-03274893 and by a Helmholtz University Young Investigator Grant (VH-NG-617). The authors would like to thank Holger Fillinger (KIT) for his assistance with CAD software and Alexandra Jung (KIT) for discussion.

References [1] Selvamanickam V and Xie Y 2009 IEEE Trans. Appl. Supercond. 19 3225 [2] Maiorov B, Gibbons J B, Kreiskott S, Matias V, Holesinger T G and Civale L 2005 Appl. Phys. Lett. 86 132504 [3] Thakur K P, Jiang Z, Staines M P, Long N J, Badcock R A and Raj A 2011 Physica C 471 42–7 [4] Goldacker W, Frank A, Heller R, Kling A, Terzieva S and Schmidt C 2009 Supercond. Sci Technol. 22 034003 [5] Greco M, Fabricatore P, Farinon S and Musenich R 2004 Physica C 401 124 [6] Rostila L, Lehtonen J and Mikkonen R 2007 Physica C 451 66 [7] Bean C P 1962 Phys. Rev. Lett. 8 250 [8] Comsol Multiphysics 2010 http://www.comsol.com [9] Gömöry F and Klincok B 2006 Supercond. Sci. Technol. 19 732 [10] Pardo E, Vojenciak M, Gömöry F and Souc J 2011 Supercond. Sci. Technol. 24 065007 [11] Anderson P W 1962 Phys. Rev. Lett. 9 309 [12] SuperPower, Inc. 2009 http://www.superpower-inc.com/ [13] Harrington S A, MacManus-Driscoll J L and Durrell J H 2009 Appl. Phys. Lett. 95 022518 [14] Terzieva S, Vojenˇciak M, Pardo E, Grilli F, Drechsler A, Kling A, Kudymow A, Gömöry F and Goldacker W 2010 Supercond. Sci. Technol. 23 014023 [15] Jiang Z, Amemiya N, Maruyama O and Shiohara Y 2007 Physica C 463–465 790

5. Conclusion Our calculations can predict the critical current of a Roebel cable from the known JC (B) dependence of a single tape. A non-homogeneous distribution of the magnetic field in the cable cross-section causes a variation of the critical current along the length of each strand. A further consequence of the critical current variation is that the overall critical current of the cable depends also on the resistance of the normal metal connections between the individual strands. The critical current of the cable is lowest if the strands are completely insulated. The critical current increases with decreasing resistance between the strands. Our calculations for the three Roebel cable samples show that the difference between the lowest and the highest possible critical currents can be more than 7%.

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