STUDY OF AC LOSSES IN SUPERCONDUCTING ...eii.unex.es/aes/Events/LucasPhD/Thesis Summary.pdfYBCO were replacing BSCO tape in superconducting electrical system designs. To complete the

STUDY OF AC LOSSES IN SUPERCONDUCTING ELECTRICAL COMPONENTS FOR ELECTRICAL SYSTEM DESIGN José María Ceballos SUMMARY The work presented here was conducted within the framework of one of the research lines of the "Benito Mahedero" Group of Electrical Applications of Superconductors, at the Industrial Engineering School of the University of Extremadura (Badajoz, Spain). The work was mainly carried out at the Group's Laboratory in Badajoz, but a part was carried out at the Institute of Electrical Engineering of the Slovak Academy of Science of Bratislava (Slovakia). Partial financial support for the work was given by the Extremadura Government through a research project (ref. 1PR98A045). The purpose of the work was to study, characterize, and measure the different components of AC losses in superconductors that are part of such electrical systems as transformers, electrical motors, etc. The reason for such a study is because, if the study of losses is an important part of the design of any electrical application, in superconducting electrical systems losses determine not only their efficiency but also the capacity of the corresponding cooling system. A difference from most of the previous AC loss studies published by other workers is that the focus of interest is not a single tape carrying current in a possible external magnetic field. Rather, our interest is in the tape as part of a multilayer coil, because this is the most usual way that the tape is used in electrical systems. The behaviour of each section of tape is different from that of an isolated piece because of the influence of the superconducting layer wound just next to it. In order to analyze the different components of the AC loss including the influence one one section of tape of another wound together with it, we made a comparative study of an isolated tape and of the same tape in the same conditions except for the proximity of another tape, with and without current, located just over the first one. This study was done with different tapes during the last year: first we used multifilament BSCO tape, then YBCO tape with ferromagnetic substrate, and finally YBCO tape with non-ferromagnetic substrate. In a first stage of the work (with BSCO tape) we studied the coil forming part of a multilayer magnetic coupling, investigating the dependence of the losses on the coil's geometric parameters. A practical formulation for the calculation of the parameters was proposed. Experimentally, the parallel field in the coil was observed to have a greater effect on the losses than the perpendicular field [1, 2]. But this effect is also observed to be different in the different layers of the coil. In the

second stage (with BSCO tapes) we therefore studied the behaviour of isolated tapes under different conditions of current and magnetic field with the aim of determining, in the third stage, the variation of this behaviour when another tape is located nearby. The results of the second stage [3-5] showed the losses to have a strong dependence on the phase difference between the transport current and the magnetic field. The third step was the design of a procedure to evaluate the influence of nearby tapes on the losses in a section of tape, comparing the results with the known behaviour of isolated tape. Two pieces of BSCO tape close together were used, carrying the same current (amplitude and phase) as in a multilayer coil. One of the pieces was cut longer than the other in order to take some measurements in the part of the longer tape not in the immediate proximity of the shorter one. This thus provided new data with which to add further precision to our conclusions. One of the most interesting results of this stage was the revelation of how the proximity of tapes carrying the same current causes a reduction of the practical critical current in them [6]. During the time in which the foregoing work was being carried out, new tapes based on YBCO were replacing BSCO tape in superconducting electrical system designs. To complete the study of AC losses, we began the study of this type of tape in the fourth and last stage of the thesis work. Samples of 2G (second generation) YBCO tape were tested in the same way as the BSCO tape in stages 1-3 of this work. The first results and conclusions of this study were presented in [7]. The thesis document presented here was closed after this publication, but the study of YBCO tape is now the focus of one of our Group's research lines. Further work with this tape includes:

− Study of the differences between losses in tapes with and without magnetic substrate.

− Study of the influence of the magnetic substrate on nearby tapes and coil AC

losses.

− Study of the anisotropy of YBCO tape with and without magnetic substrate.

− Study of the influence of the tape's anisotropy on the losses and practical critical current of a coil, depending on the bending curvature of the tape.

The results of the work described have led to our participation in 5 international conferences in applied superconductivity, and to the publication of the articles [1-7] referenced in this summary and attached to the document.

REFERENCES [1] B. Pérez, A. Álvarez, P. Suárez, J.M. Ceballos, X. Obradors, X. Granados, R. Bosch.

“Ac losses in a toroidal superconductor transformer”. IEEE Transactions on Applied Superconductivity, 13, pp. 2341-2343 (2003).

[2] P. Suárez, A. Álvarez, B. Pérez, D. Cáceres, E. Cordero, J.M. Ceballos. “Influence of

the shape in the losses of solenoidal air-core transformers”. IEEE Transactions on Applied Superconductivity,15, pp. 1855-1858 (2005).

[3] F Gomöry, J. Souč, M. Vojenciak, E. Seiler, B. Klincŏk, J.M. Ceballos, E. Pardo, A.

Sánchez, C. Navau, S. Farinon, P. Fabbricatore. “Predicting ac loss in practical superconductors”. Supercond. Sci. Technol., 19, pp. 60-66 (2006).

[4] M. Vojenciak, J. Souč, J.M. Ceballos, F Gomöry, B. Klincŏk, F. Grilli. “Study of ac

loss in Bi-2223/Ag tape under the simultaneous action of ac transport current and ac magnetic field shifted in phase”. Supercond. Sci. Technol., 19, pp. 397-404 (2006).

[5] E. Pardo, F Gomöry, J. Souč, J.M. Ceballos. “Current distribution and ac loss for a superconducting rectangular strip with in-phase alternating current and applied field”. Supercond. Sci. Technol., 20, pp. 351-364 (2007).

[6] P. Suárez , A. Álvarez, B. Pérez And J.M. Ceballos. “Influence of the current through

one turn of a multilayer coil on the nearest turn in a consecutive layer”. Journal Of Physics: Conference Series, 97 (2008).

[7] P. Suárez, A. Álvarez, J.M. Ceballos and B. Pérez “Losses in 2G tapes wound close

together: Comparison with similar 1G tape configurations”. (in press).

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 13, NO. 2, JUNE 2003 2341

AC Losses in a Toroidal Superconducting TransformerB. Pérez, A. Álvarez, Member, IEEE, P. Suárez, D. Cáceres, J. M. Ceballos, X. Obradors, X. Granados, and R. Bosch

Abstract—In order to study the viability of coreless AC coupledcoils, a superconductor transformer based on BSCCO-2223 PITtapes was constructed. To achieve the minimum flux leakage, atoroidal geometry was selected. Both secondary and primary coilswere wound around a glass fiber reinforced epoxy torus, obtaininga solid system. The field inside the transformer, the coupling factor,and the losses in the system were computed and measured, pro-viding suitable parameters for new improvements in these systems.

Index Terms—AC losses, Bi-2223 tape, superconductor trans-former.

I. INTRODUCTION

H IGH temperature superconducting transformers arelighter, smaller and have a higher efficiency than conven-

tional transformers [1]. The windings of most superconductingtransformer prototypes have been built with Bi-2223 tapes[1]–[4]. These prototypes have used very different geometries,but when the ferromagnetic material is taken out and one wantsto maintain a high coupling factor it is necessary to look fora geometry to confine the magnetic field lines. Examples areannular [5] and solenoidal transformers [6].

We propose an alternative geometry to get a high couplingfactor without an iron core: a single-phase superconductingtorus. We made this transformer with an air core and determinedits coupling factor and its AC losses. The test of AC losses wasperformed by means of the electrical method using a lock-inamplifier [3], [7], [8].

II. DESIGN OF THEPROTOTYPE

The transformer was wound with Bi-2223 tape. In order toreinforce the transformer structure, the Bi-2223 windings werewound onto a glass fiber torus [3], [9]. The minimum coil radiusbefore the coil loses its superconductor characteristics has beenevaluated previously [10] and based on this work a torus with30 cm inner diameter and 36 cm outer diameter was chosen.The cross section diameter of the torus is thus less than 10%of its major diameter aiming at a geometry close to the ideal,

Manuscript received August 6, 2002. This work was supported in part by theInter-ministerial Commission of Science and Technology of Spain and Govern-ment of Extremadura.

A. Álvarez, B. Pérez, P. Suárez, J. M. Ceballos, and R. Bosch are withthe Electrical Engineering Department, University of Extremadura, Apdo382, 06071 Badajoz, Spain (e-mail: [email protected], [email protected],[email protected], [email protected]).

D. Cáceres is with the Applied Mathematic Department, University of Ex-tremadura, Apdo 382, 06071 Badajoz, Spain (e-mail: [email protected]).

X. Obradors and X. Granados are with the Institute of Material ScienceICMAB (CSIC), Barcelona, Spain.

Digital Object Identifier 10.1109/TASC.2003.813122

TABLE ISPECIFICATIONS OF THEHTS TAPE

TABLE IICHARACTERISTIES OF THETRANSFORMER

Fig. 1. The prototype transformer.

when the field is constant inside and null outside. Similarly, wetried to wind the tape as close together as possible to achieve ahomogeneous current distribution. To accomplish this, 341 turnswere needed for the inner coil. For the outer coil we used 447turns to get a transformation ratio different from unity. The finalstructure consists of 5 layers (A, B for the inner coil and C, D,E for the outer) that we connected properly to get the desiredrating.

The characteristics of the HTS tape and the windings are pre-sented in Tables I and II, respectively, and Fig. 1 shows the com-pleted toroidal transformer. Coils were cooled using liquid ni-trogen at 77 K.

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2342 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 13, NO. 2, JUNE 2003

TABLE IIIEXPERIMENTAL COUPLINCE FACTOR

III. COUPLING FACTOR OF THETRANSFORMER

A. Theoretical Coupling Factor

The transformer coupling factor is used to verify that the mag-netic field is shared enough to suppose that the magnetic lossesare produced by the same distribution of magnetic field in bothof the coils.

To obtain the theoretical coupling factor of this transformerwe calculated the inner magnetic field from each toroidal cur-rent using a numerical integration procedure based on the Biot-Savart law and the superposition principle (since the system islinear). Then the linked and leakage fluxes and the winding cou-pling factors were computed.

The evaluated coupling factor was 0.88 for the inner windingand 0.94 for the outer winding. Therefore the evaluated theoret-ical coupling factor was 0.91 for the transformer.

B. Measurement of the Transformer Coupling Factor

The prototype was tested without load, feeding both thehigh-tension and low-tension coils to determine the experi-mental coupling factor. The results are summarized in Table III.They show that the measured transformer coupling factor isadequate to assume that most magnetic flux is shared by bothcoils.

IV. AC L OSSES

A. Measurement Method

An electrical technique was used to measure the AC lossesin the transformer. We used a lock-in amplifier with four in-puts: two of them to measure the primary voltage and currentand to calculate the power fed into the transformer,, and theother inputs to measure the secondary voltage and current andto calculate the power that the transformer gives out,. Eachpower was evaluated by integration of the product of currentand voltage over an integer number of periods. The transformerlosses, , were calculated as the difference betweenand ,so that, .

The electric circuit used is shown in Fig. 2. Various tests onthe transformer were carried out, in short-circuit, without load,and with different values of the load. In all cases, the low-tensioncoil was used as the primary and the potential difference wasmeasured in all layers. The frequency was 50 Hz.

Fig. 2. Electric circuit for ac loss measurement of the transformer.

Fig. 3. Magnetization losses versus(V=N) for the transformer layers testedin short-circuit. Linear dependence can be seen.

B. Theoretical Method

In electric power applications like the present case, thetotal AC losses, , are the result of two contributions: thealternating transport current losses, , and the magnetizationlosses, . The first are dominated by hysteresis losses[11] that can be evaluated theoretically by the elliptic modelformulated by Norris:

(1)

where , is the peak current, is the frequency, andis the magnetic permeability H/m . The primary

and secundary Norris losses were evaluated by (1) and the totalNorris losses calculated as their sum.

Subtracting these from the total measured losses, an estimateof the total magnetization losses can be obtained. For low fre-quencies, it has been shown that there must be a proportionalitybetween magnetization losses per unit of length in each windingand the square of the magnetic field, which can be written:

PÉREZet al.: AC LOSSES IN A TOROIDAL SUPERCONDUCTING TRANSFORMER 2343

Fig. 4. Magnetization losses versus(V=N) for the transformer layers testedwithout load. Linear dependence can be seen.

Fig. 5. AC losses of the transformer to test in short-circuit. Comparison withtheoretical curve.

This linear proportionality betwen and is shownin Fig. 3 for primary and secondary windings in the test of thetransformer in short-circuit. Fig. 4 shows this relation in the testswithout load. The similarity of the slopes reinforces the idea thatthe proportionality constant, , only depends on details ofthe construction of the tape.

An alternative formulation is proposed to find the AC lossesin the transformer by means of the following expression:

(2)

where the subscripts 1 and 2 represent the primary and sec-ondary of the transformer.

Fig. 6. AC losses of the transformer to test without load. Comparison withtheoretical curve.

When this expression was applied to the losses seen in shortcircuit and no load tests, good agreement between experimentand theory was achieved as shown in Figs. 5 and 6.

V. CONCLUSION

AC losses in a superconducting transformer were measuredby the electrical method with a 4-channel lock-in amplifier. Thetheoretical AC losses were evaluated and an alternative formula-tion proposed. The experimental results show good agreementwith the proposed theoretical expression. For this reason, webelieve that (2) adequately represents the AC losses in a trans-former working at low frequencies. Further testing is requiredto validate (2) for higher frequencies.

REFERENCES

[1] H. J. Lee, G. Cha, J. Lee, K. D. Choi, K. W. Ryu, and S. Y. Hahn, “Testand characteristic analysis of an HTS power transformer,”IEEE Trans.Appl. Superconduct., vol. 11, no. 1, pp. 1486–1489, 2001.

[2] K. Funakiet al., “Preliminary tests of a 500 kVA-class oxide supercon-ducting transformer cooled by subcooled nitrogen,”IEEE Trans. Appl.Superconduct., vol. 7, no. 2, pp. 824–827, 1997.

[3] M. Iwakumaet al., “AC loss properties of a 1 MVA single-phase HTSpower transformer,”IEEE Trans. Appl. Superconduct., vol. 11, no. 1, pp.1482–1485, 2001.

[4] G. Donnier-Valentin, P. Tixador, and E. Vinot, “Considerations aboutHTS superconducting transformers,”IEEE Trans. Appl. Superconduct.,vol. 11, no. 1, pp. 1498–1501, 2001.

[5] G. Fontana, “Coreless transformers with high coupling factor,”Rev. Sci.Instrum., vol. 66, no. 3, pp. 2641–2643, 1995.

[6] M. Polàk, P. Usák, J. Pitel, L. Jansák, Z. Timoranský, F. Zìzek, and H.Piel, “Comparison of solenoidal and pancake model windings for a su-perconducting transformers,”IEEE Trans. Appl. Superconduct., vol. 11,no. 1, pp. 1478–1481, 2001.

[7] S. K. Olsen, C. Træholt, A. Kühle, O. Tønnesen, M. Däumling, and J.Østergaard, “Loss and inductance investigations in a 4-layer supercon-ducting prototypes cable conductor,”IEEE Trans. Appl. Superconduct.,vol. 9, no. 2, pp. 833–836, 1999.

[8] S. Mukoyama, K. Miyoshi, H. Tsubouti, H. Tanaka, A. Takagi, K. Wada,S. Megro, K. Matsuo, S. Honjo, T. Mimura, and Y. Takahashi, “AC lossesof HTS power transmission cables using Bi-2223 tapes with twisted fila-ments,”IEEE Trans. Appl. Superconduct., vol. 11, no. 1, pp. 2192–2195,2001.

[9] S. P. Hornfeldt, “HTS in electric power applications, transformers,”Physica C, vol. 341–348, pp. 2531–2533, 2000.

[10] A. Álvarez, P. Suárez, D. Cáceres, B. Pérez, E. Cordero, and A. Castaño,“Superconducting tape characterization under flexion,”Physica C, vol.372–376, pp. 851–853, 2002.

[11] J. W. Lue, M. S. Lubell, and M. J. Tomsic, “AC losses of HTS tapes andbundles with de-coupling barriers,”IEEE Trans. Appl. Superconduct.,vol. 9, no. 2, pp. 793–796, 1999.

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 2, JUNE 2005 1855

Influence of the Shape in the Losses of SolenoidalAir-Core Transformers

Pilar Suárez, Alfredo Álvarez, Member, IEEE, Belén Pérez, Dolores Cáceres, Eduardo Cordero, andJosé-María Ceballos

Abstract—The losses in an HTS tape depend strongly on the per-pendicular magnetic field. In order to avoid this magnetic fieldcomponent in an air core transformer, a toroidal geometry wasproposed and studied in previous work. Due to the difficulties thatone finds in constructing toroidal coils, the straight solenoidal ge-ometry is now under study. In this case, the magnetic field closeto the ends of the coil is not parallel to the axis and a perpendic-ular component appears. In the present work, the losses due to thiscomponent are studied as a function of the coil geometry—i.e., theratio between length and diameter—and a practical formulation isfound.

Index Terms—Bi-2223 coil, magnetization losses, supercon-ducting transformer.

I. INTRODUCTION

H IGH temperature superconducting transformers arelighter, smaller and have a higher efficiency than conven-

tional transformers [1]. The windings of most superconductingtransformer prototypes have been built with Bi-2223 tapes[1]–[4]. These prototypes have used very different geometries,but when the ferromagnetic material is taken out and one wantsto maintain a high coupling factor it is necessary to look fora geometry to confine the magnetic field. In a previous paper[5], we studied a superconducting toroidal transformer anddescribed a method to measure AC losses.

Due to the difficulties that one finds in constructing toroidalcoils, the straight solenoidal geometry is now under study. Inthis case, the magnetic field close to the ends of the coil is notparallel to the axis and a perpendicular component appears.

But when a transformer is constructed from these coils, thereare also losses due to the parallel magnetic field. The presentwork analyzes the losses due to these components, taking theinfluence of the coil geometry into account.

II. DESIGN OF THE SOLENOIDAL TRANSFORMER

A prior step to constructing the solenoidal transformer is toshow the behavior of a coil alone. The minimum coil radius be-fore it loses its superconducting characteristics had been evalu-ated previously [6], and based on that work several coils wereconstructed by winding with Bi-2223 tape. The specifications ofthe HTS tape are presented in Table I. In order to reinforce the

Manuscript received October 4, 2004. This research is funded in part by theInter-Ministerial Commission of Science and Technology of Spain and Govern-ment of Extremadura.

The authors are with the Electrical Engineering Department, University ofExtremadura, 06071 Badajoz, Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TASC.2005.849312

TABLE ISPECIFICATIONS OF THE HTS TAPE

TABLE IICHARACTERISTICS OF SOLENOIDAL COILS

coil structures, the Bi-2223 coils were wound onto glass-fibersolenoids [7], [8] and several prototypes of transformers wereformed by placing these coils concentrically. The characteris-tics of the two sets of coils constructed are given in Table II,where N is the number of turns of each coil, R the respectiveradius, and L the respective length. Fig. 1 shows one of the pro-totypes ready to be tested.

III. MEASUREMENT OF THE LOSSES

A. Measurement of Total Losses

The transformer losses were measured by the electricalmethod. Fig. 2 shows the circuit used. Since the only energyentering the system comes from the power supply, this methodgives the total AC losses of the transformer as the mean valueof the product of and during an integer number ofperiods. The readings of the voltages and currents were takenby means of a DAQ card. Voltages were read directly from thetaps on the transformers, and currents were read by means ofa Hall probe as is shown in Fig. 2. The data were processedby a routine written in LabVIEW, evaluating the losses byintegrating the product of voltage and current. Fig. 3 shows thetotal losses in each coil of prototype 2. Similar curves wereobtained with prototype 1.

1051-8223/$20.00 © 2005 IEEE

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1856 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 2, JUNE 2005

Fig. 1. Prototype of solenoidal superconducting transformer.

Fig. 2. Diagram to test the transformer to measure the total losses. Here coils1 (primary) and 4 (secondary) are under test. Similar trials were done with theother coils. In all the cases one coil was connected to the supply, and the otherthree were without load.

Fig. 3. Total losses in each coil of prototype 2. Similar curves were observedwith prototype 1.

B. Measurement of Transport Losses

In order to measure the transport losses, the same setup wasused. In this case pairs of consecutive coils were connected inseries and supplied with current in opposite senses to annul themagnetic field and therefore the losses due to it. The measure-ments thus give us twice the transport losses in one of the coils.The results were similar in all the coils. Fig. 4 shows the meanresults.

Fig. 4. Transport losses in each coil of a solenoidal superconductingtransformer.

Fig. 5. Scheme of the expected magnetic field lines in a solenoidal transformer.Top: the inner coil is supplied; bottom: the outer coil is supplied.

IV. THEORETICAL HYPOTHESIS

When magnetization losses are measured in a sample of su-perconducting tape the contribution of the perpendicular mag-netic field is much greater than that of the parallel field. Butwhen losses are studied in solenoidal transformers, there aresome differences. The total losses in the transformer depend onwhich coil is supplied (Fig. 3). This effect can be explained withthe aid of Fig. 5, where the expected magnetic field is shown.An observation of the figure suggests the following hypothesis:When the inner coil (coil 1) is supplied, the main contributions

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SUÁREZ et al.: INFLUENCE OF THE SHAPE IN THE LOSSES OF SOLENOIDAL AIR-CORE TRANSFORMERS 1857

Fig. 6. Influence of the coil radius, R, on the magnetization losses. V is thevoltage in the coil and N the number of turns. V=N is proportional to the fluxof the magnetic field. Different slopes are observed for each coil. In practice,P is proportional to (V=N) and the constant, k (the slope) is a function ofR. In [9] we studied this function and show that k(R) = k (1=R) .

to the losses are due to the transport current and the perpendic-ular magnetic field at the ends of all the coils:

(1)

where is the quantity of total losses, the transport currentcontribution and the perpendicular magnetic field contribu-tion in all the coils.

But an additional contribution appears when another coil (in-termediate or outer) is supplied.

This new contribution is due to the parallel magnetic fieldalong the remaining of the coils inside the supplied one. I.e.,when coil 3 is supplied the effects of the parallel magnetic fieldare twice those when coil 2 is supplied. Similarly, when coil 4is supplied the effects of the parallel component are three timesthose when coil 2 is supplied. Assuming that the contributionto the losses of the perpendicular magnetic field is similar in allthe cases, one has:

(2)

where is the contribution of the parallel magnetic field in onecoil.

The contribution of the parallel magnetic field due to the sup-plied coil in the others, , can be obtained from (2):

(3)

The assumption that the contribution to the losses of the per-pendicular magnetic field is similar in all the cases is based ona previous work [9]. In this, we studied the dependence of themagnetization losses in solenoidal coils on the coil geometry(coil radius, density of turns, coil length, ). Figs. 6 and 7 show

Fig. 7. Representation of the practical equation of P for low frequencyelectrical applications: P = k (1=R) (V=N) .

Fig. 8. Comparison of the losses due to the parallel magnetic field betweencoils 2 and 3. The straight line fits to these points has a slope of 1.92, close to 2(theoretical value), and a regression coefficient of 0.93.

the main results of that work for a set of three solenoidal coilswith different radii and numbers of turns.

Applying these results to the values in Table II shows themagnetization losses to be similar.

V. RESULTS

The results of the measurements obtained in Section III con-firm the hypothesis proposed in Section IV.

Figs. 8 and 9 show the results corresponding to the casesof and , respectively. Both cases fitted a linearregression:

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1858 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 2, JUNE 2005

Fig. 9. Comparison of the losses due to the parallel magnetic field betweencoils 2 and 4. The straight line fits to these points has a slope of 2.90, close to 3(theoretical value), and a regression coefficient of 0.97.

VI. CONCLUSION

The losses in solenoidal superconducting transformers de-pend on which coil is supplied. The magnetization losses in-clude a very important contribution from the parallel magneticfield, because this component affects much more of the lengthof the superconducting tape than the perpendicular component.

Furthermore, the influence of geometric factors (coil radius andlength) has also to be taken into account.

REFERENCES

[1] H. J. Lee, G. Cha, J. Lee, K. D. Choi, K. W. Ryu, and S. Y. Hahn, “Testand characteristic analysis of an HTS power transformer,” IEEE Trans.Appl. Supercond., vol. 11, no. 1, pp. 1486–1489, Mar. 2001.

[2] K. Funaki et al., “Preliminary tests of a 500 kVA-class oxide supercon-ducting transformer cooled by subcooled nitrogen,” IEEE Trans. Appl.Supercond., vol. 7, no. 2, pp. 824–827, Jun. 1997.

[3] M. Iwakuma et al., “AC loss properties of a 1 MVA single-phase HTSpower transformer,” IEEE Trans. Appl. Supercond., vol. 11, no. 1, pp.1482–1485, Mar. 2001.

[4] G. Donnier-Valentin, P. Tixador, and E. Vinot, “Considerations aboutHTS superconducting transformers,” IEEE Trans. Appl. Supercond., vol.11, no. 1, pp. 1498–1501, Mar. 2001.

[5] B. Pérez, A. Álvarez, P. Suárez, D. Cáceres, M. Ceballos, X. Obradors,X. Granados, and R. Bosch, “AC losses in a toroidal superconductingtransformer,” IEEE Trans. Appl. Supercond., vol. 13, pp. 2341–2343,2003.

[6] A. Álvarez, P. Suárez, D. Cáceres, B. Pérez, E. Cordero, and A. Castaño,“Superconducting tape characterization under flexion,” Physica C, vol.372–376, pp. 851–853, 2002.

[7] M. Iwakuma et al., “AC loss properties of a 1 MVA single-phaseHTS power transformer,” IEEE Trans. Appl. Supercond., vol. 11, pp.1482–1485, 2001.

[8] S. Hörnfeldt, “HTS in electric power applications, transformers,”Physica C, vol. 341–348, pp. 2531–2533, 2000.

[9] P. Suárez, A. Álvarez, B. Pérez, and D. Cáceres, “Practical formula-tion of low frequency AC losses for superconducting coils,” presented atthe 6th European Conf. Applied Superconductivity, Sorrento, Italy, Sep.2003.

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INSTITUTE OF PHYSICS PUBLISHING SUPERCONDUCTOR SCIENCE AND TECHNOLOGY

Supercond. Sci. Technol. 19 (2006) S60–S66 doi:10.1088/0953-2048/19/3/009

Predicting AC loss in practicalsuperconductors

F Gomory1, J Souc1, M Vojenciak1, E Seiler1, B Klincok1,J M Ceballos1,2, E Pardo1,3, A Sanchez3, C Navau3, S Farinon4 andP Fabbricatore4

1 Institute of Electrical Engineering, Slovak Academy of Sciences, Dubravska cesta 9,842 39 Bratislava, Slovakia2 Department of Electrical Engineering, University of Extremadura, Badajoz, E-06071, Spain3 Grup d’Electromagnetisme, Departament de Fisica, Universitat Autonoma Barcelona,08193 Bellaterra (Barcelona), Catalonia, Spain4 Istituto Nazionale di Fisica Nucleare, Via Dodecaneso 33, Genoa, I-16146, Italy

Received 3 October 2005, in final form 21 November 2005Published 20 January 2006Online at stacks.iop.org/SUST/19/S60

AbstractRecent progress in the development of methods used to predict AC loss insuperconducting conductors is summarized. It is underlined that the loss isjust one of the electromagnetic characteristics controlled by the timeevolution of magnetic field and current distribution inside the conductor.Powerful methods for the simulation of magnetic flux penetration, likeBrandt’s method and the method of minimal magnetic energy variation,allow us to model the interaction of the conductor with an external magneticfield or a transport current, or with both of them. The case of a coincidentaction of AC field and AC transport current is of prime importance forpractical applications. Numerical simulation methods allow us to expand theprediction range from simplified shapes like a (infinitely high) slab or(infinitely thin) strip to more realistic forms like strips with finite rectangularor elliptic cross-section. Another substantial feature of these methods is thatthe real composite structure containing an array of superconductingfilaments can be taken into account. Also, the case of a ferromagneticmatrix can be considered, with the simulations showing a dramatic impacton the local field. In all these circumstances, it is possible to indicate howthe AC loss can be reduced by a proper architecture of the composite. Onthe other hand, the multifilamentary arrangement brings about a presence ofcoupling currents and coupling loss. Simulation of this phenomenonrequires 3D formulation with corresponding growth of the problemcomplexity and computation time.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Since the discovery of superconductivity, the disappearanceof electrical resistivity motivated many scientists to thinkabout the application of superconductors in electric powerengineering. When hard superconductors as materialsreaching the demand for competitive current transportcapability appeared in the early 1960s, a new problem emerged:it was soon realized that the ability to lead persistent electricalcurrents is linked with the appearance of dissipation in the

AC regime, i.e. at transporting AC current or during exposureto AC magnetic field. This phenomenon, called the AC lossin superconductors, has occupied a significant sector of thesuperconducting research until now.

In the era of low-temperature superconductors (LTSs), atypical structure of a superconducting wire was established. Itcontained superconducting filaments in a metallic matrix. Thecross-section of such a composite wire had the form of a circleor a rectangle with the aspect ratio of the sides rarely exceedingtwo, i.e. close to a square. For this metal–superconductor

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AC loss in practical superconductors

structure, the fundamental principles of the AC loss mechanismwere revealed. It was found that the heat generation canbe attributed to electrical currents induced in the compositestructure. Several significant paths for current loops have beenidentified, and two loss dissipation mechanisms distinguished:in the case of screening current forming a loop entirely withinthe superconducting filament, its creation will cost energy dueto the pinning of magnetic flux in the superconductor. This losscomponent is called the hysteresis loss. Another mechanismis the so-called coupling loss, ascribed to the current flowingin a loop that is mostly formed by superconducting paths, butalso contains portions with normal resistance [1–5].

This terminology remains valid in the investigation ofwires made from high-temperature superconductors (HTSs).However, several factors have pushed the research of AC losssignificantly forward in the recent period. First, the most usedHTS wire has the form of a tape, to achieve good alignmentof HTS grains. The aspect ratio of the sides of its rectangularcross-section typically exceeds 10. Then, the approximationof an infinite slab in parallel magnetic field, fruitfully appliedto the wires and windings from LTS wires, becomes ratherdoubtful. Also, the composite structure of HTS wires is farless regular than in the case of LTSs. Then, instead of theeffective medium approach that successfully explained manyof the effects observed in LTS wires, one has to take intoconsideration the real structure of the composite. On the otherhand, tremendous increase of the computing power accessibleon a personal computer made available numerical methods thatare adequate to cope with the aforementioned complications.It is the main purpose of this paper to encourage non-specialistsin numerical simulations to utilize these methods in theinvestigation of AC applications of superconductors.

In section 2, the properties of simple shapes will besummarized in order to show the general rules that governthe AC loss. The importance of the demagnetizing field tothe AC loss will be illustrated. In section 3, the use oftwo powerful numerical methods allowing us to simulate thebehaviour of a superconducting wire in arbitrary conditionswill be demonstrated for several cases of practical importance.We will show how the treatment of superconducting materialas a conductor with a highly non-linear current–voltage curveallows us to solve the problem of current distribution intoparallel paths and to simulate the flux distribution in a cableconsisting of a single layer of superconducting tapes. Theinfluence of tape arrangement, in particular the width ofgaps between tapes, on the AC loss can be investigated inthis way. Also, the method of minimum magnetic energyvariation (MMEV) will be presented as a tool to understandthe dissipation in the case of simultaneous action of AC fieldand AC current for a wire with rectangular cross-section. Wealso show that this method allows us to simulate the criticalstate and accompanied dissipation in a superconducting wirewith ferromagnetic matrix. In section 4 we briefly summarizethe presented results.

2. AC loss in hard superconductors with simpleshapes

The appearance of dissipation in hard superconductors exposedto a changing magnetic field was recognized simultaneously

with the formulation of the critical-state model. This model hasserved as an excellent approximation of the electromagneticbehaviour of these materials since then [6]. The simplest shapeallowing us to derive analytical expressions for the distributionof local magnetic field, current density and electrical field isthat of a slab (infinitely high) in a parallel magnetic field.Dissipation in the cyclic regime of external magnetic fieldBac = Ba sin ωt is easily calculated, showing the followingfundamental features.

The dissipation depends on the volume of the sampleaffected by the movement of flux lines. In the beginning partof the magnetization process, there remains a portion of thesample untouched by the field change because the screeningcurrents, starting from the sample surface, are able to shield thechange of applied field completely. This shielding capability isexhausted when the applied field has reached the value calledthe penetration field Bp. For low fields, i.e. smaller than Bp,the AC loss is proportional to B3

a . Beyond Bp, the pattern ofinduced currents is saturated, and the loss increase with Ba isjust linear, creating in this way a kink in the AC loss dependenceon Ba. The value of penetration field depends on the samplethickness, thus the AC loss is not solely a material property.This was further underlined when samples in perpendicularmagnetic field (e.g. single crystals of HTS) started to attractattention. Interestingly, the value of Bp remained roughlythe same as that found in parallel field [7], but the lossesincreased dramatically. Taking into account the enhancementof local magnetic field in perpendicular geometry due to thedemagnetizing effect, this observation can be easily explained.The results achieved for AC loss in superconductors of simpleshapes exposed to a cyclic AC field can be generally expressedby the following formula [8]:

q = Sπ

µ0χ0 B2

a χ ′′int(y) (1)

where q is the loss per metre length of a superconductingwire, S is its cross-section, µ0 = 4π × 10−7 H m−1, χ0 isthe initial susceptibility that will be discussed later on andχ ′′

int is the imaginary part of the internal complex magneticsusceptibility. As indicated in the formula, it depends on thevariable y = Ba

Bmax, i.e. the AC field amplitude scaled by the

value where the susceptibility curve reaches its maximum.For the slab in parallel field Bmax ≈ Bp; however, in theperpendicular geometry, e.g. that of a thin strip or disc in atransversal field, the AC susceptibility reaches its maximumwell before the complete saturation of the sample cross-section by the critical current density. In theoretical papersdealing with magnetic flux penetration into superconductorsof various shapes [6, 9–11], the corresponding χ ′′

int(Ba

Bmax) curve

can be found as well as the value of χ0. Interestingly, thisdependence is rather similar for all the investigated simpleshapes, indicating that a significant loss reduction cannot bereached by a simple shape optimization of the conductor [8].This is illustrated in figure 1. However, the shape factor χ0 forall the simple shapes can be roughly estimated by the formula

χ0 = 1 +a

b(2)

where a is the wire dimension perpendicular to the appliedfield and b is the dimension parallel to the applied field. In the

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F Gomory et al

0.001

0.01

0.1

1

0.01 0.1 1 10 100Ba /Bmax

χχ χχ"/χ/χ /χ/χ

" max

stripdiskslabelliptic strip

Figure 1. Theoretical prediction of AC loss behaviour—in appliedmagnetic field—for hard superconductors of various simple shapes.The loss is characterized by the imaginary part of AC susceptibilityaccording to the formula (1). To allow direct comparison of thedependences, the both axes are normalized to make the curves meetat (1, 1).

case of a slab in a parallel field b � a, thus χ0 ≈ 1. Whenthe field is rotated by 90◦, the same slab in a perpendicularfield will exhibit χ0 ≈ a

b � 1. Thus, the rule of thumbfor the magnetization loss reduction is ‘avoid perpendicularmagnetic field’. This conclusion also remains valid for arough estimation of magnetization hysteresis loss in an arrayof filaments [12]. Rigorous analytical derivation of AC lossformulae for an infinite stack or a horizontal array of filamentscan be found in [13]. Extensive numerical calculations ofAC loss in two-dimensional matrices of filaments have beenpublished [14]. Valuable comparison of AC loss measuredon a horizontal array with theoretical predictions obtained bydifferent methods has been published recently [15].

In the case of transporting AC current, the result is evensimpler: save very thin strips, with the side aspect ratioexceeding 100, the shape of the superconducting wire does notinfluence the loss notably [16]. Nevertheless, the distributionof critical current density—or the density of filaments in thecase of a multifilamentary wire—would lead to the deviationof this unique behaviour. Generally, the conductor with betterproperties of the surface filaments will exhibit at low currentsa lower AC loss then predicted by the Norris formula [17, 18].

3. Flux penetration into complex shapes

The approximation of superconductors in real windings byone of the models mentioned in the previous paragraph,successfully applied in the LTS era, faces serious problemsin the case of HTS wires. The perpendicular geometrywas found as the critical arrangement, and the study oftape conductors in perpendicular field became the standardexperiment. Fortunately, at the same time the performance ofcommon personal computers increased in a way that allows usto carry out the numerical simulations taking into account thereal structure of composite wires.

Systematic investigation for a series of Bi-2223/Agcomposites demonstrated the possibility of determining χ0

with the help of a finite element simulation of the diamagneticstate of superconducting filaments as a linear problem [19].However, the substantial progress in understanding the lossbehaviour of composite HTS wires has been reached thanks tothe use of two simulation methods, allowing us to develop thedistribution of local current density, electrical and magneticfield in the whole range of applied magnetic fields and/ortransport currents.

The first method treats the hard superconductor as aconductive medium with highly non-linear current–voltagerelation, and allows us to predict nicely the flux penetration.This approach is in agreement with experimental evidence thatthe relation between the electrical field, E , and the currentdensity can be fairly approximated by the power law relationE = E0(

jj0)n , where E0 is the conventional criterion for

the determination of the critical current density, j0, and theexponent n reflects the smoothness of the transition. For anHTS at 77 K, typical values of n are in the range between 15and 30. This idea was successfully used to predict the influenceof the conductor shape on the AC loss [20], to simulatethe distribution of currents and AC loss in multifilamentarywires [21] and to calculate the AC loss at in a wire transportingAC current with simultaneous exposure to AC magneticfield [22]. The state equation

∇ × 1

µ0∇ × A = σ(E) · E (3)

where

E = −∂A∂t

− ∇V

is to be resolved for the magnetic vector potential A andthe electric potential V . In the case of two-dimensionalproblems—e.g. for an infinite wire exposed to a perpendicularmagnetic field—the cross-section of a superconducting wire isdivided into rectangular regions where the conductivity of thesuperconductor obeys the power law

σ(E) = j0E0

(E

E0

) 1−nn

. (4)

An alternative formulation [23] uses the current vectorpotential T and magnetic scalar potential �. Thoroughdiscussion about the advantages and drawbacks of variousformulations for the nonlinear-resistivity approach tothe simulation of electromagnetic phenomena in hardsuperconductors has been published recently [24].

Another numerical simulation method of great practicaluse is the minimum magnetic energy variation (MMEV)method [25]. Its application is quite straightforward for two-dimensional problems; however, the original formulation [26]is three dimensional [27]. Similarly to the previous method,the cross-section S of the superconductor is divided intorectangular regions; each of them should be filled either with+ jc, − jc, or left empty to minimize the functional

F[ j ] = 12

∫S

j (r)A j(r) dS −∫

Sj (r)A�j(r) dS

+∫

Sj (r)(Aa(r) − A

�a(r)) dS (5)

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AC loss in practical superconductors

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10-1

100

101

102

Ia/I

c

2πQ

µ0Ic2

- fill 85%- fill 63%- fill 48%

Figure 2. Calculated normalized AC loss, i.e. divided by the lossprefactor proportional to I 2

c , for three cores of superconductingcable differing in the percentage of HTS occupation of the perimeterof the core former. As expected, larger gaps between neighbouringtapes, i.e. lower fill of the core perimeter by HTS tapes, lead tohigher AC loss when transporting AC. Kim’s dependence of jc(B)with jc0 = 9.37 × 107 A m−2 and B0 = 14 mT was assumed.

where A j is the magnetic vector potential created by thecurrents in the superconductor and Aa is that from the appliedfield. The quantities with caps correspond to those obtainedby the solution in the previous time step. The minimizationshould obey the constraints

I =∫

Sj (r) dS

| j | � jc.

Thus, the MMEV follows the original idea of the critical state,i.e. the availability of just one value for the current density. Inspite of the fact that this is a notable simplification for HTSs,the method offers a distinct shape of the boundaries betweenpositive and negative current density, as well as the shape ofthe current-free core being precisely found. Then, one canunderstand the underlying physics of the flux penetration in agiven configuration.

3.1. Influence of gaps in transmission cable on its AC loss

Here we show how one can assess the importance of permittingminimum gaps between neighbouring tapes of the single-layercable as a model arrangement for the power transmissioncable. High packing factor—defined as the percentage ofthe perimeter of the central cylindrical mandrel occupied bythe tapes—is not reached easily in the factory production,leading to a significant increase of the manufacturing cost whentrying to minimize the gaps. Simulations following Brandt’smethod [20] allowed us to calculate the quantitative predictionpresented in figure 2. We considered 14 tapes placed straightin parallel on the central mandrel of 21 mm diameter. Threecables made from tapes of the same superconducting materialbut different widths have been compared. The decrease of thetape width leads to a reduction of the cable critical current dueto two independent mechanisms: the first is a simple reductionof the superconductor’s cross-section, the second is the changeof local magnetic field distribution. Therefore, we comparethe normalized values of transport AC loss, i.e. divided by

0.2 0.6 1

0.6 -0.2 -1

-0.6 0.2 1

Figure 3. Distribution of current density in the rectangularcross-section of wire made from hard superconductor duringsimultaneous transport of AC current and application of AC field,calculated with the help of the minimum magnetic energy variation(MMEV) method. AC current amplitude is 60% of the wire criticalcurrent, and the magnetic field amplitude is 72% of the penetrationfield. The sequence of distributions, each characterized by thenumber indicating the ratio of actual transport current with respectto the amplitude value of AC current, goes from left to right andfrom top to bottom.

the factor proportional to I 2c . Also, the transport current is

normalized with respect to the critical current of the cablein figure 2. In the simulations, the critical current densitydependence on local magnetic field was assumed to followthe Kim’s relation jc(x, y) = jc0

1+ |B(x,y)|B0

. As one can see, the

gaps between tapes can significantly influence the AC lossperformance of a power transmission cable.

3.2. AC–AC case for a wire with a rectangular cross-section

The circumstance that commonly a superconducting wireexperiences in a cable (made from more wires connectedin parallel) or a winding of an electromagnet is that ofAC transport under the simultaneous action of the AC fieldproduced by the currents in other wires. The transportcurrent and the magnetic field change in phase. Somebasic predictions for these conditions have been derived, butonly for the slab geometry [28] or that of an infinitely thinstrip [9, 29]. For HTS tapes, the empirical engineeringapproach based on the fits of experimental data [30] is ofpractical use; on the other hand, it does not explain why theformulae seem to require modification in some cases. Withthe help of the MMEV method, it is possible to visualizethe movement of flux fronts as the boundaries between zoneswith different current densities. In figure 3 is shown theevolution of current density distribution in rectangular cross-section of a superconducting conductor with rectangular cross-section (aspect ratio 1:5), calculated with the help of MMEV.Details of the computation can be found elsewhere [31].The overall picture resembles the results obtained by othermethods [22]. Because the boundaries between positive andnegative current density, as well as the shape of the current-freecore, are clearly determined, one can easily understand whythe predictions based on the assumption of a one-dimensionalflux penetration—where these boundaries are planar—wouldnot lead to satisfactory agreement with experiment. Thepredictions of our calculations, performed under assumptionof magnetic-field-independent critical current density, havebeen compared with experimental AC loss data obtained onBi-2223/Ag multifilamentary tape. In figure 4 is the result

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F Gomory et al

0,1 1

0,1

1

10

100

k

Ia/I

c

Γ=2π

Q/(

µ 0I a2 )

Figure 4. AC loss in Bi-2223 tape transporting AC current with theamplitude Ia under simultaneous action of AC magnetic field withthe amplitude Ba. The loss is shown in terms of the loss factor,i.e. the total AC loss divided by the electromagnetic energy, that isproportional to I 2

a , on the AC current amplitude Ia (normalized toIc) at the condition that Ba increases in proportion to Ia . Theoreticalcurves (lines) calculated for the proportionality coefficients k in theformula Ba = k

wµ0 Ia equal to 0, 0.05, 0.25, 0.5, 0.99, 2, and 4 (from

bottom to top) are compared with experimental data plotted bysquares.

of this comparison plotted as the normalized loss (sometimescalled the loss factor) versus the normalized current, assumingthat the applied AC field with the amplitude Ba increases inproportion to the AC current amplitude, Ia. In other words,Ba = k

wµ0 Ia, where w is the tape width and k is the constant

of proportionality. This is the condition representing the mostimportant practical cases, with k � 1 characterizing thecase of prevailing self-field (as in the transmission cable) andk > 1 corresponding to the situation met in an electromagnetwinding. An interesting feature of the theoretical prediction isthat, at low excitations, the loss factor always increases withthe first power of the current (thus also the first power of theapplied field), then in turn predicts a universal loss dependenceof a superconducting device at low energizing current as ∝I 3

a .Save for the very low k, i.e. the nearly self-field case, theexperimental data obey this prediction quite well.

Another interesting situation to investigate in the AC–ACcase is the cyclic change of transport current and magneticfield with a certain phase shift between them. This couldhappen in some important applications like the three-phasetransmission line or a transformer winding. We have employedboth of the simulation methods mentioned in section 3 tocheck the AC loss measured on the same Bi-2223/Ag tape.The results are presented in figure 5. A certain deviationbetween the theoretical predictions based on two differentmodels for superconducting media is not surprising: the losspredicted for the critical-state-like simulation performed bythe MMEV method is systematically below the experimentaldata, while the prediction of the calculation starting from asmooth non-linear E( j ) curve with the power exponent n = 25is mostly above them. Interestingly, for all three curves theabsolute loss maximum is not found for the in-phase condition.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 50 100 150

Phase difference [degrees]

P [

W/m

]

Experiment

Brandt's method

Minimum Magnetic Energy Variation(MMEV)

Figure 5. Dependence of AC loss on the phase shift between ACtransport current and AC magnetic field of the same frequency.Simulations for a tape 4 mm wide, 0.2 mm thick with the criticalcurrent of 38 A, exposed to 10 mT AC field when transporting17.7 A of AC current, are compared with experimental data. Themaximum of dissipation obtained by two different simulationmethods as well as that found experimentally is clearly differentfrom the in-phase case.

We have found the shift of the loss maximum from zerodegrees reproducibly in both the experiments and simulationsperformed for a wide range of current/field ratios [32].

3.3. Effect of ferromagnetic cover on AC loss

Another important problem that can be investigated with thehelp of the MMEV method is that of a superconductor coveredby a ferromagnetic material. Several predictions have beenmade that such a cover should decrease the AC loss [33, 34],though none of these works has rigorously calculated the fluxpenetration into hard superconductor put in such a compositestructure. Because the MMEV procedure of finding thesuccession of flux front movements is equivalent to the critical-state approach, the implementation of this method would allowus to perform such a refinement. We have adopted the finiteelement code FEMLAB to calculate the magnetic field inthe simulation box with dimensions ten times exceeding thetape width. The consequence of placing a ferromagnetic stripon a superconducting wire with rectangular cross-section isillustrated in figure 6. The shape of the flux lines and hencethe flux penetration front completely changes with respect toa bare superconducting wire. It seems that the ferromagneticsheath acts as a magnetic mirror, straightening the flux linesinside the superconductor. This would influence the valuesof penetration field Bp, the diamagnetic susceptibility χ0 andalso the shape of the χ ′′

int(y) dependence in formula (1). Incomparison to the previous work, performed using the ANSYScode with constant permeability of iron [35], we have used thenon-linear dependence

µr = µmax

1 +(

BBc

)2+ 1 (6)

to approximate the non-linear magnetic permeability of theferromagnetic sheath. At this stage we have not been

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AC loss in practical superconductors

Figure 6. Effect of a ferromagnetic cover on the behaviour of a stripfrom a hard superconductor exposed to perpendicular magnetic fieldequal to 50% of the penetration field. A dramatic change of theshape of the flux penetration front as well as a significant reductionof magnetic field inside the superconductor is clearly visible.Non-linear but reversible permeability of the ferromagnetic materialwas approximated by expression (6) with µmax = 1000 andBc = 0.1 T.

able to insert the hysteresis of ferromagnetic material in thecalculations, and this remains one of the big challenges forthis kind of simulations. As an illustration of the predictionswe have achieved, we present here the influence of the width ofthe ferromagnetic cover on the AC susceptibility χ ′′ = χ0χ

′′int

dependence on the applied magnetic field. One can expectthat, depending on how much surface will be covered by a softmagnetic material with necessary thickness, the effect will bemore or less visible. This is indeed what we have found, asshown in figure 7. With the narrowing of the ferromagneticcover, the susceptibility increases approaching that of the baresuperconducting wire.

We should underline that, because no hysteresis isconsidered for the ferromagnetic material, this part of theconductor is not accounted for in the total loss. Such anassumption should be modified when a comparison withexperimental data will is carried out. Otherwise, a netreduction of the AC loss results from the covering of thesuperconductor by a ferromagnetic material.

4. Conclusions

The numerical simulation methods available at the presenttime to model the electromagnetic behaviour of a hardsuperconductor represent a significant step forward withrespect to the analytical models. Nowadays, two-dimensionalproblems can be tackled successfully using the Brandt’smethod or the minimum magnetic energy variation (MMEV)method. This means that the hysteresis loss in single-corewires from hard superconductors of any shape can be predicted

0

0.5

1

1.5

2

2.5

0.000001 0.00001 0.0001 0.001Ba [T]

χχ χχ"

Figure 7. Effect of the completeness of the ferromagnetic sheathcover on the AC loss in an external magnetic field, calculated by theMMEV method. Assuming no hysteresis in the ferromagneticmaterial, the AC loss—expressed through the imaginary part of ACsusceptibility, χ ′′—is reduced when the ferromagnetic materialbetter covers the superconducting strip.

at any combination of transport AC current and applied ACmagnetic field. In the case of arrays of filaments, the hysteresisloss—i.e. that prevailing at low frequencies—can be calculatedas well.

When the coupling current flowing across the metallicmatrix in direction perpendicular to the filaments cannot beneglected, the problem becomes three dimensional. For thissituation, the approach of representing the hard superconductoras a normal conductor with non-linear current–voltage curveshould work as well, as the results achieved with the help ofphysically justified simplifications have demonstrated [36, 37].However, the requirements on computing power are still severeand further development for 3D calculations is necessary.

Another field where significant progress is required isthe investigation of composites containing superconductingfilaments and ferromagnetic parts. Surprisingly, the simulationmethods for magnetic hysteresis in superconductors are nowbetter developed than those for ferromagnetic shapes when theexact distribution of local magnetic field is regarded. It wouldbe interesting to see whether a general hysteresis simulationmethod like the Preisach model [38] could be helpful inresolving this problem.

Acknowledgments

Financial support of this work by the Science and TechnologyAssistance Agency (contract APVT-20-012902) and theNATO Science programme (grant PST.CLG 980001) isacknowledged.

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INSTITUTE OF PHYSICS PUBLISHING SUPERCONDUCTOR SCIENCE AND TECHNOLOGY

Supercond. Sci. Technol. 19 (2006) 397–404 doi:10.1088/0953-2048/19/4/026

Study of ac loss in Bi-2223/Ag tape underthe simultaneous action of ac transportcurrent and ac magnetic field shifted inphase

M Vojenciak1,5, J Souc1,6, J M Ceballos2, F Gomory1, B Klincok1,E Pardo3 and F Grilli4

1 Institute of Electrical Engineering, Centre of Excellence CENG, Slovak Academy ofSciences, Bratislava, Slovak Republic2 Industrial Engineering School, University of Extremadura, Badajoz, Extremadura, Spain3 Grup d’Electromagnetisme, Departament de Fısica, Universitat Autonoma de Barcelona,08193 Bellaterra, Barcelona, Catalonia, Spain4 Superconductivity Technology Center, Los Alamos National Laboratory, Los Alamos,NM 87545, USA5 University of Zilina, Zilina, Slovak Republic

E-mail: [email protected]

Received 28 November 2005, in final form 7 February 2006Published 7 March 2006Online at stacks.iop.org/SUST/19/397

AbstractInvestigation of ac loss under the simultaneous action of the transportac current and the external ac magnetic field is of prime importance forthe reliable prediction of dissipation in electric power devices such asmotors/generators, transformers and transmission cables. An experimentalrig allowing us to perform ac loss measurements in such conditions, on short(10 cm) tape samples of high-temperature superconductor Bi-2223/Ag, wasdesigned and tested. Both the electromagnetic and thermal methods wereincorporated, allowing us to combine the better sensitivity of the former andthe higher reliability of the latter. Our main aim was to see how the ac lossdepends on the phase shift between the transport current and the externalmagnetic field. Such a shift could have different values in variousapplications. While in a transformer winding, the maximum phase shift atfull load will probably not exceed a few degrees, in a three phasetransmission cable in tri-axial configuration it is around 120◦. Therefore, weexplored the whole range of phase shifts from 0 to 360◦. Surprisingly, themaxima of dissipation did not coincide with zero shift as expected fromqualitative considerations.

1. Introduction

An understanding of ac loss is of prime importance whenelectric power applications of superconductors are underconsideration. In the past four decades, an extensiveknowledge has been gathered about the behaviour of wiresand windings under the action of external ac magnetic fields.

6 Author to whom any correspondence should be addressed.

As a consequence, the ac magnetization loss is now quitewell understood as a result of the interaction between themagnetic field and the composite superconducting wires. Inthe case of ac transport current, the dissipation is controlledby the same principles. However, the driving magnetic fieldis generated by the transport current itself. Such self-fieldloss (also called the ac transport loss) has attracted particularinterest in the last decade, oriented on the high temperaturesuperconducting (HTSC) tapes. Similarly, as in the case of

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ac magnetization loss, the theoretical understanding and theexperimental techniques are now quite well established for actransport loss studies.

However, in any power application, the superconductingwire transports ac current and experiences an additional acmagnetic field due to the currents in other wires at the sametime. Therefore, for the forecast of ac loss in superconductingdevices, the knowledge of dissipation under the combinedaction of ac current and ac field is essential. Theoreticalpredictions derived for the simplest case of an infinite slabor strip [1, 2] have been refined for more realistic geometriesby numerical simulations [3]. The collection of reliableexperimental data is quite laborious, because there are manypitfalls in the measurement procedures. Nevertheless, ageneral consensus has been reached about the possibilityof determining the total ac loss, Ptot, taking into accountthat the electromagnetic energy interacting with the samplecomes from two independent sources of energy: one powersupply provides the transport ac current to the sampleand covers the part of the dissipation called the transportloss, PT, while another feeds the winding that producesthe external ac field, also balancing the magnetization lossPM in the sample. An important conclusion of severalpapers dealing with this issue is that the transport lossand the magnetization loss can be determined separately byelectromagnetic measurements [4–8].

The advantages of using the electromagnetic measurementtechnique for determining ac loss are its better sensitivity andlower time consumption compared to the thermal method. Onthe other hand, the thermal measurement was found to beindispensable when the ac field is shifted in phase with respectto the ac current [9–11]. Such conditions, when the currentIT = √

2Irms cos(2π f t) combines with the magnetic fieldBext = √

2Brms cos(2π f t + ϕ), are met in the windings oftransformers, generators and power transmission cables. Thecrucial question of the ac loss investigation for ϕ �= 0 is:could the ac loss at a given phase shift, ϕ, be deduced fromthe value measured at zero phase shift? The experimental datashowing a 180◦ periodicity with a maximum at zero phaseshift, obtained on Bi-2223 tape transporting ac current withan amplitude equal to the dc critical current [10] favour sucha hypothesis. Similarly, the analytical calculations for a bifilarcoil from Bi-2223 [12] found the maximum of dissipation at accurrent in phase with the ac field. In the case when this werea general feature, one could develop an empirical formula ofthe form Ptot(ϕ) = Ptot(0) (1 − k sin 2ϕ) where Ptot(0) can bedetermined by the sensitive and time-effective electromagneticmeasurement for any value of current and field, and the onlyfeature to explore would be the dependence of the constant kon Brms and Irms. On the other hand, the results of numericalcalculations for a coil geometry [13] as well as the recentexperiments on a single straight Bi-2223 tape raised somedoubts in this regard. As concluded in the latter work, thechange of total loss due to ϕ is difficult to see because of theinsufficient sensitivity of the thermal method.

This work presents the measured ac losses of a Bi2223/Agtape under the simultaneous action of transport current andexternal magnetic field, shifted in phase. In particular, we havedetermined the values of the phase-shift angle correspondingto the maximum of the ac losses, for different combinations ofcurrent and field amplitudes.

sample

Cu magnet

I

current leads

Figure 1. Schematic diagram of the transport current circuit andexternal magnetic field circuit. The design of the current leads allowsus to change their position with respect to the Cu magnet in order toachieve zero mutual inductance between both circuits.

The paper is organized as follows. Section 2 describesthe experimental set-up and the procedure for determining theac losses both with thermal and electromagnetic methods; thethermal method is used to validate the electromagnetic one(used later in this paper), which needs particular attention dueto the simultaneous presence of transport current and externalfield, shifted in phase. Section 3 contains the experimentalresults of the ac losses measured with the electromagneticmethod. Section 4 contains a comparison of the experimentalresults and the predictions of three different numerical models.Finally, section 5 draws the conclusions of this work.

2. Experimental details

The main aim of our work was to determine, by anelectromagnetic method, the ac loss in a superconducting wiretransporting the ac current IT = √

2Irms cos(2π f t) whileexposed to the ac magnetic field Bext = √

2Brms cos(2π f t+ϕ).In particular, the dependence on the phase shift ϕ between IT

and Bext has been investigated. The electromagnetic methodwas chosen because of the known limitations of the thermalmethod that nevertheless was used as a reference. To carry outthe electromagnetic measurement, one has to resolve two mainproblems:

(1) An independent supply of current into the sample andin the ac field magnet winding is necessary to keep thevalues of Irms and Brms constant while changing the phasedifference.

(2) The distinction between the dissipation covered by thepower supply for IT and the one delivered by theenergizing system of the ac magnet is necessary to avoid adouble count of dissipation in loss registration.

2.1. Supply of ac transport current in ac applied field shiftedin phase

The independent operation of two power supplies wasguaranteed by the design and construction of the apparatusshown in figure 1. The racetrack shaped magnet made ofcopper wire was used to generate the ac external magneticfield. The construction of the transport current leads supplyingthe sample allow us to change their relative position withrespect to the Cu magnet winding. Thanks to this concept,the accomplishment of zero mutual inductance between bothcircuits was possible. In practice this was achieved bysearching for that particular position of the loop, composedof the current leads together with the sample, at which no

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differentialthermocouple

wiring for transportloss measurements

Bext

IT

0.E+00

4.E-06

8.E-06

1.E-05

2.E-05

2.E-05

0 0.2 0.4 0.6 0.8 1

PT [W/m]

Utc

[V]

a b

Figure 2. (a) Simplified set-up for the calibration of the thermal method by the standard electromagnetic measurement of the ac transport lossin self field, (b) calibration curve—thermocouple dc voltage in dependence on the ac transport loss measured by the standard electrical lock-inmethod.

voltage is induced in it at any value of external ac magneticfield. In such a configuration, no influence of the magneticfield on the transport current (and no influence of the transportcurrent on the magnetic field) occurs regardless of the phaseshift between them. Then, after setting of the desired constanttransport current and constant magnetic field, the measurementof the ac loss dependence on the phase shift between them canbe carried out without the necessity of adjusting the settings ofthe power supplies.

The multifilamentary Bi-2223/Ag tape sample7 withcritical current Ic = 38 A was immersed in liquid nitrogenduring the measurements. A 2-channel generator allowingus to set the phase shift between two sinusoidal signals ofadjustable amplitude but with the same frequency was used.After being amplified by a two-channel audio amplifier, thesesignals were used for generating the ac external magnetic fieldand the ac transport current flowing in the sample, respectively.The current to the Cu magnet was delivered from the amplifieroutput directly. To achieve the required amplitude of thetransport current, a toroidal transformer connected to thesecond output of the amplifier was used. The measurementwas carried out at the frequency f = 72 Hz both for thetransport current and the external magnetic field. The magneticfield was oriented perpendicular to the wide face of the sample.The Rogowski coils were used for measurement of both thecurrent of the Cu magnet and the sample transport current.To improve the stability of the impedances of the individualcircuits and to reduce the drift of the phase difference in time,the transport current leads including transformer as well as theCu coil were immersed in a liquid nitrogen bath. Two double-channel lock-in voltmeters were used in the apparatus, onedetecting the sample current and voltage, and the second onefor the measurement of the magnetization loss. The values ofall of the electromagnetic quantities considered in this work arerms.

2.2. Thermal method

The measurement set-up for the thermal method, togetherwith the calibration curve is illustrated in figure 2. Thebasic principle is described in [10]. For thermal insulationof the sample, two polyethylene foam blocks were used. A

7 Australian Superconductor No 2001-3-A/MF.

differential method using two E-type thermocouples connectedin series was used to probe the increase of the sampletemperature due to ac losses. One thermocouple was placed onthe sample surface isolated by Teflon tape, and the second onewas immersed in liquid nitrogen to serve as a reference. Thethermocouple wires were tightly twisted to reduce the voltageinduced by the external ac magnetic field. The calibrationconsists of measurement of the thermocouple voltage Utc bya Keithley 2700 voltmeter as a function of the transport currentloss PT. The standard electrical measurement of PT in transportconditions (pure self field) was used for calibration using thelock-in technique.

The thermal method can only be used in a limited rangeof currents and fields. To achieve a measurable increase ofthe sample temperature, an ac loss exceeding 0.1 W m−1 isrequired in our set-up. The upper limit was 1 W m−1, whenthe decrease in the critical current due to the temperature risebecomes significant. After calibration, the measurement of thetotal ac loss Ptot dependence on the phase shift between acexternal magnetic field Bext and ac transport current IT can becarried out. The results of this measurement are presented insection 2.3.

2.3. Electromagnetic method

In the electromagnetic measurement, two loss componentshave been determined independently. In the following, theparts of the dissipation covered by the amplifier feeding thesample with ac current and the one supplying the current in theCu magnet will be called transport loss (PT) and magnetizationloss (PM), respectively.

The approach of determining the loss from the point ofview of the energy source [14, 15] is very profitable. As shownin several studies [16, 17], the total loss Ptot is obtained as thesum of the losses covered by individual sources of deliveredpower:

Ptot = PT + PM. (1)

In our procedure, PT and PM were determined separately fromthe signals registered by the pair of taps and the pick-up coil,respectively, using the lock-in technique. The measurementset-up is illustrated in figure 3.

The procedure to evaluatePT and PM was as follows.Transport loss was measured with the pair of voltage taps

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1 23 4

LN2

2 channel audioamplifier

2 channel generatorref out

B ext

IT

Lock-in T ref inLock-in M ref inchA chBchA chB

∆∆∆ ∆

Rogovski coil M

Rogovski coil T

transformer

Pick-up coil and compensation coilfor magnetization loss measurement

loop for transportloss measurement

IM

Figure 3. Measurement set-up for transport loss and magnetization loss measurement by the electromagnetic method as a function of thephase shift between the transport current and the external magnetic field. Two types of operational amplifier are used: the amplifiers numbered1, 2, 3, and 4 have variable gain, and are used for fine cancellation of the unwanted signals in the differential units indicated by the symboldelta.

separated by a distance L = 0.05 m. The wires leading thesignal from these contacts were first guided in the transversaldirection to the tape axis to the distance of 1 cm, then bent by90◦ and put together forming a loop with the plane orientedparallel with the external magnetic field and perpendicularto the wide side of the sample. The value of the transportcurrent was measured by a Rogowski coil (Rogowski coil T infigure 3) connected to channel A of the lock-in (Lock-in T infigure 3). This channel was also used to set the reference phaseof channel B, where the voltage signal was connected. Thephase setting is necessary because only the voltage in phasewith the transport current represents the loss voltage, UTre.

For the faultless determination of transport loss in thepresence of an external ac magnetic field shifted in phase, thevoltage induced in the signal wires by the external ac fieldshould be zero. This was not required when the transportcurrent and external magnetic field were in phase. In that case,the induced signal is perfectly out of phase with respect tothe transport current, and thus it does not interfere with theloss voltage UTre. However, in our experiments with phaseshifts, any signal induced by the ac field will add a componentthat is not distinguishable from the true UTre signal. Thereduction of the false signal was attained using two steps.First, a coarse reduction by adjusting the position of the signalwires; therefore, a fine reduction by using the voltage derivedfrom another Rogowski coil (Rogowski coil M in figure 3)coupled with the current supplied to the ac magnet. Properlyadjusted by the wide band operational amplifier with variablegain OA1 (in figure 3 indicated by 1), the correction signalis subtracted from the measured voltage. We found that asingle zeroing procedure at Irms = 0 was valid for the wholerange of investigated external magnetic fields. After such acompensation and after applying the desired transport current,magnetic field and phase shift, the in-phase component ofthe signal measured by channel B of the lock-in is the lossvoltage UTre. To increase the measurement sensitivity, theinductive part of the measured voltage (induced by the self

field from the transport current) is to be compensated as well.For this purpose the signal derived from the Rogowski coil T,and adequately adjusted by the other wide band operationalamplifier OA2 (in figure 3 indicated by 2), was subtracted fromthe measured loop signal. The loss voltage UTre was used forevaluation of the transport loss according to the formula

PT = IrmsUTre/L . (2)

To measure the magnetization loss, the pick-up coilwas used (see figure 3). A compensation coil of the samedimensions but wound in the opposite direction is connected toimprove the sensitivity. The magnet current IM was measuredby the Rogowski coil M connected to channel A of the secondlock-in. Also, the phase setting for both of the channels ofthis lock-in is derived from this signal. The pick-up loopsystem was calibrated using the superconducting sample withknown magnetization loss measured by the calibration freemethod [15]. Magnetization loss is then determined by theformula:

PM = C IMUM/ l (3)

where C is the calibration constant, IM is the rms value of themagnet current, l is the length of the sample and UM is thepart of the pick-up coil voltage, which is in phase with themagnet current. This voltage is measured by the input channelB of the second lock-in (Lock-in M in figure 3). The series ofcorrections similar to the one described for the transport lossmeasurement was used. The signal derived from the transportcurrent was amplified by the operational amplifier OA3 withvariable gain and subsequently used to cancel out the falsesignal induced in the pick-up coil by the transport current.The gain adjustment carried out at Brms = 0 was sufficientto achieve the signal correction proper for the whole rangeof ac fields. Another compensation signal derived from theRogowski coil M and adjusted by OA4 was used to reduce theout-of-phase part of the measured signal.

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ac loss in Bi-2223/Ag tape

0

0.2

0.4

0.6

0.8

0 60 120 180 240 300 360 420 480phase shift [deg]

P[W

/m]

Ptot,thermalPtot,elmagPTPM

Figure 4. Comparison of total ac loss measured by the thermal (fullsquares) and the electromagnetic (empty squares) method at atransport current of 23.5 A and a magnetic field of 26.2 mT.Transport loss (triangles) and magnetization loss (diamonds) are alsoshown.

2.4. Test of the electromagnetic method by the thermal method

The electromagnetic measurement of ac loss described inthe previous section represents a complex task. To confirmthe correctness of the suggested experimental procedures, theresults have been checked by the thermal method. The result,as shown in figure 4, was more than satisfactory. For thesake of this comparison, the tape was exposed to the externalmagnetic field Brms = 26.2 mT when carrying the ac transportcurrent Irms = 23.5 A. The ac loss measurement was carriedout in the whole phase shift range with 10◦ step. In the figure,the dependences of loss constituents PT and PM together withPtot = PT + PM are shown. Because the thermal method isbased on a physical principle completely different from theelectromagnetic method, the excellent coincidence shown infigure 4 confirms the correctness of the results obtained by theelectromagnetic measurement in our experimental apparatus.

3. Experimental results

All data presented in this section were measured by theelectromagnetic method. In the following we present theac loss dependences on the phase shift measured at Irms =11.8, 17.7, 24.8 A and Brms = 5, 10 and 15 mT as parameters.The step of the phase shift was 10◦.

In figure 5, the dependence of transport loss PT as afunction of the phase shift is displayed. The results obtainedwith three different currents are gathered in one graph. Notethe different scale for the three plots in the figure. As onecan see, both the ac current as well as the ac field cause anincrease in the transport loss in the whole range of phase shifts.The maximum ac loss is observed at phase shifts ϕmax > 0,whose value is strongly dependent on Irms and Brms. Whileat Brms = 5 mT ϕmax is about 30◦ for all three considered accurrents, at higher fields it moves from ∼45◦ at low currentsdown to ∼15◦ at the highest current.

In figure 6, the magnetization loss PM as a function ofthe phase shift is displayed. Although the effect of Brms is amonotonous increase in the magnetization loss, the influenceof Irms on the magnetization loss is not always in the samedirection: at Brms = 5 mT, the rise of current increasesthe loss at ϕ = 0, but reduces at ϕ = 60. On the other

0.00

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PT

[W/m

]

24.8 A, 5mT17.7 A, 5mT11.8 A, 5mT

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0.30

-180 -120 -60 0 60 120 180 240phase shift [deg]

PT

[W/m

]

24.8 A, 15mT17.7 A, 15mT11.8 A, 15mT

Figure 5. Transport loss under the combination of the ac field andthe ac current shifted in phase, measured on Bi-2223 tape withIc = 38 A at 77 K, f = 72 Hz.

hand, at Brms = 15 mT the magnetization loss is reducedby the transport current at any phase shift. Interestingly, atBrms = 10 mT and the phase shift ranging from 0 to 30◦a non-monotonous dependence of the ac magnetization losson IT was observed. For zero phase shift this result is ingood qualitative agreement with the data published in [17],where magnetization loss was measured for this case by anelectromagnetic method as well. In contrast to the transportloss, the magnetization loss maximum is found at ϕmax < 0 forall investigated combinations of the parameters Irms and Brms.

It is not obvious to deduce a general rule for the positionof the maximum in the total loss. From one side, the plot oftotal ac loss Ptot in figure 7—which is nothing more than thesum of the magnetization loss and the transport loss—showsa common increase of the total loss with both the Irms andBrms. However, for Brms = 5 mT the small movement of themaximum with the increase of Irms is towards higher ϕ, whileat Brms = 15 mT the value of ϕmax reduces significantly withincreasing Irms. The summary of ϕmax dependence on the Irms

and Brms combination is illustrated in figure 8. One conclusionis clear from these data: to find the maximum of total loss atϕ = 0 is more an exception than a rule.

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0.00

0.01

0.02

0.03

-180 -120 -60 0 60 120 180 240phase shift [deg]

PM

[W/m

]

24.8 A, 5mT17.7 A, 5mT11.8 A, 5mT

0.00

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-180 -120 -60 0 60 120 180 240phase shift [deg]

PM

[W/m

]

24.8 A, 10mT17.7 A, 10mT11.8 A, 10mT

0.00

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-180 -120 -60 0 60 120 180 240phase shift [deg]

PM

[W/m

]

24.8 A, 15mT

17.7 A, 15mT

11.8 A, 15mT

Figure 6. Magnetization loss under the combination of the ac fieldand the ac current shifted in phase, measured on Bi-2223 tape withIc = 38 A at 77 K, f = 72 Hz.

4. Comparison with numerical simulations

The most interesting feature found in our experimentalobservations is that the maximum of the total loss does notoccur for the zero phase shift between the ac current andthe ac field. To check whether such behaviour could beplausible, three independent techniques have been used fornumerical simulations. Because of the long tape lengthand the compact arrangement of the filaments, one shouldexpect strong coupling currents between the filaments. As aconsequence, the tape electromagnetic behaviour is mainly thatof a single filament [18].

Two of the calculation techniques assume that thesuperconductor’s electrical behaviour is described by meansof a non-linear resistivity, derived from the E(J ) power-lawE(J ) = Ec(J/Jc)

n . The current–voltage curve with n = 25was used in the simulation in Matlab code based on the Brandtmethod [19], founded on solving Maxwell’s equations in theform of integral equations. A sinusoidal current is imposedto flow in the cross-section of the tape divided into 60 × 60superconducting elements of rectangular cross-section. Also,

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t[W

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0.50

-180 -120 -60 0 60 120 180 240phase shift [deg]

Pto

t[W

/m]

24.8 A, 15mT17.7 A, 15mT11.8 A, 15mT

Figure 7. Total ac loss under the combination of the ac field and theac current shifted in phase, measured on Bi-2223 tape withIc = 38 A at 77 K, f = 72 Hz.

the replacement of the superconductor by a media with non-linear resistivity was used in two-dimensional finite elementsimulations with the commercial software Flux3D [20]. Thistechnique allows us to obtain detailed information aboutthe current density and magnetic field distributions insideconductors, as well as to compute the ac losses. The sinusoidaltransport current is imposed by means of a current source,whereas the magnetic field (which has variable phase withrespect to the current) is imposed by means of appropriateconditions for the magnetic vector potential A on the domainboundary [21].

For these two techniques, the ac losses at a givenfrequency f , expressed in W m−1, are computed as follows:

PSC = f∫ 1/ f

0

∫S

J · E dS dt (4)

where S is the cross-section of the superconductor.Another approach is to assume the critical state model [22]

and calculate the current distribution in the tape by meansof the minimum magnetic energy variation (MMEV) method.This technique was introduced in [23, 24] for cylinders in a

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0

10

20

30

40

50

10 15 20 25IT [A]

ϕ max

[deg

]20mT15mT10mT5mT

0

10

20

30

40

50

0 5 10 15 20 25Bext [mT]

ϕ max

[deg

]

24 A17 A11 A

a b

Figure 8. Position of the total loss maximum as a function of (a) transport current at Brms = 20, 15, 10 and 5 mT as a parameter and on (b)magnetic field at Irms = 11, 17, 24 A.

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phase shift [deg]

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/m]

experiment 20mTexperiment 15mTexperiment 10mTexperiment 5mTMMEV 20mTMMEV 15mTMMEV 10mTMMEV 5mTBrandt 20 mTBrandt 15 mTBrandt 10 mTBrandt 5mTFlux3D 20 mTFlux3D 15 mTFlux3D 10mTFlux3D 5 mT

IT = 17,7 A

Figure 9. Ptot dependence on phase shift at IT = 17.7 A and Bext = 5, 10, 15, 20 mT. Squares: experimental results (full line); diamonds:simulation by minimum magnetic energy variation method; triangles: simulation in Matlab based on the Brandt method; circles: simulation inFlux3D software. The colour of the points represent the measurement at constant magnetic field (colour online only).

magnetic field and later applied to the infinitely long geometryin a transverse magnetic field or transporting current [25, 26].For those situations, the current distribution can be foundby magnetic energy minimization, while for the case ofac transport in alternating applied field it is necessary touse Prigozhin’s minimization principle [27]. The numericalprocedure for calculating the current distribution and the acloss is mainly the same as in [28], where we assumed a uniformcritical current density Jc. The main difference is that now thestarting situation for the initial stage (first increase of currentfrom 0 to the maximum) is the case of only magnetic fieldwith Bext = √

2Brms cos ϕ, calculated by magnetic energyminimization. The rest of the time evolution is calculated byMMEV as in [28], obtaining a stationary cyclic state beyondthe end of the first reverse stage (IT decreasing from themaximum to the minimum).

In figure 9, the dependences of the total loss Ptot on phaseshift obtained by simulations are compared with experimentaldata achieved by the electromagnetic method. The results oftwo numerical methods based on the smooth current–voltagecurve are systematically above the estimation calculated by theMMEV. Experimental data fall in between these predictions. Itis also encouraging that the absolute values of the predictedloss agree quite well with the experimental ones. Theobserved agreement is surprising taking into account that noneof the simulations used here considered the critical current

dependence on the magnetic field. This probably also explainswhy the theoretical curves exhibit weaker dependence of theac loss on the phase shift compared with the experimentalones. Anyway, all three methods predict qualitatively similarbehaviour; in particular that the loss maximum is not found atzero phase shift.

5. Conclusion

An experimental set-up for the measurement of ac lossunder the simultaneous action of the transport current andthe magnetic field shifted in phase was developed andtested. Experimental results obtained by the electromagneticmeasurement are in excellent agreement with those obtainedby the thermal method. Moreover, the results obtained ona standard Bi-2223/Ag tape are in good agreement withtheoretical predictions of three numeric calculations. Ourinvestigations clearly show that the maximum loss is not atzero phase shift, and its position depends on the magnitude ofthe current and field.

Acknowledgments

This work was supported in part by the APVT-20-012902project, by the European Commission (Project ENK6-CT-2002-80658 ‘ASTRA’) and in part by the US Department ofEnergy Office of Electricity Delivery.

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[22] Bean C P 1962 Phys. Rev. Lett. 8 250[23] Sanchez A and Navau C 2001 Phys. Rev. B 64 214506[24] Sanchez A and Navau C 2001 Supercond. Sci. Technol. 14 444[25] Pardo E, Sanchez A and Navau C 2003 Phys. Rev. B 67 104517[26] Pardo E, Sanchez A, Chen D-X and Navau C 2005 Phys. Rev. B

71 134517[27] Prigozhin L 1996 J. Comput. Phys. 129 190[28] Pardo E, Gomory F, Souc J and Ceballos J M 2005 Preprint

cond-mat/0510314

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IOP PUBLISHING SUPERCONDUCTOR SCIENCE AND TECHNOLOGY

Supercond. Sci. Technol. 20 (2007) 351–364 doi:10.1088/0953-2048/20/4/009

Current distribution and ac loss for asuperconducting rectangular strip within-phase alternating current and appliedfieldE Pardo1,2, F Gomory1, J Souc1 and J M Ceballos1,3

1 Institute of Electrical Engineering, Centre of Excellence CENG, Slovak Academy ofSciences, 841 04 Bratislava, Slovakia2 Grup d’Electromagnetisme, Departament de Fısica, Universitat Autonoma Barcelona, 08193Bellaterra, Barcelona, Catalonia, Spain3 Laboratorio Benito Mahedero de Aplicaciones Electricas de los Superconductores, Escuelade Ingenierıas Industriales, Universidad de Extremadura, Apartado 382, Avenida de Elvas s/n06071 Badajoz, Spain

Received 3 October 2006, in final form 4 December 2006Published 5 March 2007Online at stacks.iop.org/SUST/20/351

AbstractThe case of ac transport at in-phase alternating applied magnetic fields for asuperconducting rectangular strip with finite thickness is investigated. Theapplied magnetic field is considered to be perpendicular to the current flow.We present numerical calculations assuming the critical-state model of thecurrent distribution and ac loss for various values of aspect ratio, transportcurrent and applied field amplitude. A rich phenomenology is obtained due tothe highly nonlinear nature of the critical state. We perform a detailedcomparison with the analytical limits and we discuss their applicability forthe actual geometry of superconducting conductors. A dissipation factor isdefined, which allows a more detailed analysis of the ac behaviour than the acloss. Finally, we measure the ac loss and compare it with the calculations,showing a significant qualitative and quantitative agreement without anyfitting parameters.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The behaviour of a superconductor transporting an alternatingcurrent or exposed to a magnetic field varying in time hasbeen widely studied since the early 1960s [1–4]. However,the case of a simultaneous alternating transport current andapplied magnetic field remains unclear. This situation isfound, for example, in superconductor windings where eachturn feels the magnetic field of all the others. Windings arepresent in many applications, such as ac magnets, transformersand motors [5–8]. From a practical point of view, it is offundamental importance to understand, predict and, eventually,reduce the energy loss (or ac loss) in the superconductor. Thestudy of the ac loss is also interesting for material science, asit can be used to characterize superconducting samples [9–12].Apart from its applications, the ac loss is the main alternating

quantity under the simultaneous application of alternatingcurrent and field and, thus, its study is significant in itself.

The superconductors suitable for electrical applicationsare hard type II ones [13]. Nowadays there is a great scientificeffort in the development of silver sheathed Bi2Sr2Ca2Cu3O10

(Ag/Bi-2223) tapes and YBa2Cu3O7−δ (YBCO) coatedconductors, which are high-temperature superconductors, andMgB2 wires [5, 8]. Superconducting tapes and wires have across-section that is roughly rectangular or elliptical. In thiswork, we will consider wires with a rectangular cross-section(or rectangular bars). We also restrict our work to the situationwhen the ac applied field is uniform and in phase with thetransport current.

Hard type II superconductors can be well describedby the critical-state model (CSM) proposed by Bean and

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London [1, 2], which assumes that the magnitude of the localcurrent density cannot be higher than a certain critical value Jc.

For the situation of only transport current, the CSM wasfirst applied by London and Hancox in the early 1960s in orderto analytically describe simple geometries, such as infinitecylinders and slabs [2, 14]. Later, an important step forwardswas made by Norris, who analytically deduced the currentdistribution and the ac loss for an infinitely thin strip by meansof conformal mapping transformations [15]. The case of a stripwith finite thickness can only be solved numerically, as done byseveral authors [16–19].

The first analysis of the CSM with only ac applied fieldwas done by Bean for a slab with an applied field parallel tothe surface [13]. The case of a thin strip with a perpendicularapplied field was analytically solved by Brandt et al [20]following the Norris’ technique [15]. The current distributionfor a strip with finite thickness was numerically calculated byBrandt [21] and Prigozhin [22], and the ac loss by Pardo et al[23].

Concerning the case of simultaneous alternating transportcurrent and applied field, the most significant publishedcalculations within the CSM are the following. In the late1970s, Carr analytically derived the ac loss for an infiniteslab in a parallel applied field [24]. In the 1990s, Brandt andZeldov et al analytically calculated the current distribution ina thin strip using conformal transformations for the situationwhere the transport current and the applied field increasemonotonically [25, 26]. Although these works providedifferent formulae, they are actually equivalent4. Moreover,Brandt studied the values of the transport current and appliedfield for which these formulae are valid, finding that they arenot applicable for high fields and low currents [25]. From thatcurrent distribution, Schonborg analytically calculated the acloss for a thin strip [27].

For strips with finite thickness, there are no publishedworks dealing with the simultaneous application of alternatingcurrents and magnetic fields in a superconductor in the CSM,according to our knowledge. However, there are severaltheoretical works assuming a relation between the electricalfield E and the current density J as E(J) = Ec(|J|/Jc)

nJ/|J|,where Ec is an arbitrary value and n is a positive exponent. Thecurrent distribution and the ac loss are calculated in [28–32]and [29, 28, 33, 31], respectively, for several values ofthe alternating transport current and applied field. Thesepublished results are incomplete, not covering the whole rangeof combinations of ac current and ac field. This contrastswith the extensive experimental studies that have been donefor Ag/Bi-2223 tapes [34–37, 31, 38] and YBCO coatedconductors [39, 40].

The objective of this paper is to rigorously study theresponse of a superconducting strip of finite thickness undersimultaneous application of an alternating transport current andfield, within the assumption of the CSM. The effect of threemain factors are considered: the aspect ratio of the cross-section and the amplitudes of the transport current and theapplied magnetic field. We also study the applicability of theCSM to actual superconducting tapes and wires by comparingthe calculations with experiments.

4 It can be seen after doing some algebra using arcsin(ix) = i arcsinh(x),arctan(ix) = i arctanh(x) and the definition of arcsinh and arctanh.

b

Δx

a

y

x

Ha

I

b

Δx

a

y

x

Ha

I

Figure 1. Sketch of the tape cross-section and division into elementsfor the calculations. The transport current I and the applied field Ha

are directed in the positive z and y directions, respectively.

This paper is structured as follows. In section 2, wepresent the numerical method used for the calculations and wediscuss some general features. The results and their discussionare presented in section 3. In section 4, the comparison withdata measured in a high-temperature superconducting tape isreported. Finally, in section 5 we present our conclusions.

2. Numerical method and general considerations

Let us consider an infinitely long superconductor along the zaxis with a rectangular cross-section with dimensions 2a × 2bin the x and y directions, respectively, figure 1. The origin ofcoordinates is taken in the centre of the strip. We study here thesituation where the superconductor carries a sinusoidal time-varying current I (t) = Im cos ωt simultaneously immersed ina uniform in-phase ac applied field Ha(t) = Hm cos ωt in they direction. It is shown below (sections 2.3 and 2.4) that ourresults are not only independent of ω but also of the specifictime waveform of I and Ha, similar to the case of only transportcurrent or magnetic field [13, 2].

In our calculations, we will consider that first I and Ha areincreased from zero to their maximum, starting from the zero-field cooled state of the superconductor. We call this processthe initial stage. Following this stage, we consider the reversecase, where the current and applied field are decreased fromHm and Im, respectively, to −Hm and −Im. Next, the appliedfield and current are increased back to their maximum, closingthe ac cycle. We refer to this latter stage as the returning one.

2.1. The critical-state model in strips

We assume that the superconductor obeys the CSM withconstant critical-current density Jc [1]. The CSM correspondsto assuming a multivalued relation of electrical field E againstcurrent density J, such that E = E(|J|)J/|J| with an E(|J|)that only takes finite values for |J | = Jc, being zero for |J | <

Jc and infinity for |J | > Jc [41]. For an infinitely long stripalong the z direction, the current density and the electrical fieldinside the superconductor are also in the z direction and theycan be considered as the scalar quantities J and E , respectively.Although in principle |J | in the CSM can be lower than Jc, ina superconducting strip J only takes the values 0 or ±Jc [42].

Let us start with the introduction of the main featuresof the current distribution in the initial stage for the case oftransport current only (i.e. Ha = 0) or when only the magneticfield is applied (I = 0).

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Current distribution and ac loss for a superconducting rectangular strip with in-phase ac and applied field

The behaviour of a superconducting strip in the critical-state model with Ha = 0 and uniform Jc is detailedin [15, 43, 19]. In the initial stage, for any I > 0 lower thanthe critical current, Ic = 4abJc, there exists a zone with J = 0surrounded by another one with J = Jc. The region withJ = 0 is usually called the current-free core. In this zone, theelectrical field is zero because in the CSM E(J = 0) = 0.With increasing I , the region with J = Jc monotonicallypenetrates from the whole surface inwards and the current-freecore shrinks, until it disappears when I reaches Ic. In the CSM,I cannot overcome Ic as it is assumed that |J | � Jc.

The situation when a magnetic field is applied to asuperconducting strip that is not transporting any net currentis described in [21, 22]. For this case, the current distributionis antisymmetric to the yz plane. In the initial stage withHa > 0, there is a zone with J = Jc in the right half andanother one with J = −Jc in the left half expanding fromthe surface to a current-free core between them. Throughoutthis paper, we call the border between regions with differentJ the current front. It is important to notice that in the currentfronts J vanishes and, then, so does E . With increasing Ha, thecross-section of the current-free core shrinks until it becomesa point at (x, y) = (0, 0) at the characteristic field Hp, thatis called the penetration field. At fields higher than Hp, thecurrent distribution is the same as for Ha = Hp.

When we now consider the simultaneous action of anapplied magnetic field on a superconductor transporting anonzero current, one can expect that the qualitative behaviourof the current distribution is similar to that for only transportcurrent or applied field. However, for some situations of I andHa the current distribution presents a different behaviour. Wediscuss this aspect in more detail below (section 3.1).

2.2. Minimization principle for the critical-state model

As discussed by several authors, such as Prigozhin [22, 41],Badia and Lopez [44, 42], Bhagwat et al [45], and Sanchezet al [46–48], the distribution of current density for asuperconductor assuming the critical-state model is such thatit minimizes a certain functional. The functionals introducedin [22, 41, 44, 45] are equivalent, whereas in [46] the magneticenergy is proposed as the quantity to be minimized. As shownin [22, 41], the principle of minimization of the functional,F , can be derived from fundamental considerations. Inappendix A, we demonstrate that the minimization of F isequivalent to minimizing the magnetic energy provided thatin the initial stage the current front penetrates monotonicallyfrom the surface inwards. Some of the situations presented inthis paper do not satisfy this condition; therefore we use theminimization of F as follows.

Let us consider the case of an infinitely long superconduc-tor extended along the z direction carrying a transport current Iand immersed in a uniform applied field in the y direction Ha,figure 1. With this geometry, the current density is in the z di-rection and, therefore, so is the vector potential A if we assumethe gauge ∇ · A = 0. Then, we can regard these quantities asscalar. Following the notation of Prigozhin [41], the current ata certain time distributes in such a way that it minimizes thefunctional

F[J ] = 12

∫S

J (r)AJ (r) dS −∫

SJ (r) AJ (r) dS

+∫

SJ (r)[Aa(r) − Aa(r)] dS, (1)

with the constraints

I =∫

SJ (r) dS (2)

|J | � Jc, (3)

where S is the superconductor cross-section, AJ is the vectorpotential created by J , Aa is the vector potential from theexternal field, and the quantities with hat correspond to thoseat the previous discretized time point. For infinitely longgeometry, AJ can be calculated from

AJ (r) = − μ0

4π

∫S

J (r′) ln[(y − y ′)2 + (x − x ′)2

]dS ′. (4)

Defining the current density variation δJ ≡ J − J , we obtainfrom equations (1) and (4) that the current density whichminimizes the functional F also minimizes the functional F ′,defined as

F ′[δJ ] ≡ 12

∫SδJ (r)δAJ (r) dS +

∫SδJ (r)δAa(r) dS, (5)

where δAJ is the vector potential created by δJ and δAa ≡Aa − Aa.

2.3. Calculation of the current distribution

We calculate the current distribution by minimization ofF ′[δJ ] of equation (5) for each time as follows.

As done in [19, 23, 48], each superconducting strip isdivided into N = 2nx × 2ny elements with dimensionsa/nx ×b/ny ; current density is assumed to be uniform in eachelement. In order to obtain a smoother current front, we allowthe current density magnitude to have discrete values below Jc,that is, |J | = k Jc/m with k being an integer number from 1to a maximum value m. As discussed in [19, 48], this reducesthe discretization error in our ac loss calculations but does notcontradict the CSM assumption. Indeed, as justified in [41],the CSM is compatible with allowing |J | � Jc, although forthe tape geometry, the local current density that minimizes Falways has the maximum magnitude, Jc [42]. As shown below(section 3.1), our results only present |J | < Jc on the currentfronts, with the physical interpretation that the elements sectionare partially filled with critical-current density. In this paper,we use between N = 12 000 and 16 000 elements and m = 20current steps.

Given a current I , applied vector potential Aa and a currentdensity J , we calculate the current density variation δJ if thecurrent is changed into I and the applied vector potential intoAa, as follows.

First, we find the element with s J < Jc, being s =sgn(I − I ), where increasing the current density by �J =s Jc/m produces the minimum increase of F ′, increasing thecurrent by �J ab/nx ny in this process. Then, we repeatthe procedure until the total current reaches I . Afterwards,the algorithm makes a current redistribution. Elements are

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found where changing the current density by �J and −�J ,respectively, reduces the most F ′ and |J | does not exceedJc; the process is repeated until varying the current in anypair of elements will increase F ′ instead of lowering it. Thisallows the creation of regions with current density opposite toI . Setting the current in this way, we ensure that the constraintsof equations (2) and (3) are fulfilled. In this procedure, the timedoes not play any role, so the resulting current distribution isindependent of the specific current (and field) waveform.

In appendix B, we discuss the fundamental aspects ofthe minimization procedure, showing that it finds the correctminimum of F ′.

The variation of F ′, �F ′, due to a variation of current�I in the element j can be calculated from equation (5) andAa = −μ0 Hax taking into account the division into elementsof the tape, with the result

�F ′j =

N∑k=1

δ Ik�IC jk + 12(�I)2C j j

− μ0(Ha − Ha)�I x j , (6)

where δ Ik is the current flowing through the element k inducedafter the change of I and Ha, x j is the x coordinate of the centreof element j , and C jk are geometrical parameters calculated inappendix A of [48].

For simplicity, we consider a constant variation of I (andHa) between different times inside each half cycle. In thispaper, we use between 80 and 320 time points per cycle.

2.4. Calculation of the ac loss

The power loss per unit volume in a conductor is J · E. Then,the ac loss per unit length and cycle Q in the superconductingstrip is

Q =∮

dt∫

SJ (x, y; t)E(x, y; t) dx dy, (7)

where the time integral is over one period.As the electrical field inside the superconductor is in the z

direction, we obtain

E = −∂zφ − ∂t A, (8)

where φ is the electrical scalar potential, ∂zφ ≡ ∂φ/∂z and∂t A ≡ ∂ A/∂t . The electrical field and the vector potential havezero components in the x and y directions, so that ∂φ/∂x =∂φ/∂y = 0. Then, as for infinitely long conductors E doesnot depend on z, ∂zφ is uniform in the whole conductor. Thequantity ∂zφ can be calculated taking one point where E = 0and using equation (8), obtaining

∂zφ(t) = −∂t A(x0, y0; t), (9)

where x0 and y0 are the x and y coordinates at some pointwhere E = 0. For the critical-state model, E always vanisheson the current-free core or on current fronts, where J = 0(section 2.1).

Inserting equations (8) and (9) into equation (7) and using∂t A = ∂t I∂I A and dI = ∂t I dt , we obtain

Q = 2∫ Im

−Im

dI∫

SJret(x, y; I)

× [∂I A(x0, y0; I) − ∂I A(x, y; I)] dx dy, (10)

where the current integration is performed in the returningstage. From equation (10), we see that Q is independent on thespecific I (t) dependence, as long as I increases or decreasesmonotonically with time in a half cycle. From this feature, itis directly deduced that the ac loss due to a sinusoidal currentand applied field is independent of their frequency.

The vector potential can be easily calculated from J ,obtained by means of the numerical procedure described insections 2.2 and 2.3. Then, we calculate ∂I A at a certain timek from the numerically obtained A as

∂I A(x, y; Ik) ≈ A(x, y; Ik+1) − A(x, y; Ik−1)

Ik+1 − Ik−1, (11)

where Ik is the current in the time k. Equation (11) yieldsmuch more accurate results of ∂I A than using finite differencesbetween consecutive time points. Indeed, according to themean value theorem, there must exist some current betweenIk−1 and Ik+1 where the derivative is exactly the right-sidepart of equation (11). Equation (11) cannot be used at theboundaries of a half cycle, I = ±Im, as ∂I A is not continuousthere. Therefore, we use finite differences between k and k + 1or k and k − 1 for I = −Im and I = Im, respectively.

The quantity ∂I A(x0, y0; Ik) is calculated as follows.Although in some situations (x0, y0) depends on I (and time),such an I dependence cannot be taken into account forcalculating ∂I A(x0, y0; I) because it is a partial derivativeand, thus, the spatial coordinates must be taken as parameters.This derivative, ∂I A(x0, y0; I), can be easily done once∂I A(x, y; Ik) is calculated for every element position, justtaking (x0, y0) as the centre of an element in the current-freecore or next to a flux front at the time k. For I = Ic andI = −Ic, (x0, y0) is approximated as that at the following andprevious time points, respectively5.

2.5. Monotonic penetration of current fronts

In many practical situations, the current front in the initialstage monotonically penetrates from the surface inwardswith increasing I and/or Ha. Some examples are a stripand arrays of strips with only applied field or transportcurrent [20, 21, 15, 19, 49, 47, 48], or a cylinder in uniformHa [50, 51]. In fact, this assumption has been taken forcalculating the current profiles for thin strips with simultaneousapplied field and transport current [25, 26], although, asdiscussed below, it is only fulfilled for high current and lowapplied field. A system with monotonic penetration of currentfronts has special properties, as follows.

If a system presents monotonic penetration of currentfronts, the current distribution for all of the cycle, and thus allthe electromagnetic properties, can be calculated from those inthe initial stage [48]. The current distribution in the reverse andreturning stages, Jrev and Jret, are, respectively,

Jrev(I) = Jin(Im) − 2Jin[(Im − I)/2], (12)

5 Even though for I = ±Ic there is neither a current-free core nor a currentfront, there still is at least one point where E = 0. For a differentially smaller(or larger) time, there must be a point where J = 0 and, thus, E = 0. Then,for I = ±Ic the electrical field vanishes at the same point for continuity in thetime dependence. As E(x, y; t) must be continuous with both the previous andthe following times, there could be two points with E = 0 for I = ±Ic. Thelatter situation appears for large enough applied field amplitudes, section 3.1,where E = 0 close to the centre of both of the strip’s vertical sides.

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Current distribution and ac loss for a superconducting rectangular strip with in-phase ac and applied field

Jret(I) = −Jin(Im) + 2Jin[(Im + I)/2], (13)

where Jin is the current distribution in the initial stage6.In fact, for this situation the ac loss can be evaluated from

the current distribution at the peak value of I and Ha and, thus,the calculation of the derivatives in the vector potential can beskipped. Specifically, from equations (10), (12) and (13) andfollowing the same deduction as Carr for the pure transportsituation [43], it can be obtained that

Q = 4Jc

∫S

s(x, y)[

Akm − Am(x, y)

]dx dy, (14)

where Akm and Am(x, y) are those corresponding to the peak

values of I and Ha and s(x, y) is a function giving the sgn ofJrev. Equation (14) with s = 1 corresponds to that obtainedby Norris for the transport case [15] and with s = x/|x | itcorresponds to the magnetic one given by Rhyner [52].

Moreover, for monotonic current front penetration, Jin

minimizes the magnetic energy and, thus, it only dependson the final I and Ha, as demonstrated in appendix A.Then, Jin (and Q) can be obtained by energy minimization(MEM). Whenever possible, it is recommended to use MEMfor calculating Jin and Q because this procedure requires asingle minimization for each Im and Hm value, whereas usingF minimization requires a large number, nt , of them. Inaddition, for F minimization, the error due to the cross-sectiondiscretization accumulates for each minimization, whereas notfor the MEM.

However, in our case, the condition of monotonic currentfront movement is not always fulfilled. Then, in order to useone single procedure, we apply F minimization for all of thestudied Im and Hm combinations.

3. Results and discussion

In this section, we present our results for the currentdistribution and the ac loss and we discuss the existinganalytical approximations for low and high b/a aspect ratios.We also introduce the dissipation factor � = 2π Q/(μ0 I 2

m),characterizing the loss behaviour better than the ac loss itself.

3.1. Current distribution

In the following, we present the current distribution for arectangular strip with aspect ratio b/a = 0.2, although thenumerical procedure gives accurate results for b/a between0.001 and 100. We consider several situations of field andcurrent.

First, we study the case of low applied fields. As anexample, we plot the current distribution for Im/Ic = 0.8 andHm/Hp = 0.08 in figure 2, where Hp is the full penetrationfield for a rectangular strip [53, 21]

Hp = Jcb

π

[2a

barctan

b

a+ ln

(1 + a2

b2

)]. (15)

The current profiles in figure 2 are qualitatively similar to thosefor transport current [19] with the difference that the current-free core is shifted to the left. In this situation, the current

6 Unfortunately, in section IIC of [48] there is a typing error in the equationfor Jret.

(a)

(b)

(c)

Figure 2. Current distribution in the initial stage for the low-field andhigh-current regime. Specific parameters are b/a = 0.2,Hm/Hp = 0.08, Im/Ic = 0.8 and I/Im = Ha/Hm = 0.2 (a), 0.6 (b)and 1 (c). The local current density is +Jc for the black region andzero for the white region.

fronts monotonically penetrate from the surface inwards. Thus,the current distribution for the whole cycle can be constructedfrom that in the initial stage using equations (12) and (13). Forthis case, the ac loss can be calculated using equation (14), sothat the evaluation of E can be skipped.

The most representative situation of the combined actionof the ac field and the ac current is that of higher applied fields,such as Im/Ic = 0.6 and Hm/Hp = 0.72, presented in figure 3.The most significant issue is that the current distribution inthe reverse stage is not always a superposition of that for theinitial stage. Not even the returning stage is related to thereverse one. However, as can be seen in the figure, the currentfronts for I/Im = 1 (and Ha/Hm = 1) in the reverse stagecorresponds to that for I/Im = −1 (and Ha/Hm = −1) forthe returning one, except some numerical deviation. Then, thecurrent distribution for the following reverse stage for a certaintransport current I (and applied field Ha) is the same but withopposite sgn with respect to those for the returning stage fortransport current −I (and applied field −Ha), being currentdistribution periodic in time after the first cycle.

The above specific case (figure 3) presents a current-freecore, but it is not always the case for higher Hm or Im. Forexample, the current-free core is not present for Im/Ic = 0.6and Hm/Hp = 1.2 (figure 4), as well as for any case withIm = Ic, as shown in figure 5 for Im = Ic and Hm/Hp = 2. Inaddition, for all of the cases with Im = Ic, the electromagnetichistory is erased at the end of one half cycle (figure 5). Thus,for this current amplitude, the returning profiles correspond tothe reverse ones with inverted sgn of the current density, so thatthe electromagnetic behaviour is simplified. Another issue isthat the current distribution for Im = Ic has only one boundarybetween the zones of positive and negative current, whereasthere can be two or more for lower Im (figures 3 and 4).

For other aspect ratios, we found the same qualitativebehaviour as for b/a = 0.2 described above. As an example,in figure 6 we present the current distribution in the returningcurve for b/a = 5. This corresponds to the situation of thesame strip as for figures 2–5 but with the applied field parallelto the wide direction.

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 3. Current distribution for b/a = 0.2, Hm/Hp = 0.72, andIm/Ic = 0.6 at several instants of the ac cycle. ((a)–(c)) are for theinitial stage with I/Im = 0.2, 0.6, and 1, respectively, ((d)–(f)) arefor the reverse stage with I/Im = 0.6, −0.2 and −1, respectively,and ((g)–(i)) are for the returning stage with I/Im = −0.6, 0.2, and1, respectively. The local current density is +Jc in the black regions,−Jc in the light grey zones, and zero in the white regions.

3.1.1. Comparison with analytical limits. It is interestingto compare the sheet current density K in a thin filmfrom [25, 26], where it is assumed that current fronts penetratemonotonically, with our results for finite thickness. Wecalculated K by integrating the current distribution over thethickness. For this situation, [25, 26] distinguishes betweenthe low-field high-current regime, for which all current hasthe same sgn, and the high-field low-current regime, whencurrent density with both sgns exists. These regimes appear

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4. Current distribution for b/a = 0.2, Hm/Hp = 1.2 andIm/Ic = 0.6 at several instants of the ac cycle. ((a)–(c)) are for theinitial stage with I/Im = 0.2, 0.6, and 1, respectively, and ((d)–(f))are for the reverse stage with I/Im = 0.6, −0.2 and −1, respectively.The local current density is +Jc in the black regions and −Jc in thegrey zones.

(a)

(b)

(c)

Figure 5. Current distribution at the reverse stage for b/a = 0.2,Hm/Hp = 2, Im/Ic = 1 and I/Im = 0.6 (a), −0.2 (b), and −1 (c),respectively. The local curent density is +Jc in the black regions and−Jc in the grey zones.

in the initial stage for I/Ic � tanh(Ha/Hc) and I/Ic <

tanh(Ha/Hc), respectively, being Hc ≡ 2Jcb/π .In figure 7, we present our numerical calculations of K

for the initial stage together with the analytical results fora thin film for b/a = 0.01, Hm/Hc = 0.6 and Im/Ic =1 (a), belonging to the low-field high-current regime for allHa and I up to their maximum, and Hm/Hc = 6 and

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Current distribution and ac loss for a superconducting rectangular strip with in-phase ac and applied field

(a) (b) (c)

Figure 6. Current distribution at the reverse stage for b/a = 5,Hm/Hp = 0.72, Im/Ic = 0.6 and I/Im = 0.6 (a), −0.2 (b), and −1(c), respectively. The local current density is +Jc in the black regionsand −Jc in the grey zones.

Im/Ic = 0.9999 (b), as an example for the high-field low-current case. As can be seen in figure 7(a), for low appliedfields all numerical results fall on the analytical curve withinthe computation error, whereas for higher fields, figure 7(b),there is only coincidence for the profiles corresponding to lowcurrent penetration. The discrepancy for higher penetrationappears because the assumption of monotonic penetration ofcurrent fronts is no longer valid for the analytical solution.As can be seen in figures 4((a)–(c)), for high I/Ic there is arecession of the zone with negative current density in favourof that with positive current density, this effect being moreimportant for higher I . For simultaneous alternating Ha andI , such a current front regression will always be present whenregions with both J = Jc and J = −Jc coexist, dueto the current penetration asymmetry. Thus, the thin filmapproximation in [25, 26] is only strictly valid for the high-current low-field regime.

For alternating applied fields and transport currents, thehigh-current low-field condition must be followed for all Ha

and I up to their maximum. Taking into account that for in-phase applied field and transport current Ha = (I/Im)Hm andusing that the first-order Taylor expansion of tanh x for low xis higher than tanh x , the high-current regime for alternatingconditions becomes

Im/Ic � Hm/Hc. (16)

Using a similar argument, it can be seen that the condition formonotonic current front penetration of equation (66) in [26]also reduces to equation (16).

We can also compare the numerically obtained currentdistribution to the analytical solution for a slab in a parallelfield, for which the current fronts are planar [24, 26]. Forstrips with high b/a in the high-field low-current regime, thecalculated current fronts approach planar ones (figure 6), theapproximation being better for higher b/a. This behaviour isin agreement to the pure magnetic case [21]. However, forthe low-field high-current regime, current fronts are similar

Figure 7. Sheet current density K in the initial stage as a function ofx for b/a = 0.01. The plots are for the external parameters Im = Ic

and Hm/Hc = 0.6 (a) and Im/Ic = 0.9999 and Hm/Hc = 6 (b) atseveral instantaneous I (and Ha). The lines depict the thin strip limitfrom [25, 26] and the symbols are for our numerical calculations. Forthe numerical results, K is the integral of J over the samplethickness.

to the ones for a thin strip with only transport current, whichare nonplanar [15, 19] and, thus, the slab approximation is nolonger valid.

We have performed numerical simulations for very highapplied fields, Hm > 5Hp, and have shown that thecurrent fronts approach vertical planes for any aspect ratio, inaccordance with the slab approximation. This is because whenthe applied field variation is much higher than the field createdby the variation of J , the first term of F ′ in equation (5) can beneglected. As Aa is proportional to x , F ′ of the new inducedcurrent density is independent of its y location, and the currentdensity profiles must be planar.

3.2. Total ac loss

First, we study the ac loss for several b/a aspect ratios andtheir possible analytical approximations. For this purpose, wepresent our results of the normalized ac loss q ≡ 2π Q/(μ0 I 2

c )

as a function of Im and constant Hm and q as a function of Hm

and constant Im in figures 8(a), 9(a), 10(a) and 8(b), 9(b), 10(b),respectively. Figures 8, 9 and 10 are for aspect ratiosb/a = 0.001, 100 and 0.1, respectively. The aspect ratiosof b/a = 0.001 and 0.1 can be used to qualitatively describeYBCO coated conductors and Ag/Bi-2223 tapes, respectively,in a perpendicular field. An aspect ratio b/a = 100 isrepresentative for a parallel applied field. For all figures,we consider Im normalized to Ic, while Hm is normalizedto Hc = 2Jcb/π in figure 8 and to Hp from equation (15)in figures 9 and 10. First, we present our results in thisconservative normalization for the sake of comparison withpublished theoretical and experimental data.

The numerical error in the ac loss has been analysed usingseveral numbers of elements, current steps and time points,showing an insignificant variation for the axis scale of allfigures below.

From figures 8–10, we see that the ac loss monotonicallyincreases with increasing either the current or the applied field

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Figure 8. Normalized ac loss 2π Q/(μ0 I 2c ) for b/a = 0.001 as a function of Im/Ic for several Hm/Hc, with Hc = 2Jcb/π (a) and as a

function of Hm/Hc for several Im/Ic (b). The lines with symbols correspond to our numerically calculated results, the dashed lines (red)correspond to the thin strip limit from [27], the thick solid line separates the low-field and high-current regime from the high-field one in a thinstrip (calculated using equation (16) and [27]), and the dotted lines (blue) (for Hm/Hc = 10 and 20) correspond to the high-field limit forslabs (equation (20)).

Figure 9. Normalized ac loss 2π Q/(μ0 I 2c ) for b/a = 100 as a function of Im/Ic (a) and as a function of Hm/Hp (b). The solid lines with

symbols (black) correspond to numerically calculated results and the dotted lines (blue) correspond to the slab approximation fromequations (17)–(18) [24].

Figure 10. Normalized ac loss 2π Q/(μ0 I 2c ) for b/a = 0.1 as a function of Im/Ic (a) and as a function of Hm/Hp (b). The lines with symbols

(black) correspond to our numerically calculated results, the dashed lines (red) correspond to the thin strip limit from [27], and the dotted lines(blue) (for Hm/Hp = 2 and 5) correspond to the high-field limit for slabs (equation (20)).

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Current distribution and ac loss for a superconducting rectangular strip with in-phase ac and applied field

amplitudes. For high applied field, Q increases linearly withHm for constant Im (figures 8(b), 9(b) and 10(b)). The ac lossfor the low-current limit in figures (a) and the low-applied-field limit in figures (b) is constant, corresponding to the puretransport and pure magnetic case, respectively. This qualitativebehaviour is consistent with experiments for YBCO coatedconductors [39, 40] and Ag/Bi-2223 tapes [34–37, 31, 38].

As expected, the loss results for zero applied field and zerotransport current are the same to the results for pure transportand pure magnetic situations calculated using MEM in [19]and [23], respectively.

3.2.1. Analytical limits for the ac loss. First, we study thevalidity of the analytical limits for thin strips and slabs in aparallel applied field.

In figure 8, we compare our numerical results of2π Q/(μ0 I 2

c ) for b/a = 0.001 (line plus symbols) with the acloss for an infinitely thin strip calculated by Schonborg [27](dashed line) from the sheet current distribution obtainedin [25] and [26]. Schonborg’s expression for Hm = 0corresponds to the Norris formula for a thin strip with puretransport current [15]. The thick continuous line plotted infigure 8 separates the low-field high-current regime from thehigh-current low-field regime, section 3.1.1. As can be seenin figures 8((a), (b)), the ac loss for the low-field high-currentregime is well described by the analytical expressions for a thinstrip. However, there is a significant deviation for the high-field regime, increasing with increasing field or current. This isbecause, as discussed in [25, 26], the current density formulaefor thin strips are only valid for monotonic penetration ofcurrent fronts, which appears only for the low-field high-current regime, figure 2. The current front penetration deviatesmore from the monotonic case for higher field and current, sothe formulae for thin strips are less applicable.

In figure 10, we plot our numerically calculated2π Q/(μ0 I 2

c ) for b/a = 0.1 (line with symbols) together withthat for a thin strip (dashed line). In this figure, we can see thatthe thin-film approximation is not valid for b/a for any caseexcept Im = Ic and low applied field.

It is also interesting to compare our numerical results tothe formulae for the ac loss obtained by Carr for a slab ina parallel applied field assuming planar current fronts [24],which in SI are

2π Q

μ0 I 2c

= πa

3bi3m

[1 + 3

h2m

i2m

], hm � im (17)

2π Q

μ0 I 2c

= πa

3bh3

m

[1 + 3

i2m

h2m

], im < hm � 1 (18)

2π Q

μ0 I 2c

= πa

bhm

[1 + i2

m

3

]− 2πa

3b(1 − im)(1 + im + i2

m)

+ 2πa

bi2m

(1 − i2m)

hm − im

− 4πb

3ai2m

(1 − im)3

(hm − im)2, hm > 1, (19)

where im = Im/Ic and hm = Hm/(Jca). The high-field limitof equation (19) is

2π Q

μ0 I 2c

= πa

bhm

[1 + i2

m

3

], hm � 1. (20)

In figure 9, we plot our numerical results of 2π Q/(μ0 I 2c )

for b/a = 100 (line with symbols) together with those for aslab calculated from equations (17)–(19) (dashed line). We seethat the above formulae for slabs agree well with the numericalresults for high fields and low currents, although they do not forlow fields and high currents. In figure 9, we also see that forHm much above Hp, Carr’s results approach the actual loss forany current. These features can be explained from the currentdistribution, discussed in section 3.1.1.

The Carr formula can also be compared to numericalresults for any b/a. In figures 8(a) and 10(a), we include thehigh-field limit of the ac loss in a slab, equation (20), for thehighest values of Hm/Hp in those graphs (dotted lines). It canbe seen that the analytical limit of equation (20) approaches thenumerical results for high Hm for Hm/Hp � 1 and Hm/Hp � 5for b/a = 0.001 and b/a = 0.1, respectively. Numericalcalculations for other b/a, such as b/a = 1, also agree withequation (20) for high applied field amplitudes. This featurecan be explained as follows. For high Hm, the current frontsare planar, like those for a slab, section 3.1.1. Moreover,if Hm is high enough, the only relevant contribution to thevector potential, and to E (equations (8) and (9)), is fromHa for any aspect ratio. Then, the high-field limit for a slabmust be valid for any aspect ratio. In fact, equation (20)can be easily deduced from equations (7)–(9) assuming that∂t A ≈ ∂t Aa = −μ0x∂t H a.

3.3. Dissipation factor

Usually ac loss under alternating field and current havebeen studied as a function of Im and fixing Hm or viceversa [34–37, 33, 31, 38–40, 27]. Here, we underline thesignificance of the ac-loss dependence when simultaneouslyincreasing Im and Hm with both parameters proportional toeach other along the curve. This situation is found in actualac devices, such as an alternating magnet.

As explained below, for Im ∝ Hm we can see more detailsof the ac loss behaviour if we plot Q normalized to I 2

m insteadto I 2

c . Indeed, the quantity 2π Q/(μ0 I 2m) ≡ � is proportional

to the ac loss of a winding per the stored magnetic energyaveraged during the cycle duration. Thus, � can be regarded asa dissipation factor. Moreover, � for only transport currentis proportional to the imaginary part of the self-inductance,defined in [54], and for only applied magnetic field � is relatedto the imaginary part of the ac susceptibility [55, 9].

In figure 11, we present our � numerical results for b/a =0.1 as a function of Im when Hm is varied proportionally toIm as Hm/Hp = α Im/Ic, where α is a constant (line withsymbols). This figure shows that for the low Im (and Hm) limit,� increases proportionally with Im (or Hm), which correspondsto a dependence proportional to I 3

m for the ac loss. Moreover,for high α, � decreases with increasing Im with a slope in log–log scale slightly higher than −1 (and a slope around 1 for theac loss), presenting a peak at a certain value of Im (or Hm).We notice that in a log–log plot of q against Im, the ac lossalways increases with Im, appearing as curves very similar tostraight lines with a slight change in the slope. However, for �

the qualitative behaviour of the loss with varying Im and α ismore evident.

The linear dependence of the dissipation factor withIm (and Hm) for the low-field limit is characteristic of the

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E Pardo et al

Figure 11. Dissipation factor � ≡ 2π Q/(μ0 I 2m) as a function of

Im/Ic with Hm proportional to Im as Hm/Hp = α Im/Ic for several α.The solid lines with symbols correspond to our numericalcalculations, the dash–dot line corresponds to only transport current,the dashed lines correspond to neglecting the effect of the transportcurrent, and the dotted lines correspond to the high-α approximationfrom equations (22) and (23).

critical-state model, also found for the pure transport and puremagnetic situations [19, 23]. This is because for low enoughIm, the current front is approximately parallel to the surface,as well as the magnetic field in the region with nonzero currentdensity, similarly to a slab [23]. This means that the dissipationfactor will increase as Im (and Q as I 3

m) at low levels ofexcitation in a superconductor winding of any shape or numberof turns. However, for strips with very small b/a, such asb/a = 0.001, the linear dependence of � with Im appears onlyfor very small Im, presenting for higher Im the I 2

m dependencetypical for thin films [23].

For comparison, in figure 11 we also include thedissipation factor for only transport current (dash–dot line),extracted from the tables in [19] and interpolating forintermediate values of Im when needed. As can be seen, forlow α the dissipation factor approaches that for only transportfor any Im. It is also interesting to consider � for the limitof high α and low Im/Ic, where the effect of the transportcurrent is negligible compared to that of the applied field. Forthis situation, � can be evaluated from the imaginary partof the ac susceptibility χ ′′ using that for only applied fieldQ = μ0π H2

mχ ′′ [55], with the result

�(Im/Ic) = 2π2(Hp/Ic

)2α2χ ′′(Hm/Hp = α Im/Ic). (21)

According to equation (21), we see that �(Im/Ic) isproportional to χ ′′(Hm/Hp) and with increasing α it shiftsupwards as α2 and to the left as α. Using equation (21) andthe χ ′′(Hm/Hp) calculated in [23], � is plotted in figure 11 forα = 2, 5, showing a good agreement with the numerical resultsfor low enough Im.

For a finer approximation for high α, we can consider thefollowing dissipation factor

�(Im, Hm) ≈ 2π

μ0 I 2m

[Qα→∞(Im, Hm)] , (22)

where Qα→∞ is an approximated ac loss as

Figure 12. Applicability conditions diagram for equations (20), (24)and Schonborg’s formula [27] for a thin strip, b/a � 0.001. In thelined regions, the analytical limits error is below 10% compared withour calculations. The areas in horizontal (red), vertical (blue) anddiagonal (black) lines correspond to equations (20), (24) andSchonborg’s formula, respectively.

Qα→∞(Im, Hm) ≡ Q(Im = 0, Hm) + 2aμ0 Hm I 2m

3Ic. (23)

The first term of equation (23) is the ac loss with only appliedmagnetic field, whereas the second term is the high-field limitfor a slab, equations (20)–(23) subtracting the ac loss forIm = 0. In figure 11, we plot the dissipation factor ofequations (22)–(23) for α = 2 and 5, obtained using thetables of numerically calculated ac susceptibility in [23]. Wefound that the approximation of equation (22) improves withincreasing α, almost overlapping our numerical results forα � 5.

For low b/a, such as b/a = 0.001 or lower, the ac lossfor α = 0 approaches the Norris formula for thin strips [15], ifIm is not very low [19]. For the high-α limit, we can obtain ananalytical solution of � by inserting the formula for the ac lossin a thin strip with Im = 0, [25], into equation (22), obtaining

� = 2Hm

Hc

[1

3+ I 2

c

I 2m

(2Hc

Hmln cosh

Hm

Hc− tanh

Hm

Hc

)], (24)

where Hc = 2bJc/π . For intermediate α, � can beapproximated from Schonborg’s formula for the ac loss in thinstrips [27], as long as Im/Ic � Hm/Hc, section 3.2.1.

3.4. Applicability conditions diagram for the analytical limits

The applicability conditions for the analytical limits of Q and� discussed in section 3.2.1 and 3.3 can be summarized in aHm–Im diagram. Such a diagram for thin strips (b/a � 0.001)is presented in figure 12, where the shaded areas show theregions where Q or � calculated from equations (20), (24)or Schonborg’s formula [27] differ by less than 10% from ournumerical calculations. If a more strict error criterion is taken,for example 1%, the applicability regions are considerablysmaller.

360

Current distribution and ac loss for a superconducting rectangular strip with in-phase ac and applied field

4. Comparison with experiments

The results of our ac loss calculations presented in figures 8–10 qualitatively agree with published measurements forAg/Bi-2223 tapes [34–37, 31, 33, 38] and YBCO coatedconductors [39, 40]. It is interesting to analyse in detailfigure 10 of [39]. It shows a comparison between the measuredac loss in a YBCO coated conductor and the theoretical onefor a thin strip in the critical-state model, evaluated fromthe current distribution in [25, 26]. It can be seen that themeasured ac loss lies below the thin strip approximation, inagreement with our numerical results in figure 8. As discussedin section 3.2.1, this is because the thin strip calculationsin [39, 27] are not valid for high applied fields. This showsthat our numerical calculations can be used to simulate the acloss in YBCO coated conductors.

In order to perform a more detailed comparison, wemeasured the dissipation factor for a commercial Ag/Bi-2223 tape with 37 filaments manufactured by AustralianSuperconductor. The sample was of 8 cm length and 3.2 ×0.31 mm cross-section with a critical current of 38 A in selffield at 77 K. The superconducting core cross-section wasroughly elliptical with dimensions 2a × 2b = 3.0 × 0.13 mm.The measurements were performed at a frequency of 72 Hzand a temperature of 77 K; the details of the experimentaltechnique are presented in [58]. We present the measuredresults in figure 13 (dotted line with symbols) together withnumerical calculations for a rectangular strip with the samethickness, width and critical current (solid lines). We noticethat for the theoretical curves we do not fit any parameter tothe measured ones. In figure 13, we label the curves with theparameter 2aHm/Im instead of α in order to avoid assumingany model a priori for performing the measurements. Indeed,α = (Hm/Hp)/(Im/Ic) contains Hp for a rectangular strip inthe CSM, whereas the tape superconducting core can be eitherof a different shape or it may not be successfully describedby the CSM. For comparison, in figure 13 we also include thedissipation factor assuming an elliptical cross section for onlytransport current [15] (dashed curve) and a negligible effectof the transport current at 2aHm/Im = 4.0 (dotted curve),calculated using equation (21) and the data for χ ′′ for onlyapplied field in [56].

From figure 13, we see that the main qualitative featuresof the measurements correspond to the behaviour for a stripassuming the critical-state model, except close to Ic for high2aHm/Im. This can be explained from the magnetic fieldB dependence of Jc, for which Jc decreases with increasing|B|. Then, for higher Hm, Ic is lower and a normal resistivecurrent appears in the silver for Im < Ic(Ba = 0), adding acertain contribution to �. Figure 13 shows that there is a betteragreement between the measured � and that for an elliptical barassuming the CSM than for the rectangular one, explained bythe overall shape of the tape superconducting core. Moreover,the fact that the multifilamentary superconducting core behavesas a single solid wire for any Hm suggests that for thisfrequency the interfilamentary coupling currents in the tapeare saturated due to the high length of the sample [57]. Thiscontrasts with magnetic measurements with shorter samples,for which the behaviour is clearly multifilamentary [59].

Figure 13. Calculated dissipation factor � together withexperimental data from a commercial Ag/Bi-2223 tape. The lineswith open symbols correspond to measurements, the solid linescorrespond to numerical calculations assuming a rectangularcross-section, and the dashed and dotted lines correspond to anelliptical cross-section at Hm = 0 and neglecting the effect of Im,respectively, using [15, 56] and equation (21).

5. Conclusions

In this paper, we have presented a rigorous theoretical studyfor the current distribution and ac loss in a rectangular striptransporting an alternating transport current I in phase with anapplied field Ha perpendicular to the current flow. We assumedthat the superconductor follows the critical-state model witha constant Jc. With this assumption, we have developeda numerical procedure which takes into account the finitethickness of the strip. General features of the critical-statemodel in such circumstances have been discussed. In order tounderstand the macroscopic physical processes in this system,we have performed extensive numerical calculations for severalaspect ratios and current and applied field amplitudes, Hm andIm respectively. Finally, we have performed measurementson Ag/Bi-2223 tapes to be compared with calculations.Good qualitative and quantitative agreement without fittingparameters has been found.

The results for the current distribution have shown a richphenomenology due to the highly nonlinear nature of theelectrical currents flowing in the superconductor. For lowHa and high I , the current distribution is qualitatively similarto the pure transport situation. Then, J at the reverse andreturning stages are a superposition of J in the initial one(equations (12) and (13)). However, it is not the same for highHa or low I due to the nonmonotonic penetration of currentfronts. In general, the returning stage cannot be deduced fromthat in the first reverse stage. The behaviour becomes periodiconly after the first cycle.

The ac loss Q has been accurately calculated for thethickness-to-width aspect ratios, b/a = 0.001, 0.1 and 100,in order to qualitatively describe YBCO coated conductorsand Ag/Bi-2223 tapes with applied fields in the transversedirection (b/a = 0.001 and 0.1, respectively) and in theparallel one (b/a = 100). Their current and applied

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E Pardo et al

field dependence is in accordance with published measureddata for YBCO coated conductors [39, 40] and Ag/Bi-2223 tapes [34–37, 31, 38]. We have shown that the acloss behaviour can be better characterized by means of thedissipation factor � = 2π Q/(μ0 I 2

m) studied as a functionof Im with Hm proportional to Im. We have measured thedissipation factor in actual Ag/Bi-2223 tapes, obtaining a goodagreement with the calculations.

We have also presented a detailed study of the analyticallimits for Q and � and their applicability. For thin samplessuch as YBCO coated conductors, b/a � 0.001, the currentprofiles and the ac loss only approach those for the analyticallimit for thin strips [27] for the low-field high-current regime.The thin film approximation is never valid for b/a ∼ 0.1 suchas for Ag/Bi-2223 tapes. We have also studied the slab limit,finding that for the situation of a parallel field, b/a = 100, theslab approximation is not valid for the transport-like regime(low-Hm and high-Im). However, the high-field limit for theslab approximation can be used for any aspect ratio providedthat Hm is high enough.

Acknowledgments

We acknowledge M Vojenciak for valuable technical support inthe measurements and D-X Chen and A Sanchez for commentson writing the paper. This work was supported in partby the European Commission (Project ENK6-CT-2002-80658‘ASTRA’).

Appendix A. Minimization of F and magnetic energy

In this appendix, we demonstrate that the current distributionin the initial stage minimizes the magnetic energy, providedthat the current front penetrates monotonically from the surfaceinwards and I is proportional to Ha (and Aa). The lattercondition is always satisfied for in-phase I and Ha. A similardemonstration for Ha = 0 can be found in [48].

For calculating the magnetic energy, we assume thatthe transport current in the strip of figure 1 returns throughanother identical one at a large distance D (D � a, b).In the following, we consider that the returning strip iscentred at (x, y) = (D, 0) [48]. Using the general formulafor the magnetic energy in an infinitely long circuit W =(1/2)

∫Sxy

J (r)AJ (r) + ∫Sxy

J (r)Aa(r), where Sx y refers to thewhole xy plane area, we find that the magnetic energy per strip,W ′, is

W ′ = 12

∫S

J (r)AJ (r) +∫

SJ (r)Aa(r), (A.1)

ignoring constant terms for a fixed I and Ha. W ′ ofequation (A.1) is independent of the position of the returningstrip.

We next demonstrate that if F ′ is minimized at every timepoint, the magnetic energy is also minimized [48].

If the current front penetrates monotonically, any physicalJ (r) in the initial stage is a composition of differential δJi(r)induced at each discretized time point i , for which F ′[δJi]of equation (5) is minimized. Thus, J (r) ≈ ∑n

i=1 δJi(r),where n is the number of time points. This decompositionof J into δJi is exact when n → ∞. For monotonic

current front penetration with increasing I , each δJi enclosesa current-free and field-free core, where the vector potential isconstant. Provided that n is high enough, δJi is nonzero in athin layer only so that the vector potential variation at time ti ,δAi ≡ δAJ,i + δAa,i , is uniform in the layer. Therefore, F ′ inequation (5) becomes

F ′[δJi ] ≈ 12δAc

i (Ii − Ii−1) + 12

∫SδJiδAa,i , (A.2)

where δAci is the value of δAi in the current-free core and Ii is

the transport current at time ti with Ii=0 = 0. Equation (A.2)is exact when n → ∞.

In the following, we decompose W ′ in terms of δJi , δAci

and the external parameters. In order to do this, we defineJi ≡ J − ∑i

j=1 δJ j , AJ,i ≡ AJ − ∑ij=1 δAJ, j and Aa,i ≡

Aa − Aa,i , where Aa,i = Aa(t = ti), and decompose thefollowing integral, γi , as

γi ≡ 12

∫S

dS Ji( AJ,i + 2 Aa,i)

= 12

∫S

dS ( Ji+1 + δJi+1)

× ( AJ,i + δAJ,i+1 + 2 Aa,i + 2δAa,i+1)

= γi+1 +∫

SdS Ji+1δAi+1

+ 12

∫S

dS δJi+1(δAi+1 + 2Aa − Aa,i+1 − Aa,i )

≈ γi+1 + 12δAc

i+1(2I − Ii+1 − Ii )

+ 12

∫S

dS δJi+1(2Aa − Aa,i+1 − Aa,i+1). (A.3)

In order to reach the expression beyond the fourth linein the deduction above, we used, from equation (4),∫

S dS Ji+1δAi+1 = ∫S dS δJi+1 AJ,i+1. The approximation

symbol in equation (A.3) corresponds to assuming that theregion where δJi+1 exists is narrow enough to consider thatδAi+1 is constant there (with a value δAc

i+1). In this step,we also took into account that the region where Ji+1 �= 0 iscontained in the flux-free zone of δJi+1 and, thus, δAi+1 isuniform with value δAc

i+1.From the decomposition of the integral γi in equa-

tion (A.3), it is straightforward to see that

W ′ ≈ γ1 + 12δAc

1(2I − I1 − I0)

+ 12

∫S

dS δJ1(2Aa − Aa,1 − Aa,0)

≈n∑

i=1

[12δAc

i (2I − Ii − Ii−1)

+ 12

∫SδJi(2Aa − Aa,i − Aa,i−1) dS

](A.4)

because γn = 0. As we assumed that I and Aa are proportionalto each other, 2Aa − Aa,i − Aa,i−1 = δAa,i (2I − Ii −Ii−1)/(Ii − Ii−1). Inserting this into equation (A.4) and usingequation (A.2), we obtain

W ′ =n∑

i=1

F ′[δJi ](

2I − Ii − Ii−1

Ii − Ii−1

). (A.5)

From equation (A.5) we directly deduce that whenminimizing F ′[δJi ] for each time, W ′ is also minimized

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Current distribution and ac loss for a superconducting rectangular strip with in-phase ac and applied field

Figure B.1. Sketch of the minimization procedure for N = 2 and noconstraints. The arrows show a change of magnitude �I in a systemvariable and the lines are possible level curves of a function of thetwo variables, F(I1, I2).

because I and Ii are fixed external parameters. As the δJi thatminimizes F ′[δJi ] is unique, the J = ∑n

i=1 δJi minimizingW ′ is also unique.

Appendix B. Fundamental aspects of theminimization procedure

In the following, we discuss the basis of the procedure in 2.2,justifying why it finds the correct minimum of the functionalF ′.

The quantity to be minimized in our problem, F ′, isa function of the current in each element i , Ii , with theconstraints |Ii | � Jcab and

∑Ni=1 Ii = I , where I is

the total current. Then, we have to minimize a scalarfunction of N variables F(I1, I2, . . . , IN ), where the N -vector(I1, I2, . . . , IN ) defines a state of the system. Ignoring theconstraints, if we ‘move’ several times the N -vector a distance�I in the Ii -axis which minimizes most of the function F ,the state falls ‘downhill’ towards the nearest local minimum,figure B.1. The procedure stops, finding a state at a distancelower than �I to the minimum, when any change of magnitude�I in any Ii increases the F value. The CSM constraint|Ii | � Jcab, just fixes a region in the N -space in which oursystem must remain. As a consequence, if the unconstrainedminimum is outside the allowed region, the possible minimumwill be on the boundary of that region.

The other constraint,∑N

i=1 Ii = I , can be set by forcing�I to be positive until the total current is I and afterwardsimpose that any possible change of �I must be followedby another one of −�I . For our case, the first part of theprocedure may increase F , although its increase would be theminimum possible, and the second one applies a correctiontowards the lowest F situation. We can imagine that if weapply a too large I variation between time points, we may‘move’ our state too much in the ‘wrong’ direction and thenit would be difficult to correct it successfully. This effectshould not be important for small enough I variations between

consecutive times. Our results obtained for several numbers oftime points confirm this hypothesis, as we found a negligiblevariation in the final results for a large enough number of timepoints.

Until now, we justified that the procedure finds the closestlocal minimum. In fact, from a physical point of view thesystem must be in a minimum of F ′, whether it is a globalor a local one. What is important is that the system stays in theappropriate minimum corresponding to the initial conditionsand the external parameters history. We ensure this by startingfrom the zero-field cool state (J = 0 everywhere) for I =Ha = 0 and increasing I and Ha in small steps, so thatthe former minimum moves slightly in the variables spaceand the system state follows it. However, according to ourexperience, the procedure finds the correct minimum even forlarge variations in the external parameters, at least for thespecific cases of Ha = 0 and I = 0 [47, 48].

References

[1] Bean C P 1962 Phys. Rev. Lett. 8 250[2] London H 1963 Phys. Lett. 6 162[3] Wilson M N 1983 Superconducting Magnets (Oxford: Oxford

University Press)[4] Carr W J Jr 1983 AC Loss and Macroscopic Theory of

Superconductors (New York: Gordon and Breach SciencePublishers Inc.)

[5] Hull J R 2003 Rep. Prog. Phys. 66 1865[6] Oomen M P, Nanke R and Leghissa M 2003 Supercond. Sci.

Technol. 16 339[7] Oomen M P, Rieger J, Hussennether V and Leghissa M 2004

Supercond. Sci. Technol. 17 S394[8] Larbalestier D, Gurevich A, Feldmann D M and

Polyanskii A 2001 Nature 414 368[9] Gomory F 1997 Supercond. Sci. Technol. 10 523

[10] Chen D-X, Pardo E, Sanchez A, Palau A, Puig T andObradors X 2004 Appl. Phys. Lett. 85 5646

[11] Gherardi L, Gomory F, Mele R and Coletta G 1997 Supercond.Sci. Technol. 10 909

[12] Miyagi D and Tsukamoto O 2002 IEEE Trans. Appl.Supercond. 12 1628

[13] Bean C P 1964 Rev. Mod. Phys. 36 31[14] Hancox R 1966 Proc. IEE 113 1221[15] Norris W T 1970 J. Phys. D: Appl. Phys. 3 489[16] Norris W T 1971 J. Phys. D: Appl. Phys. 4 1358[17] Fukunaga T, Inada R and Oota A 1998 Appl. Phys. Lett.

72 3362[18] Daumling M 1998 Supercond. Sci. Technol. 11 590[19] Pardo E, Chen D-X, Sanchez A and Navau C 2004 Supercond.

Sci. Technol. 17 83[20] Brandt E H, Indebom M and Forkl A 1993 Europhys. Lett. 22

735[21] Brandt E H 1996 Phys. Rev. B 54 4246[22] Prigozhin L 1996 J. Comput. Phys. 129 190[23] Pardo E, Chen D-X, Sanchez A and Navau C 2004 Supercond.

Sci. Technol. 17 537[24] Carr W J 1979 IEEE Trans. Magn. 15 240[25] Brandt E H and Indenbom M 1993 Phys. Rev. B 48 12893[26] Zeldov E, Clem J R, McElfresh M and Darwin M 1994 Phys.

Rev. B 49 9802[27] Schonborg N 2001 J. Appl. Phys. 90 2930[28] Amemiya N, Miyamoto K, Murasawa S, Mukai H and

Ohmatsu K 1998 Physica C 310 30[29] Yazawa T, Rabbers J J, ten Haken B, ten Kate H H J and

Yamada Y 1998 Physica C 310 36[30] Stavrev S, Dutoit B and Nibbio N 2002 IEEE Trans. Appl.

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[31] Tonsho H, Fukui S, Sato T, Yamaguchi M, Torii S, Takao T andUeda K 2003 IEEE Trans. Appl. Supercond. 13 2368

[32] Enomoto N and Amemiya N 2004 Physica C 412–414 1050[33] Zannella S, Montelatici L, Grenci G, Pojer M, Jansak L,

Majoros M, Coletta G, Mele R, Tebano R andZanovello F 2001 IEEE Trans. Appl. Supercond. 11 2441

[34] Rabbers J J, van der Laan D C, ten Haken B andten Kate H H J 1999 IEEE Trans. Appl. Supercond.9 1185

[35] Magnusson N and Hornfeldt S 1999 IEEE Trans. Appl.Supercond. 9 785

[36] Ashworth S P and Suenaga M 1999 Physica C 313 175[37] Ashworth S P and Suenaga M 2000 Physica C 329 149[38] Amemiya N, Jiang Z, Ayai N and Hayashi K 2003 Physica C

392–396 1083[39] Ashworth S P, Maley M, Suenaga M, Foltyn S R and Willis J O

2000 J. Appl. Phys. 88 2718[40] Ogawa J, Shiokawa M, Ciszek M and Tsukamoto O 2003 IEEE

Trans. Appl. Supercond. 13 1735[41] Prigozhin L 1997 IEEE Trans. Appl. Supercond. 7 3866[42] Badia A and Lopez C 2002 Phys. Rev. B 65 104514[43] Carr W J 2004 Physica C 402 293[44] Badia A and Lopez C 2001 Phys. Rev. Lett. 87 127004

[45] Bhagwat K V, Nair S V and Chaddah P 1994 Physica C227 176

[46] Sanchez A and Navau C 2001 Phys. Rev. B 64 214506Navau C and Sanchez A 2001 Phys. Rev. B 64 214507

[47] Pardo E, Sanchez A and Navau C 2003 Phys. Rev. B 67 104517[48] Pardo E, Sanchez A, Chen D-X and Navau C 2005 Phys. Rev. B

71 134517[49] Mawatari Y 1996 Phys. Rev. B 54 13215[50] Clem J R and Sanchez A 1994 Phys. Rev. B 50 9355[51] Brandt E H 1998 Phys. Rev. B 58 6506[52] Rhyner J 2002 Physica C 377 56[53] Forkl A 1993 Phys. Scr. T 49 148[54] Gomory F and Tebano R 1998 Physica C 310 116[55] Chen D-X and Sanchez A 1991 J. Appl. Phys. 70 5463[56] Chen D-X, Pardo E and Sanchez A 2005 Supercond. Sci.

Technol. 18 997[57] Fukumoto Y, Wiesmann H J, Garber M, Suenaga M and

Haldar P 1995 Appl. Phys. Lett. 67 3180[58] Vojenciak M, Souc J, Ceballos J M, Gomory F, Klincok B,

Pardo E and Grilli F 2006 Supercond. Sci. Technol. 19 397[59] Chen D-X, Pardo E, Navau C, Sanchez A, Fang J, Zhu Q,

Luo X-M and Han Z-H 2004 Supercond. Sci. Technol.17 1477

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Influence of the current through one turn of a multilayer coil on the nearest turn in a consecutive layer

P. Suárez *, A. Alvarez, B. Pérez and J. M. Ceballos

Industrial Engineering School, University of Extremadura, Apdo 382, 06071 Badajoz, Spain

* E-mail: [email protected]

Abstract. Many references on AC losses can be found for straight superconducting tapes with or without an external magnetic field. There are fewer references on AC losses for bent tapes such as we find it in a spire or solenoid. But even fewer are the references on the study of AC losses in multilayer coils or magnetically coupled coils wound close together. In these cases, the loss in each piece of tape depends on three factors: the transport current in it, the global magnetic field due to the complete coil, and the local magnetic field due to the current in the tape wound just over or under the piece in question –the main difference between multilayer coils and magnetically coupled coils is that the current in the former is the same in all the layers and the currents in magnetically coupled coils are different in amplitude and phase. In order to determine the losses due to the third factor above, the local magnetic fields, we propose in this paper an experiment that consists of the measurement of losses in two straight insulated superconducting tapes located one over the other as close together as possible. In this way, the effect of the global magnetic field of the coil disappears because the coil does not exist. Furthermore, one of the tapes is made to be twice as long as the other so that we can measure the part of the transport losses in the part of the tape independent on the influence of the other. This permits us to distinguish the component of the losses due to the interaction between the pair of tapes. BSCCO tape was used and the pieces were fed with two different power supplies each one giving a current adjustable in amplitude. Measurements of the voltages between taps and in contact-less loops were taken both between the tapes and, in the longer tape, away from the influence of the shorter one. The losses were calculated from the wave forms of the contact and contact-less voltages and the currents. The influence of the proximity of the tapes was determined.

1. Introduction In many applications of superconducting tape in electrical devices, the tape must be wound in multilayer coils as in figure 1. In such a case, every piece of tape is located very close to some other piece, in the next layer, along the coil. The proximity of these two parts of the circuit adds a new component (not necessarily positive) in the total loss of the multilayer coil that does not exist in the tape or single-layer coil loss. Therefore, we can divide the loss into 3 components:

• The transport loss, PN , that can be calculated by the Norris equation [1]. • The magnetic loss, Pmag , due to the global magnetic field created by the complete coil.

8th European Conference on Applied Superconductivity (EUCAS 2007) IOP PublishingJournal of Physics: Conference Series 97 (2008) 012176 doi:10.1088/1742-6596/97/1/012176

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• The local loss, Ploc , due to the proximity of turns in the same position of consecutive layers. So, the total loss, PT , can be written as follows:

T N mag locP P P P= + + (1)

To evaluate the new component of the loss, Ploc, we designed and carried out the experiment described in the next paragraph. The results of the experiment are presented from different points of view in the following paragraphs.

Figure 1. Multi-layer coils carrying the same or different current in each layer can be found in electrical designs. In these cases the proximity of the tapes in the same position of consecutive layers makes the AC loss different from in a single layer coil or a simple tape.

2. Experimental Figure 2 shows the arrangement of the tapes for the measurement of the losses. In this case, the tape is not bent as in a coil, and therefore Pmag = 0 (no global magnetic field has to be taken into account).

The electrical method is used to determine the losses in the longer tape through the measurement of the voltage between taps on the tape (see figure 2, circuits CI and CO) or the emf in a contact-less loop (circuits CLI and CLO) [2].

Figure 2. Experimental arrangement of the tape (the longer) for the measurement of the losses both under and outside the influence of another tape (the shorter) very close to the former.

The shorter tape is located over the CI and CLI circuits, leaving the CO a CLO circuits outside its

influence. The current IL in the longer tape and IS in the shorter one are independent but in phase for this

study. The measuring equipment picks up the waveforms of the currents through two Hall current probes,

and the waveforms of the tap and loop voltages through four measurement amplifiers that filter and adapt the signals to be read by a data acquisition board (DAQ). All the waveforms have a whole number of periods (typically 5).

The process is controlled and the data analyzed by a program based on the software Labview. The measurements were made at a frequency, f, of 100 Hz. The working temperature was 77K.

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The characteristics of the tape under test are summarized in table 1. Note that we include an estimated value of the critical current. This is the value obtained by the AC losses analysis method [3].

Table 1. Characteristics of the tape.

Tape reference NTS Superconductor Bi(Pb)-2223 Matrix Silver alloy Thickness (µm) 261 Width (mm) 3.77 Rated Ic (A) 36.5 Estimated Ic (A) 27.2

3. Data processing The waveforms collected by the DAQ are converted to real values by multiplication by the corresponding factors. The resulting data are given in the table 2 together with there equations.

Table 2. Real waveforms recorded by the system. All the waveforms contain the same whole number of periods. The phases ϕ in the equations correspond to the value of the parameters in the sample initial time. The functions H(t) include the harmonics of the voltage functions.

Waveform Equation Long tape current ( ) 2 cos( )L L ii t I tω ϕ= − Short tape current ( ) 2 cos( )S S ii t I tω ϕ= − Contact tap voltage outside the short tape ( ) 2 cos( ) ( )CO CO CO COv t V t H tω ϕ= − + Contactless loop voltage outside the short tape ( ) 2 cos( ) ( )CLO CLO CLO CLOv t V t H tω ϕ= − + Contact tap voltage under the short tape ( ) 2 cos( ) ( )CI CI CI CIv t V t H tω ϕ= − + Contactless loop voltage under the short tape ( ) 2 cos( ) ( )CLI CLI CLI CLIv t V t H tω ϕ= − +

The power loss per meter of tape in the different probes, x, was calculated in two ways (x = CO,

CLO, CI or CLI, and Lx is the length of the probe): • As the average value of the instant power over a whole number of periods:

( ) ( ) /x x L xnTP v t i t L=

• From the current and the voltage first harmonic RMS values:

cos( ) /x x L x i xP V I Lϕ ϕ= −

The results of the power calculated by means of these equations were very similar, so no differentiation is necessary.

4. Results and discussion The first verification we have to do is to check the independence of the loss outside the short tape with respect to the current IL. Figure 3 shows clearly this independence.

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Figure 3. Loss in the probe CO as a function of IS for different currents IL (from 5 to 40 A). The losses are constant and don’t depend on IS.

Figure 4 shows the loss in the long tape, outside the influence of the shorter, as a function of the

transport current IL. This corresponds to the expected transport loss in the tape, PN. The Norris theoretical estimate of this loss for the estimated critical current is included in the graph, and there were no significant differences with the measurement.

Figure 4. Loss outside the influence of the short tape (probe CO) for different currents IS (from 0 to 45 A). This loss corresponds to the transport loss in a single tape.

Minor differences between the curves can be observed in a closer view as in figure 5. This is

probably due to the influence of the short tape current leads. We take the loss for IS = 0 in figure 4 as the experimental transport loss in the single tape.

Figures 6 and 7 show the losses measured from probe CI under the short tape. The scales in these figures are the same as in figures 4 and 5, respectively. The measurements from CI and CO were taken simultaneously.

In this case, the loss curves spread in separate ways. Two opposite effects are observed: • The measurement under a low current or no current in the short tape (IS ≤ 10 A) is lower than

outside. • The measurement under a high current in the short tape (IS ≥ 15 A) is higher than outside.

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Figure 5. Detail of the loss outside the influence of the short tape.

Figure 6. Loss under the influence of the short tape (probe CI) for different currents IS (from 0 to 40 A).

Figure 7. Detail of the measured loss under the influence of the short tape.

As an explanation of this behaviour we propose: • In the case of a low current or no current in the short tape, the screening effect over the long

tape modifies the distribution of the field between their filaments in such a way that it reduces the self-field due to the transport current, increasing the effective value of the critical current,

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Ic. A higher value of Ic reduces the transport loss (see Norris equation in [1]) and the matrix loss due to the reduced excess current over the critical value.

• When the current in the short tape is high enough, the magnetic field it creates acts as an external magnetic field on the long tape, increasing its magnetic loss, Pmag, and reducing the effective critical current. Transport and matrix losses increase because of the reduction of Ic.

For a coil as in figure 1, the loss in each tape is due to the current in it and the same current in the

adjacent tape. The interest in this case is in the loss measured under the short tape with IS = IL. But there has to be

an extra consideration in this case. The loss measured through the probe CI, PCI , contains not only the loss in the long tape, but also a fraction of the loss in the short one measured by the contactless method [2] by means of a loop formed with the taps wires of the probe CI and the segment of the tape between the contacts.

We assume that this fraction can be estimated as one half of the power measured by the probe CLI (the shape and size of the loops were made equal for this propose). Therefore, the total loss per meter in the tape can be written as:

PT = PCI – ½ PCLI , with IS = IL

Figures 8 and 9 show these results. One observes that below the critical current (although we know that the effective value of Ic varies, we use 27.2 A as a reference for the critical current, that corresponds to an RMS value of 19.2 A) the estimated total loss is slightly higher than the loss in a single tape (figure 9). On the contrary, for currents higher than Ic the loss is very much higher than in a single tape.

Figure 8. Measured loss in the tape under the influence of another tape carrying the same current. Above the critical current (27.2 A, that correspond to 19.2 A RMS) the loss is very much higher than the loss in a single tape. See figure 9 for details below the critical current.

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Figure 9. Detail of the measure of loss in the tape under the influence of another tape carrying the same current. Below the critical current the estimated total loss is slightly higher than the loss in a single tape.

5. Conclusions A method to determine the total loss in a tape located very close to another tape carrying the same current has been proposed.

The presence of the second tape makes Ic very depending on the transport current in the two tapes. The total loss in the tape in a configuration as in figure 1 is higher than in a single tape and very

much higher when matrix losses appear. The dependence of Ic on the transport currents is being studied by our group.

References [1] Norris W T 1970 J. Phys. D 3 489 [2] Gömöry F, Frolek L, Souc J, Laudis A, Kovác P and Husek I 2003 IEE Trans. Appl. Supercond.

11 2967 [3] Alvarez A, Suarez P, Perez B and Bosch R 2004 Physica C 401 206

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Abstract—In multilayer and magnetically coupled coils made

from tape, the loss in each segment of tape in a coil depends on the parallel segments in the adjacent layers. In the case of a single multilayer coil, the current in all the layers is the same, but in magnetically coupled coils, the current in adjacent layer from different coils can be different both in amplitude and phase – usually 180º out of phase one with respect to the other.

In previous work, we have studied the influence of the proximity between tapes by considering the total loss in a segment as the sum of three components: the transport current in it, the global magnetic field due to the complete coil (or coils), and the local magnetic field due to the current in the tape wound just over or under the segment in question.

To measure the last component, an experimental method has been proposed and carried out with Bi-2223 tape, showing that the loss in the tape can be increased or reduced by the proximity of another tape, depending on the current, if any, that the latter carries. By means of the loss variation, we have shown how the variation of transport currents (and, therefore, of the associated magnetic fields) influences the practical critical current of the tape under test. Advances in YBCO tape (2G tape) fabrication have led to increases in the field tolerance of the tape, and the dependences of loss and practical critical current on the proximity of an adjacent tape needed to be revised. In the present work, we study the behavior of the loss in 2G tapes under the influence of other tapes carrying zero or different currents. A comparison between Bi-2223 and YBCO tapes is shown.

Index Terms—AC losses, YBCO tapes, transport current.

I. INTRODUCTION

INCE the 90’s decade HTS tapes have being using in electric power applications. Many of these applications,

such as fault current limiters, power cables, motors or transformers can contain multilayer and magnetically coupled coils made from superconducting tape. In previous work [1,

Manuscript received 19 August 2008. (Write the date on which you submitted your paper for review.) This research is funded in part by the Government of Extremadura (SPAIN).

P. Suárez is with the Applied Physics Department, University of Extremadura, 06071 Badajoz, SPAIN (corresponding author, phone: 0034-924-289646, fax: 0034-924-289601, e-mail: [email protected])

A. Álvarez is with the Electrical Engineering Department, University of Extremadura, 06071 Badajoz, SPAIN. He is an IEEE Member (email: [email protected]).

J. M. Ceballos is with the Electrical Engineering Department, University of Extremadura, Apdo 382, 06071 Badajoz, SPAIN (e-mail: [email protected]).

B. Pérez is with the Electrical Engineering Department, University of Extremadura, 06071 Badajoz, SPAIN (e-mail: [email protected]).

2], we have studied superconducting coils from BSCCO tapes and we have reported that the proximity between neighboring layers adds a new component (not necessarily positive) in the total loss of the multilayer coil that does not exist in the tape or single-layer. So, we can divide the loss into 3 components: the transport loss, PN , that can be calculated by the Norris equation [3], the magnetic loss, Pmag , due to the global magnetic field created by the complete coil and the local loss, Ploc , due to the proximity of turns in the same position of consecutive layers.

During the last years, many different 2G tapes are being developed to be used in high temperature superconductor (HTS) electric power devices due to their lower cost and better magnetic field tolerance compared to the 1G tapes, but its configuration includes a new source of losses, the ferromagnetic substrate (the eddy currents can be neglected due to its small contribution at low frequency [4]). Then the dependences of loss on the proximity of an adjacent tape need to be revised. If we add this component of losses, PFM , to the others three mentioned above, the total loss, PT , for a 2G coil, can be written as follows:

T N mag loc FMP P P P P= + + + (1)

In the present work, we study the behavior of the loss in 2G

tapes under the influence of other tapes carrying zero or different currents. A comparison between 1G and 2G tapes is shown.

II. EXPERIMENTAL

The arrangement of the tapes for the measurement of the losses is the same that in [1] and it is shown in Fig 1. In this case, the tape is not bent as in a coil, and therefore Pmag = 0 (no global magnetic field has to be taken into account). The electrical method is used to determine the losses in the longer tape through the measurement of the voltage between taps on the tape (see Fig 1, circuits CI and CO) or the emf in a contact-less loop (circuits CLI and CLO) [5]. The shorter tape is located over the CI and CLI circuits, leaving the CO a CLO circuits outside its influence.

The current IL in the longer tape and IS in the shorter one are independent but in phase for this study.

The measuring equipment picks up the waveforms of the currents through two Hall current probes, and the waveforms of the tap and loop voltages through four measurement

Losses in 2G tapes wound close together: Comparison with similar 1G tape

configurations P. Suárez, A. Álvarez Member IEEE, J.M. Ceballos and B. Pérez

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amplifiers that filter and adapt the signals to be read by a data acquisition board (DAQ). All the waveforms have a whole number of periods (typically 5). The process is controlled and the data analyzed by a program based on the software Labview.

Fig. 1. Experimental arrangement of a 12 cm tape (the longer) for the measurement of the losses both under and outside the influence of another piece of the same tape (the shorter, of 6 cm in length) laid directly on the former.

The measurements were made at a frequency, f, of 100 Hz and the working temperature was 77K.

The tested 1G tape has been fabricated with PIT technique by InnoST and the tested 2G tape comes from American Superconductor manufactured by Metal Organic Deposition (MOD)/Rolling Assisted Biaxially Textured Substrates (RABiTS) approach and it is exactly labeled as 344 Superconductor [6]. The characteristics of the tapes under test are summarized in Table I.

TABLE I CHARACTERISTICS OF TAPES UNDER TEST

1G Samples 2G Samples

Tape Reference InnoST American Superconductor Superconductor Bi-2223 (Pb) YBCO Fabrication Tech. PIT MOD/RABiTS Matrix Silver Alloy ----- Substrate ----- Ni-W alloy Coating ----- Stainless Steel Thickness (μm) 230 ± 10 150 ± 20 Width (mm) 4.20 ± 0.10 4.40 ± 0.15 Ic(A) 95 75

III. RESULTS AND DISCUSSION

A. 2G Single Tapes. Comparison with 1G Single Tapes The first verification we have to do is to check the

independence of the loss outside the short tape with respect to the current IS. Fig. 2 shows clearly this independence. So the measurements from probe CO represent the behavior of a single tape.

Fig. 3 shows the measured loss in the 2G long tape, outside the influence of the shorter, as a function of the transport current IL. This loss corresponds to the transport loss in a 2G single tape at currents lower than critical current. The sample shows significantly higher losses compared with the theoretical values at currents lower than the critical current. This indicates that the losses in the 2G tapes are affected by the magnetic losses in the ferromagnetic substrates [4, 7-9].

We have estimated losses in ferromagnetic substrates at currents lower than critical current from [4, 7, 8]. Fig. 4 shows this estimation for two different substrates with quite differences between their losses: Ni (FM1) and Ni-5%at.W (FM2). Then, in Fig. 5 we have plotted the experimental data

of losses in 2G sample and the sum of Norris’s models with FM losses. In general, the addition of the ferromagnetic losses FM1 or FM2 to the Norris Elliptical losses improves considerably the agreement with our experimental measurements as in [8]. However, our measurements fit to “FM2 + Norris Elliptic” for all values of IL but only fit to “FM1 + Norris Elliptic” for IL > 25 A.

Fig 2. Loss in the probe CO as a function of IS for different currents IL (from 5 to 45 A). The losses are constant and don’t depend on IS.

Fig. 3. Loss outside the influence of the short tape (probe CO) for different currents IS (from 0 to 75 A), in 2G sample. It is been that the experimental measurements follow an IL2 dependence [10].

Fig. 4. Experimental losses in the ferromagnetic substrate. Dashed line is for

Ni tapes (FM1) and full line is for Ni-5%at.W tapes (FM2) [4, 7, 8].

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Fig. 5. Measured loss in the 2G tape, and theoretical loss calculated taking into account the estimated ferromagnetic losses and the Norris’ models.

In Fig. 6 a comparison of losses between 1G and 2G tapes

for I < Ic is shown and we can see a significant difference between them but the most important one is the different slopes because of FM losses as it is seen in Fig. 7.

Fig. 6. Comparison between losses in 1G and 2G tapes outside the influence of the short tape (probe CO) for different currents IS (from 0 to 45 A). These losses correspond to the transport losses in each single tape.

Fig. 7. Measured losses in 1G and 2G single tapes (probe CO, IS = 0 A), and transport loss in 2G tape evaluated by subtraction of ferromagnetic loss from the measured loss (dashed line, 2G-FM1, is for Ni substrate, and full line, 2G-FM2, is for Ni-5%at.W substrate).

In this figure we have plotted the results of the loss measurement from CO probe, for 1G tape (transport loss) and 2G tape (transport and ferromagnetic losses). Furthermore, transport loss in 2G tape has been estimated subtracting FM1 and FM2 losses from the measurement, and represented in Fig. 7 too. One can see that 2G curve has a slope equal to 2 while

the others have a slope close to 3 as it is predicted by Norris elliptic model [3]. Of course, the higher values of 2G-FM1 or 2G-FM2 curves with respect to those in 1G curve are due to the lower critical current of the 2G tape.

B. 2G Assembled Tapes. Comparison with 1G Assembled Tapes In [1] we demonstrated for 1G superconductors that the

presence of transport currents through neighboring tapes to the tested sample had influence in losses. Also some authors [10, 11] have predicted that the influence of the self-field of the neighboring 1G and 2G tapes and the adjacent layers cannot be neglected and needs further investigation. So we have studied this question in 2G tapes. The results were obtained from probe CI (Fig. 1) and are shown in Fig. 8 and the corresponding details for 1G and 2G setups are shown in Figs. 9 and 10 respectively.

Fig. 8. Loss under the influence of the short tape (probe CI) for 1G and 2G tapes and for different currents IS (from 0 to 60 A).

Fig. 9. Detail of the measured loss under the influence of the short tape, for 1G tape and different currents IS (from 0 to 105 A).

Fig. 10. Detail of the measured loss under the influence of the short tape, for 2G tape and different currents IS (from 0 to 85 A).

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4

The results for 1G setup (Figs. 8 and 9) show an expected

behavior that is, the losses in probe CI increase when IS increase concluding that the presence of consecutive superconducting layers affect to AC losses of the neighboring layers due to the dependence of the critical current with the transport current through the two tapes [1], but for 2G setup (Figs. 8 and 10) we can see a different and interesting behavior. When IS increase below Ic the losses in the long tape decrease but when IS increase above Ic the losses in the long tape increase. Fig. 11 shows a scheme of our arrangement to give a possible explanation of this effect. The short tape FM substrate and the conducting layers (Ag layer and stainless steel coverts) are located between short and long tapes YBCO layers. So when IS < Ic, the magnetic field cause by IS in the long tape FM substrate reduces the magnetic field in the same substrate due to IL. This effect is stronger when IS increases producing a reduction of the losses in the long tape. However, when IS > Ic, the current (IS – Ic) goes through the conducting layers increasing the magnetic field in the long tape FM substrate and the losses in the long tape.

SS/ Ag

FM susbtrate

YBCO layerIS

IL BS BL

IS < Ic

SS/ Ag

FM susbtrate

YBCO layerIS - Ic

BS + BS-c BL

Ic

IL

IS > Ic

Fig. 11. Scheme of our arrangement. Above is shown the case for IS < Ic and below is drawn the case for IS > Ic. In both of them are the short and the long tapes. Spots represent Ag layers and stain steel covers, grey color represent FM substrates and white color represent YBCO layers. The layers are not at scale.

IV. CONCLUSIONS

In present work, two similar arrangements for 1G and 2G assembled tapes have been constructed and studied in order to establish a comparison between them.

We have found a good agreement between our measurements and those estimated from bibliography for 2G single tape but it is necessary to carry on the study taking our own losses measurements in the ferromagnetic substrate.

Also we have shown a different behavior between 1G and 2G assembled tapes demonstrating that the existence and the location of ferromagnetic substrates are highly influent on the losses of the tapes. However, experiences with different 2G samples must be realized.

In order to complete and clarify the behavior of neighboring tapes data from probes CLO and CLI are being analyzing now; also measurements with short and long tapes currents out of phase have been taken from our arrangements and are being studied.

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