IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 23, NO. 3, JUNE 2013 5900406

AC Loss in Pancake Coil Made From12 mm Wide REBCO Tape

Fedor Gömöry, Ján Šouc, Enric Pardo, Eugen Seiler, Mykola Soloviov, Lubomír Frolek,Michal Skarba, Pavol Konopka, Marcela Pekarcíková, and Jozef Janovec

(Invited Paper)

Abstract—The design of a superconducting coil from high-performance REBCO coated conductors is often complicated be-cause of complex anisotropy of the critical current density Jc.It is important to understand how much detail of this featuremust be taken into consideration in the prediction of maximumachievable current and the expected ac loss. We present the resultsof investigation performed with a small (ten turns, 60 mm innerdiameter) coil made from SuperPower tape of 12 mm width.The knowledge of Jc(B, θ) determined on short sample allowedprediction of the maximum achievable current of the coil and theac loss behavior. We have also investigated the effect of the tapenonuniformity. Our results confirm that the lateral nonuniformitywhen Jc at tape edges is lower than in its center leads to significantincrease of ac loss. A longitudinal nonuniformity, in particulara reduction of critical current in some portion along the tapelength, is hardly observable in the ac loss result. On the otherhand, using a piece of tape with lower Jc in the innermost coilturn would significantly reduce the maximum current. We alsopresent calculations showing the change in current–voltage curveand redistribution of ac dissipation in the case of nonuniform tapequality.

Index Terms—AC losses, high-temperature superconductors,numerical computations, superconducting magnets.

I. INTRODUCTION

COATED conductor (CC) tapes from high temperaturesuperconductors (HTS) are available in lengths and

amperages that allow considering a replacement of copperelectromagnets in different kinds of electric power devices(transformers, motors) by coils from HTS CC. There are severalsignificant differences between these two materials (e.g., work-ing temperature, shape and sensitivity to mechanical strain, . . .).From the electromagnetic point of view the HTS material iscompletely dissimilar to a normal metal that is well charac-

Manuscript received October 5, 2012; accepted January 7, 2013. Date ofpublication January 9, 2013; date of current version February 15, 2013. Thiswork was supported in part by the VEGA grant agency under Contract N.1/0162/11 and by the European Commission EURATOM project FU07-CT-2007-00051.

F. Gömöry, J. Šouc, E. Pardo, E. Seiler, M. Soloviov, and L. Frolek are withthe Institute of Electrical Engineering, Slovak Academy of Sciences, 84104Bratislava, Slovakia (e-mail: elekgomo@savba.sk).

M. Skarba, P. Konopka, M. Pekarcíková, and J. Janovec are with the Facultyof Materials Science and Technology in Trnava, Slovak Technical University,917 24 Trnava, Slovakia.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TASC.2013.2238986

Fig. 1. Photograph of the coil (top) assembled with current leads and voltagetaps and (bottom) its geometry used in calculations: Rin = 60 mm, dR =0.4 mm, wSC = 12 mm. Number of turns is 10.

terized by its electrical resistivity. The conductivity of HTSdepends in a highly non-linear way on the electric and magneticfield, and thence the distribution of current is generally non-uniform. Then the design of a coil from HTS as well as theinterpretation of results from its experimental testing requireparticular procedures. We illustrate this approach on the exam-ple of a small coil (10 turns, inner diameter 60 mm) made from12 mm wide CC tape [1]. Individual turns were insulated by tex-tile tape 0.25 mm thick and then the winding was impregnatedby polyesterimide resin dissolved in styrene [2]. The windingwas self-supported—shown in Fig. 1—but to some extent de-formable without damage. We assume that CC tapes did notexperience significant mechanical load during thermal cycling.

Coated conductor coils have been the object of study ofseveral works, either by measurements [3]–[7] or computations[4], [6]–[10]. However, the effect of degradation of the tapeacross its width or its length has not been studied. This ar-ticle not only presents such study but also a full systematicapproach for the characterization and numerical simulation ofpancake coils.

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5900406 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 23, NO. 3, JUNE 2013

Fig. 2. Short-sample data (symbols) of the critical current dependence on the magnetic field magnitude (different symbol for each value explained in the legend)and its orientation with respect to the CC tape wide face, defined by the angle �. Empirical fits with the expression (1) are shown by lines. The measureddependence has full 360◦ periodicity at low magnetic fields; however, beyond 100 mT, the values for 90◦–270◦ could be considered a repetition of those obtainedin the −90◦ − 90◦ range.

In the first part of the paper we show how the coil per-formance can be predicted from the data obtained on a shortsample. Because of highly non-linear properties of HTS tape itis necessary to employ numerical computation technique in thedesign of the coil and analysis of the observed behavior. Nu-merical methods also allow to calculate the quantities that arenot easily accessible in experimental way like the distributionof heat generated by AC current in the coil.

II. TAPE PROPERTIES

Basic information in the design of a superconducting coilis the dependence of critical current on magnetic field andits orientation. In the case of SuperPower tape SCS12050-AP this dependence is rather complicated as shown in Fig. 2.Explanation of this complex dependence is beyond the scopeof our investigation. However its description is essential inthe coil design and performance prediction. We have usedseveral approximations [11], the results presented here havebeen obtained by using the empirical description:

Jc(B, θ) =I1(B)

SSC

×{1 +

(θ−θ0(B)

Δθ

)2 (I2(B)I1(B) − 1

)for |θ − θ0| < Δθ

1 for |θ − θ0| ≥ Δθ

(1)

where Δθ = 0.75 radians and

I1(B) = Imin0

[1 + kmin

(1− B

B0

)β]

I2(B) = Imax0

[1 + kmax

(1− B

B0

)β]

θ0(B) = θ00

[1 + kθ

(1− B

B0

)β]

with the parameters B0 = 0.5 T, β = 4.5, Imin0 = 136 A,Imax0 = 218 A, θ00 = 33 degrees and SSC is the cross-section

of the superconducting layer that in our calculations was repre-sented by the rectangle of width wSC = 12 mm and thicknesshSC = 1 μm. The expression (1) with these parameters fits theIc data for magnetic fields higher than 0.1 T in the whole rangeof angles. At lower fields it works for angles θ from −90◦ to90◦ and, the critical currents slightly higher have been found inthe range of angles θ between 90◦ and 270◦.

III. MAXIMUM DC CURRENT OF THE COIL

Knowledge of the tape critical current described by theexpression (1) allows predicting the maximal DC current thecoil could carry. For this purpose we created the finite-elementmodel based on the geometry shown in Fig. 1. Rotationalsymmetry is used i.e. the real spiral turns have been replaced bycircular ones. The approximation of uniform current density inthe winding was used to calculate the magnetic field distributionin the coil. Then in an iterative procedure the level of coilcurrent that would generate the electrical field Ec = 1 μV/cmin various turns has been identified. The result—which could bewith certain simplification called the critical currents of individ-ual turns—is shown in Fig. 3. As expected, the innermost turn(assigned No. 1) limits the maximum DC current the coil couldcarry. Then it does not make much sense to calculate the currentthat would create 1 μV/cm electrical field in the whole coil.Instead we estimated the current-voltage curve of the wholecoil. A rigorous calculation would require in the finite elementcalculation a dense mesh of current-voltage dependences takenon short sample at different field values and orientations. Weused a simplified procedure instead: First, the measured shortsample curves were approximated by the power-law functionE(I) = Ec(I/Ic)

n (from which the values of Ic establishedat Ec = 1 μV/cm were used in Fig. 2) where the exponent nwas found to vary between 17 and 25 in the range of magneticfields 0.08–0.14 T that would appear in the coil winding atcarrying maximum current. Then we took the “critical currents”of individual turns, Ici, and the power-law exponent in the

GÖMÖRY et al.: AC LOSS IN PANCAKE COIL MADE FROM 12 mm WIDE REBCO TAPE 5900406

Fig. 3. Critical currents of coil turns estimated with the help of (1). Turns arenumbered from inside towards outside. The innermost turn has number 1.

Fig. 4. Current–voltage dependence calculated for the coil (full line) com-pared with the experimental data (circles). Better agreement is reached assum-ing the critical current data lowered by 2% (dotted). Simulations for a 7%degradation of the innermost turn (long dashes) and of all the coil turns (shortdashes) were carried out to see how much such damage could be detected in theE(I) tests.

mid of its range, n = 21 to estimate the current-voltage curveof the coil:

Ecoil(Icoil) =1

N∑i=1

2πri

N∑i=1

2πriEc

(IcoilIci

)n

. (2)

In this expression N = 10 is the number of turns, each of themwith the radius ri. The result was compared with experimen-tal data as shown in Fig. 4. Calculated prediction is slightlyhigher than the experimental finding, better agreement wouldbe achieved assuming a 2% reduction of the critical currentdensity in all the coil winding. The key role of the innermostturn is illustrated by E(I) curve calculated considering thedegradation of critical current by e.g., 7%: The increase ofthe coil voltage obtained assuming a uniform reduction of Iciin all the turns is close to the result obtained assuming suchdegradation in the innermost turn only.

IV. AC LOSS BEHAVIOR

In the case of a coil considered for power applications the dis-sipation with AC transport (AC loss) is an important parameter.We performed numerical calculations based on the A− ϕ− jformulation of critical state utilizing the finite-element soft-ware package Comsol Multiphysics [12]. The original stepwisecurrent-voltage curve proposed for hard superconductor [13] isreplaced by its smooth approximation

Js,θ(r, z) = Jc tanh

(Eφ(r, z)

Ec

). (3)

where the critical current density Jc is now the upper limit forthe current density in superconductor, Js, and the cylindricalgeometry shown in Fig. 1 is introduced. Because the electricalfield is generated by the electrostatic gradient ∇ϕ as wellas the time variation of the magnetic vector potential, �A, theexpression (3) can be rewritten [6] as

Js,θ(r, z) = Jc(B, θ) tanh

(−∇ϕ(r, z)− ∂Aφ(r,z)

∂t

Ec

)(4)

where we also indicated that Jc depends on the local value ofthe magnetic field and its orientation with respect to the tape. Inthe finite element calculation the distribution of A and ∇ϕ thatobeys the Maxwell equations with the additional constraint thatthe total current in each turn is equal to the current delivered bythe power supply is found

∫Si

Js,θ(r, z)

∣∣∣∣∣i=1...N

= Icoil. (5)

Series of distributions are calculated for

Icoil,j = Imax sin(ωtj) = Imax sin(αj) (6)

whereω is the angle frequency and αj = 0, ω.Δtj , 2ω.Δtj,,. . . 2π are the angles corresponding to the time instants tj whenthe distributions are resolved. We have found that the calcula-tion of M = 40 distributions in one AC period is sufficient toobtain the representative result within reasonable computationtime. The power dissipated in the coil at the time instant tj isthe sum of contributions from individual turns

Pj =

N∑i=1

∫Si

Js(r, z)Ei,j(r, z)dS (7)

where Js is calculated using (4) and the electrical field in thei-th turn at the instant defined by αj is

Ei,j(r, z) = −∇ϕi,j − ωAφ(r, z)j −Aφ(r, z)j−1

αj − αj−1. (8)

In the last expression we took into account that ∇ϕ is constantin the cross-section of one turn and it thus takes discrete values∇ϕi for different turns [14], [15]. The power dissipated in allthe turns is integrated over one cycle to obtain the total loss inthe winding. In order to allow comparison with results obtained

5900406 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 23, NO. 3, JUNE 2013

Fig. 5. AC loss in the coil calculated by finite-element method (lines) com-pared with experimental data (symbols). Taking into account the Jc(B, θ) isnot enough to reproduce the measurements (dashed line). Sufficient agreementwas reached when the lateral nonuniformity in the critical current density wasincorporated as well (full line). When assuming a field-independent Jc that isnonuniform across the tape width, the short-dashed prediction was obtained.

on different coils we prefer to present the dissipation divided bythe total length of the tape in Joule/meter units:

Q =1

Lcoil

M∑j=1

PjΔt =1

N∑i=1

2πri

2π

ωM

M∑j=1

Pj . (9)

In Fig. 5 the prediction obtained by such calculation incor-porating the Jc(B, θ) expression (1) is compared with exper-imental data determined by standard lock-in technique. Theloss per cycle measured at frequencies of 36, 72 and 144 Hz,respectively, was identical which testifies pure hysteretic char-acter consistent with the critical state model. As one can see,the calculated prediction (dashed line) was below the experi-mental data. The observed discrepancy was removed when wetook into account the fact that the tape edges exhibit reducedcurrent transport capability compared to the central part. Suchreduction was observed in a separate experiment mapping themagnetic field distribution above the tape transporting DCcurrent. The variation of Jc across the tape width shown inFig. 6 has been derived by an inversion calculation [16]. Alsothe structural investigations we performed by scanning electronmicroscopy revealed that there is significant difference in thesuperconducting layer morphology across the tape width. Thestructural analysis suggests that the volume ratio of inhomo-geneities in the superconducting layer increased towards bothedges of the tape. Such non-uniformity cannot be deduced fromthe Ic(B, θ) data. On the other hand at AC transport the dissipa-tion depends on details of the flux front movement resulting insignificant difference between two calculated curves in Fig. 5.

However, the coincidence of the measured AC loss data withthe calculation is not always the proof of a valid descriptionof superconductor properties. We performed the series of ACloss calculations neglecting the magnetic-field dependence ofthe critical current density. Main motivation for such exercise

Fig. 6. Nonuniformity of the tape across its width illustrated by (top) profileof the critical current density and (bottom) morphology of the superconductinglayer. The profile of Jc (full line) was obtained from the measured distributionof magnetic field above the tape carrying critical current at dc background fieldof 250 mT. Its linear approximation is shown by the dashed line. Electronmicroscope pictures were taken after removing the protecting Ag and stabi-lizing Cu layer. The inhomogenities—seen as “islands” in the superconductinglayer—occupy a bigger portion at tape edges.

was to check how much the improvement (or degradation) ofthe critical current would affect the AC loss. Here we show theresults for three values of Jc that correspond to the critical cur-rents equal to 260, 334 and 390 A, respectively. The lowest andhighest of these values are the extreme levels of critical currentsthat were calculated for individual turns in the coil taking thecomplex Ic(B, θ) of the tape into account. The middle value isobtained from an average of the values shown in Fig. 2.

The main conclusion is that the coil AC loss calculatedassuming a field-independent critical current density Jc=constant—see Fig. 7—is inversely proportional to Jc. Suchbehavior is typical for the self-field (transport) regime [17]. It ispossible that in a coil with significantly larger number of turnsthis conclusion would no more apply. Rather surprising how-ever is that the result calculated in such simplified manner—assuming that the critical current density in all the wind-ing is equal to that of the innermost turn—does very wellmatch the experimental data. We believe this is a mere coin-cidence: As shown in Fig. 5 the loss curve obtained assumingonly the Jc(B) dependence—without lateral non-uniformity—underestimates the experimentally found data, and also the

GÖMÖRY et al.: AC LOSS IN PANCAKE COIL MADE FROM 12 mm WIDE REBCO TAPE 5900406

Fig. 7. AC loss in the coil calculated in simplified manner assuming a constantcritical current density Jc (lines) compared with experimental data (symbols).The results are shown for three values of Jc corresponding to the tape criticalcurrents shown in the legend.

slope of its dependence on the coil current is higher thanobserved in experiments. Both these defects can be repaired bytaking into account the lateral Jc non-uniformity. The short-dashed loss curve in Fig. 5 calculated for a field-independentbut non-uniform Jc exhibits the slope that is the lowest amongthe compared numerical results. Thus the opposite effects ofJc(B) and non-uniformity on the loss curve led to the behaviorvery similar to that predicted by the simplest possible modelwith constant Jc.

As we have found, the lateral non-uniformity of Jc withrespect to the tape width has significant influence on the coilloss. This effect has already been found in the case of powertransmission cable [18] or fault current limiter [19]. In thefollowing we present the results of our analysis about thepossible consequence of Jc variation along the tape length. Inparticular we calculated the AC loss and its distribution in thewinding for the case when Jc in one of the turns is lower thanin the rest.

V. INFLUENCE OF LOCAL IC DEGRADATION

In this section we show the results of AC loss calculationperformed following the procedure described by the formulas(4)–(9) i.e. taking into account the complex Jc(B, θ) estab-lished on the short sample assuming however that the super-conductor in some turns has Jc that is proportionally lowered atall the field magnitudes and orientations.

First notable feature is that the AC loss is far less sensitiveto a single turn degradation than the current-voltage curve.This is obvious when comparing the effect of reducing thecritical current of the innermost turn to 93% of original valueon the current-voltage curve in Fig. 4 with the AC loss resultsshown in Fig. 8. Even a drop to 75% of the original value doesnot produce an increase of AC loss in the coil that would bedistinctively beyond the experimental noise.

This result can be better understood when we analyze thedistribution of dissipation in the coil winding. In Fig. 9 thedissipation is plotted as calculated for individual turns in the AC

Fig. 8. Calculated effect of a reduced critical current in the innermost turn onac loss: Loss in the coil (full line from Fig. 5) compared with the result obtainedassuming critical current of the turn N.1 reduced to 93% and 75%, respectively.As one can see, such degradation produces an effect that is hardly visible.

Fig. 9. Distribution of dissipation in the coil turns at ac amplitude of 70 A(27% of undegraded Ic of inner turn): Line with diamonds shows the valuescalculated for the coil made of identical turns from the tape with propertiesshown in Fig. 2. The full squares have been obtained, assuming that in theinnermost turn, the critical current has been lowered by 25%.

cycle with amplitude of 70 A. The dissipation is rather uni-formly distributed inside the coil, the extreme turns (the inner-most and the outermost one, respectively) producing heat about50% above the average.

Then one can better understand why a reduction of thecritical current in one turn causes a minor increase of thedissipation in the whole coil. What is interesting is that anincrease of dissipation is predicted also for the neighboringturns in spite of the fact that their properties have not changed.Surprising is the result we have found for higher AC amplitudesas shown in Fig. 10: In the case of coil made of tape with thesame properties in all the turns the dissipation pattern is similarto that found at lower currents. Striking is the prediction thatusing a degraded tape in the innermost turn would reduce thedissipation in that turn at higher currents. The elevated AC lossin the whole coil should be borne by nearly all the remainingturns, the additional dissipation peaking at the neighboringturn N.2. We have checked this prediction also by a differentnumerical method based on the critical state principles [10]with identical results. Quite opposite i.e. the increase of loss at

5900406 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 23, NO. 3, JUNE 2013

Fig. 10. Same as Fig. 9, but the ac amplitude is now 140 A (54% ofundegraded Ic of inner turn). It is surprising that the turn with lowered criticalcurrent exhibits lower dissipation than in the case if it were undegraded. Theincreased dissipation is distributed in the remaining coil turns.

lowered Jc would be expected for a single conductor, becausethe dissipating portion of superconductor volume would belarger. In the coil, however, the flux penetration is controlledby net currents in all the turns. Then at some circumstancesthe increase of flux penetration into degraded turn is smallerthan expected from single tape considerations and in the sametime it is higher in the neighboring turns. Because the localdissipation rate is proportional to Jc an observed loss reductioncould happen. Another explanation could be based on the factthat a lower Jc leads to a reduction of the field lines curvature.This caused the field to be more parallel to the tape surface inthe innermost turn when its Jc is reduced. However one shouldnot forget that the angle dependence of Jc is rather uncommonfor the used tape and the origin of the loss reduction in thedegraded turn could be linked with this property. Unfortunatelythere is no easy way to detect the local dissipation in coil turnsin experiment thus we could only speculate about the cause ofthis striking effect.

VI. CONCLUSION

Properties of a small coil made from 12 mm wide CC tapehave been investigated experimentally and by finite elementnumerical models. Taking into account the complexity of Jc de-pendence on magnetic field and its orientation in the calculationwe achieved rather precise prediction of the DC current-voltagecurve of the coil as well as the dissipation in AC regime.

Similar to the results of other groups analyzing the AC loss insuperconducting cables or fault current limiters we have foundthat a non-uniform transport capability of the HTS materialacross the tape width leads to significant increase in AC dis-sipation. Our structural investigations provide strong indicationthat this phenomenon is caused by increased share of “islands”not participating in the current flow at the tape edges.

Longitudinal non-uniformity of tape critical current can havesevere consequences on the maximum current it is able to reach.On the other hand the AC loss is rather insensitive to this matter.This could maybe explain surprising similarity of AC loss ofcoils with rather different critical current [20]. Our calculationsconfirmed that for a coil with constant Jc the AC loss would beinversely proportional to Jc.

Rather interesting is the result of numerical calculationshowing that a turn from tape with reduced Jc could transferthe surplus dissipation to neighboring turns. At AC currentswith amplitude approaching the critical current this would evenmake the actual dissipation in weaker turn lower comparedto the case when all the turns are made from identical tape.Experimental verification of this theoretical prediction wouldbe very desirable.

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