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 Transformer
B. Pérez, A. Álvarez, Member, IEEE, P. Suárez, D. Cáceres, J. M. Ceballos, X. Obradors, X. Granados, and R. Bosch
TABLE I
SPECIFICATIONS OF THE HTS TAPE
Abstract—In order to study the viability of coreless AC coupled
coils, a superconductor transformer based on BSCCO-2223 PIT
tapes was constructed. To achieve the minimum flux leakage, a
toroidal geometry was selected. Both secondary and primary coils
were wound around a glass fiber reinforced epoxy torus, obtaining
a solid system. The field inside the transformer, the coupling factor,
and the losses in the system were computed and measured, providing suitable parameters for new improvements in these systems.
Index Terms—AC losses, Bi-2223 tape, superconductor transformer.
TABLE II
CHARACTERISTIES OF THE TRANSFORMER
I. INTRODUCTION
H
IGH temperature superconducting transformers are
lighter, smaller and have a higher efficiency than conventional transformers [1]. The windings of most superconducting
transformer 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 wants
to maintain a high coupling factor it is necessary to look for
a geometry to confine the magnetic field lines. Examples are
annular [5] and solenoidal transformers [6].
We propose an alternative geometry to get a high coupling
factor without an iron core: a single-phase superconducting
torus. We made this transformer with an air core and determined
its coupling factor and its AC losses. The test of AC losses was
performed by means of the electrical method using a lock-in
amplifier [3], [7], [8].
II. DESIGN OF THE PROTOTYPE
The transformer was wound with Bi-2223 tape. In order to
reinforce the transformer structure, the Bi-2223 windings were
wound onto a glass fiber torus [3], [9]. The minimum coil radius
before the coil loses its superconductor characteristics has been
evaluated previously [10] and based on this work a torus with
30 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 the
Inter-ministerial Commission of Science and Technology of Spain and Government of Extremadura.
A. Álvarez, B. Pérez, P. Suárez, J. M. Ceballos, and R. Bosch are with
the Electrical Engineering Department, University of Extremadura, Apdo
382, 06071 Badajoz, Spain (e-mail: aalvarez@unex.es, belenpc@unex.es,
psuarez@unex.es, jmceballos@terra.es).
D. Cáceres is with the Applied Mathematic Department, University of Extremadura, Apdo 382, 06071 Badajoz, Spain (e-mail: dcaceres@unex.es).
X. Obradors and X. Granados are with the Institute of Material Science
ICMAB (CSIC), Barcelona, Spain.
Digital Object Identifier 10.1109/TASC.2003.813122
Fig. 1. The prototype transformer.
when the field is constant inside and null outside. Similarly, we
tried to wind the tape as close together as possible to achieve a
homogeneous current distribution. To accomplish this, 341 turns
were needed for the inner coil. For the outer coil we used 447
turns to get a transformation ratio different from unity. The final
structure consists of 5 layers (A, B for the inner coil and C, D,
E for the outer) that we connected properly to get the desired
rating.
The characteristics of the HTS tape and the windings are presented in Tables I and II, respectively, and Fig. 1 shows the completed toroidal transformer. Coils were cooled using liquid nitrogen at 77 K.
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TABLE III
EXPERIMENTAL COUPLINCE FACTOR
Fig. 2. Electric circuit for ac loss measurement of the transformer.
III. COUPLING FACTOR OF THE TRANSFORMER
A. Theoretical Coupling Factor
The transformer coupling factor is used to verify that the magnetic field is shared enough to suppose that the magnetic losses
are produced by the same distribution of magnetic field in both
of the coils.
To obtain the theoretical coupling factor of this transformer
we calculated the inner magnetic field from each toroidal current using a numerical integration procedure based on the BiotSavart law and the superposition principle (since the system is
linear). Then the linked and leakage fluxes and the winding coupling factors were computed.
The evaluated coupling factor was 0.88 for the inner winding
and 0.94 for the outer winding. Therefore the evaluated theoretical coupling factor was 0.91 for the transformer.
B. Measurement of the Transformer Coupling Factor
The prototype was tested without load, feeding both the
high-tension and low-tension coils to determine the experimental coupling factor. The results are summarized in Table III.
They show that the measured transformer coupling factor is
adequate to assume that most magnetic flux is shared by both
coils.
IV. AC LOSSES
A. Measurement Method
An electrical technique was used to measure the AC losses
in the transformer. We used a lock-in amplifier with four inputs: two of them to measure the primary voltage and current
and to calculate the power fed into the transformer, , and the
other inputs to measure the secondary voltage and current and
to calculate the power that the transformer gives out, . Each
power was evaluated by integration of the product of current
and voltage over an integer number of periods. The transformer
losses, , were calculated as the difference between and ,
so that,
.
The electric circuit used is shown in Fig. 2. Various tests on
the transformer were carried out, in short-circuit, without load,
and with different values of the load. In all cases, the low-tension
coil was used as the primary and the potential difference was
measured in all layers. The frequency was 50 Hz.
Fig. 3. Magnetization losses versus (V =N ) for the transformer layers tested
in short-circuit. Linear dependence can be seen.
B. Theoretical Method
In electric power applications like the present case, the
total AC losses,
, are the result of two contributions: the
alternating transport current losses,
, and the magnetization
losses,
. The first are dominated by hysteresis losses
[11] that can be evaluated theoretically by the elliptic model
formulated by Norris:
(1)
where
, is the peak current, is the frequency, and
is the magnetic permeability
H/m . The primary
and secundary Norris losses were evaluated by (1) and the total
Norris losses calculated as their sum.
Subtracting these from the total measured losses, an estimate
of the total magnetization losses can be obtained. For low frequencies, it has been shown that there must be a proportionality
between magnetization losses per unit of length in each winding
and the square of the magnetic field, which can be written:
PÉREZ et al.: AC LOSSES IN A TOROIDAL SUPERCONDUCTING TRANSFORMER
2343
Fig. 6. AC losses of the transformer to test without load. Comparison with
theoretical curve.
When this expression was applied to the losses seen in short
circuit and no load tests, good agreement between experiment
and theory was achieved as shown in Figs. 5 and 6.
Fig. 4. Magnetization losses versus (V =N ) for the transformer layers tested
without load. Linear dependence can be seen.
V. CONCLUSION
AC losses in a superconducting transformer were measured
by the electrical method with a 4-channel lock-in amplifier. The
theoretical AC losses were evaluated and an alternative formulation proposed. The experimental results show good agreement
with the proposed theoretical expression. For this reason, we
believe that (2) adequately represents the AC losses in a transformer working at low frequencies. Further testing is required
to validate (2) for higher frequencies.
REFERENCES
Fig. 5. AC losses of the transformer to test in short-circuit. Comparison with
theoretical curve.
This linear proportionality betwen
and
is shown
in Fig. 3 for primary and secondary windings in the test of the
transformer in short-circuit. Fig. 4 shows this relation in the tests
without load. The similarity of the slopes reinforces the idea that
the proportionality constant,
, only depends on details of
the construction of the tape.
An alternative formulation is proposed to find the AC losses
in the transformer by means of the following expression:
(2)
where the subscripts 1 and 2 represent the primary and secondary of the transformer.
[1] H. J. Lee, G. Cha, J. Lee, K. D. Choi, K. W. Ryu, and S. Y. Hahn, “Test
and characteristic analysis of an HTS power transformer,” IEEE Trans.
Appl. Superconduct., vol. 11, no. 1, pp. 1486–1489, 2001.
[2] K. Funaki et al., “Preliminary tests of a 500 kVA-class oxide superconducting transformer cooled by subcooled nitrogen,” IEEE Trans. Appl.
Superconduct., vol. 7, no. 2, pp. 824–827, 1997.
[3] M. Iwakuma et al., “AC loss properties of a 1 MVA single-phase HTS
power transformer,” IEEE Trans. Appl. Superconduct., vol. 11, no. 1, pp.
1482–1485, 2001.
[4] G. Donnier-Valentin, P. Tixador, and E. Vinot, “Considerations about
HTS 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 superconducting 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 superconducting 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 losses
of HTS power transmission cables using Bi-2223 tapes with twisted filaments,” 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 and
bundles 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 Solenoidal
Air-Core Transformers
Pilar Suárez, Alfredo Álvarez, Member, IEEE, Belén Pérez, Dolores Cáceres, Eduardo Cordero, and
José-María Ceballos
TABLE I
SPECIFICATIONS OF THE HTS TAPE
Abstract—The losses in an HTS tape depend strongly on the perpendicular magnetic field. In order to avoid this magnetic field
component in an air core transformer, a toroidal geometry was
proposed and studied in previous work. Due to the difficulties that
one finds in constructing toroidal coils, the straight solenoidal geometry is now under study. In this case, the magnetic field close
to the ends of the coil is not parallel to the axis and a perpendicular component appears. In the present work, the losses due to this
component are studied as a function of the coil geometry—i.e., the
ratio between length and diameter—and a practical formulation is
found.
Index Terms—Bi-2223 coil, magnetization losses, superconducting transformer.
I. INTRODUCTION
TABLE II
CHARACTERISTICS OF SOLENOIDAL COILS
H
IGH temperature superconducting transformers are
lighter, smaller and have a higher efficiency than conventional transformers [1]. The windings of most superconducting
transformer 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 wants
to maintain a high coupling factor it is necessary to look for
a geometry to confine the magnetic field. In a previous paper
[5], we studied a superconducting toroidal transformer and
described a method to measure AC losses.
Due to the difficulties that one finds in constructing toroidal
coils, the straight solenoidal geometry is now under study. In
this case, the magnetic field close to the ends of the coil is not
parallel to the axis and a perpendicular component appears.
But when a transformer is constructed from these coils, there
are also losses due to the parallel magnetic field. The present
work analyzes the losses due to these components, taking the
influence of the coil geometry into account.
coil structures, the Bi-2223 coils were wound onto glass-fiber
solenoids [7], [8] and several prototypes of transformers were
formed by placing these coils concentrically. The characteristics of the two sets of coils constructed are given in Table II,
where N is the number of turns of each coil, R the respective
radius, and L the respective length. Fig. 1 shows one of the prototypes ready to be tested.
III. MEASUREMENT OF THE LOSSES
II. DESIGN OF THE SOLENOIDAL TRANSFORMER
A prior step to constructing the solenoidal transformer is to
show the behavior of a coil alone. The minimum coil radius before it loses its superconducting characteristics had been evaluated previously [6], and based on that work several coils were
constructed by winding with Bi-2223 tape. The specifications of
the HTS tape are presented in Table I. In order to reinforce the
Manuscript received October 4, 2004. This research is funded in part by the
Inter-Ministerial Commission of Science and Technology of Spain and Government of Extremadura.
The authors are with the Electrical Engineering Department, University of
Extremadura, 06071 Badajoz, Spain (e-mail: psuarez@unex.es; aalvarez@
unex.es; belenpc@unex.es; dcaceres@unex.es; educorde@unex.es; jmceballos@unex.es).
Digital Object Identifier 10.1109/TASC.2005.849312
A. Measurement of Total Losses
The transformer losses were measured by the electrical
method. Fig. 2 shows the circuit used. Since the only energy
entering the system comes from the power supply, this method
gives the total AC losses of the transformer as the mean value
of the product of
and
during an integer number of
periods. The readings of the voltages and currents were taken
by means of a DAQ card. Voltages were read directly from the
taps on the transformers, and currents were read by means of
a Hall probe as is shown in Fig. 2. The data were processed
by a routine written in LabVIEW, evaluating the losses by
integrating the product of voltage and current. Fig. 3 shows the
total losses in each coil of prototype 2. Similar curves were
obtained with prototype 1.
1051-8223/$20.00 © 2005 IEEE
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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 2, JUNE 2005
Fig. 1. Prototype of solenoidal superconducting transformer.
Fig. 4. Transport losses in each coil of a solenoidal superconducting
transformer.
Fig. 2. Diagram to test the transformer to measure the total losses. Here coils
1 (primary) and 4 (secondary) are under test. Similar trials were done with the
other coils. In all the cases one coil was connected to the supply, and the other
three were without load.
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.
Fig. 3. Total losses in each coil of prototype 2. Similar curves were observed
with prototype 1.
IV. THEORETICAL HYPOTHESIS
B. Measurement of Transport Losses
In order to measure the transport losses, the same setup was
used. In this case pairs of consecutive coils were connected in
series and supplied with current in opposite senses to annul the
magnetic field and therefore the losses due to it. The measurements thus give us twice the transport losses in one of the coils.
The results were similar in all the coils. Fig. 4 shows the mean
results.
When magnetization losses are measured in a sample of superconducting tape the contribution of the perpendicular magnetic field is much greater than that of the parallel field. But
when losses are studied in solenoidal transformers, there are
some differences. The total losses in the transformer depend on
which coil is supplied (Fig. 3). This effect can be explained with
the 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
Fig. 6. Influence of the coil radius, R, on the magnetization losses. V is the
voltage in the coil and N the number of turns. V=N is proportional to the flux
of 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 of
R. In [9] we studied this function and show that k(R) = k (1=R) .
1857
Fig. 7. Representation of the practical equation of P
= k (1=R) (V=N ) .
electrical applications: P
for low frequency
to the losses are due to the transport current and the perpendicular magnetic field at the ends of all the coils:
(1)
where
is the quantity of total losses,
the transport current
contribution and
the perpendicular magnetic field contribution in all the coils.
But an additional contribution appears when another coil (intermediate or outer) is supplied.
This new contribution is due to the parallel magnetic field
along the remaining of the coils inside the supplied one. I.e.,
when coil 3 is supplied the effects of the parallel magnetic field
are twice those when coil 2 is supplied. Similarly, when coil 4
is supplied the effects of the parallel component are three times
those when coil 2 is supplied. Assuming that the contribution
to the losses of the perpendicular magnetic field is similar in all
the cases, one has:
(2)
where
is the contribution of the parallel magnetic field in one
coil.
The contribution of the parallel magnetic field due to the sup, can be obtained from (2):
plied coil in the others,
(3)
The assumption that the contribution to the losses of the perpendicular magnetic field is similar in all the cases is based on
a previous work [9]. In this, we studied the dependence of the
magnetization losses in solenoidal coils on the coil geometry
(coil radius, density of turns, coil length, ). Figs. 6 and 7 show
Fig. 8. Comparison of the losses due to the parallel magnetic field between
coils 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 coils
with different radii and numbers of turns.
Applying these results to the values in Table II shows the
magnetization losses to be similar.
V. RESULTS
The results of the measurements obtained in Section III confirm the hypothesis proposed in Section IV.
Figs. 8 and 9 show the results corresponding to the cases
and
, respectively. Both cases fitted a linear
of
regression:
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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 15, NO. 2, JUNE 2005
Furthermore, the influence of geometric factors (coil radius and
length) has also to be taken into account.
REFERENCES
Fig. 9. Comparison of the losses due to the parallel magnetic field between
coils 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 depend on which coil is supplied. The magnetization losses include a very important contribution from the parallel magnetic
field, because this component affects much more of the length
of the superconducting tape than the perpendicular component.
[1] H. J. Lee, G. Cha, J. Lee, K. D. Choi, K. W. Ryu, and S. Y. Hahn, “Test
and 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 superconducting 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 HTS
power transformer,” IEEE Trans. Appl. Supercond., vol. 11, no. 1, pp.
1482–1485, Mar. 2001.
[4] G. Donnier-Valentin, P. Tixador, and E. Vinot, “Considerations about
HTS 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 superconducting
transformer,” 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-phase
HTS 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 formulation of low frequency AC losses for superconducting coils,” presented at
the 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 practical
superconductors
F Gömöry1 , J Šouc1 , M Vojenčiak1 , E Seiler1 , B Klinčok1 ,
J M Ceballos1,2 , E Pardo1,3 , A Sanchez3 , C Navau3 , S Farinon4 and
P Fabbricatore4
1
Institute of Electrical Engineering, Slovak Academy of Sciences, Dúbravská cesta 9,
842 39 Bratislava, Slovakia
2
Department of Electrical Engineering, University of Extremadura, Badajoz, E-06071, Spain
3
Grup d’Electromagnetisme, Departament de Fisica, Universitat Autonoma Barcelona,
08193 Bellaterra (Barcelona), Catalonia, Spain
4
Istituto Nazionale di Fisica Nucleare, Via Dodecaneso 33, Genoa, I-16146, Italy
Received 3 October 2005, in final form 21 November 2005
Published 20 January 2006
Online at stacks.iop.org/SUST/19/S60
Abstract
Recent progress in the development of methods used to predict AC loss in
superconducting conductors is summarized. It is underlined that the loss is
just one of the electromagnetic characteristics controlled by the time
evolution of magnetic field and current distribution inside the conductor.
Powerful methods for the simulation of magnetic flux penetration, like
Brandt’s method and the method of minimal magnetic energy variation,
allow us to model the interaction of the conductor with an external magnetic
field or a transport current, or with both of them. The case of a coincident
action of AC field and AC transport current is of prime importance for
practical applications. Numerical simulation methods allow us to expand the
prediction range from simplified shapes like a (infinitely high) slab or
(infinitely thin) strip to more realistic forms like strips with finite rectangular
or elliptic cross-section. Another substantial feature of these methods is that
the real composite structure containing an array of superconducting
filaments can be taken into account. Also, the case of a ferromagnetic
matrix can be considered, with the simulations showing a dramatic impact
on the local field. In all these circumstances, it is possible to indicate how
the AC loss can be reduced by a proper architecture of the composite. On
the other hand, the multifilamentary arrangement brings about a presence of
coupling currents and coupling loss. Simulation of this phenomenon
requires 3D formulation with corresponding growth of the problem
complexity and computation time.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Since the discovery of superconductivity, the disappearance
of electrical resistivity motivated many scientists to think
about the application of superconductors in electric power
engineering.
When hard superconductors as materials
reaching the demand for competitive current transport
capability appeared in the early 1960s, a new problem emerged:
it was soon realized that the ability to lead persistent electrical
currents is linked with the appearance of dissipation in the
0953-2048/06/030060+07$30.00 © 2006 IOP Publishing Ltd
AC regime, i.e. at transporting AC current or during exposure
to AC magnetic field. This phenomenon, called the AC loss
in superconductors, has occupied a significant sector of the
superconducting research until now.
In the era of low-temperature superconductors (LTSs), a
typical structure of a superconducting wire was established. It
contained superconducting filaments in a metallic matrix. The
cross-section of such a composite wire had the form of a circle
or a rectangle with the aspect ratio of the sides rarely exceeding
two, i.e. close to a square. For this metal–superconductor
Printed in the UK
S60
AC loss in practical superconductors
structure, the fundamental principles of the AC loss mechanism
were revealed. It was found that the heat generation can
be attributed to electrical currents induced in the composite
structure. Several significant paths for current loops have been
identified, and two loss dissipation mechanisms distinguished:
in the case of screening current forming a loop entirely within
the superconducting filament, its creation will cost energy due
to the pinning of magnetic flux in the superconductor. This loss
component is called the hysteresis loss. Another mechanism
is the so-called coupling loss, ascribed to the current flowing
in a loop that is mostly formed by superconducting paths, but
also contains portions with normal resistance [1–5].
This terminology remains valid in the investigation of
wires made from high-temperature superconductors (HTSs).
However, several factors have pushed the research of AC loss
significantly forward in the recent period. First, the most used
HTS wire has the form of a tape, to achieve good alignment
of HTS grains. The aspect ratio of the sides of its rectangular
cross-section typically exceeds 10. Then, the approximation
of an infinite slab in parallel magnetic field, fruitfully applied
to the wires and windings from LTS wires, becomes rather
doubtful. Also, the composite structure of HTS wires is far
less regular than in the case of LTSs. Then, instead of the
effective medium approach that successfully explained many
of the effects observed in LTS wires, one has to take into
consideration the real structure of the composite. On the other
hand, tremendous increase of the computing power accessible
on a personal computer made available numerical methods that
are adequate to cope with the aforementioned complications.
It is the main purpose of this paper to encourage non-specialists
in numerical simulations to utilize these methods in the
investigation of AC applications of superconductors.
In section 2, the properties of simple shapes will be
summarized in order to show the general rules that govern
the AC loss. The importance of the demagnetizing field to
the AC loss will be illustrated. In section 3, the use of
two powerful numerical methods allowing us to simulate the
behaviour of a superconducting wire in arbitrary conditions
will be demonstrated for several cases of practical importance.
We will show how the treatment of superconducting material
as a conductor with a highly non-linear current–voltage curve
allows us to solve the problem of current distribution into
parallel paths and to simulate the flux distribution in a cable
consisting of a single layer of superconducting tapes. The
influence of tape arrangement, in particular the width of
gaps between tapes, on the AC loss can be investigated in
this way. Also, the method of minimum magnetic energy
variation (MMEV) will be presented as a tool to understand
the dissipation in the case of simultaneous action of AC field
and AC current for a wire with rectangular cross-section. We
also show that this method allows us to simulate the critical
state and accompanied dissipation in a superconducting wire
with ferromagnetic matrix. In section 4 we briefly summarize
the presented results.
2. AC loss in hard superconductors with simple
shapes
The appearance of dissipation in hard superconductors exposed
to a changing magnetic field was recognized simultaneously
with the formulation of the critical-state model. This model has
served as an excellent approximation of the electromagnetic
behaviour of these materials since then [6]. The simplest shape
allowing us to derive analytical expressions for the distribution
of local magnetic field, current density and electrical field is
that of a slab (infinitely high) in a parallel magnetic field.
Dissipation in the cyclic regime of external magnetic field
Bac = Ba sin ωt is easily calculated, showing the following
fundamental features.
The dissipation depends on the volume of the sample
affected by the movement of flux lines. In the beginning part
of the magnetization process, there remains a portion of the
sample untouched by the field change because the screening
currents, starting from the sample surface, are able to shield the
change of applied field completely. This shielding capability is
exhausted when the applied field has reached the value called
the penetration field Bp . For low fields, i.e. smaller than Bp ,
the AC loss is proportional to Ba3 . Beyond Bp , the pattern of
induced currents is saturated, and the loss increase with Ba is
just linear, creating in this way a kink in the AC loss dependence
on Ba . The value of penetration field depends on the sample
thickness, thus the AC loss is not solely a material property.
This was further underlined when samples in perpendicular
magnetic field (e.g. single crystals of HTS) started to attract
attention. Interestingly, the value of Bp remained roughly
the same as that found in parallel field [7], but the losses
increased dramatically. Taking into account the enhancement
of local magnetic field in perpendicular geometry due to the
demagnetizing effect, this observation can be easily explained.
The results achieved for AC loss in superconductors of simple
shapes exposed to a cyclic AC field can be generally expressed
by the following formula [8]:
q=S
π
χ0 Ba2 χint
(y)
µ0
(1)
where q is the loss per metre length of a superconducting
wire, S is its cross-section, µ0 = 4π × 10−7 H m−1 , χ0 is
the initial susceptibility that will be discussed later on and
χint
is the imaginary part of the internal complex magnetic
susceptibility. As indicated in the formula, it depends on the
a
variable y = BBmax
, 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 the
perpendicular geometry, e.g. that of a thin strip or disc in a
transversal field, the AC susceptibility reaches its maximum
well before the complete saturation of the sample crosssection by the critical current density. In theoretical papers
dealing with magnetic flux penetration into superconductors
a
of various shapes [6, 9–11], the corresponding χint
( BBmax
) curve
can be found as well as the value of χ0 . Interestingly, this
dependence is rather similar for all the investigated simple
shapes, indicating that a significant loss reduction cannot be
reached by a simple shape optimization of the conductor [8].
This is illustrated in figure 1. However, the shape factor χ0 for
all the simple shapes can be roughly estimated by the formula
χ0 = 1 +
a
b
(2)
where a is the wire dimension perpendicular to the applied
field and b is the dimension parallel to the applied field. In the
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F Gömöry et al
1
/χ
/χ"max
χ "/χ
0.1
strip
disk
slab
elliptic strip
0.01
0.001
0.01
0.1
1
Ba /Bmax
10
100
Figure 1. Theoretical prediction of AC loss behaviour—in applied
magnetic field—for hard superconductors of various simple shapes.
The loss is characterized by the imaginary part of AC susceptibility
according to the formula (1). To allow direct comparison of the
dependences, the both axes are normalized to make the curves meet
at (1, 1).
case of a slab in a parallel field b a, thus χ0 ≈ 1. When
the field is rotated by 90◦ , the same slab in a perpendicular
field will exhibit χ0 ≈ ab 1. Thus, the rule of thumb
for the magnetization loss reduction is ‘avoid perpendicular
magnetic field’. This conclusion also remains valid for a
rough estimation of magnetization hysteresis loss in an array
of filaments [12]. Rigorous analytical derivation of AC loss
formulae for an infinite stack or a horizontal array of filaments
can be found in [13]. Extensive numerical calculations of
AC loss in two-dimensional matrices of filaments have been
published [14]. Valuable comparison of AC loss measured
on a horizontal array with theoretical predictions obtained by
different methods has been published recently [15].
In the case of transporting AC current, the result is even
simpler: save very thin strips, with the side aspect ratio
exceeding 100, the shape of the superconducting wire does not
influence the loss notably [16]. Nevertheless, the distribution
of critical current density—or the density of filaments in the
case of a multifilamentary wire—would lead to the deviation
of this unique behaviour. Generally, the conductor with better
properties of the surface filaments will exhibit at low currents
a lower AC loss then predicted by the Norris formula [17, 18].
3. Flux penetration into complex shapes
The approximation of superconductors in real windings by
one of the models mentioned in the previous paragraph,
successfully applied in the LTS era, faces serious problems
in the case of HTS wires. The perpendicular geometry
was found as the critical arrangement, and the study of
tape conductors in perpendicular field became the standard
experiment. Fortunately, at the same time the performance of
common personal computers increased in a way that allows us
to carry out the numerical simulations taking into account the
real structure of composite wires.
S62
Systematic investigation for a series of Bi-2223/Ag
composites demonstrated the possibility of determining χ0
with the help of a finite element simulation of the diamagnetic
state of superconducting filaments as a linear problem [19].
However, the substantial progress in understanding the loss
behaviour of composite HTS wires has been reached thanks to
the use of two simulation methods, allowing us to develop the
distribution of local current density, electrical and magnetic
field in the whole range of applied magnetic fields and/or
transport currents.
The first method treats the hard superconductor as a
conductive medium with highly non-linear current–voltage
relation, and allows us to predict nicely the flux penetration.
This approach is in agreement with experimental evidence that
the relation between the electrical field, E, and the current
density can be fairly approximated by the power law relation
E = E 0 ( jj0 )n , where E 0 is the conventional criterion for
the determination of the critical current density, j0 , and the
exponent n reflects the smoothness of the transition. For an
HTS at 77 K, typical values of n are in the range between 15
and 30. This idea was successfully used to predict the influence
of the conductor shape on the AC loss [20], to simulate
the distribution of currents and AC loss in multifilamentary
wires [21] and to calculate the AC loss at in a wire transporting
AC current with simultaneous exposure to AC magnetic
field [22]. The state equation
∇×
1
∇ × A = σ (E) · E
µ0
where
E=−
(3)
∂A
− ∇V
∂t
is to be resolved for the magnetic vector potential A and
the electric potential V . In the case of two-dimensional
problems—e.g. for an infinite wire exposed to a perpendicular
magnetic field—the cross-section of a superconducting wire is
divided into rectangular regions where the conductivity of the
superconductor obeys the power law
j0
σ (E) =
E0
E
E0
1−n
n
.
(4)
An alternative formulation [23] uses the current vector
potential T and magnetic scalar potential . Thorough
discussion about the advantages and drawbacks of various
formulations for the nonlinear-resistivity approach to
the simulation of electromagnetic phenomena in hard
superconductors has been published recently [24].
Another numerical simulation method of great practical
use is the minimum magnetic energy variation (MMEV)
method [25]. Its application is quite straightforward for twodimensional problems; however, the original formulation [26]
is three dimensional [27]. Similarly to the previous method,
the cross-section S of the superconductor is divided into
rectangular regions; each of them should be filled either with
+ jc , − jc, or left empty to minimize the functional
F[ j ] = 12
j (r)A j (r) dS −
j (r)A j (r) dS
S
S
j (r)(Aa (r) − Aa (r)) dS
(5)
+
S
AC loss in practical superconductors
10
2
10
1
10
0
- fill 85%
- fill 63%
- fill 48%
0.2
0.6
1
0.6
-0.2
-1
-0.6
0.2
1
2πQ
2
µ0Ic
10
-1
0.3
0.4
0.5
0.6
I /I
0.7 0.8 0.9 1
a c
Figure 2. Calculated normalized AC loss, i.e. divided by the loss
prefactor proportional to Ic2 , for three cores of superconducting
cable differing in the percentage of HTS occupation of the perimeter
of the core former. As expected, larger gaps between neighbouring
tapes, i.e. lower fill of the core perimeter by HTS tapes, lead to
higher 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 the
currents in the superconductor and Aa is that from the applied
field. The quantities with caps correspond to those obtained
by the solution in the previous time step. The minimization
should obey the constraints
j (r) dS
I =
S
| 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. In
spite of the fact that this is a notable simplification for HTSs,
the method offers a distinct shape of the boundaries between
positive and negative current density, as well as the shape of
the current-free core being precisely found. Then, one can
understand the underlying physics of the flux penetration in a
given configuration.
3.1. Influence of gaps in transmission cable on its AC loss
Here we show how one can assess the importance of permitting
minimum gaps between neighbouring tapes of the single-layer
cable as a model arrangement for the power transmission
cable. High packing factor—defined as the percentage of
the perimeter of the central cylindrical mandrel occupied by
the tapes—is not reached easily in the factory production,
leading to a significant increase of the manufacturing cost when
trying to minimize the gaps. Simulations following Brandt’s
method [20] allowed us to calculate the quantitative prediction
presented in figure 2. We considered 14 tapes placed straight
in parallel on the central mandrel of 21 mm diameter. Three
cables made from tapes of the same superconducting material
but different widths have been compared. The decrease of the
tape width leads to a reduction of the cable critical current due
to two independent mechanisms: the first is a simple reduction
of the superconductor’s cross-section, the second is the change
of local magnetic field distribution. Therefore, we compare
the normalized values of transport AC loss, i.e. divided by
Figure 3. Distribution of current density in the rectangular
cross-section of wire made from hard superconductor during
simultaneous 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 critical
current, and the magnetic field amplitude is 72% of the penetration
field. The sequence of distributions, each characterized by the
number indicating the ratio of actual transport current with respect
to the amplitude value of AC current, goes from left to right and
from top to bottom.
the factor proportional to Ic2 . Also, the transport current is
normalized with respect to the critical current of the cable
in figure 2. In the simulations, the critical current density
dependence on local magnetic field was assumed to follow
jc0
the Kim’s relation jc (x, y) = 1+ |B(x,y)|
. As one can see, the
B0
gaps between tapes can significantly influence the AC loss
performance of a power transmission cable.
3.2. AC–AC case for a wire with a rectangular cross-section
The circumstance that commonly a superconducting wire
experiences in a cable (made from more wires connected
in parallel) or a winding of an electromagnet is that of
AC transport under the simultaneous action of the AC field
produced by the currents in other wires. The transport
current and the magnetic field change in phase. Some
basic predictions for these conditions have been derived, but
only for the slab geometry [28] or that of an infinitely thin
strip [9, 29]. For HTS tapes, the empirical engineering
approach based on the fits of experimental data [30] is of
practical use; on the other hand, it does not explain why the
formulae seem to require modification in some cases. With
the help of the MMEV method, it is possible to visualize
the movement of flux fronts as the boundaries between zones
with different current densities. In figure 3 is shown the
evolution of current density distribution in rectangular crosssection of a superconducting conductor with rectangular crosssection (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 other
methods [22]. Because the boundaries between positive and
negative current density, as well as the shape of the current-free
core, are clearly determined, one can easily understand why
the predictions based on the assumption of a one-dimensional
flux penetration—where these boundaries are planar—would
not lead to satisfactory agreement with experiment. The
predictions of our calculations, performed under assumption
of magnetic-field-independent critical current density, have
been compared with experimental AC loss data obtained on
Bi-2223/Ag multifilamentary tape. In figure 4 is the result
S63
F Gömöry et al
0.14
100
0.12
0.1
P [W/m]
k
2
Γ=2πQ/(µ0Ia )
10
1
0.08
0.06
Experiment
0.04
Minimum Magnetic Energy Variation
(MMEV)
0.02
0,1
0
0,1
0
50
100
Phase difference [degrees]
150
1
Ia/Ic
Figure 4. AC loss in Bi-2223 tape transporting AC current with the
amplitude Ia under simultaneous action of AC magnetic field with
the amplitude Ba . The loss is shown in terms of the loss factor,
i.e. the total AC loss divided by the electromagnetic energy, that is
proportional to Ia2 , on the AC current amplitude Ia (normalized to
Ic ) at the condition that Ba increases in proportion to Ia . Theoretical
curves (lines) calculated for the proportionality coefficients k in the
formula Ba = wk µ0 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 by
squares.
of this comparison plotted as the normalized loss (sometimes
called the loss factor) versus the normalized current, assuming
that the applied AC field with the amplitude Ba increases in
proportion to the AC current amplitude, Ia . In other words,
Ba = wk µ0 Ia , where w is the tape width and k is the constant
of proportionality. This is the condition representing the most
important practical cases, with k 1 characterizing the
case of prevailing self-field (as in the transmission cable) and
k > 1 corresponding to the situation met in an electromagnet
winding. An interesting feature of the theoretical prediction is
that, at low excitations, the loss factor always increases with
the first power of the current (thus also the first power of the
applied field), then in turn predicts a universal loss dependence
of a superconducting device at low energizing current as ∝Ia3 .
Save for the very low k, i.e. the nearly self-field case, the
experimental data obey this prediction quite well.
Another interesting situation to investigate in the AC–AC
case is the cyclic change of transport current and magnetic
field with a certain phase shift between them. This could
happen in some important applications like the three-phase
transmission line or a transformer winding. We have employed
both of the simulation methods mentioned in section 3 to
check the AC loss measured on the same Bi-2223/Ag tape.
The results are presented in figure 5. A certain deviation
between the theoretical predictions based on two different
models for superconducting media is not surprising: the loss
predicted for the critical-state-like simulation performed by
the MMEV method is systematically below the experimental
data, while the prediction of the calculation starting from a
smooth non-linear E( j ) curve with the power exponent n = 25
is mostly above them. Interestingly, for all three curves the
absolute loss maximum is not found for the in-phase condition.
S64
Brandt's method
Figure 5. Dependence of AC loss on the phase shift between AC
transport current and AC magnetic field of the same frequency.
Simulations for a tape 4 mm wide, 0.2 mm thick with the critical
current of 38 A, exposed to 10 mT AC field when transporting
17.7 A of AC current, are compared with experimental data. The
maximum of dissipation obtained by two different simulation
methods as well as that found experimentally is clearly different
from the in-phase case.
We have found the shift of the loss maximum from zero
degrees reproducibly in both the experiments and simulations
performed 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 the
help of the MMEV method is that of a superconductor covered
by a ferromagnetic material. Several predictions have been
made that such a cover should decrease the AC loss [33, 34],
though none of these works has rigorously calculated the flux
penetration into hard superconductor put in such a composite
structure. Because the MMEV procedure of finding the
succession of flux front movements is equivalent to the criticalstate approach, the implementation of this method would allow
us to perform such a refinement. We have adopted the finite
element code FEMLAB to calculate the magnetic field in
the simulation box with dimensions ten times exceeding the
tape width. The consequence of placing a ferromagnetic strip
on a superconducting wire with rectangular cross-section is
illustrated in figure 6. The shape of the flux lines and hence
the flux penetration front completely changes with respect to
a bare superconducting wire. It seems that the ferromagnetic
sheath acts as a magnetic mirror, straightening the flux lines
inside the superconductor. This would influence the values
of penetration field Bp , the diamagnetic susceptibility χ0 and
also the shape of the χint
(y) dependence in formula (1). In
comparison to the previous work, performed using the ANSYS
code with constant permeability of iron [35], we have used the
non-linear dependence
µr =
µmax
2 + 1
1 + BBc
(6)
to approximate the non-linear magnetic permeability of the
ferromagnetic sheath. At this stage we have not been
AC loss in practical superconductors
2.5
2
χ"
1.5
1
0.5
0
0.000001
0.00001
Ba [T]
0.0001
0.001
Figure 7. Effect of the completeness of the ferromagnetic sheath
cover on the AC loss in an external magnetic field, calculated by the
MMEV method. Assuming no hysteresis in the ferromagnetic
material, the AC loss—expressed through the imaginary part of AC
susceptibility, χ —is reduced when the ferromagnetic material
better covers the superconducting strip.
Figure 6. Effect of a ferromagnetic cover on the behaviour of a strip
from a hard superconductor exposed to perpendicular magnetic field
equal to 50% of the penetration field. A dramatic change of the
shape of the flux penetration front as well as a significant reduction
of magnetic field inside the superconductor is clearly visible.
Non-linear but reversible permeability of the ferromagnetic material
was approximated by expression (6) with µmax = 1000 and
Bc = 0.1 T.
able to insert the hysteresis of ferromagnetic material in the
calculations, and this remains one of the big challenges for
this kind of simulations. As an illustration of the predictions
we have achieved, we present here the influence of the width of
the ferromagnetic cover on the AC susceptibility χ = χ0 χint
dependence on the applied magnetic field. One can expect
that, depending on how much surface will be covered by a soft
magnetic material with necessary thickness, the effect will be
more or less visible. This is indeed what we have found, as
shown in figure 7. With the narrowing of the ferromagnetic
cover, the susceptibility increases approaching that of the bare
superconducting wire.
We should underline that, because no hysteresis is
considered for the ferromagnetic material, this part of the
conductor is not accounted for in the total loss. Such an
assumption should be modified when a comparison with
experimental data will is carried out. Otherwise, a net
reduction of the AC loss results from the covering of the
superconductor by a ferromagnetic material.
4. Conclusions
The numerical simulation methods available at the present
time to model the electromagnetic behaviour of a hard
superconductor represent a significant step forward with
respect to the analytical models. Nowadays, two-dimensional
problems can be tackled successfully using the Brandt’s
method or the minimum magnetic energy variation (MMEV)
method. This means that the hysteresis loss in single-core
wires from hard superconductors of any shape can be predicted
at any combination of transport AC current and applied AC
magnetic field. In the case of arrays of filaments, the hysteresis
loss—i.e. that prevailing at low frequencies—can be calculated
as well.
When the coupling current flowing across the metallic
matrix in direction perpendicular to the filaments cannot be
neglected, the problem becomes three dimensional. For this
situation, the approach of representing the hard superconductor
as a normal conductor with non-linear current–voltage curve
should work as well, as the results achieved with the help of
physically justified simplifications have demonstrated [36, 37].
However, the requirements on computing power are still severe
and further development for 3D calculations is necessary.
Another field where significant progress is required is
the investigation of composites containing superconducting
filaments and ferromagnetic parts. Surprisingly, the simulation
methods for magnetic hysteresis in superconductors are now
better developed than those for ferromagnetic shapes when the
exact distribution of local magnetic field is regarded. It would
be interesting to see whether a general hysteresis simulation
method like the Preisach model [38] could be helpful in
resolving this problem.
Acknowledgments
Financial support of this work by the Science and Technology
Assistance Agency (contract APVT-20-012902) and the
NATO Science programme (grant PST.CLG 980001) is
acknowledged.
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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 under
the simultaneous action of ac transport
current and ac magnetic field shifted in
phase
M Vojenčiak1,5 , J Šouc1,6 , J M Ceballos2 , F Gömöry1, B Klinčok1 ,
E Pardo3 and F Grilli4
1
Institute of Electrical Engineering, Centre of Excellence CENG, Slovak Academy of
Sciences, Bratislava, Slovak Republic
2
Industrial Engineering School, University of Extremadura, Badajoz, Extremadura, Spain
3
Grup d’Electromagnetisme, Departament de Fı́sica, Universitat Autònoma de Barcelona,
08193 Bellaterra, Barcelona, Catalonia, Spain
4
Superconductivity Technology Center, Los Alamos National Laboratory, Los Alamos,
NM 87545, USA
5
University of Žilina, Žilina, Slovak Republic
E-mail: eleksouc@savba.sk
Received 28 November 2005, in final form 7 February 2006
Published 7 March 2006
Online at stacks.iop.org/SUST/19/397
Abstract
Investigation of ac loss under the simultaneous action of the transport
ac current and the external ac magnetic field is of prime importance for
the reliable prediction of dissipation in electric power devices such as
motors/generators, transformers and transmission cables. An experimental
rig allowing us to perform ac loss measurements in such conditions, on short
(10 cm) tape samples of high-temperature superconductor Bi-2223/Ag, was
designed and tested. Both the electromagnetic and thermal methods were
incorporated, allowing us to combine the better sensitivity of the former and
the higher reliability of the latter. Our main aim was to see how the ac loss
depends on the phase shift between the transport current and the external
magnetic field. Such a shift could have different values in various
applications. While in a transformer winding, the maximum phase shift at
full load will probably not exceed a few degrees, in a three phase
transmission cable in tri-axial configuration it is around 120◦ . Therefore, we
explored the whole range of phase shifts from 0 to 360◦ . Surprisingly, the
maxima of dissipation did not coincide with zero shift as expected from
qualitative considerations.
1. Introduction
An understanding of ac loss is of prime importance when
electric power applications of superconductors are under
consideration.
In the past four decades, an extensive
knowledge has been gathered about the behaviour of wires
and windings under the action of external ac magnetic fields.
6 Author to whom any correspondence should be addressed.
0953-2048/06/040397+08$30.00
As a consequence, the ac magnetization loss is now quite
well understood as a result of the interaction between the
magnetic field and the composite superconducting wires. In
the case of ac transport current, the dissipation is controlled
by the same principles. However, the driving magnetic field
is generated by the transport current itself. Such self-field
loss (also called the ac transport loss) has attracted particular
interest in the last decade, oriented on the high temperature
superconducting (HTSC) tapes. Similarly, as in the case of
© 2006 IOP Publishing Ltd Printed in the UK
397
M Vojenčiak et al
ac magnetization loss, the theoretical understanding and the
experimental techniques are now quite well established for ac
transport loss studies.
However, in any power application, the superconducting
wire transports ac current and experiences an additional ac
magnetic field due to the currents in other wires at the same
time. Therefore, for the forecast of ac loss in superconducting
devices, the knowledge of dissipation under the combined
action of ac current and ac field is essential. Theoretical
predictions derived for the simplest case of an infinite slab
or strip [1, 2] have been refined for more realistic geometries
by numerical simulations [3]. The collection of reliable
experimental data is quite laborious, because there are many
pitfalls in the measurement procedures. Nevertheless, a
general consensus has been reached about the possibility
of determining the total ac loss, Ptot , taking into account
that the electromagnetic energy interacting with the sample
comes from two independent sources of energy: one power
supply provides the transport ac current to the sample
and covers the part of the dissipation called the transport
loss, PT , while another feeds the winding that produces
the external ac field, also balancing the magnetization loss
PM in the sample. An important conclusion of several
papers dealing with this issue is that the transport loss
and the magnetization loss can be determined separately by
electromagnetic measurements [4–8].
The advantages of using the electromagnetic measurement
technique for determining ac loss are its better sensitivity and
lower time consumption compared to the thermal method. On
the other hand, the thermal measurement was found to be
indispensable when the ac field is shifted in phase with respect
to the ac
√ current [9–11]. Such conditions, when the current
IT = √2 Irms cos(2π f t) combines with the magnetic field
Bext = 2 Brms cos(2π f t + ϕ), are met in the windings of
transformers, generators and power transmission cables. The
crucial question of the ac loss investigation for ϕ = 0 is:
could the ac loss at a given phase shift, ϕ , be deduced from
the value measured at zero phase shift? The experimental data
showing a 180◦ periodicity with a maximum at zero phase
shift, obtained on Bi-2223 tape transporting ac current with
an amplitude equal to the dc critical current [10] favour such
a hypothesis. Similarly, the analytical calculations for a bifilar
coil from Bi-2223 [12] found the maximum of dissipation at ac
current in phase with the ac field. In the case when this were
a general feature, one could develop an empirical formula of
the form Ptot (ϕ) = Ptot (0) (1 − k sin 2ϕ) where Ptot (0) can be
determined by the sensitive and time-effective electromagnetic
measurement for any value of current and field, and the only
feature to explore would be the dependence of the constant k
on Brms and Irms . On the other hand, the results of numerical
calculations for a coil geometry [13] as well as the recent
experiments on a single straight Bi-2223 tape raised some
doubts in this regard. As concluded in the latter work, the
change of total loss due to ϕ is difficult to see because of the
insufficient sensitivity of the thermal method.
This work presents the measured ac losses of a Bi2223/Ag
tape under the simultaneous action of transport current and
external magnetic field, shifted in phase. In particular, we have
determined the values of the phase-shift angle corresponding
to the maximum of the ac losses, for different combinations of
current and field amplitudes.
398
current leads
I
sample
Cu magnet
Figure 1. Schematic diagram of the transport current circuit and
external magnetic field circuit. The design of the current leads allows
us to change their position with respect to the Cu magnet in order to
achieve zero mutual inductance between both circuits.
The paper is organized as follows. Section 2 describes
the experimental set-up and the procedure for determining the
ac losses both with thermal and electromagnetic methods; the
thermal method is used to validate the electromagnetic one
(used later in this paper), which needs particular attention due
to the simultaneous presence of transport current and external
field, shifted in phase. Section 3 contains the experimental
results of the ac losses measured with the electromagnetic
method. Section 4 contains a comparison of the experimental
results 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 an
electromagnetic method, the ac loss in√a superconducting wire
transporting the ac current IT = √
2 Irms cos(2π f t) while
exposed to the ac magnetic field Bext = 2 Brms cos(2π f t +ϕ).
In particular, the dependence on the phase shift ϕ between IT
and Bext has been investigated. The electromagnetic method
was chosen because of the known limitations of the thermal
method that nevertheless was used as a reference. To carry out
the electromagnetic measurement, one has to resolve two main
problems:
(1) An independent supply of current into the sample and
in the ac field magnet winding is necessary to keep the
values of Irms and Brms constant while changing the phase
difference.
(2) The distinction between the dissipation covered by the
power supply for IT and the one delivered by the
energizing system of the ac magnet is necessary to avoid a
double count of dissipation in loss registration.
2.1. Supply of ac transport current in ac applied field shifted
in phase
The independent operation of two power supplies was
guaranteed by the design and construction of the apparatus
shown in figure 1. The racetrack shaped magnet made of
copper wire was used to generate the ac external magnetic
field. The construction of the transport current leads supplying
the sample allow us to change their relative position with
respect to the Cu magnet winding. Thanks to this concept,
the accomplishment of zero mutual inductance between both
circuits was possible. In practice this was achieved by
searching for that particular position of the loop, composed
of the current leads together with the sample, at which no
ac loss in Bi-2223/Ag tape
a
b
Bext
2.E-05
Utc [V]
2.E-05
IT
1.E-05
8.E-06
4.E-06
wiring for transport
loss measurements
differential
thermocouple
0.E+00
0
0.2
0.4
0.6
0.8
1
PT [W/m]
Figure 2. (a) Simplified set-up for the calibration of the thermal method by the standard electromagnetic measurement of the ac transport loss
in self field, (b) calibration curve—thermocouple dc voltage in dependence on the ac transport loss measured by the standard electrical lock-in
method.
voltage is induced in it at any value of external ac magnetic
field. In such a configuration, no influence of the magnetic
field on the transport current (and no influence of the transport
current on the magnetic field) occurs regardless of the phase
shift between them. Then, after setting of the desired constant
transport current and constant magnetic field, the measurement
of the ac loss dependence on the phase shift between them can
be carried out without the necessity of adjusting the settings of
the power supplies.
The multifilamentary Bi-2223/Ag tape sample7 with
critical current Ic = 38 A was immersed in liquid nitrogen
during the measurements. A 2-channel generator allowing
us to set the phase shift between two sinusoidal signals of
adjustable amplitude but with the same frequency was used.
After being amplified by a two-channel audio amplifier, these
signals were used for generating the ac external magnetic field
and the ac transport current flowing in the sample, respectively.
The current to the Cu magnet was delivered from the amplifier
output directly. To achieve the required amplitude of the
transport current, a toroidal transformer connected to the
second output of the amplifier was used. The measurement
was carried out at the frequency f = 72 Hz both for the
transport current and the external magnetic field. The magnetic
field was oriented perpendicular to the wide face of the sample.
The Rogowski coils were used for measurement of both the
current of the Cu magnet and the sample transport current.
To improve the stability of the impedances of the individual
circuits and to reduce the drift of the phase difference in time,
the transport current leads including transformer as well as the
Cu coil were immersed in a liquid nitrogen bath. Two doublechannel lock-in voltmeters were used in the apparatus, one
detecting the sample current and voltage, and the second one
for the measurement of the magnetization loss. The values of
all of the electromagnetic quantities considered in this work are
rms.
2.2. Thermal method
The measurement set-up for the thermal method, together
with the calibration curve is illustrated in figure 2. The
basic principle is described in [10]. For thermal insulation
of the sample, two polyethylene foam blocks were used. A
7
Australian Superconductor No 2001-3-A/MF.
differential method using two E-type thermocouples connected
in series was used to probe the increase of the sample
temperature due to ac losses. One thermocouple was placed on
the sample surface isolated by Teflon tape, and the second one
was immersed in liquid nitrogen to serve as a reference. The
thermocouple wires were tightly twisted to reduce the voltage
induced by the external ac magnetic field. The calibration
consists of measurement of the thermocouple voltage Utc by
a Keithley 2700 voltmeter as a function of the transport current
loss PT . The standard electrical measurement of PT in transport
conditions (pure self field) was used for calibration using the
lock-in technique.
The thermal method can only be used in a limited range
of currents and fields. To achieve a measurable increase of
the sample temperature, an ac loss exceeding 0.1 W m−1 is
required in our set-up. The upper limit was 1 W m−1 , when
the decrease in the critical current due to the temperature rise
becomes significant. After calibration, the measurement of the
total ac loss Ptot dependence on the phase shift between ac
external magnetic field Bext and ac transport current IT can be
carried out. The results of this measurement are presented in
section 2.3.
2.3. Electromagnetic method
In the electromagnetic measurement, two loss components
have been determined independently. In the following, the
parts of the dissipation covered by the amplifier feeding the
sample with ac current and the one supplying the current in the
Cu magnet will be called transport loss (PT ) and magnetization
loss (PM ), respectively.
The approach of determining the loss from the point of
view of the energy source [14, 15] is very profitable. As shown
in several studies [16, 17], the total loss Ptot is obtained as the
sum of the losses covered by individual sources of delivered
power:
Ptot = PT + PM .
(1)
In our procedure, PT and PM were determined separately from
the signals registered by the pair of taps and the pick-up coil,
respectively, using the lock-in technique. The measurement
set-up is illustrated in figure 3.
The procedure to evaluate PT and PM was as follows.
Transport loss was measured with the pair of voltage taps
399
M Vojenčiak et al
IM
2 channel audio
amplifier
2 channel generator
ref out
transformer
Rogovski coil M
B ext
IT
loop for transport
loss measurement
LN2
Rogovski coil T
4
3
∆
Pick-up coil and compensation coil
for magnetization loss measurement
2
1
∆
chA
chB
Lock-in M
ref in
∆
chA
Lock-in T
∆
chB
ref in
Figure 3. Measurement set-up for transport loss and magnetization loss measurement by the electromagnetic method as a function of the
phase shift between the transport current and the external magnetic field. Two types of operational amplifier are used: the amplifiers numbered
1, 2, 3, and 4 have variable gain, and are used for fine cancellation of the unwanted signals in the differential units indicated by the symbol
delta.
separated by a distance L = 0.05 m. The wires leading the
signal from these contacts were first guided in the transversal
direction to the tape axis to the distance of 1 cm, then bent by
90◦ and put together forming a loop with the plane oriented
parallel with the external magnetic field and perpendicular
to the wide side of the sample. The value of the transport
current was measured by a Rogowski coil (Rogowski coil T in
figure 3) connected to channel A of the lock-in (Lock-in T in
figure 3). This channel was also used to set the reference phase
of channel B, where the voltage signal was connected. The
phase setting is necessary because only the voltage in phase
with the transport current represents the loss voltage, UTre .
For the faultless determination of transport loss in the
presence of an external ac magnetic field shifted in phase, the
voltage induced in the signal wires by the external ac field
should be zero. This was not required when the transport
current and external magnetic field were in phase. In that case,
the induced signal is perfectly out of phase with respect to
the transport current, and thus it does not interfere with the
loss voltage UTre . However, in our experiments with phase
shifts, any signal induced by the ac field will add a component
that is not distinguishable from the true UTre signal. The
reduction of the false signal was attained using two steps.
First, a coarse reduction by adjusting the position of the signal
wires; therefore, a fine reduction by using the voltage derived
from another Rogowski coil (Rogowski coil M in figure 3)
coupled with the current supplied to the ac magnet. Properly
adjusted by the wide band operational amplifier with variable
gain OA1 (in figure 3 indicated by 1), the correction signal
is subtracted from the measured voltage. We found that a
single zeroing procedure at Irms = 0 was valid for the whole
range of investigated external magnetic fields. After such a
compensation and after applying the desired transport current,
magnetic field and phase shift, the in-phase component of
the signal measured by channel B of the lock-in is the loss
voltage UTre . To increase the measurement sensitivity, the
inductive part of the measured voltage (induced by the self
400
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 operational
amplifier OA2 (in figure 3 indicated by 2), was subtracted from
the measured loop signal. The loss voltage UTre was used for
evaluation of the transport loss according to the formula
PT = Irms UTre /L .
(2)
To measure the magnetization loss, the pick-up coil
was used (see figure 3). A compensation coil of the same
dimensions but wound in the opposite direction is connected to
improve the sensitivity. The magnet current IM was measured
by the Rogowski coil M connected to channel A of the second
lock-in. Also, the phase setting for both of the channels of
this lock-in is derived from this signal. The pick-up loop
system was calibrated using the superconducting sample with
known magnetization loss measured by the calibration free
method [15]. Magnetization loss is then determined by the
formula:
PM = C IM UM /l
(3)
where C is the calibration constant, IM is the rms value of the
magnet current, l is the length of the sample and UM is the
part of the pick-up coil voltage, which is in phase with the
magnet current. This voltage is measured by the input channel
B of the second lock-in (Lock-in M in figure 3). The series of
corrections similar to the one described for the transport loss
measurement was used. The signal derived from the transport
current was amplified by the operational amplifier OA3 with
variable gain and subsequently used to cancel out the false
signal induced in the pick-up coil by the transport current.
The gain adjustment carried out at Brms = 0 was sufficient
to achieve the signal correction proper for the whole range
of ac fields. Another compensation signal derived from the
Rogowski coil M and adjusted by OA4 was used to reduce the
out-of-phase part of the measured signal.
ac loss in Bi-2223/Ag tape
0.4
0.03
0.02
0.01
0.2
0
24.8 A, 5mT
17.7 A, 5mT
11.8 A, 5mT
0.04
0.6
PT [W/m]
P [W/m]
0.05
Ptot,thermal
Ptot,elmag
PT
PM
0.8
0
60
120
180
240
300
360
420
0.00
-180
480
-120
-60
0.15
60
120
180
240
120
180
240
120
180
24.8 A, 10mT
17.7 A, 10mT
11.8 A, 10mT
0.12
PT [W/m]
Figure 4. Comparison of total ac loss measured by the thermal (full
squares) and the electromagnetic (empty squares) method at a
transport current of 23.5 A and a magnetic field of 26.2 mT.
Transport loss (triangles) and magnetization loss (diamonds) are also
shown.
0
phase shift [deg]
phase shift [deg]
0.09
0.06
0.03
2.4. Test of the electromagnetic method by the thermal method
0.00
-180
-120
-60
0
60
phase shift [deg]
0.30
0.25
PT [W/m]
The electromagnetic measurement of ac loss described in
the previous section represents a complex task. To confirm
the correctness of the suggested experimental procedures, the
results have been checked by the thermal method. The result,
as shown in figure 4, was more than satisfactory. For the
sake of this comparison, the tape was exposed to the external
magnetic field Brms = 26.2 mT when carrying the ac transport
current Irms = 23.5 A. The ac loss measurement was carried
out in the whole phase shift range with 10◦ step. In the figure,
the dependences of loss constituents PT and PM together with
Ptot = PT + PM are shown. Because the thermal method is
based on a physical principle completely different from the
electromagnetic method, the excellent coincidence shown in
figure 4 confirms the correctness of the results obtained by the
electromagnetic measurement in our experimental apparatus.
24.8 A, 15mT
17.7 A, 15mT
11.8 A, 15mT
0.20
0.15
0.10
0.05
0.00
-180
-120
-60
0
60
240
phase shift [deg]
Figure 5. Transport loss under the combination of the ac field and
the ac current shifted in phase, measured on Bi-2223 tape with
Ic = 38 A at 77 K, f = 72 Hz.
3. Experimental results
All data presented in this section were measured by the
electromagnetic method. In the following we present the
ac 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 a
function of the phase shift is displayed. The results obtained
with three different currents are gathered in one graph. Note
the different scale for the three plots in the figure. As one
can see, both the ac current as well as the ac field cause an
increase 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 . While
at Brms = 5 mT ϕmax is about 30◦ for all three considered ac
currents, at higher fields it moves from ∼45◦ at low currents
down to ∼15◦ at the highest current.
In figure 6, the magnetization loss PM as a function of
the phase shift is displayed. Although the effect of Brms is a
monotonous increase in the magnetization loss, the influence
of Irms on the magnetization loss is not always in the same
direction: at Brms = 5 mT, the rise of current increases
the loss at ϕ = 0, but reduces at ϕ = 60. On the other
hand, at Brms = 15 mT the magnetization loss is reduced
by the transport current at any phase shift. Interestingly, at
Brms = 10 mT and the phase shift ranging from 0 to 30◦
a non-monotonous dependence of the ac magnetization loss
on IT was observed. For zero phase shift this result is in
good qualitative agreement with the data published in [17],
where magnetization loss was measured for this case by an
electromagnetic method as well. In contrast to the transport
loss, the magnetization loss maximum is found at ϕmax < 0 for
all investigated combinations of the parameters Irms and Brms .
It is not obvious to deduce a general rule for the position
of the maximum in the total loss. From one side, the plot of
total ac loss Ptot in figure 7—which is nothing more than the
sum of the magnetization loss and the transport loss—shows
a common increase of the total loss with both the Irms and
Brms . However, for Brms = 5 mT the small movement of the
maximum with the increase of Irms is towards higher ϕ , while
at Brms = 15 mT the value of ϕmax reduces significantly with
increasing Irms . The summary of ϕmax dependence on the Irms
and Brms combination is illustrated in figure 8. One conclusion
is clear from these data: to find the maximum of total loss at
ϕ = 0 is more an exception than a rule.
401
M Vojenčiak et al
0.08
0.02
24.8 A, 5mT
17.7 A, 5mT
11.8 A, 5mT
0.06
Ptot [W/m]
PM [W/m]
0.03
0.01
24.8 A, 5mT
17.7 A, 5mT
11.8 A, 5mT
0.04
0.02
0.00
-180
-120
-60
0
60
120
180
0.00
-180
240
-120
-60
phase shift [deg]
0.20
Ptot [W/m]
PM [W/m]
0.06
0.04
0.00
-180
60
120
180
240
120
180
240
120
180
240
0.25
0.08
0.02
0
phase shift [deg]
24.8 A, 10mT
17.7 A, 10mT
11.8 A, 10mT
-120
-60
24.8 A, 10mT
17.7 A, 10mT
11.8 A, 10mT
0.15
0.10
0.05
0
60
120
180
0.00
-180
240
-120
-60
0
60
phase shift [deg]
phase shift [deg]
0.50
0.12
0.40
Ptot [W/m]
PM [W/m]
0.09
0.06
0.03
0.00
-180
24.8 A, 15mT
17.7 A, 15mT
11.8 A, 15mT
0.30
0.20
0.10
0.00
-180
-120
-60
24.8 A, 15mT
17.7 A, 15mT
11.8 A, 15mT
0
60
120
180
-120
-60
0
60
phase shift [deg]
240
phase shift [deg]
Figure 6. Magnetization loss under the combination of the ac field
and the ac current shifted in phase, measured on Bi-2223 tape with
Ic = 38 A at 77 K, f = 72 Hz.
4. Comparison with numerical simulations
The most interesting feature found in our experimental
observations is that the maximum of the total loss does not
occur for the zero phase shift between the ac current and
the ac field. To check whether such behaviour could be
plausible, three independent techniques have been used for
numerical simulations. Because of the long tape length
and the compact arrangement of the filaments, one should
expect strong coupling currents between the filaments. As a
consequence, the tape electromagnetic behaviour is mainly that
of a single filament [18].
Two of the calculation techniques assume that the
superconductor’s electrical behaviour is described by means
of a non-linear resistivity, derived from the E(J ) power-law
E(J ) = E c (J /Jc )n . The current–voltage curve with n = 25
was used in the simulation in Matlab code based on the Brandt
method [19], founded on solving Maxwell’s equations in the
form of integral equations. A sinusoidal current is imposed
to flow in the cross-section of the tape divided into 60 × 60
superconducting elements of rectangular cross-section. Also,
402
Figure 7. Total ac loss under the combination of the ac field and the
ac current shifted in phase, measured on Bi-2223 tape with
Ic = 38 A at 77 K, f = 72 Hz.
the replacement of the superconductor by a media with nonlinear resistivity was used in two-dimensional finite element
simulations with the commercial software Flux3D [20]. This
technique allows us to obtain detailed information about
the current density and magnetic field distributions inside
conductors, as well as to compute the ac losses. The sinusoidal
transport current is imposed by means of a current source,
whereas the magnetic field (which has variable phase with
respect to the current) is imposed by means of appropriate
conditions for the magnetic vector potential A on the domain
boundary [21].
For these two techniques, the ac losses at a given
frequency f , expressed in W m−1 , are computed as follows:
PSC = f
0
1/ f
J · E d S dt
(4)
S
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 means
of the minimum magnetic energy variation (MMEV) method.
This technique was introduced in [23, 24] for cylinders in a
ac loss in Bi-2223/Ag tape
a 50
30
ϕmax [deg]
40
ϕmax [deg]
b 50
20mT
15mT
10mT
5mT
20
10
24 A
17 A
11 A
40
30
20
10
0
10
15
20
0
25
0
5
IT [A]
10
15
20
25
Bext [mT]
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.
0.30
experiment 20mT
experiment 15mT
experiment 10mT
experiment 5mT
MMEV 20mT
MMEV 15mT
MMEV 10mT
MMEV 5mT
Brandt 20 mT
Brandt 15 mT
Brandt 10 mT
Brandt 5mT
Flux3D 20 mT
Flux3D 15 mT
Flux3D 10mT
Flux3D 5 mT
IT = 17,7 A
Ptot [W/m]
0.25
0.20
0.15
0.10
0.05
0.00
-30
0
30
60
90
120
150
180
phase shift [deg]
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 in
Flux3D 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 geometry
in a transverse magnetic field or transporting current [25, 26].
For those situations, the current distribution can be found
by magnetic energy minimization, while for the case of
ac transport in alternating applied field it is necessary to
use Prigozhin’s minimization principle [27]. The numerical
procedure for calculating the current distribution and the ac
loss is mainly the same as in [28], where we assumed a uniform
critical current density Jc . The main difference is that now the
starting situation for the initial stage (first increase of current
from 0 to the √
maximum) is the case of only magnetic field
with Bext =
2 Brms cos ϕ , calculated by magnetic energy
minimization. The rest of the time evolution is calculated by
MMEV as in [28], obtaining a stationary cyclic state beyond
the end of the first reverse stage ( IT decreasing from the
maximum to the minimum).
In figure 9, the dependences of the total loss Ptot on phase
shift obtained by simulations are compared with experimental
data achieved by the electromagnetic method. The results of
two numerical methods based on the smooth current–voltage
curve are systematically above the estimation calculated by the
MMEV. Experimental data fall in between these predictions. It
is also encouraging that the absolute values of the predicted
loss agree quite well with the experimental ones. The
observed agreement is surprising taking into account that none
of the simulations used here considered the critical current
dependence on the magnetic field. This probably also explains
why the theoretical curves exhibit weaker dependence of the
ac loss on the phase shift compared with the experimental
ones. Anyway, all three methods predict qualitatively similar
behaviour; in particular that the loss maximum is not found at
zero phase shift.
5. Conclusion
An experimental set-up for the measurement of ac loss
under the simultaneous action of the transport current and
the magnetic field shifted in phase was developed and
tested. Experimental results obtained by the electromagnetic
measurement are in excellent agreement with those obtained
by the thermal method. Moreover, the results obtained on
a standard Bi-2223/Ag tape are in good agreement with
theoretical predictions of three numeric calculations. Our
investigations clearly show that the maximum loss is not at
zero phase shift, and its position depends on the magnitude of
the current and field.
Acknowledgments
This work was supported in part by the APVT-20-012902
project, by the European Commission (Project ENK6-CT2002-80658 ‘ASTRA’) and in part by the US Department of
Energy Office of Electricity Delivery.
403
M Vojenčiak et al
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cond-mat/0510314
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 a
superconducting rectangular strip with
in-phase alternating current and applied
field
E Pardo1,2 , F Gömöry1 , J Šouc1 and J M Ceballos1,3
1
Institute of Electrical Engineering, Centre of Excellence CENG, Slovak Academy of
Sciences, 841 04 Bratislava, Slovakia
2
Grup d’Electromagnetisme, Departament de Fı́sica, Universitat Autònoma Barcelona, 08193
Bellaterra, Barcelona, Catalonia, Spain
3
Laboratorio Benito Mahedero de Aplicaciones Eléctricas de los Superconductores, Escuela
de Ingenierı́as Industriales, Universidad de Extremadura, Apartado 382, Avenida de Elvas s/n
06071 Badajoz, Spain
Received 3 October 2006, in final form 4 December 2006
Published 5 March 2007
Online at stacks.iop.org/SUST/20/351
Abstract
The case of ac transport at in-phase alternating applied magnetic fields for a
superconducting rectangular strip with finite thickness is investigated. The
applied magnetic field is considered to be perpendicular to the current flow.
We present numerical calculations assuming the critical-state model of the
current distribution and ac loss for various values of aspect ratio, transport
current and applied field amplitude. A rich phenomenology is obtained due to
the highly nonlinear nature of the critical state. We perform a detailed
comparison with the analytical limits and we discuss their applicability for
the actual geometry of superconducting conductors. A dissipation factor is
defined, which allows a more detailed analysis of the ac behaviour than the ac
loss. Finally, we measure the ac loss and compare it with the calculations,
showing a significant qualitative and quantitative agreement without any
fitting parameters.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The behaviour of a superconductor transporting an alternating
current or exposed to a magnetic field varying in time has
been widely studied since the early 1960s [1–4]. However,
the case of a simultaneous alternating transport current and
applied magnetic field remains unclear. This situation is
found, for example, in superconductor windings where each
turn feels the magnetic field of all the others. Windings are
present in many applications, such as ac magnets, transformers
and motors [5–8]. From a practical point of view, it is of
fundamental importance to understand, predict and, eventually,
reduce the energy loss (or ac loss) in the superconductor. The
study of the ac loss is also interesting for material science, as
it can be used to characterize superconducting samples [9–12].
Apart from its applications, the ac loss is the main alternating
0953-2048/07/040351+14$30.00
quantity under the simultaneous application of alternating
current and field and, thus, its study is significant in itself.
The superconductors suitable for electrical applications
are hard type II ones [13]. Nowadays there is a great scientific
effort in the development of silver sheathed Bi2 Sr2 Ca2 Cu3 O10
(Ag/Bi-2223) tapes and YBa2 Cu3 O7−δ (YBCO) coated
conductors, which are high-temperature superconductors, and
MgB2 wires [5, 8]. Superconducting tapes and wires have a
cross-section that is roughly rectangular or elliptical. In this
work, we will consider wires with a rectangular cross-section
(or rectangular bars). We also restrict our work to the situation
when the ac applied field is uniform and in phase with the
transport current.
Hard type II superconductors can be well described
by the critical-state model (CSM) proposed by Bean and
© 2007 IOP Publishing Ltd Printed in the UK
351
E Pardo et al
London [1, 2], which assumes that the magnitude of the local
current density cannot be higher than a certain critical value Jc .
For the situation of only transport current, the CSM was
first applied by London and Hancox in the early 1960s in order
to analytically describe simple geometries, such as infinite
cylinders and slabs [2, 14]. Later, an important step forwards
was made by Norris, who analytically deduced the current
distribution and the ac loss for an infinitely thin strip by means
of conformal mapping transformations [15]. The case of a strip
with finite thickness can only be solved numerically, as done by
several authors [16–19].
The first analysis of the CSM with only ac applied field
was done by Bean for a slab with an applied field parallel to
the surface [13]. The case of a thin strip with a perpendicular
applied field was analytically solved by Brandt et al [20]
following the Norris’ technique [15]. The current distribution
for a strip with finite thickness was numerically calculated by
Brandt [21] and Prigozhin [22], and the ac loss by Pardo et al
[23].
Concerning the case of simultaneous alternating transport
current and applied field, the most significant published
calculations within the CSM are the following. In the late
1970s, Carr analytically derived the ac loss for an infinite
slab in a parallel applied field [24]. In the 1990s, Brandt and
Zeldov et al analytically calculated the current distribution in
a thin strip using conformal transformations for the situation
where the transport current and the applied field increase
monotonically [25, 26]. Although these works provide
different formulae, they are actually equivalent4. Moreover,
Brandt studied the values of the transport current and applied
field for which these formulae are valid, finding that they are
not applicable for high fields and low currents [25]. From that
current distribution, Schönborg analytically calculated the ac
loss for a thin strip [27].
For strips with finite thickness, there are no published
works dealing with the simultaneous application of alternating
currents and magnetic fields in a superconductor in the CSM,
according to our knowledge. However, there are several
theoretical works assuming a relation between the electrical
field E and the current density J as E(J) = E c (|J|/Jc )n J/|J|,
where E c is an arbitrary value and n is a positive exponent. The
current distribution and the ac loss are calculated in [28–32]
and [29, 28, 33, 31], respectively, for several values of
the alternating transport current and applied field. These
published results are incomplete, not covering the whole range
of combinations of ac current and ac field. This contrasts
with the extensive experimental studies that have been done
for Ag/Bi-2223 tapes [34–37, 31, 38] and YBCO coated
conductors [39, 40].
The objective of this paper is to rigorously study the
response of a superconducting strip of finite thickness under
simultaneous application of an alternating transport current and
field, within the assumption of the CSM. The effect of three
main factors are considered: the aspect ratio of the crosssection and the amplitudes of the transport current and the
applied magnetic field. We also study the applicability of the
CSM to actual superconducting tapes and wires by comparing
the calculations with experiments.
4 It can be seen after doing some algebra using arcsin(i x) = i arcsinh(x),
arctan(ix) = i arctanh(x) and the definition of arcsinh and arctanh.
352
Ha
Δx
y
I
x
b
a
Figure 1. Sketch of the tape cross-section and division into elements
for 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, we
present the numerical method used for the calculations and we
discuss some general features. The results and their discussion
are presented in section 3. In section 4, the comparison with
data measured in a high-temperature superconducting tape is
reported. Finally, in section 5 we present our conclusions.
2. Numerical method and general considerations
Let us consider an infinitely long superconductor along the z
axis with a rectangular cross-section with dimensions 2a × 2b
in the x and y directions, respectively, figure 1. The origin of
coordinates is taken in the centre of the strip. We study here the
situation where the superconductor carries a sinusoidal timevarying current I (t) = Im cos ωt simultaneously immersed in
a uniform in-phase ac applied field Ha (t) = Hm cos ωt in the
y direction. It is shown below (sections 2.3 and 2.4) that our
results are not only independent of ω but also of the specific
time waveform of I and Ha , similar to the case of only transport
current or magnetic field [13, 2].
In our calculations, we will consider that first I and Ha are
increased from zero to their maximum, starting from the zerofield cooled state of the superconductor. We call this process
the initial stage. Following this stage, we consider the reverse
case, where the current and applied field are decreased from
Hm and Im , respectively, to −Hm and −Im . Next, the applied
field and current are increased back to their maximum, closing
the 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 with
constant critical-current density Jc [1]. The CSM corresponds
to assuming a multivalued relation of electrical field E against
current 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 strip
along the z direction, the current density and the electrical field
inside the superconductor are also in the z direction and they
can be considered as the scalar quantities J and E , respectively.
Although in principle |J | in the CSM can be lower than Jc , in
a superconducting strip J only takes the values 0 or ±Jc [42].
Let us start with the introduction of the main features
of the current distribution in the initial stage for the case of
transport current only (i.e. Ha = 0) or when only the magnetic
field is applied ( I = 0).
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 criticalstate model with Ha = 0 and uniform Jc is detailed
in [15, 43, 19]. In the initial stage, for any I > 0 lower than
the critical current, Ic = 4ab Jc , there exists a zone with J = 0
surrounded by another one with J = Jc . The region with
J = 0 is usually called the current-free core. In this zone, the
electrical field is zero because in the CSM E(J = 0) = 0.
With increasing I , the region with J = Jc monotonically
penetrates from the whole surface inwards and the current-free
core 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 a
superconducting strip that is not transporting any net current
is described in [21, 22]. For this case, the current distribution
is antisymmetric to the yz plane. In the initial stage with
Ha > 0, there is a zone with J = Jc in the right half and
another one with J = −Jc in the left half expanding from
the surface to a current-free core between them. Throughout
this paper, we call the border between regions with different
J the current front. It is important to notice that in the current
fronts J vanishes and, then, so does E . With increasing Ha , the
cross-section of the current-free core shrinks until it becomes
a point at (x, y) = (0, 0) at the characteristic field Hp , that
is called the penetration field. At fields higher than Hp , the
current distribution is the same as for Ha = Hp .
When we now consider the simultaneous action of an
applied magnetic field on a superconductor transporting a
nonzero current, one can expect that the qualitative behaviour
of the current distribution is similar to that for only transport
current or applied field. However, for some situations of I and
Ha the current distribution presents a different behaviour. We
discuss 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 Sanchez
et al [46–48], the distribution of current density for a
superconductor assuming the critical-state model is such that
it minimizes a certain functional. The functionals introduced
in [22, 41, 44, 45] are equivalent, whereas in [46] the magnetic
energy is proposed as the quantity to be minimized. As shown
in [22, 41], the principle of minimization of the functional,
F , can be derived from fundamental considerations. In
appendix A, we demonstrate that the minimization of F is
equivalent to minimizing the magnetic energy provided that
in the initial stage the current front penetrates monotonically
from the surface inwards. Some of the situations presented in
this paper do not satisfy this condition; therefore we use the
minimization of F as follows.
Let us consider the case of an infinitely long superconductor extended along the z direction carrying a transport current I
and immersed in a uniform applied field in the y direction Ha ,
figure 1. With this geometry, the current density is in the z direction and, therefore, so is the vector potential A if we assume
the gauge ∇ · A = 0. Then, we can regard these quantities as
scalar. Following the notation of Prigozhin [41], the current at
a certain time distributes in such a way that it minimizes the
functional
F [J ] =
J (r)A J (r) d S − J (r) Â J (r) d S
S
S
+
J (r)[Aa (r) − Âa (r)] d S,
1
2
(1)
S
with the constraints
I =
J (r) d S
(2)
S
|J | Jc ,
(3)
where S is the superconductor cross-section, A J is the vector
potential created by J , Aa is the vector potential from the
external field, and the quantities with hat correspond to those
at the previous discretized time point. For infinitely long
geometry, A J can be calculated from
μ0
A J (r) = −
J (r ) ln (y − y )2 + (x − x )2 d S . (4)
4π S
Defining the current density variation δ J ≡ J − Jˆ, we obtain
from equations (1) and (4) that the current density which
minimizes the functional F also minimizes the functional F ,
defined as
F [δ J ] ≡ 12
δ J (r)δ A J (r) d S + δ J (r)δ Aa (r) d S, (5)
S
S
where δ A J is the vector potential created by δ J and δ Aa ≡
Aa − Âa .
2.3. Calculation of the current distribution
We calculate the current distribution by minimization of
F [δ J ] of equation (5) for each time as follows.
As done in [19, 23, 48], each superconducting strip is
divided into N = 2n x × 2n y elements with dimensions
a/n x × b/n y ; current density is assumed to be uniform in each
element. In order to obtain a smoother current front, we allow
the current density magnitude to have discrete values below Jc ,
that is, |J | = k Jc /m with k being an integer number from 1
to a maximum value m . As discussed in [19, 48], this reduces
the discretization error in our ac loss calculations but does not
contradict the CSM assumption. Indeed, as justified in [41],
the CSM is compatible with allowing |J | Jc , although for
the tape geometry, the local current density that minimizes F
always has the maximum magnitude, Jc [42]. As shown below
(section 3.1), our results only present |J | < Jc on the current
fronts, with the physical interpretation that the elements section
are partially filled with critical-current density. In this paper,
we use between N = 12 000 and 16 000 elements and m = 20
current steps.
Given a current Iˆ, applied vector potential Âa and a current
density Jˆ, we calculate the current density variation δ J if the
current is changed into I and the applied vector potential into
Aa , as follows.
First, we find the element with s J < Jc , being s =
sgn(I − Iˆ), where increasing the current density by J =
s Jc /m produces the minimum increase of F , increasing the
current by J ab/n x n y in this process. Then, we repeat
the procedure until the total current reaches I . Afterwards,
the algorithm makes a current redistribution. Elements are
353
E Pardo et al
found where changing the current density by J and −J ,
respectively, reduces the most F and |J | does not exceed
Jc ; the process is repeated until varying the current in any
pair of elements will increase F instead of lowering it. This
allows the creation of regions with current density opposite to
I . Setting the current in this way, we ensure that the constraints
of equations (2) and (3) are fulfilled. In this procedure, the time
does not play any role, so the resulting current distribution is
independent of the specific current (and field) waveform.
In appendix B, we discuss the fundamental aspects of
the minimization procedure, showing that it finds the correct
minimum of F .
The variation of F , F , due to a variation of current
I in the element j can be calculated from equation (5) and
Aa = −μ0 Ha x taking into account the division into elements
of the tape, with the result
F j =
N
δ Ik I C j k + 12 (I )2 C j j
k=1
− μ0 (Ha − Ĥa )I x j ,
(6)
where δ Ik is the current flowing through the element k induced
after the change of I and Ha , x j is the x coordinate of the centre
of element j , and C j k are geometrical parameters calculated in
appendix A of [48].
For simplicity, we consider a constant variation of I (and
Ha ) between different times inside each half cycle. In this
paper, 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 superconducting
strip is
Q = dt
J (x, y; t)E(x, y; t) dx d y,
(7)
where the current integration is performed in the returning
stage. From equation (10), we see that Q is independent on the
specific I (t) dependence, as long as I increases or decreases
monotonically with time in a half cycle. From this feature, it
is directly deduced that the ac loss due to a sinusoidal current
and applied field is independent of their frequency.
The vector potential can be easily calculated from J ,
obtained by means of the numerical procedure described in
sections 2.2 and 2.3. Then, we calculate ∂ I A at a certain time
k 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) yields
much more accurate results of ∂ I A than using finite differences
between consecutive time points. Indeed, according to the
mean value theorem, there must exist some current between
Ik−1 and Ik+1 where the derivative is exactly the right-side
part of equation (11). Equation (11) cannot be used at the
boundaries of a half cycle, I = ±Im , as ∂ I A is not continuous
there. Therefore, we use finite differences between k and k + 1
or 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 for
calculating ∂ I A(x0 , y0 ; I ) because it is a partial derivative
and, 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, just
taking (x0 , y0 ) as the centre of an element in the current-free
core or next to a flux front at the time k . For I = Ic and
I = −Ic , (x0 , y0 ) is approximated as that at the following and
previous time points, respectively5.
2.5. Monotonic penetration of current fronts
S
where the time integral is over one period.
As the electrical field inside the superconductor is in the z
direction, we obtain
E = −∂z φ − ∂t A,
(8)
where φ is the electrical scalar potential, ∂z φ ≡ ∂φ/∂z and
∂t A ≡ ∂ A/∂t . The electrical field and the vector potential have
zero components in the x and y directions, so that ∂φ/∂x =
∂φ/∂y = 0. Then, as for infinitely long conductors E does
not depend on z , ∂z φ is uniform in the whole conductor. The
quantity ∂z φ can be calculated taking one point where E = 0
and using equation (8), obtaining
∂z φ(t) = −∂t A(x0 , y0 ; t),
(9)
where x0 and y0 are the x and y coordinates at some point
where E = 0. For the critical-state model, E always vanishes
on 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 d I = ∂t I dt , we obtain
Im
Q=2
dI
Jret (x, y; I )
−Im
S
× [∂ I A(x0 , y0 ; I ) − ∂ I A(x, y; I )] dx d y,
354
(10)
In many practical situations, the current front in the initial
stage monotonically penetrates from the surface inwards
with increasing I and/or Ha . Some examples are a strip
and arrays of strips with only applied field or transport
current [20, 21, 15, 19, 49, 47, 48], or a cylinder in uniform
Ha [50, 51]. In fact, this assumption has been taken for
calculating the current profiles for thin strips with simultaneous
applied field and transport current [25, 26], although, as
discussed below, it is only fulfilled for high current and low
applied field. A system with monotonic penetration of current
fronts has special properties, as follows.
If a system presents monotonic penetration of current
fronts, the current distribution for all of the cycle, and thus all
the electromagnetic properties, can be calculated from those in
the initial stage [48]. The current distribution in the reverse and
returning stages, Jrev and Jret , are, respectively,
Jrev (I ) = Jin (Im ) − 2 Jin [(Im − I )/2],
(12)
Even though for I = ±Ic there is neither a current-free core nor a current
front, 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 the
time dependence. As E(x, y; t ) must be continuous with both the previous and
the following times, there could be two points with E = 0 for I = ±Ic . The
latter 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.
5
Current distribution and ac loss for a superconducting rectangular strip with in-phase ac and applied field
Jret (I ) = −Jin (Im ) + 2 Jin [(Im + I )/2],
(13)
6
where Jin is the current distribution in the initial stage .
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 be
skipped. Specifically, from equations (10), (12) and (13) and
following the same deduction as Carr for the pure transport
situation [43], it can be obtained that
Q = 4 Jc s(x, y) Akm − Am (x, y) dx d y,
(14)
(a)
(b)
S
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 of
Jrev . Equation (14) with s = 1 corresponds to that obtained
by Norris for the transport case [15] and with s = x/|x| it
corresponds to the magnetic one given by Rhyner [52].
Moreover, for monotonic current front penetration, Jin
minimizes the magnetic energy and, thus, it only depends
on 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 MEM
for calculating Jin and Q because this procedure requires a
single minimization for each Im and Hm value, whereas using
F minimization requires a large number, n t , of them. In
addition, for F minimization, the error due to the cross-section
discretization accumulates for each minimization, whereas not
for the MEM.
However, in our case, the condition of monotonic current
front movement is not always fulfilled. Then, in order to use
one single procedure, we apply F minimization for all of the
studied Im and Hm combinations.
3. Results and discussion
In this section, we present our results for the current
distribution and the ac loss and we discuss the existing
analytical approximations for low and high b/a aspect ratios.
We also introduce the dissipation factor = 2π Q/(μ0 Im2 ),
characterizing the loss behaviour better than the ac loss itself.
3.1. Current distribution
In the following, we present the current distribution for a
rectangular strip with aspect ratio b/a = 0.2, although the
numerical procedure gives accurate results for b/a between
0.001 and 100. We consider several situations of field and
current.
First, we study the case of low applied fields. As an
example, we plot the current distribution for Im /Ic = 0.8 and
Hm /Hp = 0.08 in figure 2, where Hp is the full penetration
field for a rectangular strip [53, 21]
a2
b 2a
b
Hp = Jc
arctan + ln 1 + 2
(15)
.
π b
a
b
The current profiles in figure 2 are qualitatively similar to those
for transport current [19] with the difference that the currentfree 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 equation
for Jret .
(c)
Figure 2. Current distribution in the initial stage for the low-field and
high-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 and
zero for the white region.
fronts monotonically penetrate from the surface inwards. Thus,
the current distribution for the whole cycle can be constructed
from that in the initial stage using equations (12) and (13). For
this case, the ac loss can be calculated using equation (14), so
that the evaluation of E can be skipped.
The most representative situation of the combined action
of 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 in
the reverse stage is not always a superposition of that for the
initial stage. Not even the returning stage is related to the
reverse one. However, as can be seen in the figure, the current
fronts for I/Im = 1 (and Ha /Hm = 1) in the reverse stage
corresponds to that for I/Im = −1 (and Ha /Hm = −1) for
the returning one, except some numerical deviation. Then, the
current distribution for the following reverse stage for a certain
transport current I (and applied field Ha ) is the same but with
opposite sgn with respect to those for the returning stage for
transport current −I (and applied field −Ha ), being current
distribution periodic in time after the first cycle.
The above specific case (figure 3) presents a current-free
core, but it is not always the case for higher Hm or Im . For
example, the current-free core is not present for Im /Ic = 0.6
and Hm /Hp = 1.2 (figure 4), as well as for any case with
Im = Ic , as shown in figure 5 for Im = Ic and Hm /Hp = 2. In
addition, for all of the cases with Im = Ic , the electromagnetic
history is erased at the end of one half cycle (figure 5). Thus,
for this current amplitude, the returning profiles correspond to
the reverse ones with inverted sgn of the current density, so that
the electromagnetic behaviour is simplified. Another issue is
that the current distribution for Im = Ic has only one boundary
between the zones of positive and negative current, whereas
there can be two or more for lower Im (figures 3 and 4).
For other aspect ratios, we found the same qualitative
behaviour as for b/a = 0.2 described above. As an example,
in figure 6 we present the current distribution in the returning
curve for b/a = 5. This corresponds to the situation of the
same strip as for figures 2–5 but with the applied field parallel
to the wide direction.
355
E Pardo et al
(a)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
(e)
(e)
(f)
(f)
(g)
(h)
(i)
Figure 3. Current distribution for b/a = 0.2, Hm /Hp = 0.72, and
Im /Ic = 0.6 at several instants of the ac cycle. ((a)–(c)) are for the
initial stage with I /Im = 0.2, 0.6, and 1, respectively, ((d)–(f)) are
for 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, and
1, 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 interesting
to compare the sheet current density K in a thin film
from [25, 26], where it is assumed that current fronts penetrate
monotonically, with our results for finite thickness. We
calculated K by integrating the current distribution over the
thickness. For this situation, [25, 26] distinguishes between
the low-field high-current regime, for which all current has
the same sgn, and the high-field low-current regime, when
current density with both sgns exists. These regimes appear
356
Figure 4. Current distribution for b/a = 0.2, Hm /Hp = 1.2 and
Im /Ic = 0.6 at several instants of the ac cycle. ((a)–(c)) are for the
initial 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 the
grey 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 ≡ 2 Jc b/π .
In figure 7, we present our numerical calculations of K
for the initial stage together with the analytical results for
a 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 all
Ha and I up to their maximum, and Hm /Hc = 6 and
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 regions
and −Jc in the grey zones.
Im /Ic = 0.9999 (b), as an example for the high-field lowcurrent case. As can be seen in figure 7(a), for low applied
fields all numerical results fall on the analytical curve within
the computation error, whereas for higher fields, figure 7(b),
there is only coincidence for the profiles corresponding to low
current penetration. The discrepancy for higher penetration
appears because the assumption of monotonic penetration of
current fronts is no longer valid for the analytical solution.
As can be seen in figures 4((a)–(c)), for high I/Ic there is a
recession of the zone with negative current density in favour
of that with positive current density, this effect being more
important for higher I . For simultaneous alternating Ha and
I , such a current front regression will always be present when
regions with both J = Jc and J = −Jc coexist, due
to the current penetration asymmetry. Thus, the thin film
approximation in [25, 26] is only strictly valid for the highcurrent low-field regime.
For alternating applied fields and transport currents, the
high-current low-field condition must be followed for all Ha
and I up to their maximum. Taking into account that for inphase applied field and transport current Ha = (I/Im )Hm and
using that the first-order Taylor expansion of tanh x for low x
is higher than tanh x , the high-current regime for alternating
conditions becomes
Im /Ic Hm /Hc .
(16)
Using a similar argument, it can be seen that the condition for
monotonic current front penetration of equation (66) in [26]
also reduces to equation (16).
We can also compare the numerically obtained current
distribution to the analytical solution for a slab in a parallel
field, for which the current fronts are planar [24, 26]. For
strips with high b/a in the high-field low-current regime, the
calculated current fronts approach planar ones (figure 6), the
approximation being better for higher b/a . This behaviour is
in agreement to the pure magnetic case [21]. However, for
the low-field high-current regime, current fronts are similar
Figure 7. Sheet current density K in the initial stage as a function of
x 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) at
several instantaneous I (and Ha ). The lines depict the thin strip limit
from [25, 26] and the symbols are for our numerical calculations. For
the numerical results, K is the integral of J over the sample
thickness.
to the ones for a thin strip with only transport current, which
are nonplanar [15, 19] and, thus, the slab approximation is no
longer valid.
We have performed numerical simulations for very high
applied fields, Hm > 5 Hp , and have shown that the
current fronts approach vertical planes for any aspect ratio, in
accordance with the slab approximation. This is because when
the applied field variation is much higher than the field created
by the variation of J , the first term of F in equation (5) can be
neglected. As Aa is proportional to x , F of the new induced
current density is independent of its y location, and the current
density profiles must be planar.
3.2. Total ac loss
First, we study the ac loss for several b/a aspect ratios and
their possible analytical approximations. For this purpose, we
present our results of the normalized ac loss q ≡ 2π Q/(μ0 Ic2 )
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 ratios
b/a = 0.001, 100 and 0.1, respectively. The aspect ratios
of b/a = 0.001 and 0.1 can be used to qualitatively describe
YBCO coated conductors and Ag/Bi-2223 tapes, respectively,
in a perpendicular field. An aspect ratio b/a = 100 is
representative for a parallel applied field. For all figures,
we consider Im normalized to Ic , while Hm is normalized
to Hc = 2 Jc b/π in figure 8 and to Hp from equation (15)
in figures 9 and 10. First, we present our results in this
conservative normalization for the sake of comparison with
published theoretical and experimental data.
The numerical error in the ac loss has been analysed using
several numbers of elements, current steps and time points,
showing an insignificant variation for the axis scale of all
figures below.
From figures 8–10, we see that the ac loss monotonically
increases with increasing either the current or the applied field
357
E Pardo et al
Figure 8. Normalized ac loss 2π Q/(μ0 Ic2 ) for b/a = 0.001 as a function of Im /Ic for several Hm /Hc , with Hc = 2 Jc b/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 thin
strip (calculated using equation (16) and [27]), and the dotted lines (blue) (for Hm /Hc = 10 and 20) correspond to the high-field limit for
slabs (equation (20)).
Figure 9. Normalized ac loss 2π Q/(μ0 Ic2 ) 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 from
equations (17)–(18) [24].
Figure 10. Normalized ac loss 2π Q/(μ0 Ic2 ) 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)).
358
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 with
Hm for constant Im (figures 8(b), 9(b) and 10(b)). The ac loss
for the low-current limit in figures (a) and the low-appliedfield limit in figures (b) is constant, corresponding to the pure
transport and pure magnetic case, respectively. This qualitative
behaviour is consistent with experiments for YBCO coated
conductors [39, 40] and Ag/Bi-2223 tapes [34–37, 31, 38].
As expected, the loss results for zero applied field and zero
transport current are the same to the results for pure transport
and pure magnetic situations calculated using MEM in [19]
and [23], respectively.
3.2.1. Analytical limits for the ac loss. First, we study the
validity of the analytical limits for thin strips and slabs in a
parallel applied field.
In figure 8, we compare our numerical results of
2π Q/(μ0 Ic2 ) for b/a = 0.001 (line plus symbols) with the ac
loss for an infinitely thin strip calculated by Schönborg [27]
(dashed line) from the sheet current distribution obtained
in [25] and [26]. Schönborg’s expression for Hm = 0
corresponds to the Norris formula for a thin strip with pure
transport current [15]. The thick continuous line plotted in
figure 8 separates the low-field high-current regime from the
high-current low-field regime, section 3.1.1. As can be seen
in figures 8((a), (b)), the ac loss for the low-field high-current
regime is well described by the analytical expressions for a thin
strip. However, there is a significant deviation for the highfield regime, increasing with increasing field or current. This is
because, as discussed in [25, 26], the current density formulae
for thin strips are only valid for monotonic penetration of
current fronts, which appears only for the low-field highcurrent regime, figure 2. The current front penetration deviates
more from the monotonic case for higher field and current, so
the formulae for thin strips are less applicable.
In figure 10, we plot our numerically calculated
2π Q/(μ0 Ic2 ) for b/a = 0.1 (line with symbols) together with
that for a thin strip (dashed line). In this figure, we can see that
the thin-film approximation is not valid for b/a for any case
except Im = Ic and low applied field.
It is also interesting to compare our numerical results to
the formulae for the ac loss obtained by Carr for a slab in
a parallel applied field assuming planar current fronts [24],
which in SI are
2π Q
πa 3
h 2m
i 1+3 2 ,
=
(17)
h m im
μ0 Ic2
3b m
im
2π Q
πa 3
i2
h m 1 + 3 m2 ,
=
(18)
im < h m 1
2
μ0 Ic
3b
hm
2π Q
πa
i m2
2πa
h
(1 − i m )(1 + i m + i m2 )
=
1
+
−
m
μ0 Ic2
b
3
3b
2πa 2 (1 − i m2 )
i
b m h m − im
4πb 2 (1 − i m )3
i
−
,
h m > 1,
(19)
3a m (h m − i m )2
where i m = Im /Ic and h m = Hm /(Jc a). The high-field limit
of equation (19) is
πa
i m2
2π Q
h
=
1
+
(20)
,
h m 1.
m
μ0 Ic2
b
3
+
In figure 9, we plot our numerical results of 2π Q/(μ0 Ic2 )
for b/a = 100 (line with symbols) together with those for a
slab calculated from equations (17)–(19) (dashed line). We see
that the above formulae for slabs agree well with the numerical
results for high fields and low currents, although they do not for
low fields and high currents. In figure 9, we also see that for
Hm much above Hp , Carr’s results approach the actual loss for
any current. These features can be explained from the current
distribution, discussed in section 3.1.1.
The Carr formula can also be compared to numerical
results for any b/a . In figures 8(a) and 10(a), we include the
high-field limit of the ac loss in a slab, equation (20), for the
highest values of Hm /Hp in those graphs (dotted lines). It can
be seen that the analytical limit of equation (20) approaches the
numerical results for high Hm for Hm /Hp 1 and Hm /Hp 5
for b/a = 0.001 and b/a = 0.1, respectively. Numerical
calculations for other b/a , such as b/a = 1, also agree with
equation (20) for high applied field amplitudes. This feature
can be explained as follows. For high Hm , the current fronts
are planar, like those for a slab, section 3.1.1. Moreover,
if Hm is high enough, the only relevant contribution to the
vector potential, and to E (equations (8) and (9)), is from
Ha for any aspect ratio. Then, the high-field limit for a slab
must be valid for any aspect ratio. In fact, equation (20)
can be easily deduced from equations (7)–(9) assuming that
∂t A ≈ ∂t Aa = −μ0 x∂t H a .
3.3. Dissipation factor
Usually ac loss under alternating field and current have
been studied as a function of Im and fixing Hm or vice
versa [34–37, 33, 31, 38–40, 27]. Here, we underline the
significance of the ac-loss dependence when simultaneously
increasing Im and Hm with both parameters proportional to
each other along the curve. This situation is found in actual
ac devices, such as an alternating magnet.
As explained below, for Im ∝ Hm we can see more details
of the ac loss behaviour if we plot Q normalized to Im2 instead
to Ic2 . Indeed, the quantity 2π Q/(μ0 Im2 ) ≡ is proportional
to the ac loss of a winding per the stored magnetic energy
averaged during the cycle duration. Thus, can be regarded as
a dissipation factor. Moreover, for only transport current
is proportional to the imaginary part of the self-inductance,
defined in [54], and for only applied magnetic field is related
to 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 to
Im as Hm /Hp = α Im /Ic , where α is a constant (line with
symbols). This figure shows that for the low Im (and Hm ) limit,
increases proportionally with Im (or Hm ), which corresponds
to a dependence proportional to Im3 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 the
ac 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 loss
always increases with Im , appearing as curves very similar to
straight lines with a slight change in the slope. However, for the qualitative behaviour of the loss with varying Im and α is
more evident.
The linear dependence of the dissipation factor with
Im (and Hm ) for the low-field limit is characteristic of the
359
E Pardo et al
Figure 11. Dissipation factor ≡ 2π Q/(μ0 Im2 ) 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 numerical
calculations, the dash–dot line corresponds to only transport current,
the dashed lines correspond to neglecting the effect of the transport
current, and the dotted lines correspond to the high-α approximation
from equations (22) and (23).
critical-state model, also found for the pure transport and pure
magnetic situations [19, 23]. This is because for low enough
Im , the current front is approximately parallel to the surface,
as well as the magnetic field in the region with nonzero current
density, similarly to a slab [23]. This means that the dissipation
factor will increase as Im (and Q as Im3 ) at low levels of
excitation in a superconductor winding of any shape or number
of turns. However, for strips with very small b/a , such as
b/a = 0.001, the linear dependence of with Im appears only
for very small Im , presenting for higher Im the Im2 dependence
typical for thin films [23].
For comparison, in figure 11 we also include the
dissipation factor for only transport current (dash–dot line),
extracted from the tables in [19] and interpolating for
intermediate values of Im when needed. As can be seen, for
low α the dissipation factor approaches that for only transport
for any Im . It is also interesting to consider for the limit
of high α and low Im /Ic , where the effect of the transport
current is negligible compared to that of the applied field. For
this situation, can be evaluated from the imaginary part
of the ac susceptibility χ using that for only applied field
Q = μ0 π Hm2 χ [55], with the result
2
(Im /Ic ) = 2π 2 Hp /Ic α 2 χ (Hm /Hp = α Im /Ic ). (21)
According to equation (21), we see that (Im /Ic ) is
proportional to χ (Hm /Hp ) and with increasing α it shifts
upwards as α 2 and to the left as α . Using equation (21) and
the χ (Hm /Hp ) calculated in [23], is plotted in figure 11 for
α = 2, 5, showing a good agreement with the numerical results
for low enough Im .
For a finer approximation for high α , we can consider the
following dissipation factor
(Im , Hm ) ≈
2π
[ Q α→∞ (Im , Hm )] ,
μ0 Im2
where Q α→∞ is an approximated ac loss as
360
(22)
Figure 12. Applicability conditions diagram for equations (20), (24)
and Schönborg’s formula [27] for a thin strip, b/a 0.001. In the
lined regions, the analytical limits error is below 10% compared with
our calculations. The areas in horizontal (red), vertical (blue) and
diagonal (black) lines correspond to equations (20), (24) and
Schönborg’s formula, respectively.
Q α→∞ (Im , Hm ) ≡ Q(Im = 0, Hm ) +
2aμ0 Hm Im2
.
3 Ic
(23)
The first term of equation (23) is the ac loss with only applied
magnetic field, whereas the second term is the high-field limit
for a slab, equations (20)–(23) subtracting the ac loss for
Im = 0. In figure 11, we plot the dissipation factor of
equations (22)–(23) for α = 2 and 5, obtained using the
tables of numerically calculated ac susceptibility in [23]. We
found that the approximation of equation (22) improves with
increasing α , almost overlapping our numerical results for
α 5.
For low b/a , such as b/a = 0.001 or lower, the ac loss
for α = 0 approaches the Norris formula for thin strips [15], if
Im is not very low [19]. For the high-α limit, we can obtain an
analytical solution of by inserting the formula for the ac loss
in a thin strip with Im = 0, [25], into equation (22), obtaining
I 2 2 Hc
2 Hm 1
Hm
Hm
=
+ c2
ln cosh
− tanh
, (24)
Hc 3
Im Hm
Hc
Hc
where Hc = 2b Jc /π . For intermediate α , can be
approximated from Schönborg’s formula for the ac loss in thin
strips [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 a
Hm – Im diagram. Such a diagram for thin strips (b/a 0.001)
is presented in figure 12, where the shaded areas show the
regions where Q or calculated from equations (20), (24)
or Schönborg’s formula [27] differ by less than 10% from our
numerical calculations. If a more strict error criterion is taken,
for example 1%, the applicability regions are considerably
smaller.
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 for
Ag/Bi-2223 tapes [34–37, 31, 33, 38] and YBCO coated
conductors [39, 40]. It is interesting to analyse in detail
figure 10 of [39]. It shows a comparison between the measured
ac loss in a YBCO coated conductor and the theoretical one
for a thin strip in the critical-state model, evaluated from
the current distribution in [25, 26]. It can be seen that the
measured ac loss lies below the thin strip approximation, in
agreement with our numerical results in figure 8. As discussed
in section 3.2.1, this is because the thin strip calculations
in [39, 27] are not valid for high applied fields. This shows
that our numerical calculations can be used to simulate the ac
loss in YBCO coated conductors.
In order to perform a more detailed comparison, we
measured the dissipation factor for a commercial Ag/Bi2223 tape with 37 filaments manufactured by Australian
Superconductor. The sample was of 8 cm length and 3.2 ×
0.31 mm cross-section with a critical current of 38 A in self
field at 77 K. The superconducting core cross-section was
roughly elliptical with dimensions 2a × 2b = 3.0 × 0.13 mm.
The measurements were performed at a frequency of 72 Hz
and a temperature of 77 K; the details of the experimental
technique are presented in [58]. We present the measured
results in figure 13 (dotted line with symbols) together with
numerical calculations for a rectangular strip with the same
thickness, width and critical current (solid lines). We notice
that for the theoretical curves we do not fit any parameter to
the measured ones. In figure 13, we label the curves with the
parameter 2a Hm /Im instead of α in order to avoid assuming
any model a priori for performing the measurements. Indeed,
α = (Hm /Hp )/(Im /Ic ) contains Hp for a rectangular strip in
the CSM, whereas the tape superconducting core can be either
of a different shape or it may not be successfully described
by the CSM. For comparison, in figure 13 we also include the
dissipation factor assuming an elliptical cross section for only
transport current [15] (dashed curve) and a negligible effect
of the transport current at 2a Hm /Im = 4.0 (dotted curve),
calculated using equation (21) and the data for χ for only
applied field in [56].
From figure 13, we see that the main qualitative features
of the measurements correspond to the behaviour for a strip
assuming the critical-state model, except close to Ic for high
2a Hm /Im . This can be explained from the magnetic field
B dependence of Jc , for which Jc decreases with increasing
|B|. Then, for higher Hm , Ic is lower and a normal resistive
current appears in the silver for Im < Ic (Ba = 0), adding a
certain contribution to . Figure 13 shows that there is a better
agreement between the measured and that for an elliptical bar
assuming the CSM than for the rectangular one, explained by
the overall shape of the tape superconducting core. Moreover,
the fact that the multifilamentary superconducting core behaves
as a single solid wire for any Hm suggests that for this
frequency the interfilamentary coupling currents in the tape
are saturated due to the high length of the sample [57]. This
contrasts with magnetic measurements with shorter samples,
for which the behaviour is clearly multifilamentary [59].
Figure 13. Calculated dissipation factor together with
experimental data from a commercial Ag/Bi-2223 tape. The lines
with open symbols correspond to measurements, the solid lines
correspond to numerical calculations assuming a rectangular
cross-section, and the dashed and dotted lines correspond to an
elliptical 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 study
for the current distribution and ac loss in a rectangular strip
transporting an alternating transport current I in phase with an
applied field Ha perpendicular to the current flow. We assumed
that the superconductor follows the critical-state model with
a constant Jc . With this assumption, we have developed
a numerical procedure which takes into account the finite
thickness of the strip. General features of the critical-state
model in such circumstances have been discussed. In order to
understand the macroscopic physical processes in this system,
we have performed extensive numerical calculations for several
aspect ratios and current and applied field amplitudes, Hm and
Im respectively. Finally, we have performed measurements
on Ag/Bi-2223 tapes to be compared with calculations.
Good qualitative and quantitative agreement without fitting
parameters has been found.
The results for the current distribution have shown a rich
phenomenology due to the highly nonlinear nature of the
electrical currents flowing in the superconductor. For low
Ha and high I , the current distribution is qualitatively similar
to the pure transport situation. Then, J at the reverse and
returning stages are a superposition of J in the initial one
(equations (12) and (13)). However, it is not the same for high
Ha or low I due to the nonmonotonic penetration of current
fronts. In general, the returning stage cannot be deduced from
that in the first reverse stage. The behaviour becomes periodic
only after the first cycle.
The ac loss Q has been accurately calculated for the
thickness-to-width aspect ratios, b/a = 0.001, 0.1 and 100,
in order to qualitatively describe YBCO coated conductors
and Ag/Bi-2223 tapes with applied fields in the transverse
direction (b/a = 0.001 and 0.1, respectively) and in the
parallel one (b/a = 100). Their current and applied
361
E Pardo et al
field dependence is in accordance with published measured
data for YBCO coated conductors [39, 40] and Ag/Bi2223 tapes [34–37, 31, 38]. We have shown that the ac
loss behaviour can be better characterized by means of the
dissipation factor = 2π Q/(μ0 Im2 ) studied as a function
of Im with Hm proportional to Im . We have measured the
dissipation factor in actual Ag/Bi-2223 tapes, obtaining a good
agreement with the calculations.
We have also presented a detailed study of the analytical
limits for Q and and their applicability. For thin samples
such as YBCO coated conductors, b/a 0.001, the current
profiles and the ac loss only approach those for the analytical
limit for thin strips [27] for the low-field high-current regime.
The thin film approximation is never valid for b/a ∼ 0.1 such
as for Ag/Bi-2223 tapes. We have also studied the slab limit,
finding that for the situation of a parallel field, b/a = 100, the
slab approximation is not valid for the transport-like regime
(low- Hm and high- Im ). However, the high-field limit for the
slab approximation can be used for any aspect ratio provided
that Hm is high enough.
current front penetration with increasing I , each δ Ji encloses
a current-free and field-free core, where the vector potential is
constant. Provided that n is high enough, δ Ji is nonzero in a
thin layer only so that the vector potential variation at time ti ,
δ Ai ≡ δ A J,i + δ Aa,i , is uniform in the layer. Therefore, F in
equation (5) becomes
F [δ Ji ] ≈ 12 δ Aci (Ii − Ii−1 ) + 12
δ Ji δ Aa,i ,
(A.2)
S
δ Aci
where
is the value of δ Ai 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 δ Ji , δ Aci
and the external
parameters. In order
to do this, we define
J¯i ≡ J − ij =1 δ J j , Ā J,i ≡ A J − ij =1 δ A J, j and Āa,i ≡
Aa − Aa,i , where Aa,i = Aa (t = ti ), and decompose the
following integral, γi , as
γi ≡ 12 d S J¯i ( Ā J,i + 2 Āa,i )
S
= 12 d S ( J¯i+1 + δ Ji+1 )
S
Acknowledgments
We acknowledge M Vojenčiak for valuable technical support in
the measurements and D-X Chen and A Sanchez for comments
on writing the paper. This work was supported in part
by 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 distribution
in the initial stage minimizes the magnetic energy, provided
that the current front penetrates monotonically from the surface
inwards and I is proportional to Ha (and Aa ). The latter
condition is always satisfied for in-phase I and Ha . A similar
demonstration for Ha = 0 can be found in [48].
For calculating the magnetic energy, we assume that
the transport current in the strip of figure 1 returns through
another identical one at a large distance D ( D a, b).
In the following, we consider that the returning strip is
centred at (x, y) = (D, 0) [48]. Using the general formula
for the magnetic energy
in an infinitely long circuit W =
(1/2) Sx y J (r)A J (r) + Sx y J (r)Aa (r), where Sx y refers to the
whole x y plane area, we find that the magnetic energy per strip,
W , is
W =
1
2
J (r)A J (r) +
S
J (r)Aa (r),
(A.1)
S
≈ γi+1 + 12 δ Aci+1 (2 I − Ii+1 − Ii )
+ 12 d S δ Ji+1 (2 Aa − Aa,i+1 − Aa,i+1 ).
ignoring constant terms for a fixed I and Ha . W of
equation (A.1) is independent of the position of the returning
strip.
We next demonstrate that if F is minimized at every time
point, the magnetic energy is also minimized [48].
If the current front penetrates monotonically, any physical
J (r) in the initial stage is a composition of differential δ Ji (r)
induced at each discretized time point i , for which
n F [δ Ji ]
of equation (5) is minimized. Thus, J (r) ≈
i=1 δ Ji (r),
where n is the number of time points. This decomposition
of J into δ Ji is exact when n → ∞. For monotonic
(A.3)
S
In order to reach the expression beyond the fourth line
in
we used, from equation (4),
the deduction above,
¯
S d S Ji+1 δ A i+1 =
S d S δ Ji+1 Ā J,i+1 . The approximation
symbol in equation (A.3) corresponds to assuming that the
region where δ Ji+1 exists is narrow enough to consider that
δ Ai+1 is constant there (with a value δ Aci+1 ). In this step,
we also took into account that the region where J¯i+1 = 0 is
contained in the flux-free zone of δ Ji+1 and, thus, δ Ai+1 is
uniform with value δ Aci+1 .
From the decomposition of the integral γi in equation (A.3), it is straightforward to see that
W ≈ γ1 + 12 δ Ac1 (2 I − I1 − I0 )
+ 12 d S δ J1 (2 Aa − Aa,1 − Aa,0 )
S
n 1
δ Aci (2 I − Ii − Ii−1 )
≈
2
i=1
S
362
× ( Ā J,i + δ A J,i+1 + 2 Āa,i + 2δ Aa,i+1 )
= γi+1 + d S J¯i+1 δ Ai+1
S
1
+ 2 d S δ Ji+1 (δ Ai+1 + 2 Aa − Aa,i+1 − Aa,i )
+
1
2
δ Ji (2 Aa − Aa,i − Aa,i−1 ) d S
(A.4)
S
because γn = 0. As we assumed that I and Aa are proportional
to each other, 2 Aa − Aa,i − Aa,i−1 = δ Aa,i (2 I − Ii −
Ii−1 )/(Ii − Ii−1 ). Inserting this into equation (A.4) and using
equation (A.2), we obtain
n
2 I − Ii − Ii−1
W =
F [δ Ji ]
(A.5)
.
Ii − Ii−1
i=1
From equation (A.5) we directly deduce that when
minimizing F [δ Ji ] for each time, W is also minimized
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 no
constraints. The arrows show a change of magnitude I in a system
variable and the lines are possible level curves of a function of the
two variables, F(I1 , I2 ).
because I and Ii are fixed external parameters.
n As the δ Ji that
minimizes F [δ Ji ] is unique, the J =
i=1 δ Ji minimizing
W is also unique.
Appendix B. Fundamental aspects of the
minimization procedure
In the following, we discuss the basis of the procedure in 2.2,
justifying why it finds the correct minimum of the functional
F .
The quantity to be minimized in our problem, F , is
a function of the current in each
N element i , Ii , with the
constraints |Ii | Jc ab and
i=1 Ii = I , where I is
the total current. Then, we have to minimize a scalar
function of N variables F(I1 , I2 , . . . , I N ), where the N -vector
(I1 , I2 , . . . , I N ) defines a state of the system. Ignoring the
constraints, 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 distance
lower than I to the minimum, when any change of magnitude
I in any Ii increases the F value. The CSM constraint
|Ii | Jc ab, just fixes a region in the N -space in which our
system must remain. As a consequence, if the unconstrained
minimum is outside the allowed region, the possible minimum
will be on the boundary of
that region.
N
The other constraint, i=
1 Ii = I , can be set by forcing
I to be positive until the total current is I and afterwards
impose that any possible change of I must be followed
by another one of −I . For our case, the first part of the
procedure may increase F , although its increase would be the
minimum possible, and the second one applies a correction
towards the lowest F situation. We can imagine that if we
apply a too large I variation between time points, we may
‘move’ our state too much in the ‘wrong’ direction and then
it would be difficult to correct it successfully. This effect
should not be important for small enough I variations between
consecutive times. Our results obtained for several numbers of
time points confirm this hypothesis, as we found a negligible
variation in the final results for a large enough number of time
points.
Until now, we justified that the procedure finds the closest
local minimum. In fact, from a physical point of view the
system must be in a minimum of F , whether it is a global
or a local one. What is important is that the system stays in the
appropriate minimum corresponding to the initial conditions
and the external parameters history. We ensure this by starting
from the zero-field cool state ( J = 0 everywhere) for I =
Ha = 0 and increasing I and Ha in small steps, so that
the former minimum moves slightly in the variables space
and the system state follows it. However, according to our
experience, the procedure finds the correct minimum even for
large variations in the external parameters, at least for the
specific cases of Ha = 0 and I = 0 [47, 48].
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17 1477
8th European Conference on Applied Superconductivity (EUCAS 2007)
Journal of Physics: Conference Series 97 (2008) 012176
IOP Publishing
doi:10.1088/1742-6596/97/1/012176
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: psuarez@unex.es
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.
c 2008 IOP Publishing Ltd
1
8th European Conference on Applied Superconductivity (EUCAS 2007)
Journal of Physics: Conference Series 97 (2008) 012176
IOP Publishing
doi:10.1088/1742-6596/97/1/012176
• 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:
PT = PN + Pmag + Ploc
(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.
2
8th European Conference on Applied Superconductivity (EUCAS 2007)
Journal of Physics: Conference Series 97 (2008) 012176
IOP Publishing
doi:10.1088/1742-6596/97/1/012176
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
iL (t ) = 2 I L cos(ωt − ϕi )
Short tape current
iS (t ) = 2 I S cos(ωt − ϕi )
Contact tap voltage outside the short tape
vCO (t ) = 2 VCO cos(ωt − ϕCO ) + H CO (t )
Contactless loop voltage outside the short tape
vCLO (t ) = 2 VCLO cos(ωt − ϕCLO ) + H CLO (t )
Contact tap voltage under the short tape
vCI (t ) = 2 VCI cos(ωt − ϕCI ) + H CI (t )
Contactless loop voltage under the short tape
vCLI (t ) = 2 VCLI cos(ωt − ϕCLI ) + H CLI (t )
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:
Px = vx (t ) iL (t )
•
nT
/ Lx
From the current and the voltage first harmonic RMS values:
Px = Vx I L cos(ϕ x − ϕi ) / Lx
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.
3
8th European Conference on Applied Superconductivity (EUCAS 2007)
Journal of Physics: Conference Series 97 (2008) 012176
IOP Publishing
doi:10.1088/1742-6596/97/1/012176
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.
4
8th European Conference on Applied Superconductivity (EUCAS 2007)
Journal of Physics: Conference Series 97 (2008) 012176
IOP Publishing
doi:10.1088/1742-6596/97/1/012176
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,
5
8th European Conference on Applied Superconductivity (EUCAS 2007)
Journal of Physics: Conference Series 97 (2008) 012176
•
IOP Publishing
doi:10.1088/1742-6596/97/1/012176
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.
6
8th European Conference on Applied Superconductivity (EUCAS 2007)
Journal of Physics: Conference Series 97 (2008) 012176
IOP Publishing
doi:10.1088/1742-6596/97/1/012176
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
7
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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
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
S
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: 0034924-289646, fax: 0034-924-289601, e-mail: psuarez@unex.es)
A. Álvarez is with the Electrical Engineering Department, University of
Extremadura, 06071 Badajoz, SPAIN. He is an IEEE Member (email:
aalvarez@unex.es).
J. M. Ceballos is with the Electrical Engineering Department, University of
Extremadura, Apdo 382, 06071 Badajoz, SPAIN (e-mail: jmceba@unex.es).
B. Pérez is with the Electrical Engineering Department, University of
Extremadura, 06071 Badajoz, SPAIN (e-mail: belenpc@unex.es).
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:
PT = PN + Pmag + Ploc + PFM
(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 contactless 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
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2
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.
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. 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.
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.
TABLE I CHARACTERISTICS OF TAPES UNDER TEST
1G Samples
Tape Reference
Superconductor
Fabrication Tech.
Matrix
Substrate
Coating
Thickness (μm)
Width (mm)
Ic(A)
InnoST
Bi-2223 (Pb)
PIT
Silver Alloy
--------230 ± 10
4.20 ± 0.10
95
2G Samples
American Superconductor
YBCO
MOD/RABiTS
----Ni-W alloy
Stainless Steel
150 ± 20
4.40 ± 0.15
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
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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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.
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.
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. 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. 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. 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, 2GFM2, 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
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.
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.
REFERENCES
SS/ Ag
IS < Ic
FM susbtrate
IS
YBCO layer
BS
BL
SS/ Ag
IL
IS > Ic
FM susbtrate
Ic
YBCO layer
IS - Ic
BS + BS-c
BL
IL
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.
[1]
P. Suárez , A. Alvarez, 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,” in Journal of Physics: Conference Series, vol. 97,
2008, 012058.
[2] B. Pérez, A. Álvarez, P. Suárez, D. Cáceres, J.M. Ceballos, X. Obradors,
X. Granados and R. Bosch, “AC losses in a toroidal superconducting
transformers,” in IEEE Trans. Appl. Supercond., vol. 13, 2003, pp.
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[8] L. Gianni, M. Bindi, F. Fontana, S. Ginocchio, L. Martini, E. Perini and
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