Superconductor strip in a closedmagnetic environment:
exact analytic representation of the critical state
Y.A. Genenko ∗, H. Rauh
Institut für Materialwissenschaft, Technische Universität Darmstadt, 64287 Darmstadt, Germany
Abstract
An exact analytic representation of the critical state of a current-carrying type-II superconductor strip located inside
a cylindrical magnetic cavity of high permeability is derived. The obtained results show that, when the cavity radius
is small, penetration of magnetic flux fronts is strongly reduced as compared to the situation in an isolated strip. From
our generic representation it is possible to establish current profiles in closed cavities of various other geometries too
by means of conformal mapping of the basic configuration addressed.
Key words: Superconductor strip, Magnetic shielding, Critical state
PACS: 74.25.Ha, 74.78.Fk, 74.78.-w, 85.25.Am
Relatively high AC losses in superconductor ca-
bles and strips present a substantial problem for
the implementation of superconductors in high-
frequency and low-frequency applications. Recently,
a suggestion for improving the current-carrying
capability of superconductor strips [1,2,3] and for
reducing AC losses in superconductor cables [4]
based on the idea of magnetic shielding of applied
fields as well as current self-induced fields was put
forth. AC losses in superconductor strips caused by
the latter type of fields are anticipated to greatly
decrease when the strips are exposed to suitably
designed magnetic environments [2,3]. Exact ana-
lytic representations of sheet current distributions
in superconductor strips located between two high-
permeability magnets occupying infinite half-spaces
were derived before [1,2]; these configurations al-
lowed to find the respective current distributions
for various other topologically open shielding ge-
ometries by application of the method of conformal
∗ Corresponding author.
Email address: yugenen@tgm.tu-darmstadt.de
(Y.A. Genenko).
mapping. Utilization of the latter tool for analyz-
ing sheet current distributions and AC losses in the
presence of topologically closed magnetic environ-
ments of practical interest requires corresponding
reference results. An establishment of such results
is the focus of the present communication.
We consider an infinitely extended type-II super-
conductor strip of width 2w located inside a cylin-
drical cavity of radius a in an infinitely extended
soft magnet of relative permeability µ, the symme-
try axis of this configuration coinciding with the z-
axis of a cartesian coordinate system x, y, z. Assum-
ing the thickness of the strip to be small compared
to its width, variations of the current over the thick-
ness of the strip may be ignored and, for mathemat-
ical convenience, the state of the strip characterized
by the sheet current J alone.
When magnetic flux penetrates the superconduc-
tor strip in the critical state, the distribution of the
sheet current is controlled by the pinning of mag-
netic vortices. In conformity with Bean’s hypothe-
sis [5], the sheet current adopts its critical value J
throughout the flux-penetrated regions of the strip,
Preprint submitted to Elsevier 18 August 2021
http://arxiv.org/abs/0704.1227v1
whereas the magnetic field component normal to the
strip vanishes in the flux-free regions of the strip.
Proceeding in the spirit of previous work [6], a dis-
tribution of the sheet current prevails with magnetic
flux penetrated from the edges of the strip, but with
the central zone −b < x < b of half-width b < w left
flux free. In this zone, the distribution of the sheet
current is governed by the integral equation [2]
dx′J(x′)
x− x′
x− a2/x′
= 0 (1)
together with the requirement that the total trans-
port current equals I. Here, q = (µ − 1)/(µ + 1)
means the strength of the image current induced by
the magnetic cavity.
In the limit µ ≫ 1, i.e. for q → 1, Eq. (1) has the
exact analytic solution
J(x) = J
a2 + b2
πs2(x)
s′(x)φ(s(x)), (2)
where
φ(s) =
c2 − s2
arctan
(h2 − b2)(c2 − s2)
(b2 − s2)(c2 − h2)
− arctan
h2 − b2
b2 − s2
b2 − s2
arctan
b2w2 − b4
a4 − b2w2
with s(x) = x(a2 + b2)/(a2 + x2), c = (a2 + b2)/2a
and h = w(a2 + b2)/(a2 + w2). Herein, K and Π
denote complete elliptic integrals of the first and,
respectively, third kind.
Sheet current profiles obtained from Eq. (2) for a
range of the geometrical parameters involved, with
a fixed value of b, are shown in Fig. 1. This exhibits
a flattening of the current profiles together with an
increase in the magnitude of the total current up to
saturation, when the radius of the magnetic cavity
is reduced, precisely as in the case of topologically
open magnetic cavities [1,2].
The half-width of the flux-free zone is controlled
by the total transport current in the strip and by the
geometry of the magnetic environment. An implicit
equation for b in the chosen geometry may be found
by integrating the sheet current over the width of
the strip using Eq. (2) which yields
(π/b)(a2 + b2)
K2 (2ab/(a2 + b2))
arctan
b2(w2 − b2)
(a4 − w2b2)
. (3)
-1,0 -0,5 0,0 0,5 1,0
Fig. 1. Distribution of the sheet current over the flux-free
zone of the partly flux-filled strip delineated by b/w = 0.8,
with a/w = 1.001, 1.01, 1.1, 2 and infinity (from the upper
curve down).
The cylindrical magnetic cavity entails a reduc-
tion of the depth of penetration of magnetic flux into
the strip, ∆(I) = w − b, as compared to the depth
in the situation without a magnetic environment,
∆0(I) = w(1−
1− (I/I
)2), where I
= 2wJ
For weak flux penetration, when ∆ ≪ w and hence
I ≪ I
, the explicit approximate result
∆(I) ≃
a2 − w2
a2 + w2
a2 + w2
is seen to hold. Thus, ∆ strongly decreases with re-
spect to ∆0 ≃ (w/2)(I/Ic)2 as a → w. This also
means a reduction of AC losses to the same extent
which typically scale with ∆2 [6,7]. These losses may
be further curtailed by optimization of the shape of
the magnetic cavity using, in the limit µ ≫ 1, the
method of conformal mapping of the basic cylindri-
cal configuration addressed above.
References
[1] Y.A. Genenko, A. Usoskin, H.C. Freyhardt, Phys. Rev.
Lett. 83 (1999) 3045.
[2] Y.A. Genenko, A. Snezhko, H.C. Freyhardt, Phys. Rev.
B 62 (2000) 3453.
[3] Y.A. Genenko, H. Rauh, A. Snezhko, Physica C 372-376
(2002) 1389.
[4] M. Majoros, B.A. Glowacki, A.M. Campbell, Physica C
334 (2000) 129.
[5] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250.
[6] W.T. Norris, J. Phys. D 3 (1970) 489.
[7] E.H. Brandt, M.V. Indenbom, Phys. Rev. B 48 (1993)
1289.
	References

  An exact analytic representation of the critical state of a current-carrying
type-II superconductor strip located inside a cylindrical magnetic cavity of
high permeability is derived. The obtained results show that, when the cavity
radius is small, penetration of magnetic flux fronts is strongly reduced as
compared to the situation in an isolated strip. From our generic representation
it is possible to establish current profiles in closed cavities of various
other geometries too by means of conformal mapping of the basic configuration
addressed.

Superconductor strip in a closedmagnetic environment:
exact analytic representation of the critical state
Y.A. Genenko ∗, H. Rauh
Institut für Materialwissenschaft, Technische Universität Darmstadt, 64287 Darmstadt, Germany
Abstract
An exact analytic representation of the critical state of a current-carrying type-II superconductor strip located inside
a cylindrical magnetic cavity of high permeability is derived. The obtained results show that, when the cavity radius
is small, penetration of magnetic flux fronts is strongly reduced as compared to the situation in an isolated strip. From
our generic representation it is possible to establish current profiles in closed cavities of various other geometries too
by means of conformal mapping of the basic configuration addressed.
Key words: Superconductor strip, Magnetic shielding, Critical state
PACS: 74.25.Ha, 74.78.Fk, 74.78.-w, 85.25.Am
Relatively high AC losses in superconductor ca-
bles and strips present a substantial problem for
the implementation of superconductors in high-
frequency and low-frequency applications. Recently,
a suggestion for improving the current-carrying
capability of superconductor strips [1,2,3] and for
reducing AC losses in superconductor cables [4]
based on the idea of magnetic shielding of applied
fields as well as current self-induced fields was put
forth. AC losses in superconductor strips caused by
the latter type of fields are anticipated to greatly
decrease when the strips are exposed to suitably
designed magnetic environments [2,3]. Exact ana-
lytic representations of sheet current distributions
in superconductor strips located between two high-
permeability magnets occupying infinite half-spaces
were derived before [1,2]; these configurations al-
lowed to find the respective current distributions
for various other topologically open shielding ge-
ometries by application of the method of conformal
∗ Corresponding author.
Email address: yugenen@tgm.tu-darmstadt.de
(Y.A. Genenko).
mapping. Utilization of the latter tool for analyz-
ing sheet current distributions and AC losses in the
presence of topologically closed magnetic environ-
ments of practical interest requires corresponding
reference results. An establishment of such results
is the focus of the present communication.
We consider an infinitely extended type-II super-
conductor strip of width 2w located inside a cylin-
drical cavity of radius a in an infinitely extended
soft magnet of relative permeability µ, the symme-
try axis of this configuration coinciding with the z-
axis of a cartesian coordinate system x, y, z. Assum-
ing the thickness of the strip to be small compared
to its width, variations of the current over the thick-
ness of the strip may be ignored and, for mathemat-
ical convenience, the state of the strip characterized
by the sheet current J alone.
When magnetic flux penetrates the superconduc-
tor strip in the critical state, the distribution of the
sheet current is controlled by the pinning of mag-
netic vortices. In conformity with Bean’s hypothe-
sis [5], the sheet current adopts its critical value J
throughout the flux-penetrated regions of the strip,
Preprint submitted to Elsevier 18 August 2021
http://arxiv.org/abs/0704.1227v1
whereas the magnetic field component normal to the
strip vanishes in the flux-free regions of the strip.
Proceeding in the spirit of previous work [6], a dis-
tribution of the sheet current prevails with magnetic
flux penetrated from the edges of the strip, but with
the central zone −b < x < b of half-width b < w left
flux free. In this zone, the distribution of the sheet
current is governed by the integral equation [2]
dx′J(x′)
x− x′
x− a2/x′
= 0 (1)
together with the requirement that the total trans-
port current equals I. Here, q = (µ − 1)/(µ + 1)
means the strength of the image current induced by
the magnetic cavity.
In the limit µ ≫ 1, i.e. for q → 1, Eq. (1) has the
exact analytic solution
J(x) = J
a2 + b2
πs2(x)
s′(x)φ(s(x)), (2)
where
φ(s) =
c2 − s2
arctan
(h2 − b2)(c2 − s2)
(b2 − s2)(c2 − h2)
− arctan
h2 − b2
b2 − s2
b2 − s2
arctan
b2w2 − b4
a4 − b2w2
with s(x) = x(a2 + b2)/(a2 + x2), c = (a2 + b2)/2a
and h = w(a2 + b2)/(a2 + w2). Herein, K and Π
denote complete elliptic integrals of the first and,
respectively, third kind.
Sheet current profiles obtained from Eq. (2) for a
range of the geometrical parameters involved, with
a fixed value of b, are shown in Fig. 1. This exhibits
a flattening of the current profiles together with an
increase in the magnitude of the total current up to
saturation, when the radius of the magnetic cavity
is reduced, precisely as in the case of topologically
open magnetic cavities [1,2].
The half-width of the flux-free zone is controlled
by the total transport current in the strip and by the
geometry of the magnetic environment. An implicit
equation for b in the chosen geometry may be found
by integrating the sheet current over the width of
the strip using Eq. (2) which yields
(π/b)(a2 + b2)
K2 (2ab/(a2 + b2))
arctan
b2(w2 − b2)
(a4 − w2b2)
. (3)
-1,0 -0,5 0,0 0,5 1,0
Fig. 1. Distribution of the sheet current over the flux-free
zone of the partly flux-filled strip delineated by b/w = 0.8,
with a/w = 1.001, 1.01, 1.1, 2 and infinity (from the upper
curve down).
The cylindrical magnetic cavity entails a reduc-
tion of the depth of penetration of magnetic flux into
the strip, ∆(I) = w − b, as compared to the depth
in the situation without a magnetic environment,
∆0(I) = w(1−
1− (I/I
)2), where I
= 2wJ
For weak flux penetration, when ∆ ≪ w and hence
I ≪ I
, the explicit approximate result
∆(I) ≃
a2 − w2
a2 + w2
a2 + w2
is seen to hold. Thus, ∆ strongly decreases with re-
spect to ∆0 ≃ (w/2)(I/Ic)2 as a → w. This also
means a reduction of AC losses to the same extent
which typically scale with ∆2 [6,7]. These losses may
be further curtailed by optimization of the shape of
the magnetic cavity using, in the limit µ ≫ 1, the
method of conformal mapping of the basic cylindri-
cal configuration addressed above.
References
[1] Y.A. Genenko, A. Usoskin, H.C. Freyhardt, Phys. Rev.
Lett. 83 (1999) 3045.
[2] Y.A. Genenko, A. Snezhko, H.C. Freyhardt, Phys. Rev.
B 62 (2000) 3453.
[3] Y.A. Genenko, H. Rauh, A. Snezhko, Physica C 372-376
(2002) 1389.
[4] M. Majoros, B.A. Glowacki, A.M. Campbell, Physica C
334 (2000) 129.
[5] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250.
[6] W.T. Norris, J. Phys. D 3 (1970) 489.
[7] E.H. Brandt, M.V. Indenbom, Phys. Rev. B 48 (1993)
1289.
	References
