Physica C 289 Ž1997. 123–130
Self-field hysteresis loss in periodically arranged superconducting
strips
K.-H. Muller
¨
)
CSIRO, DiÕision of Telecommunications and Industrial Physics, P.O. Box 218, Lindfield NSW 2070, Australia
Received 16 May 1997; accepted 3 July 1997
Abstract
The magnetic field and current distributions inside superconducting strips arranged in a z-stack and an x-array are
calculated analytically using the transformation method proposed by Mawatari for the case where a transport current is
passed through the strips. The magnetic field distributions are used to calculate the self-field ac hysteresis losses of both the
z-stack and the x-array. The self-field ac hysteresis loss of the z-stack is always larger than the loss of the x-array. q 1997
Elsevier Science B.V.
PACS: 85.25Kx; 84.70.q p; 74.76.Bz
1. Introduction
The critical state model w1,2x has been used extensively to describe the electrodynamics of type II
superconductors and to calculate the ac hysteresis
loss. Calculations were first performed for the simple
cases of long superconducting cylinders and infinite
superconducting slabs in parallel magnetic fields.
The more complicated case of the critical state in a
flat superconducting strip was first investigated by
Swan w3x and Norris w4x. High-temperature superconductors ŽHTS. have renewed the interest in the critical state of flat superconductors with demagnetizing
factors close to 1 w5,6x. Examples are HTS single
crystals which are thin along the c-direction and
)
Tel.: q61 2 9313 7211; fax: q61 2 9313 7202; e-mail:
karl@dap.csiro.au.
wide in the ab-directions used in experiments on
flux vortex dynamics as well as Bi-2223rAg tapes
w7x and YBCOrNi Žhastelloy. thin-film flexible tapes
w8x to be used in power applications. In most applications, the tapes will be used in array or stack configurations were the strips Žtapes. couple electromagnetically. The ac hysteresis loss of multiple strips is
expected to be quite different from that of an isolated
strip, and it is therefore important to investigate the
behaviour of superconducting strips in array and
stack configurations.
Mawatari w9,10x recently calculated the magnetic
field distribution H Ž x . and the current density distribution J Ž x . Ž x is the coordinate along the width of
the strip. for a z-stack and an x-array of strips where
an ac magnetic field is applied perpendicular to the
strips. Mawatari w9x showed that, for both the x-array
and the z-stack, the H–J relation of Biot–Savart’s
law can be transformed into the H–J relation of an
0921-4534r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved.
PII S 0 9 2 1 - 4 5 3 4 Ž 9 7 . 0 1 5 8 3 - 9
K.-H. Mullerr
Physica C 289 (1997) 123–130
¨
124
isolated strip. H Ž x . and J Ž x . for multiple strips can
then be obtained from the H Ž x . and J Ž x . of the
isolated strip by a simple transformation.
In the following paper we adopt the transformation method developed by Mawatari w9x to calculate
the magnetic field distribution H Ž x . and the current
density distribution J Ž x . for an x-array and a z-stack
for the important cases where dc or ac transport
currents are flowing through the strips. From H Ž x .
we calculate the self-field ac hysteresis losses of the
z-stack and the x-array.
2. The self-field critical state in a periodic array
of superconducting strips
Fig. 1. z-stack of infinitely long superconducting strips Ž l ™`.
where d is the thickness of a strip, 2 a the strip width and D the
stacking spacing.
2.1. The z-stack
The magnetic field H Ž r . at position r created by
a current distribution J Ž rX . is given by Biot–Savart’s
law
HŽ r. s
1
ryr
X
X
H JŽ r . = < ryr <
4p
X 3
d3 rX .
d
2p
Hya
J Ž u.
2
Ž x y u. q z 2
d
sy
d u,
dr2
X
y
X
2
2
q Ž nD .
Ž 4.
du
a
H J Ž u . coth
2 D ya
ž
p Ž x y u.
D
/
d u.
Ž 5.
Introducing the variable transformations
where J Ž u. is the current density in the superconducting strip averaged over the thickness d of the
strip,
H J Ž x, z . d z .
d ydr2
nsy`
Hz Ž x .
x˜ s
1
xyu
a
Hya J Ž u . Ž x y u .
Ý
or w9,11x
Ž 2.
JŽ x. s
`
d
y
2p
xyu
a
Hz Ž x . s
Ž 1.
Let us consider a single infinitely long superconducting strip of width 2 a along the x-direction, thickness
d along the z-direction Ž a 4 d ., positioned at the
coordinate origin. In this case one obtains from Eq.
Ž1.
Hz Ž x , z . s y
The z-component of the magnetic field, Hz Ž x . across
the strip, is given by
D
p
tanh
and a˜ s
px
ž /
D
p
D
tanh
, u˜ s
p
tanh
pu
ž /
D
pa
ž /
D
Ž
one obtains for Eq. 5. using J Ž u. s J Žyu.
Ž 3.
Let us now consider an infinite number of infinitely long strips periodically stacked above each
other along the z-direction with a spacing D where
D ) d. Such a z-stack of strips is shown in Fig. 1.
D
Hz Ž x . s H˜z Ž x˜ . s y
d
ã
J˜Ž u˜ .
d u,
˜
H
2p yã x˜ y u˜
Ž 6.
Ž 7.
where J˜Ž u˜ . s J Ž u..
Eq. Ž7. has the same form as Eq. Ž2. for z s 0
which describes Hz Ž x . of a single strip. The current
K.-H. Mullerr
Physica C 289 (1997) 123–130
¨
and magnetic field distributions for a single strip
have been discussed by Norris w4x, Brandt et al. w5x
and Zeldov et al. w6x. The major problem solved by
these authors was to find a current distribution which
produces a zero z-component of the magnetic field
inside the superconducting strip for < x < - b and a
critical state magnetic field distribution for b - < x < a, where b represents the distance of the flux front
from the centre of the strip. From their calculations it
follows that for the z-stack
H̃z Ž x˜ .
°0,
~
˜
< x < - b,
yHd
s
x˜
< x˜ <
artanh
x˜
¢yH < x̃ < artanh
d
1r2
x˜ 2 y b˜ 2
,
a˜ 2 y b˜ 2
b˜ - < x˜ < - a,
˜
,
x˜ 2 y b˜ 2
tanh
p
pb
ž /
D
and Hd s Jc drp .
Ž 9.
J˜Ž x˜ .
c
~
p
s
arctan
b˜ 2 y x̃ 2
a˜ 2 y b˜ 2
¢0,
1r2
arcosh
p
ž
cosh Ž p arD .
cosh Ž p aIdc rDIc .
,
/
.
Ž 13 .
Fig. 2Ža. shows the magnetic field distribution
Hz Ž x .rHd ŽEqs. Ž8. and Ž9.. for different values of
Idc rIc for a z-stack with Dra s 0.5 and for an
isolated strip Ž Dra ™ `.. The field Hz increases
with decreasing Dra. In the limit D ™ d ™ 0 the
magnetic field distribution Hz Ž x . is that of an infinite slab of width 2 a i.e.
< x < - bX ,
p x
Ž < x < y bX . ,
<
<
d
x
s
x p aIdc
y
,
< x < dIc
~
y
˜
< x˜ < - b,
bX - < x < - a,
Ž 14 .
< x < ) a,
where bX s aŽ1 y Idc rIc ..
Fig. 2Žb. shows the current distribution J Ž x .rJc
ŽEq. Ž10.. for different values of Idc rIc for a z-stack
with Dra s 0.5 and for an isolated strip Ž Dra ™ `..
The smaller the vertical spacing D becomes, the
more the current density profiles resemble that of an
infinite slab of width 2 a, i.e.
JŽ x.
Jc
°0,
s~1,
¢0,
< x < - bX ,
bX - < x < - a,
< x < ) a.
Ž 15 .
2.2. Self-field hysteresis loss of the z-stack
< x˜ < ) a.
˜
Ž 10 .
Let us assume that after zero-field cooling a current
Idc is applied to each strip where Idc - Ic and Ic is
the critical current of a strip, Ic s 2 adJc . Then
a
D
b˜ - < x˜ < - a,
˜
Jc ,
Idc s d
bs
¢
Here, Jc is a field-independent critical current density. ŽThe more complicated case of a field-dependent Jc has been discussed in Ref. w12x..
For J˜ Ž x˜ . one obtains from w5x
°2 J
Ž 12 .
Because x˜ s Drp ) a,
˜ we derive from the last line
of Eq. Ž8. using Eqs. Ž9. and Ž12.
°0,
where
D
Idc s y2 DH˜z Ž x˜ s Drp . .
< x˜ < ) a,
˜
Ž 8.
b̃ s
Using Eq. Ž7. and the fact that H˜ Ž x˜ . s yH˜ Žyx˜ . we
obtain from Eq. Ž11.
Hz Ž x . rHd
1r2
a˜ 2 y b˜ 2
125
ã
J˜Ž u˜ .
Hya J Ž u . d u s d Hya˜ 1 y Ž p urD
˜ .
2
d u.
˜
When an ac current Iac Ž t . s Im cos v t is flowing
through each strip, the magnetic field and current
density distributions can be expressed in terms of Hz
of Eq. Ž7. and Ž8.. For the first half of the ac cycle
Ž0 - t - prv ., the z-component of the magnetic
field distribution Hz x Ž x,t . is given by w5x
Hz x Ž x ,t . s Hz Ž x , Jc , Idc s Im .
Ž 11 .
y Hz Ž x ,2 Jc , Idc s Im y Iac Ž t . .
Ž 16 .
126
K.-H. Mullerr
Physica C 289 (1997) 123–130
¨
and the current density distribution J x Ž x,t . is given
by
J x Ž x ,t . s J Ž x , Jc , Idc s Im .
y J Ž x ,2 Jc , Idc s Im y Iac Ž t . . .
Ž 17 .
The self-field hysteresis loss Prl per strip per
length l is defined as
vd
P
pr v
H0
s
p
l
a
dt 2
H0 E Ž x ,t . J Ž x ,t . d x
y
, Ž 18 .
where E y Ž x,t . is the electrical field inside a strip
generated by the self-field magnetic flux which
moves in and out during each ac cycle. The integral
over the time t accounts for the time averaging of
the loss.
The electric field E y is given by Faraday’s law as
E y Ž x ,t . s mo
d
x
HH
dt 0
zx
Ž u,t . d u.
Ž 19 .
Using the fact that J Ž x . s Jc for those x where
E y Ž x,t . / 0 one obtains
P
sy
4m o
p
l
a
dJc v
x
HbŽ0. HbŽ0. H
zx
Ž u,t s 0 . d u d x ,
Ž 20 .
where
b Ž 0. s
D
p
arcosh
ž
cosh Ž p arD .
cosh Ž p aIm rDIc .
/
.
Ž 21 .
Eq. Ž20. can be simplified further by partial integration
P
sy
4m o
p
l
a
dJc v
HbŽ0. Ž x y a. H
zx
Ž x ,t s 0 . d x.
Ž 22 .
Using Eq. Ž6., Eq. Ž8. and Eq. Ž9. we obtain
P
s
l
mo
Ic2
p
a2
v
2
=artanh
Fig. 2. Ža. z-stack magnetic field H z r Hd and Žb. current density
Jr Jc versus x r a at different values of Idc r Ic for Dr as 0.5
Žsolid lines. and for an isolated strip Ž Dr a™`. Ždashed lines..
ž
a
HbŽ0. Ž a y x .
tanh2 Ž p xrD . y tanh2 Ž p b Ž 0 . rD .
tanh2 Ž p arD . y tanh2 Ž p b Ž 0 . rD .
/
d x.
Ž 23 .
K.-H. Mullerr
Physica C 289 (1997) 123–130
¨
127
Žisolated strip. Eq. Ž23. becomes the Norris formula
w4x, i.e.
mo
P
s
2p
l
2
v Ic2
ž
q 1q
Im
Ic
Im
Im
ž / ž /
/ ž / ž /
1y
Ic
ln 1 q
ln 1 y
Im
y
Ic
Ic
Im
2
Ic
,
Ž 24 .
where
P
s
mo
12p
l
v
2
Im4
for Im < Ic
Ž 25 .
Ž 2 ln 2 y 1 . v Ic2 for Im s Ic .
Ž 26 .
Ic2
and
P
s
l
Fig. 3. z-stack ac hysteresis loss Pr l per strip per length versus
ac current amplitude Im r Ic for different spacings Dr a. Here
v r2p s 50 Hz, as 0.5 cm, Ic s100 A and ds1 mm.
2p 2
In contrast, for small D Ž D - a. the z-stack behaves
similar to an infinite superconducting slab. In the
limit D ™ d, the loss per strip corresponds to that of
an infinite slab of width 2 a which is, using Eq. Ž14.,
Eq. Ž16. and Eq. Ž22.
P
s
Fig. 3 shows the self-field hysteresis loss Prl of
the z-stack as a function of Im rIc for different
values of Dra using vr2p s 50 Hz, a s 0.5 cm
and Ic s 100 A. The loss per strip increases dramatically with decreasing spacing D as the magnetic
field increases rapidly inside the strip. For Dra ™ `
mo
l
mo
6p
sv
a Im3
d Ic
.
Ž 27 .
In Fig. 3, the curve for Dra s dra can be obtained
by either using Eq. Ž23. or Eq. Ž27.. As can be see in
Fig. 3, Prl changes from a Im4 behaviour at large
Dra to a Im3 behaviour at small Dra.
Fig. 4. x-array of infinitely long superconducting strips Ž l ™ `.. The x-array periodicity is L.
K.-H. Mullerr
Physica C 289 (1997) 123–130
¨
128
2.3. The x-array
Let us consider an infinite number of long superconducting strips periodically placed next to each
other along the x-direction at a spacing L where
L ) 2 a. Such an x-array is shown in Fig. 4. From
Eq. Ž1., assuming a 4 d, one finds
Hz Ž x . s y
`
d
2p
J Ž u.
a
Hya x y Ž u q nL . d u,
Ý
nsy`
Ž 28 .
which alternatively can be expressed by w9,11x
Hz Ž x . s y
d
a
H J Ž u . cot Ž p Ž x y u . rL . d u.
2 L ya
Ž 29 .
Introducing the variable transformations
x˜ s
L
p
tan
and a˜ s
px
ž /
L
L
p
tan
, u˜ s
L
p
tan
pu
ž /
L
pa
ž /
Ž 30 .
L
one obtains for Eq. Ž29.
Hz Ž x . s H˜z Ž x˜ . s y
d
ã
J˜Ž u˜ .
d u,
˜
H
2p yã x˜ y u˜
Ž 31 .
which is the same as Eq. Ž7..
It is important to realize that the transformation
D ™ i L Žwhere i s 'y 1 . causes Eqs. Ž5. and Ž6. to
become Eqs. Ž29. and Ž30.. This transformation has
also been used in Ref. w9x. From Eq. Ž13., with
D ™ i L, we obtain for the position b of the flux
front in the strips
bs
L
p
arccos
ž
cos Ž p arL .
cos Ž p aIdc rLIc .
/
.
Ž 32 .
Fig. 5Ža. shows the magnetic field distribution
Hz Ž x .rHd ŽEqs. Ž8. and Ž9.. of the x-array for
different currents Idc rIc with Lra s 2.5 as well as
for the isolated strip Ž Lra ™ `.. The smaller the
spacing L is, the weaker Hz Ž x . becomes, contrary to
the z-stack behaviour. Fig. 5Žb. shows the current
density distribution J Ž x .rJc ŽEq. Ž10.. for different
values of Idc rIc for the x-array with Lra s 2.5 and
Fig. 5. Ža. x-array magnetic field H z r Hd and Žb. current density
Jr Jc versus x r a at different values of Idc r Ic for Lr as 2.5
Žsolid lines. and for an isolated strip Ž Lr a™`. Ždashed lines..
K.-H. Mullerr
Physica C 289 (1997) 123–130
¨
129
where
b Ž 0. s
L
p
arccos
ž
cos Ž p arL .
cos Ž p aIm rLIc .
/
.
Ž 34 .
Fig. 6 shows the self-field hysteresis loss as a
function of Im rIc for different spacings Lra using
vr2p s 50 Hz, a s 0.5 cm and Ic s 100 A. As can
be seen, in the limit L ™ 2 a the loss Prl goes to
zero while for Lra ™ ` Eq. Ž33. becomes the Norris
formula w4x of Eq. Ž24..
It is important to realize that, for L close to 2 a,
the loss which is due to the x-component of the
magnetic field, H x , becomes more important than
the loss caused by the perpendicular magnetic field
Hz which has been considered so far. The loss from
the x-component of the magnetic field is
P
s
l
Fig. 6. x-array ac hysteresis loss Pr l per strip per length versus
ac current amplitude Im r Ic for different spacings Dr a. Here
v r2p s 50 Hz, as 0.5 cm and Ic s100 A. The dashed curve
represents Eq. Ž35..
for an isolated strip. Away from the edges, J Ž x .
becomes nearly constant, i.e. independent of x, when
the spacing L is small. In the limit L ™ 2 a one
obtains J Ž x .rJc s IdcrIc .
2.4. Self-field hysteresis loss of the x-array
Using the transformation D ™ i L, one obtains for
the self-field hysteresis loss Prl per strip per length
l of the x-array
P
s
l
mo
p
v
2
= artanh
Ic2
a
ž
2
a
HbŽ0. Ž a y x .
tan2 Ž p xrL . y tan2 Ž p b Ž 0 . rL .
tan2 Ž p arL . y tan2 Ž p b Ž 0 . rL .
/
d x,
Ž 33 .
mo
24p
v
d Im3
a Ic
,
Ž 35 .
which is obtained from Eq. Ž27. by replacing d by
2 a and a by dr2. Prl of Eq. Ž35. is shown as a
dashed line in Fig. 6 where vr2 p s 50 Hz, a s 0.5
cm, Ic s 100 A and d s 1 mm. The total loss cannot
become smaller than the loss of Eq. Ž35..
3. Conclusion
Our investigations show that in the case of a
z-stack made of long superconducting strips of rectangular cross-section, the z-component of the magnetic field inside the strip increases when the spacing
D is decreased. For small D Ž D - a. the z-stack
behaves magnetically similar to a superconducting
slab. The z-stack self-field ac loss per strip is smallest for the isolated strip where Prl ; Im4 rIc for
Im - Ic . For D s d the loss is maximal and corresponds to that of a slab of width 2 a where Prl ;
Ž ard .Ž Im3 rIc ..
In the case of the x-array, the z-component of the
magnetic field inside the strip decreases with decreasing spacing L. In the limit L ™ 2 a the z-component of the magnetic field vanishes and the current
density distribution J Ž x . becomes constant where
130
K.-H. Mullerr
Physica C 289 (1997) 123–130
¨
J Ž x .rJc s IdcrIc . The x-array self-field ac loss per
strip is largest for the isolated strip and decreases
with decreasing spacing L. For L , 2 a the loss from
the x-component of the magnetic field, H x , becomes
larger than the loss from the z-component. The
H x-loss behaves like Prl ; Ž dra.Ž Im3 rIc .. The selffield loss of the x-array is always smaller than the
self-field loss of the z-stack. The ratio between the
minimum loss in the x-array, which occurs when
L s 2 a, and the maximum loss in the z-stack, which
occurs when D s d, is Ž dr2 a. 2 . The behaviours of
the x-array and the z-stack are reversed when the
arrays are exposed to an ac magnetic field applied in
the z-direction with zero applied currents w9x. In this
case the loss of the x-array is always larger than the
loss of the z-stack.
After this manuscript was completed we became
aware of a preprint by Mawatari w13x, which applies
the same formalism to the self-field case. In contrast
to his work our paper gives a more detailed derivation of the essential results and discusses the self-field
ac loss in greater detail.
Acknowledgements
The author gratefully acknowledges the support of
MM Cables, Metal Manufactures Ltd, the Energy
Research and Development Corporation and DIST.
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