APPLIED PHYSICS LETTERS
VOLUME 75, NUMBER 3
19 JULY 1999
Alternating current loss in coplanar arrays of superconducting strips
with bidirectional currents
Yasunori Mawataria) and Hirofumi Yamasaki
Frontier Technology Division, Electrotechnical Laboratory, 1–1–4 Umezono, Tsukuba,
Ibaraki 305–8568, Japan
共Received 10 May 1999; accepted for publication 27 May 1999兲
Magnetic field, current density, and hysteretic alternating current loss power P in coplanar arrays of
superconducting strip lines are analyzed on the bases of the critical state model. For a simplified
model of a film-type fault current limiter, we consider a strip array in which multiple strip lines are
periodically arranged in one plane and are carrying bidirectional currents. We investigate the effect
of spacing between the strip lines on P. The P of the strip array is higher than that of a single strip,
because the magnetic field in a strip is enhanced by currents in neighboring strips. © 1999
American Institute of Physics. 关S0003-6951共99兲04329-6兴
strip at 兩 x⫺nL 兩 ⭐w carries a transport current (⫺1) n I t along
the y axis.
When 2wⰇd, the electromagnetic properties of the strip
array can be described by the mean current density along the
⫹d/2
y axis, J(x)⬅(1/d) 兰 ⫺d/2
J y (x,z)dz, and by the magnetic
field along the z axis, H(x)⬅H z (x,z⫽0). 2,3 When the strip
array carries bidirectional currents without an applied magnetic field, the magnetic-field and current-density distributions have periodicity H(x⫹nL)⫽(⫺1) n H(x) and J(x
⫹nL)⫽(⫺1) n J(x), respectively. Using the periodicity of
J(x) and the symmetry of J(⫺x)⫽J(x), we obtain the following relationship between H(x) and J(x):
High-temperature superconducting films and tapes that
have strip geometry with large aspect ratio are often used in
power and electronics devices. Macroscopic electromagnetic
properties in a current-carrying superconducting strip are
well described by the critical state model for a single strip.1–3
Because applications of superconductors involve multiple
films or tapes, the electromagnetic interaction of multiple
strip lines must be studied. A simple method has been proposed to analyze the electromagnetic properties of periodically arranged strips4 共i.e., stacks of strips and coplanar arrays of strips兲. In this method, the magnetic-field and
current-density distributions for strip arrays are easily derived by using a transformation of these distributions for an
isolated strip line. This transformation method was originally
introduced for strip arrays in an applied magnetic field,4 and
then later applied to strip arrays that carry transport current
in one common direction.5 See also Ref. 6.
Fault current limiters 共FCLs兲 have been developed to
protect electric-power systems from power failures, and resistive FCLs that use superconducting films are one of the
most promising applications of high-temperature superconductors. Because complicated meandering current paths are
patterned in film type FCLs 关see Fig. 1共a兲兴, analytical investigation of electromagnetic response of FCLs that carry alternating current 共ac兲 may be highly complicated. If we ignore the effects of both ends where current direction is
turned and edges in the outermost strips, we can simplify
FCLs as a coplanar array of strips that carry bidirectional
currents as shown in Fig. 1共b兲. This simplification to the strip
array allows us to do a theoretical analysis of the electromagnetic properties of resistive FCLs.
In this letter, we analyze the electromagnetic properties
of an idealized film-type FCL, i.e., a coplanar array of strip
lines that carry bidirectional currents.
Consider a coplanar array of superconducting-strip lines
in which an infinite number of parallel strip lines are arranged in the xy plane 关Fig. 1共c兲兴. Each strip line is 2w wide
and d thick, and is spaced at intervals of L in the xy plane.
The strip lines carry bidirectional currents; namely, the nth
FIG. 1. 共a兲 Schematic of the current paths 共shaded areas兲 in a resistive FCL
on a superconducting film. 共b兲 Simplified configuration of a FCL, neglecting
edge effects. 共c兲 Coplanar array of strip lines carrying bidirectional currents.
The nth strip occupies an area of 兩 x⫺nL 兩 ⬍w, 兩 y 兩 ⬍⬁, and 兩 z 兩 ⬍d/2 共where
L⬎2wⰇd and n⫽0, ⫾1, ⫾2,..., ⫾⬁). Spacing between strips is s⫽L
⫺2w. The nth strip carries a transport current (⫺1) n I t along the y axis.
a兲
Electronic mail: mawatari@etl.go.jp
0003-6951/99/75(3)/406/3/$15.00
406
© 1999 American Institute of Physics
Appl. Phys. Lett., Vol. 75, No. 3, 19 July 1999
冕
⫹⬁
d
H 共 x 兲 ⫽⫺
2 ␲ n⫽⫺⬁
兺
冕
冕
⫽⫺
d
2L
⫽⫺
d
L
w
0
⫺w
⫹w
du
⫺w
⫹w
du
Y. Mawatari and H. Yamasaki
407
共 ⫺1 兲 n J 共 x 兲
,
x⫺u⫺nL
J共 u 兲
,
sin关 ␲ 共 x⫺u 兲 /L 兴
duJ 共 u 兲
sin共 ␲ x/L 兲 cos共 ␲ u/L 兲
.
sin2 共 ␲ x/L 兲 ⫺sin2 共 ␲ u/L 兲
共1兲
By introducing the following variable transformation,
x̃⫽
冉 冊
␲x
L
sin
,
␲
L
共2兲
and ũ⫽(L/ ␲ )sin(␲u/L), we can reduce Eq. 共1兲 to
H̃ 共 x̃ 兲 ⫽⫺
d
␲
冕
w̃
dũ
0
˜x J̃ 共 ũ 兲
,
x̃ 2 ⫺ũ 2
共3兲
where H̃(x̃) is the transformed magnetic field, J̃(x̃) is the
transformed current density, and w̃⫽(L/ ␲ )sin(␲w/L) is the
transformed strip width. Note that Eq. 共3兲 corresponds to the
relationship between the magnetic field and current density
for an isolated strip line that carries a transport current without an applied magnetic field. In other words, by using the
transformation in Eq. 共2兲, we obtain H(x) and J(x) for the
strip array from those for a single isolated strip line. This
transformation method is applicable to the critical state
model with a constant critical current density,4,5 but is not
applicable to arbitrary current-voltage characteristics.
In the critical state model, the current density saturates to
the critical current density J c in the flux-filled region, i.e.,
J(x)⫽J c for a⬍ 兩 x 兩 ⬍w, and the magnetic field is shielded in
the flux-free region, i.e., H(x)⫽0 for 兩 x 兩 ⬍a, where a is the
position of the flux front.2,3,7 When the transport current I t is
monotonically increased from zero, expressions for H(x)
and J(x) for the strip array in the critical state are
H共 x 兲⫽
冦
再
0,
兩 x 兩 ⬍a,
J c d 共 ⫺x 兲
arctanh关 ␸ 共 x,a 兲兴 ,
␲ 兩x兩
a⬍ 兩 x 兩 ⬍w,
J c d 共 ⫺x 兲
1
arctanh
,
␲ 兩x兩
␸ 共 x,a 兲
兩 x 兩 ⬎w,
冋
冋
册
2J c
1
arctan
,
␲
␸
x,a
兲
共
J共 x 兲⫽
J c , a⬍ 兩 x 兩 ⬍w,
册
兩 x 兩 ⬍a,
共4兲
冋
兩 sin2 共 ␲ a/L 兲 ⫺sin2 共 ␲ x/L 兲 兩
sin2 共 ␲ w/L 兲 ⫺sin2 共 ␲ a/L 兲
册
冕
a
0
dx arctan关 ␸ 共 x,a 兲兴 ,
⫽
共5兲
1/2
.
共6兲
⫹w
J(x)dx is determined as a
The transport current I t ⫽d 兰 ⫺w
function of a from Eq. 共5兲:
I t共 a 兲
2
⫽1⫺
Ic
␲w
increases from I t (a⫽w)⫽0 to I t (a⫽0)⫽I c . For infinite
spacing L˜⬁, Eq. 共7兲 reduces to the expression for a single
isolated strip line,2,3 I t /I c ⫽(1⫺a 2 /w 2 ) 1/2.
Equations 共4兲–共7兲 are valid when I t is monotonically
increased from zero after zero-field cooling. The H(x) and
J(x) distributions for ac current whose amplitude is I 0
共where 0⬍I 0 ⬍I c 兲 are derived from the expressions for
monotonically increasing I t . 2 Thus, the hysteretic ac loss
power of each strip per unit length P, is calculated from the
magnetic-field distribution given by Eq. 共4兲:2
冕 冕 ⬘
冕 冉 冊
w
P 共 a 0 兲 ⫽8 ␮ 0 ␯ J c d
where J c is the constant critical current density. The function
␸ (x,a) is defined as
␸ 共 x,a 兲 ⬅
FIG. 2. Current-amplitude dependence of ac loss power P in a strip array for
various spacing between strips. Current amplitude I 0 is normalized by the
critical current I c ⫽2J c wd. 共a兲 ac loss power P per unit length of a strip in
a strip array 共solid lines兲 and of an isolated strip 共dashed line兲. Dotted4
dashed lines denote the dependence P⬀I 2.4
0 and P⬀I 0 . 共b兲 Ratio of ac loss
power P s in a strip array to ac loss power P ⬁ in an isolated strip.
␲w
a0
a0
dx 关 ⫺H 共 x ⬘ 兲兴 a⫽a 0 ,
w
dx 1⫺
a0
x
arctanh关 ␸ 共 x,a 0 兲兴 ,
w
共8兲
where ␯ is the frequency and a 0 is the flux front defined by
I t (a 0 )⫽I 0 in Eq. 共7兲. If we set I 0 ⫽I t (a 0 ) then eliminate a 0
in Eqs. 共7兲 and 共8兲, the P can be expressed as a function of
I 0 . Equations 共7兲 and 共8兲 for L˜⬁ corresponds to P of an
isolated strip line.1–3
For the lower limit of current 共i.e., I 0 /I c ˜0, or 1
⫺a 0 /w˜0兲, Eqs. 共7兲 and 共8兲 reduce to I 0 (a 0 )⬀(1
⫺a 0 /w) 1/2 and P(a 0 )⬀(1⫺a 0 /w) 2 . The P for this lower
current limit is, therefore, proportional to the biquadrate of
current,
共7兲
where I c ⫽2J c wd is the critical current. When the position of
the flux front a decreases from w to 0 in Eq. 共7兲, the I t
2 ␮ 0 ␯ I 2c
x
dx
P⯝
冉 冊
␮ 0 ␯ I 40 ␲ w
F
,
L
I 2c
F共 ␪ 兲⬅
␲ 3 ␪ 2 /24
, 共9兲
sin2 共 2 ␪ 兲关 K 共 sin ␪ 兲兴 4
where K(k) is the complete elliptic integral.8
408
Appl. Phys. Lett., Vol. 75, No. 3, 19 July 1999
FIG. 3. Dependence of ac loss power P on spacing between strips s(⫽L
⫺2w) for various current amplitudes I 0 .
Figure 2共a兲 shows the dependence of P on the normalized current amplitude I 0 /I c for various spacings between
strips, s(⫽L⫺2w). The P increases with decreasing s, because the magnetic field is enhanced by the current in neighboring strips. Whereas P for the lower current limit (I 0 /I c
˜0) is proportional to I 40 as derived in Eq. 共9兲, the exponent
n defined by n⫽ ⳵ (log P)/⳵(log I0) 共i.e., P⬀I n0 ) can be less
than 3 for sⰆ2w.
Figure 2共b兲 shows the ratio of P for the strip array to that
for an isolated strip. Compared with P for a single strip
共where s/2w˜⬁兲, P for s/2w⫽1 is about 27% larger,
whereas P for s/2w⫽0.2 is nearly 2.8 times larger for the
lower current I 0 /I c ⬍0.1. For s/2w⬎5, we can neglect the
interaction of strips, and P of a strip array is at most 2.3%
higher than that in an isolated strip.
Small P for the strip array in the superconducting state is
desirable for applications to FCLs. In practice, typical spac-
Y. Mawatari and H. Yamasaki
ing between strips in FCLs may be 0.1⬍s/2w⬍1, a range in
which the spacing dependence of P in a strip array drastically
changes 共Fig. 3兲. This dependence means that the configuration of strips in the strip array is crucial for reducing the P.
Furthermore, in practice, spacing between strips may be determined by various factors, such as electric and thermal behavior during current-limiting processes.
Although the configuration of strips considered here is
oversimplified, the theoretical expressions of P given by Eqs.
共7兲 and 共8兲 can be used in the initial design phase of FCLs.
In summary, we analyzed the electromagnetic properties
of a coplanar array of strip lines that carry bidirectional currents. The magnetic-field and current-density distributions in
the critical state were derived by using a simple transformation of these distributions for an isolated strip. This theoretical analysis shows that the P of a strip array is higher than
that of a single strip, because the H in a strip is enhanced by
current flowing in neighboring strips.
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