PHYSICAL REVIEW B
VOLUME 54, NUMBER 18
1 NOVEMBER 1996-II
Critical state of periodically arranged superconducting-strip lines in perpendicular fields
Y. Mawatari
Superconductor Technology Section, Electrotechnical Laboratory, 1–1–4 Umezono, Tsukuba, Ibaraki 305, Japan
~Received 12 June 1996!
Analytical investigations of the critical state are carried out for strip-array systems, in which thin superconducting strip lines parallel to the xy plane are arranged periodically. Two types of strip-array systems are
considered: a stack of strip lines piled along the z axis and an array of strip lines aligned in the xy plane. The
results show that the relations between the magnetic field H and the current density J of strip-array systems in
an applied magnetic field perpendicular to the strips can be transformed to the H-J relation for an isolated strip
line. Therefore, H and J of strip-array systems in the Meissner state and in the Bean’s critical state can be
easily derived by transforming the H and J for an isolated strip line. Magnetization properties and interactions
in multiple strip lines are discussed. @S0163-1829~96!06842-7#
I. INTRODUCTION
The concept of a critical state was introduced to describe
the magnetic hysteresis of type-II superconductors and was
first applied to slab and cylindrical superconductors in magnetic fields applied parallel to the superconductors.1 The
critical state in superconducting strips in perpendicular magnetic fields was considered by Swan2 and Norris.3 Recently,
the electromagnetic properties ~e.g., magnetic field, current
density, magnetization, and ac loss! of flat superconductors
in perpendicular fields, in which the demagnetization factor
is close to 1, have been extensively investigated.
The magnetic field and current density in the critical state
of strip lines have been analytically investigated as
follows.4–7 Consider an isolated strip line of width 2w and
thickness d ~the strip line fills the area u x u <w, u z u <d/2, and
u y u ,`). The width of the strip line, 2w, is assumed to be
much larger than the thickness d and the effective penetration depth in thin films8 2l 2 /d, where l is the London penetration depth. This assumption allows us to describe the
electromagnetic property of the strip array by the z component of the magnetic field at z50, H(x)[H z (x,z50), and
by the y component of the mean current density,
J(x)[(1/d) * 1d/2
2d/2J y (x,z)dz. The basic equation of the BiotSavart law for H(x) and J(x) for an isolated strip line is
given by
H ~ x ! 5H a 2
d
2p
E
1w J ~ u !
2w
x2u
du,
~1!
where the magnetic field H a is applied parallel to the z axis.
In the critical state of a strip line, H(x) and J(x) are determined so as to satisfy the integral equations H( u x u ,a)50
and u J(a, u x u ,w) u 5J c with Eq. ~1!, where a is the position
of the flux front and J c is the critical current density. The
integral equations for H(x) and J(x) can be solved analytically assuming the Bean’s critical state model,4–6 in which
J c is assumed to be field independent; on the contrary, a
numerical calculation is necessary for field-dependent J c . 7
Bean’s critical state for disk superconductors in perpendicular fields has also been investigated analytically.9–11 These
results show that the H(x) and J(x) profiles of a strip and a
0163-1829/96/54~18!/13215~7!/$10.00
54
disk in perpendicular fields are very different from those of a
slab and a cylinder in parallel fields.
The magnetic field distribution in strip lines predicted by
those previous analyses have been experimentally confirmed
by using magneto-optical Faraday effects12,13 and by using
an electron spin resonance probe.14 Magnetization properties
predicted by the theoretical works have been confirmed in
studies of the magnetization curves for YBa 2 Cu 3 O y disks.15
In these previous works, isolated strip lines or disks were
investigated. When multiple strip lines are subject to a magnetic field, on the other hand, H(x) and J(x) profiles are
affected by the interaction of the strip lines. For example, a
stack of strip lines may behave like a slab superconductor, if
the spacing between strip lines in the stack is much narrower
than the width. Multiple strip lines or wires are utilized for
device and power application of superconductors, and basic
physical investigations of the electromagnetic response of
such a strip-array system is useful for the application of superconductors.
In the present paper, analytical expressions of H(x) and
J(x) are determined for systems of multiple strip lines in
perpendicular magnetic fields on the basis of the Bean’s critical state model. The results show that the basic H-J relation
of the Biot-Savart law for strip-array systems can be transformed to the H-J relation for an isolated strip line, Eq. ~1!.
Conversely, the H(x) and J(x) for multiple strip lines can
then be obtained by a simple transformation of H(x) and
J(x) for an isolated strip line. I consider two types of striparray systems: a stack of strip lines along the z axis ~I call
this stack a Z stack! is analyzed in Sec. II and an array of
strip lines in the xy plane ~I call this array an X array! in Sec.
III.
II. STACK OF STRIP LINES ALONG THE z AXIS
„Z STACK…
In this section we consider a Z stack in which an infinite
number of strip lines are stacked along the z axis. Each strip
line has width 2w and thickness d, and is stacked at the same
interval D as shown in Fig. 1. The nth superconducting strip
occupies an area where u x u <w, u y u ,`, and u z2nD u <d/2
13 215
© 1996 The American Physical Society
13 216
54
Y. MAWATARI
M5
1
2w
E
1w
2w
~6!
xJ ~ x ! dx.
If a Z stack carries no transport current, I t 50, then
J(x) is asymmetric, J(2x)52J(x), and we have
H̃ ~ x̃5D/ p ! 5H a .
~7!
This equation can be used to determine H̃ a , because H̃ a is
implicit on the left-hand side of the equation. Since the extension to the case of I t Þ0 is straightforward,16 in this paper
we derive H(x) and J(x) for H a Þ0 and I t 50.
FIG. 1. Arrangement of strip lines in a Z stack, in which an
infinite number of strip lines are stacked along the z axis at an
interval D. Each strip line has width 2w, thickness d, and is infinitely long along the y axis.
(n50,61,62,•••,6`). We assume 2w@d, 2w@2l 2 /d,
and D@d.
B. Meissner state
In the Meissner state H̃(x̃) and J̃(x̃) for a Z stack, which
are the same expressions as H(x) and J(x) for an isolated
strip line,4–6 are given by
H̃ ~ u x̃ u .w̃ ! 5H̃ a
A. Basic equations
When H a is applied parallel to the z axis ~perpendicular to
the strips!, the Biot-Savart law for the z component of the
magnetic field H(x) and the mean current density J(x) in a
Z stack is given by
E
1`
H ~ x ! 5H a 2
5H a 2
d
2 p n52`
(
d
2D
E
1w
2w
1w
2w
J~ u !
J ~ u ! coth
S
~ x2u ! du
~ x2u ! 2 1 ~ nD ! 2
D
p ~ x2u !
du.
D
J̃ ~ u x̃ u ,w̃ ! 52
Ax̃
2H̃ a
d
2w̃ 2
~8!
,
x̃
Aw̃ 2 2x̃ 2
~9!
,
and H̃( u x̃ u ,w̃)50. Substituting x̃5D/ p in Eq. ~8! and using Eq. ~7!, we have
H̃ a 5
~2!
u x̃ u
2
Ha
.
cosh~ p w/D !
~10!
The H(x) and J(x) for a Z stack are obtained from Eqs. ~3!,
~8!, ~9!, and ~10! as
By introducing the following transformation of the variables,
S D
S D
D
px
x̃5 tanh
,
p
D
D
pw
w̃5 tanh
,
p
D
~3!
Eq. ~2! is reduced to
d
H̃ ~ x̃ ! 5H̃ a 2
2p
E
d
2p
5H a 1
d
2D
E
E
˜
1w
˜
2w
˜ J̃ ~ ũ !
1w
˜
2w
J̃ ~ ũ !
1w
2w
H a sinh~ p u x u /D !
Asinh ~ p x/D ! 2sinh2 ~ p w/D !
J ~ u x u ,w ! 52
x̃2ũ
dũ,
~4!
where H̃(x̃) is the transformed magnetic field, J̃(x̃) is the
transformed current density, and H̃ a is the transformed applied field given by
H̃ a 5H a 1
H ~ u x u .w ! 5
2
~ 2H a /d ! sinh~ p x/D !
Asinh2 ~ p w/D ! 2sinh2 ~ p x/D !
J ~ u ! tanh
S D
pu
du.
D
H ~ u x u →w10 ! ;H a
J ~ u x u →w20 ! ;2
~5!
Note that Eq. ~4! has the same form as Eq. ~1!; namely, Eq.
~4! shows the H̃(x̃)-J̃(x̃) relation for an isolated strip line of
width w̃ with a magnetic field H̃ a in x̃ space. Therefore,
H(x) and J(x) of a Z stack can be derived from those of an
isolated strip line using the transformation given by Eqs. ~3!
and ~5!. The magnetization M is then calculated from J(x)
as
,
~12!
and H( u x u ,w)50. At the edge of the strip line ( u x u ;w),
H(x) and J(x) diverge as follows:
ũdũ
~ D/ p ! 2 2 ~ ũ ! 2
~11!
,
A
tanh~ p w/D !
,
2 p ~ u x u 2w ! /D
Ha
sgn~ x !
d
A
~13!
tanh~ p w/D !
. ~14!
p ~ w2 u x u ! /2D
The magnetization M is calculated as
M 52H a
F S DG
D2
pw
ln cosh
p wd
D
.
~15!
C. Critical state
In the critical state, when a Z stack is exposed to a magnetic field increased after zero-field cooling, H( u x u ,a)50
in the shielded region, and H(a, u x u ,w).0 in the flux-
54
CRITICAL STATE OF PERIODICALLY ARRANGED . . .
filled region, where a.0 is the position of the flux front. The
current density J saturates as J(a, u x u ,w)52J c sgn(x),
where we assume that the critical current density J c is field
independent1 in this paper.
In the critical state, the H̃(x̃) and J̃(x̃) are given by4–6
H̃ ~ x̃ ! 5
H
H 0 arctanh~ 1/u w̃ ~ x̃ ! u ! ,
H 0 arctanhu w̃ ~ x̃ ! u ,
J̃ ~ x̃ ! 5
H
ã, u x̃ u ,w̃,
2 ~ 2/p ! J c arctanw̃ ~ x̃ ! ,
u x̃ u ,ã,
ã, u x̃ u ,w̃,
2J c sgn~ x̃ ! ,
~16!
u x̃ u .w̃,
~17!
where H 0 is the characteristic field given by
H 0 5J c d/ p
~18!
x̃
w̃
A
2 ~ 2/p ! J c arctanw ~ x ! ,
u x u ,a,
~26!
a, u x u ,w,
2J c sgn~ x ! ,
tanh~ p x/D !
tanh~ p w/D !
A
tanh2 ~ p w/D ! 2tanh2 ~ p a/D !
.
u tanh2 ~ p a/D ! 2tanh2 ~ p x/D ! u
~27!
At the edge of the strip line ( u x u ;w), H(x) diverges logarithmically as
S DG
F
tanh~ p w/D !
Ha
H0
ln
sinh2
2
p uu x u 2w u /2D
H0
H ~ u x u →w ! ;
.
At the flux front ( u x u ;a), J(x) and H(x) have vertical
slopes,
w̃ 2 2ã 2
u ã 2x̃ u
2
2
~19!
.
;
S DA
Ha
2
J c coth
p
H0
2 p ~ u x u 2a ! /D
.
tanh~ p a/D !
~29!
The transformed flux front ã5(D/ p )tanh(pa/D) satisfies
S D
w̃ tanh~ p w/D !
H̃ a
5 5
.
cosh
H0
ã tanh~ p a/D !
~20!
The relation between H a and H̃ a is then obtained from Eq.
~7! as
sinh
S D
H̃ a
sinh~ H a /H 0 !
5
.
H0
cosh~ p w/D !
~21!
Using Eqs. ~20! and ~21!, we have the following relation
between H a and a:
S D
pa
sinh~ p w/D !
sinh
5
.
D
cosh~ H a /H 0 !
~22!
For a low applied field @ H a !H 0 and w2a!min(w,D)#, Eq.
~22! is then approximated as
S D S D
D Ha
w2a.
2p H0
2
tanh
pw
.
D
~23!
For a high field (H a *H 0 and a!D), on the other hand, we
have
S D S D
2D
pw
Ha
a.
exp 2
sinh
.
p
H0
D
H~ x !5
~24!
H
u x u ,a,
H 0 arctanh~ 1/u w ~ x ! u ! ,
H 0 arctanhu w ~ x ! u ,
a, u x u ,w,
u x u .w,
Logarithmic divergence of H(x) at u x u ;w and vertical
slopes of H(x) and J(x) at u x u ;a appear in a Z stack as well
as in an isolated strip line. For a high field (H a .H̃ a *H 0 ),
H̃ a given by Eq. ~21! is reduced to
F S DG
~25!
pw
D
H̃ a 5H a 2H 0 ln cosh
~30!
,
and H(x) is given by
U
U
H 0 sinh2 ~ p w/D !
ln
21 .
2
sinh2 ~ p x/D !
H ~ x ! 5H a 2
~31!
Figure 2 shows H(x) and J(x) profiles for a Z stack ~solid
lines! and for an isolated strip line ~dashed lines!. This figure
clearly shows that the field shielding effect is larger and the
width of the flux-filled region w2a is smaller for a Z stack
than for an isolated strip line.
In the critical state, M for a Z stack is calculated by substituting Eqs. ~26! and ~27! into Eq. ~6!; thus we have
M 52
52
As shown in Eqs. ~23! and ~24!, w2a grows with field as
;H 2a and saturates exponentially. These field penetration
properties are similar to those for an isolated strip line.
Substituting Eq. ~3! into Eqs. ~16!, ~17!, and ~19!, we
obtain H(x) and J(x) in the critical state as
0,
~28!
2
J c 2 u J ~ u x u →a20 ! u ; H ~ u x u →a10 !
d
and w̃ (x̃) is
w̃ ~ x̃ ! 5
H
where w (x) is
w~ x !5
u x̃ u ,ã,
0,
J~ x !5
13 217
D2
2 p wd
M0
n 2z
E F
Ha
0
E F
nz
0
ln 11
ln
G
sinh2 ~ p w/D !
dH 8
cosh2 ~ H 8 /H 0 !
G
cosh~ s1 h !
ds,
cosh~ s2 h !
~32!
where M 0 , n z , and h are defined as
1
M 0 5 wJ c ,
2
n z5
pw
,
D
h5
Ha pHa
.
5
H0
J cd
~33!
For h , n z ~i.e., H a , p H 0 w/D5J c wd/D), Eq. ~32! can be
rewritten as the following series:
SD
h
2h
M
5
2
2
M 0 nz
nz
2
2
1
`
(
n 2z k51
~ 21 ! k21
k2
3 @ 12e 22k h 2e 22k n z sinh~ 2k h !# .
~34!
Y. MAWATARI
13 218
54
Hysteretic ac loss per unit volume for one field cycle,
P(H m ), is calculated as17
R
m E
P ~ H m ! 52 m 0
52
M ~ H a ! dH a
1H m
0
2H m
M ↓ ~ H a ! dH a
54 m 0 H m M zfc~ H m ! 28 m 0
E
Hm
0
M zfc~ H a ! dH a .
~41!
Substituting Eq. ~32! for M zfc(H a ), Eq. ~41! becomes
P~ Hm!5
2 m 0D 2
p wd
F
E
0
3ln 11
FIG. 2. Profiles of ~a! magnetic field H(x) and ~b! current density J(x) in a Z stack where D/w50.7 ~solid lines! and in an
isolated strip line where D/w→` ~dashed lines! at
H a /H 0 50.5,1,2,3, and 4.
On the other hand, for h . n z ~i.e., H a . p H 0 w/D), we have
`
2
M
~ 21 ! k21 22k h
512 2
e
sinh2 ~ k n z ! .
2
M0
k2
n z k51
(
~35!
In a low applied field ( h !1), Eq. ~32! is reduced to
F
G
M
1
h3
2
. 2 2 h ln~ coshn z ! 2 tanh2 n z 1••• ,
M 0 nz
3
~36!
and in a high field ( h @1), we have
2
M
sinh2 n z
.122e 22 h
1•••.
M0
n 2z
~37!
When the spacing in a Z stack is large ( n z !1), we have
2
n 2z tanhh
M
.tanhh 2
1•••.
M0
6 cosh2 h
~38!
The first term of the right-hand side of Eq. ~38! corresponds
to the magnetization for an isolated strip line.
We denote the magnetization M in an increasing field
(H a .0) after zero-field cooling as M zfc(H a ), which is given
by Eq. ~32!. Then, the magnetization M ↓ (H a ) when H a is
decreasing from a maximum field H m .0 ( u H a u ,H m ) and
the magnetization M ↑ (H a ) when H a is increasing from
2H m are given by4,15,17
M ↓ ~ H a ! 5M zfc~ H m ! 22M zfc
M ↑ ~ H a ! 52M zfc~ H m ! 12M zfc
S
S
D
H m 2H a
,
2
D
dH a ~ H m 22H a !
G
sinh2 ~ p w/D !
,
cosh2 ~ H a /H 0 !
~42!
D. Densely piled Z stack
All results presented in Secs. IIA–IIC with D→` correspond to the results for an isolated strip line, which is considered in Refs. 4–6. For a densely piled Z stack in which
spacing between strip lines is small (D&w), on the other
hand, the H(x) and J(x) profiles are significantly affected by
the interactions between multiple strip lines. Here we consider such a Z stack where D&w (e 22 p w/D !1).
For u x u 2w*D, H(x) in the Meissner state given by Eq.
~11! is approximated as
H ~ u x u .w ! .H a @ 11 21 e 22 p ~ u x u 2w ! /D # .
~43!
For u x u *D and w2 u x u *D, J(x) given by Eq. ~12! is approximated as
J ~ u x u ,w ! .2
2H a
sgn~ x ! e 2 p ~ w2 u x u ! /D .
d
~44!
These two equations show that H(x) is almost same as H a
except near the edge of the strips, and that the current flows
only near the edge.
In the critical state, Eq. ~22! is approximated as
Ha
p
2ln2,
~ w2a ! .
D
H0
~45!
for a*D and w2a*D. Here, the flux-filled region w2a
grows linearly with increasing H a , similar to growth seen in
slab superconductors with parallel fields. Note that w2a
grows as Eq. ~23! in the vicinity of zero field (w2a!D)
even in the densely piled Z stack. For w2a*D,
w.a. u x u *D, and a2 u x u *D, J(x) is given by
~39!
H m 1H a
52M ↓ ~ 2H a ! .
2
~40!
Hm
J ~ u x u ,a ! .2
2
J sgn~ x ! e 2 p ~ a2 u x u ! /D ,
p c
~46!
namely, u J( u x u ,a) u !J c . For w2a*D, H(x) outside the
strip and not so close to the edge ( u x u 2w*D) is given by
54
CRITICAL STATE OF PERIODICALLY ARRANGED . . .
13 219
FIG. 4. Arrangement of strip lines in an X array, in which an
infinite number of strip lines are aligned along the x axis at a period
L. Each strip line has width 2w and thickness d, and is infinitely
long along the y axis.
nth superconducting strip occupies an area where
u x2nL u <w (n50,61,62, . . . ,6`), u y u ,`, and u z u
<d/2.
FIG. 3. Profiles of ~a! magnetic field H(x) and ~b! current density J(x) in a densely piled Z stack where D/w50.2 at
H a /H 0 51,5,10,15, and 20.
H ~ u x u .w ! .H a 1
H 0 22 p ~ u x u 2w ! /D
e
.
2
~47!
In the flux-filled region not so close to either the edge
(w2 u x u *D) or the flux front ( u x u 2a*D), we have
F
G
1 22 p ~ w2 u x u ! /D
w2 u x u
2
H ~ a, u x u ,w ! .H a 2 p H 0
e
.
D
2p
~48!
As shown in Eqs. ~47! and ~48!, except near the flux front
and near the edge of the strip lines, H(x) in a densely piled
Z stack is approximately given by H( u x u .w).H a and
H(a, u x u ,w).H a 2(d/D)J c (w2 u x u ). Figure 3 shows
that the H(x) and J(x) profiles in a Z stack with narrow
spacing (D/w50.2) are similar to those in slab superconductors. The anomaly of H(x) at the edge ( u x u ;w), and at
the flux front ( u x u ;a), becomes small as D/w decreases.
Furthermore, at the narrow spacing limit ( n z @1), M given
by Eq. ~34! is approximated as M /M 0 ;2(2 h / n z )
1( h / n z ) 2 for low fields except in the vicinity of zero field
( h , n z and e 22 h !1), and becomes saturated as
M /M 0 ;21 in high fields ( h . n z ). These results show that
the electromagnetic properties of a densely piled Z stack
with a narrow spacing of D!w in the critical state are similar to those of a slab in which critical current density is
J c d/D.
III. PARALLEL STRIP LINES IN THE xy PLANE
„X ARRAY…
In this section we consider an X array in which an infinite
number of strip lines are aligned in the xy plane. Each superconducting strip has width 2w, thickness d (!w), and is
placed at the same interval L (.2w) as shown in Fig. 4. The
A. Basic equations
When H a is applied parallel to the z axis, the relation
between H(x) and J(x) in an X array is given by
1`
d
H ~ x ! 5H a 2
2 p n52`
(
5H a 2
E
d
2L
1w
2w
E
1w
2w
J ~ u ! du
x2 ~ u1nL !
J ~ u ! cot
S
D
p ~ x2u !
du.
L
~49!
Since J(x) and H(x) are periodic with period L as
J(x1nL)5J(x) and H(x1nL)5H(x), it is sufficient to
consider only the electromagnetic field for u x u ,L/2. By introducing the following transformation of the variables,
S D
px
L
x̃5 tan
,
p
L
S D
pw
L
w̃5 tan
,
p
L
~50!
E
~51!
Eq. ~49! is reduced to
H̃ ~ x̃ ! 5H̃ a 2
d
2p
˜ J̃ ~ ũ !
1w
˜
2w
x̃2ũ
dũ,
where H̃(x̃) is the transformed magnetic field, J̃(x̃) is the
transformed current, and H̃ a is the transformed applied field
given by
H̃ a 5H a 2
d
2p
5H a 2
d
2L
E
E
˜
1w
˜
2w
J̃ ~ ũ !
1w
2w
ũdũ
~ L/ p ! 2 1 ~ ũ ! 2
J ~ u ! tan
S D
pu
du5H ~ x5L/2! . ~52!
L
Note that Eq. ~51! has the same form as Eq. ~1!. Therefore,
H(x) and J(x) of an X array can be derived from those of an
isolated strip line using the transformation given by Eqs. ~50!
and ~52!.
Y. MAWATARI
13 220
54
Furthermore, the transformation of D→iL ~where
i5 A21) in Eqs. ~2!, ~3!, and ~5! lead to Eqs. ~49!, ~50!, and
~52!, respectively. The expressions for H, J, M , and P for an
X array are then easily obtained by the simple transformation
D→iL of the expressions for a Z stack given in Secs. II B
and II C.
B. Meissner state
In the Meissner state, H(x) and J(x) for an X array are
H a sin~ p u x u /L !
H ~ w, u x u ,L/2! 5
J ~ u x u ,w ! 5
Asin ~ p x/L ! 2sin2 ~ p w/L !
2
2 ~ 2H a /d ! sin~ p x/L !
Asin2 ~ p w/L ! 2sin2 ~ p x/L !
,
,
~53!
~54!
and H( u x u ,w)50.
For L22w!L, Eq. ~54! is approximated as
J ~ u x u ,w ! .2
S D
px
2H a
tan
,
d
L
~55!
except near the edge (w2 u x u !w;L/2). The H(x) at the
center of the spacing of strip lines for L22w!L is much
larger than H a ,
H ~ u x u 5L/2! 5
Ha
4w
.
H .
cos~ p w/L ! p ~ L22w ! a
~56!
In the Meissner state, M for an X array is
M 5H a
F S DG
L2
pw
ln cos
p wd
L
~57!
.
FIG. 5. Profiles of ~a! magnetic field H(x) and ~b! current density J(x) in an X array where L/w53 ~solid lines! and in an isolated strip line where L/w→`
~dashed lines! at
H a /H 0 50.5,1,2,3, and 4.
for an isolated strip line. The magnetic field for
w, u x u ,L/2 is enhanced due to the demagnetization effect
of neighboring strip lines.
The magnetization M for an X array in the critical state is
given by
M5
C. Critical state
The flux front a for an X array in the critical state is given
L2
2 p wd
E F
Ha
0
pa
sin~ p w/L !
5
,
L
cosh~ H a /H 0 !
~58!
and H(x) and J(x) are given by Eqs. ~25! and ~26!, where
w (x) for an X array is
A
tan2 ~ p w/L ! 2tan2 ~ p a/L !
.
u tan2 ~ p a/L ! 2tan2 ~ p x/L ! u
~59!
U
U
H 0 sin2 ~ p w/L !
ln
21 ,
2
sin2 ~ p x/L !
~60!
which is approximated for L.2w as
F S DG
H ~ u x u .a ! .H a 1H 0 ln tan
puxu
L
.
2
2
M
~ 21 ! k21 22k h 2
512 2
e
sin ~ k n x ! ,
M0
k2
n x k51
(
~63!
where M 0 5wJ c /2, n x 5 p w/L, and h 5H a /H 0 . For low
fields ( h !1), we have
2
F
M
h3
1
.2 2 2 h ln~ cosn x ! 1 tan2 n x 1•••
M0
3
nx
G
~64!
and, for high fields ( h @1),
For a high field (H a *H 0 ), H(x) is given by
H ~ u x u .a ! .H a 2
~62!
`
S D
tan~ p x/L !
w~ x !5
tan~ p w/L !
G
sin2 ~ p w/L !
dH 8
cosh2 ~ H 8 /H 0 !
and is expanded as the following series:
by
sin
ln 12
~61!
Figure 5 shows H(x) and J(x) profiles for an X array
with L/w53 ~solid lines! and for an isolated strip line
~dashed lines!. Note that the edges of the neighboring strips
are at x/w562. Both profiles for u x u ,w are similar to those
2
M
sin2 n x
.122e 22 h
1•••.
M0
n 2x
~65!
When the spacing in an X array is large ( n x !1), Eq. ~62! is
reduced to
2
n 2x tanhh
M
.tanhh 1
1•••.
M0
6 cosh2 h
~66!
Hysteretic ac loss per unit volume for one field cycle
P(H m ) is
CRITICAL STATE OF PERIODICALLY ARRANGED . . .
54
P ~ H m ! 52
2 m 0L 2
p wd
F
3ln 12
E
Hm
0
dH a ~ H m 22H a !
G
sin2 ~ p w/L !
.
cosh2 ~ H a /H 0 !
~67!
13 221
array, in perpendicular magnetic fields. The magnetic field
H and the current density J for both systems can be derived
by using simple transformations of H and J for an isolated
strip line. For a Z stack in which the distance between the
strips is much smaller than the width, electromagnetic properties in the critical state are similar to those for slab superconductors.
IV. CONCLUSIONS
The electromagnetic properties in the Meissner state and
the critical state were determined analytically for two types
of superconducting strip-array systems, a Z stack and an X
C.P. Bean, Phys. Rev. Lett. 8, 250 ~1962!; Rev. Mod. Phys. 36,
31 ~1964!.
2
G.W. Swan, J. Math. Phys. 9, 1308 ~1968!.
3
W.T. Norris, J. Phys. D 3, 489 ~1970!.
4
E.H. Brandt, M.V. Indenbom, and A. Forkl, Europhys Lett. 22,
735 ~1993!.
5
E.H. Brandt and M. Indenbom, Phys. Rev. B 48, 12 893 ~1993!.
6
E. Zeldov, J.R. Clem, M. McElfresh, and M. Darwin, Phys. Rev.
B 49, 9802 ~1994!.
7
J. McDonald and J.R. Clem, Phys. Rev. B 53, 8643 ~1996!.
8
J. Pearl, Appl. Phys. Lett. 5, 65 ~1964!.
9
P.N. Mikheenko and Yu.E. Kuzovlev, Physica C 204, 229 ~1993!.
1
ACKNOWLEDGMENT
The author would like to thank Dr. H. Yamasaki for his
valuable comments.
10
J. Zhu, J. Mester, J. Lockhart, and J. Turneaure, Physica C 212,
216 ~1993!.
11
J.R. Clem and A. Sanchez, Phys. Rev. B 50, 9355 ~1994!.
12
T. Schuster, H. Kuhn, E.H. Brandt, M. Indenbom, M.R. Koblischka, and M. Konczykowski, Phys. Rev. B 50, 16 684 ~1994!.
13
A. Forkl and H. Kronmüller, Physica C 228, 1 ~1994!.
14
R.I. Khasanov, Yu.I. Talanov, Yu.M. Vashakidze, and G.B.
Teitel’baum, Physica C 242, 333 ~1995!.
15
Y. Mawatari, A. Sawa, and H. Obara, Physica C 258, 121 ~1996!.
16
Y. Mawatari ~unpublished!.
17
E.H. Brandt, Phys. Rev. B 52, 15 442 ~1995!.