Superconductor dynamics
Fedor Gömöry
Institute of Electrical Engineering
Slovak Academy of Sciences
Dubravska cesta 9, 84101 Bratislava,
Slovakia
elekgomo@savba.sk
www.elu.sav.sk
Some useful formulas:
magnetic moment
of a current loop
[Am
2
]
magnetization
of a sample
[A/m]
alternative (preferred in
[T]
SC community)
Measurable quantities:
magnetic field
B [T]
–
Hall probe, NMR
voltage
from a pick

up coil [V]
I
S
m
S
I
m
V
m
M
V
m
M
0
t
B
NS
t
Φ
N
t
Ψ
u
i
d
d
d
d
d
d
u
i
Y
linked magnetic flux
number of turns
area of single turn
average of
magnetic field
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
Flux pinning
Coupling currents
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
Flux pinning
Coupling currents
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Superconductors used in magnets

what is essential?
type II. superconductor (critical field)
high transport current density
Superconductors used in magnets

what is essential?
type II. superconductor (critical field)
mechanism(s) hindering the change of magnetic field distribution
=>
pinning of magnetic flux
= hard superconductor
gradient in the flux density
pinning of flux quanta
distribution persists in static regime (DC field), but would
require a work to be changed
=> dissipation in dynamic regime
y
z
j
x
B
0
0
0
0
0
0
0
0
B
j
F
L
macroscopic behavior described by the
critical state model [Bean 1964]:
local density of electrical current in hard superconductor is
either 0 in the places that have not experienced any electric field
or the critical current density
,
j
c
, elsewhere
in the simplest version (first approximation)
j
c
=const.
(repulsive) interaction of flux quanta
=> flux line lattice
summation of microscopic pinning forces
+ elasticity of the flux line lattice
= macroscopic pinning force density
F
p
[N/m
3
]
nm
45
T;
1
B
Vs
10
2
0
2
0
15
0
a
B
B
a
a
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
Flux pinning
Coupling currents
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Transport of electrical current
e.g. the critical current measurement
0 A
20 A
100 A
80 A
20 A
0 A
I
j =+ j
c
j =0
Transport of electrical current
e.g. the critical current measurement
0 A
20 A
100 A
80 A
20 A
0 A
I
j =+ j
c
j =0
Transport of electrical current
e.g. the critical current measurement
0 A
20 A
100 A
80 A
20 A
0 A
I
j =+ j
c
j =0
Transport of electrical current
e.g. the critical current measurement
0 A
20 A
100 A
80 A
20 A
0 A
j =

j
c
I
Transport of electrical current
e.g. the critical current measurement
0 A
20 A
100 A
80 A
20 A
0 A
j =+ j
c
j =

j
c
j =0
I
Transport of electrical current
e.g. the critical current measurement
0 A
20 A
100 A
80 A
20 A
0 A
j =+ j
c
j =

j
c
j =0
I
p
ersistent
magnetization
current
Transport of electrical current
AC cycle with
I
a
less than
I
c
:
neutral zone
80
60

80

60
0 A
persistent
magnetization
current
T
Id
IUdt
Q
n
eutral
zone:
j =
0
, E =
0
U
t
Φ
U
check for hysteresis in
I vs.
plot
AC transport in hard superconductor is not dissipation

less (AC loss)
AC transport loss in hard superconductor
hysteresis
dissipation
AC loss

1.5E

05

1.0E

05

5.0E

06
0.0E+00
5.0E

06
1.0E

05
1.5E

05

150

100

50
0
50
100
150
[Vs/m]
I [A]
Hard superconductor in changing magnetic field
0
30
50
40
mT
0

50

40
0
50
mT
Hard superconductor in changing magnetic field
0
30
50
40
mT
0

50

40
0
50
mT
Hard superconductor in changing magnetic field
0
30
50
40
mT
0

50

40
0
50
mT
Hard superconductor in changing magnetic field
0
30
50
40
mT
0

50

40
0
50
mT
Hard superconductor in changing magnetic field
0
30
50
40
mT
0

50

40
0
50
mT
Hard superconductor in changing magnetic field
0
30
50
40
mT
0

50

40
0
50
mT
Hard superconductor in changing magnetic field
0
30
50
40
mT
0

50

40
0
50
mT
volume loss density Q [J/m
3
]
magnetization:
(
2D geometry)
Hard superconductor in changing magnetic field
d
issipation
because of flux pinning
M
B
V
Q
a
d
S
y
x
y
x
j
x
S
M
d
d
)
,
(
.
1
B
a
x
y
Round wire from hard superconductor in changing magnetic field

3
.E+
04

2
.E+
04

1
.E+
04
0
.E+
00
1
.E+
04
2
.E+
04
3
.E+
04

0
.
06

0
.
04

0
.
02
0
0
.
02
0
.
04
0
.
06
M [A/m]
B [T]
B
p
M
s
M
s
saturation magnetization
,
B
p
p
enetration
field
Round wire from hard superconductor in changing magnetic field

3
.E+
04

2
.E+
04

1
.E+
04
0
.E+
00
1
.E+
04
2
.E+
04
3
.E+
04

0
.
25

0
.
2

0
.
15

0
.
1

0
.
05
0
0
.
05
0
.
1
0
.
15
0
.
2
0
.
25
M [A/m]
B [T]
estimation
of AC loss at
B
a
>>
B
p
s
a
M
B
V
Q
4
(infinite) slab in parallel magnetic field
–
analytical solution
s
a
M
B
Q
4
j
B
penetration field
w
2
0
w
j
B
c
p
2
3
0
3
4
2
3
2
1
p
a
p
p
a
B
B
B
B
B
V
Q
0
2
4
p
c
s
B
w
j
M
for B
a
<B
p
for B
a
>B
p
Slab in parallel magnetic field
–
analytical solution
s
a
M
B
Q
4
1
.E

08
1
.E

07
1
.E

06
1
.E

05
1
.E

04
1
.E

03
1
.E

02
1
.E

01
1
.E+
00
1
.E+
01
1
.E+
02
1
.E+
03
1
.E+
04
1
.E+
05
1
.E

05
1
.E

04
1
.E

03
1
.E

02
1
.E

01
1
.E+
00
Q/V [J/m]
B
a
[T]
jc=
10
^
8
A/m
2
, w=
1
mm
(Bp =
63
mT)
jc=
10
^
8
A/m
2
, w=
0
.
1
mm
(Bp =
6
.
3
mT)
jc=
10
^
7
A/m
2
, w=
1
mm
(Bp =
6
.
3
mT)
jc=
10
^
7
A/m
2
, w=
0
.
1
mm
(Bp =
0
.
63
mT)
3
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
Flux pinning
Coupling currents
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Two parallel superconducting wires in metallic matrix
c
oupling
currents
B
a
0
20
80
60
mT
in the case of a perfect coupling:
Magnetization of two parallel wires

2
.E+
05

1
.E+
05

5
.E+
04
0
.E+
00
5
.E+
04
1
.E+
05
2
.E+
05

0
.
15

0
.
1

0
.
05
0
0
.
05
0
.
1
0
.
15
M [A/m]
B [T]
coupled:
uncoupled:
Magnetization of two parallel wires

2
.E+
05

1
.E+
05

5
.E+
04
0
.E+
00
5
.E+
04
1
.E+
05
2
.E+
05

0
.
15

0
.
1

0
.
05
0
0
.
05
0
.
1
0
.
15
M [A/m]
B [T]
coupled:
uncoupled:
how to reduce the coupling currents ?
Composite wires
–
twisted filaments
B
j
B
l
p
t
p
B
l
j
2
1
1
1
1
m
t
m
t
good interfaces
bad interfaces
m
SC
S
S
Composite wires
–
twisted filaments
coupling currents (partially) screen the applied field
B
B
B
ext
i

time constant of the magnetic flux diffusion
2
0
2
2
p
t
l
2
2
0
2
max
1
2
B
V
Q
2
2
0
0
2
max
1
B
V
Q
round wire
A.Campbell (1982) Cryogenics 22 3
K. Kwasnitza, S. Clerc (1994) Physica C 233 423
K. Kwasnitza, S. Clerc, R. Flukiger, Y. Huang (1999)
Cryogenics 39 829
B
ext
t
B
i
in AC excitation
shape factor
(
~
aspect ratio)
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
Flux pinning
Coupling currents
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Persi
s
tent currents:
at large fields proportional to
B
p
~
j
c
w
=
magn
e
tization
reduction by either lower
j
c
or reduced
w
lowering of
j
c
would mean more superconducting material
required to transport the same current
thus only plausible way is the
reduction of
w
width of superconductor
(perpendicular to the applied
magnetic field)
effect of the field orientation
s
a
M
B
V
Q
4

2
.E+
05

1
.E+
05

5
.E+
04
0
.E+
00
5
.E+
04
1
.E+
05
2
.E+
05

0
.
15

0
.
1

0
.
05
0
0
.
05
0
.
1
0
.
15
M [A/m]
B [T]
p
erpendicular
field
parallel field
Magnetization loss in strip with aspect ratio 1:1000
1.E07
1.E06
1.E05
1.E04
1.E03
1.E02
1.E01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
H
max
[A/m]
Q/V [J/m
3
]
parallel
perpendicular
H
H
s
a
M
B
V
Q
4
B
a
/
0
[A/m]
B
B
in the case of flat wire or cable the orientation is not a free parameter
= reduction of the width
e.g. striation of CC tapes
B
B
~ 6 times lower magnetization
striation of CC tapes
but in operation the filaments are connected at magnet terminations
B
B
coupling currents will appear
=> transposition necessary
Coupling currents:
at low frequencies proportional to the time constant of
magnetic flux diffusion
= filaments (in single tape) or strands (in a cable)
should be transposed
= low loss requires high inter

filament or inter

strand
resistivity
but good stability needs the opposite
transposition length
effective transverse resistivity
2
0
2
2
p
t
l
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
Flux pinning
Coupling currents
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Different methods necessary to investigate
•
Wire (strand, tape)
•
Cable
•
Magnet
relevant information can
be achieved in harmonic
regime
final testing necessary in
actual regime
shape of the excitation field (current) pulse
transition
unipolar
harmonic
ideal magnetization loss measurement:
ext
B
Area
S
)
(
)
(
d
)
(
1
)
(
d
)
(
d
d
)
(
d
)
(
0
int
t
M
t
B
S
t
B
S
t
B
t
t
B
S
t
t
t
u
ext
S
m
m
pick

up coil wrapped around the sample
induced voltage
u
m
(
t
) in one turn:
dt
t
dM
dt
t
dB
S
t
u
ext
m
)
(
)
(
)
(
0
u
m
pick

up coil voltage processed by integration
either numerical or by an electronic integrator:
)
(
d
)
(
1
)
(
0
t
B
t
t
u
S
t
M
ext
m
Method 1: double pick

up coil system with an electronic integrator :
measuring coil, compensating coil
t
t
u
S
t
M
M
d
)
(
1
)
(
0
T
t
t
M
t
B
M
B
Q
0
d
d
d
)
(
d
B
M
= 0
=
=
/2
= 3
/2
AC loss in one magnetization cycle [J/m
3
]:
ext
B
u
m
u
c
u
M
dt
1
int
U
M
Harmonic AC excitation
–
use of complex susceptibilities
1
0
sin
"
cos
'
)
(
cos
)
(
n
n
n
a
a
ext
t
n
t
n
B
t
M
t
B
t
B
fundamental component
n
=1
0
2
"
a
q
B
W
0
2
2
'
a
m
B
W
Temperature dependence:
T
”
’

1
AC loss per cycle
energy of magnetic shielding
Method 2: Lock

in amplifier
–
phase sensitive analysis of voltage signal spectrum
in

phase and out

of

phase signals
2
0
2
0
d
cos
)
(
1
d
sin
)
(
1
t
t
n
t
u
U
t
t
n
t
u
U
M
nC
M
nS
r
eference signal necessary to set the
frequency
phase
taken from
the current energizing the
AC
field coil
t
B
B
a
ext
cos
ext
B
u
m
u
c
u
M
Lock

in amplifier
reference
U
nS
U
nC
Method 2: Lock

in amplifier
–
only at harmonic AC excitation
t
B
B
a
ext
cos
1
cos
"
sin
'
sin
)
(
n
n
n
a
M
t
n
t
n
n
t
B
S
t
u
empty coil
sample magnetization
N
C
a
C
N
S
a
S
U
U
B
S
U
U
U
B
S
U
1
1
1
1
"
1
1
'
N
nC
n
N
nS
n
nU
U
nU
U
"
'
fundamental susceptibility
higher harmonic susceptibilities
Real magnetization loss measurement:
Pick

up coil
sample
Calibration necessary
t
u
C
M
d
by means of:
measurement on a sample
with known properrties
calibration coil
numerical calculation
…
AC loss can be determined from the balance of energy flows
AC power
supply
AC power flow
AC loss in
SC object
Solution 1

detection of power flow to the sample
AC power
supply
AC power flow
AC loss in
SC object
Solution 2

elimination of parasitic power flows
AC power
supply
AC power flow
AC loss in
SC object
Loss measurement from the side of AC power supply:
AMPLIFIER
LOCK

IN
channel A
channel B
generator
Rogowski
coil
transformer
LN
2
I
m
sample
B
m
sample
U
I
P
power supply
Loss measurement from the side of AC power supply:
YI
hysteresis loop registration for superconducting magnet (Wilson 1969)
Y
dt
U
I
RI
x
y
SC magnet
Conclusions:
1)
Hard superconductors in dynamic regime produce heat because of
magnetic flux pinning

> transient loss, AC loss
2)
Extent of dissipation is proportional to macroscopic magnetic
moments of currents induced because of the magnetic field change
3)
Hysteresis loss (current loop
entirely within
the superconductor)
reduced by the reduction of superconductor dimension (width)
4)
Coupling loss (currents connecting parallel superconductors)
reduced by the transposition (twisting) and the control of
transverse resistance
5)
Minimization of loss often in conflict with other requirements
6)
Basic principles are known, particular cases require clever
approach and innovative solutions
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