Superconductors for large scale applications - CAS

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15 Νοε 2013 (πριν από 4 χρόνια και 1 μήνα)

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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.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+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:

YI

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