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
m
Some useful formulas:
magnetic moment
of a current loop


m  IS [Am2]
I
S

 m
magnetization of a sample M 
[A/m]
V 

m

alternative (preferred in
[T]
M  0
V
SC community)
Measurable quantities:
magnetic field B [T] – Hall probe, NMR
average of
magnetic field
voltage from a pick-up coil [V]
dΨ
dΦ
dB
ui  
 N
  NS
dt
dt
dt
linked magnetic flux
number of turns
Y
area of single turn
ui
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Flux pinning
Coupling currents
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Flux pinning
Coupling currents
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
0
0
0
0
0
0
0
  
FL  j  B
gradient in the flux density
Bz
  0 j y
x
pinning of flux quanta
distribution persists in static regime (DC field), but would
require a work to be changed
=> dissipation in dynamic regime
(repulsive) interaction of flux quanta
=> flux line lattice
summation of microscopic pinning forces
+ elasticity of the flux line lattice
= macroscopic pinning force density Fp [N/m3]
 0  2  1015 Vs
0
B 2
a
0
a
B
B 1T;
macroscopic behavior described by the
critical state model [Bean 1964]:
a  45 nm
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, jc , elsewhere
in the simplest version (first approximation) jc =const.
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Flux pinning
Coupling currents
Transport of electrical current
I
e.g. the critical current measurement
0A
20 A
80 A
20 A
100 A
0A
Transport of electrical current
I
e.g. the critical current measurement
0A
20 A
100 A
j =+ jc
j =0
80 A
20 A
0A
Transport of electrical current
I
e.g. the critical current measurement
0A
20 A
100 A
j =+ jc
j =0
80 A
20 A
0A
Transport of electrical current
I
e.g. the critical current measurement
0A
20 A
100 A
j =+ jc
j =0
80 A
20 A
j =- jc
0A
Transport of electrical current
I
e.g. the critical current measurement
0A
20 A
100 A
j =+ jc
j =0
80 A
20 A
j =- jc
0A
Transport of electrical current
I
e.g. the critical current measurement
0A
20 A
100 A
j =+ jc
j =0
80 A
20 A
0A
j =- jc
persistent magnetization
current
Transport of electrical current
AC cycle with Ia less than Ic : neutral zone
persistent
magnetization
current
80

60

-80

-60

0 A
AC transport in hard superconductor is not dissipation-less (AC loss)
Q   IUdt   Id
T
neutral zone:
j =0, E = 0

U
check for hysteresis in I vs.  plot
Φ
U 
t
AC transport loss in hard superconductor
1.5E-05
1.0E-05
 [Vs/m]
5.0E-06
-150
0.0E+00
-100
-50
0
50
-5.0E-06
-1.0E-05
-1.5E-05
I [A]
hysteresis  dissipation  AC loss
100
150
Hard superconductor in changing magnetic field
0
0


30

50
-50

-40


40 mT
0

50 mT
Hard superconductor in changing magnetic field
0
0


30

50
-50

-40


40 mT
0

50 mT
Hard superconductor in changing magnetic field
0
0


30

50
-50

-40


40 mT
0

50 mT
Hard superconductor in changing magnetic field
0
0


30

50
-50

-40


40 mT
0

50 mT
Hard superconductor in changing magnetic field
0
0


30

50
-50

-40


40 mT
0

50 mT
Hard superconductor in changing magnetic field
0
0


30

50
-50

-40


40 mT
0

50 mT
Hard superconductor in changing magnetic field
0
0


30

50
-50

-40


40 mT
0

50 mT
Hard superconductor in changing magnetic field
dissipation because of flux pinning
Ba
y
x
volume loss density Q
magnetization:
(2D geometry)
[J/m3]
Q
  Ba dM
V
1
M    x. j ( x, y)dxdy
SS
Round wire from hard superconductor in changing magnetic field
3.E+04
Ms
2.E+04
M [A/m]
1.E+04
Bp
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
B [T]
Ms saturation magnetization,
Bp penetration field
Round wire from hard superconductor in changing magnetic field
estimation of AC loss at Ba >> Bp
3.E+04
2.E+04
M [A/m]
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
B [T]
Q
 4 Ba M s
V
0.2
0.25
(infinite) slab in parallel magnetic field – analytical solution
B
penetration field
w
B p   0 jc
2
w Bp
M s  jc 
4 20
w
j
3

2 Ba
Q 1 
3 Bp


V 0 
4 2
2 B p Ba  B p

3
Q  4Ba M s
for Ba<Bp
for Ba>Bp
Slab in parallel magnetic field – analytical solution
1.E+05
Q  4Ba M s
1.E+04
1.E+03
1.E+02
1.E+01
3
Q/V [J/m]
1.E+00
1.E-01
jc=10^8 A/m2, w=1 mm
(Bp = 63 mT)
1.E-02
1.E-03
jc=10^8 A/m2, w=0.1 mm
(Bp = 6.3 mT)
1.E-04
1.E-05
jc=10^7 A/m2, w=1 mm
(Bp = 6.3 mT)
1.E-06
jc=10^7 A/m2, w=0.1 mm
(Bp = 0.63 mT)
1.E-07
1.E-08
1.E-05
1.E-04
1.E-03
Ba [T]
1.E-02
1.E-01
1.E+00
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Flux pinning
Coupling currents
Two parallel superconducting wires in metallic matrix
Ba
coupling currents
in the case of a perfect coupling:
0

20

80

60 mT
Magnetization of two parallel wires
2.E+05
1.E+05
M [A/m]
5.E+04
uncoupled:
0.E+00
-5.E+04
-1.E+05
coupled:
-2.E+05
-0.15
-0.1
-0.05
0
B [T]
0.05
0.1
0.15
Magnetization of two parallel wires
2.E+05
1.E+05
M [A/m]
5.E+04
uncoupled:
0.E+00
-5.E+04
-1.E+05
coupled:
-2.E+05
-0.15
-0.1
-0.05
0
0.05
0.1
B [T]
how to reduce the coupling currents ?
0.15
Composite wires – twisted filaments
lp
 B
1 
good interfaces  t   m
1 
1 
t   m
bad interfaces
1 
S SC

Sm
B
j
j 
l p B
2t
Composite wires – twisted filaments
coupling currents (partially) screen the applied field
Bext
Bi
Bi  Bext  B
t
 - time constant of the magnetic flux diffusion
0

2 t
 lp

 2



2
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
shape factor
(~ aspect ratio)
in AC excitation
Q B
2

2 2
V
0 1   
2
max
round wire
 0
Q B

2 2
V
0 1   
2
max
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Flux pinning
Coupling currents
Persistent currents:
at large fields proportional to Bp ~ jc w
width of superconductor
(perpendicular to the applied
magnetic field)
= magnetization reduction by either lower jc or reduced w
lowering of jc would mean more superconducting material
required to transport the same current
thus only plausible way is the reduction of w
effect of the field orientation
perpendicular field
2.E+05
1.E+05
parallel field
M [A/m]
5.E+04
0.E+00
-5.E+04
-1.E+05
-2.E+05
-0.15
-0.1
-0.05
0
B [T]
Q
 4 Ba M s
V
0.05
0.1
0.15
Magnetization loss in strip with aspect ratio 1:1000
1.E+07
1.E+06
Q
 4 Ba M s
V
H
B
1.E+05
1.E+04
1.E+03
H
B
3
Q/V [J/m ]
1.E+02
1.E+01
1.E+00
1.E-01
1.E-02
parallel
1.E-03
1.E-04
perpendicular
1.E-05
1.E-06
1.E-07
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
HBmax
[A/m]
a/0 [A/m]
1.E+05
1.E+06
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
0  l p


2  t  2



2
transposition length
effective transverse resistivity
= 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
Outline:
1.
Hard superconductor in varying magnetic field
2.
Magnetization currents:
3.
Possibilities for reduction of magnetization currents
4.
Methods to measure magnetization and AC loss
Flux pinning
Coupling currents
Different methods necessary to investigate
•
Wire (strand, tape)
•
Cable
•
Magnet
shape of the excitation field (current) pulse
transition
unipolar
harmonic
relevant information can
be achieved in harmonic
regime
final testing necessary in
actual regime
ideal magnetization loss measurement:
pick-up coil wrapped around the sample
Bext
induced voltage um(t) in one turn:
um
Area S
dm (t )
dB (t )
um (t )  
 S
dt
dt
1
B (t )   Bint (t )dS  Bext (t )  0 M (t )
SS
dM (t ) 
 dBext (t )
um (t )  S 
 0

dt
dt


pick-up coil voltage processed by integration
either numerical or by an electronic integrator:
1
0 M (t )    um (t )dt  Bext (t )
S
Method 1: double pick-up coil system with an electronic integrator :
measuring coil, compensating coil
Bext
1
0 M (t )    u M (t )dt
S
um
uM
uc
1
 int
 Udt
M
=
M
 = /2
AC loss in one magnetization cycle [J/m3]:
T
Q   BdM   B(t )
0
dM
dt
dt
B
 = 3/2
=0
Harmonic AC excitation – use of complex susceptibilities
Bext (t )  Ba cost

0 M (t )  Ba   n ' cosnt   n " sin nt 
n 1
Temperature dependence:
fundamental component n =1
AC loss per cycle
Wq  "
”
Ba2
T
0
energy of magnetic shielding
Ba2
Wm   '
2 0
’
-1
Method 2: Lock-in amplifier
– phase sensitive analysis of voltage signal spectrum
in-phase and out-of-phase signals
U nS 
U nC 
1 2

 uM (t ) sin ntdt
0
1 2

 uM (t ) cos ntdt
Bext  Ba cost
reference signal necessary to set the
frequency
phase
taken from the current energizing the
AC field coil
0
Bext
reference
um
uM
UnS
Lock-in amplifier
uc
UnC
Method 2: Lock-in amplifier – only at harmonic AC excitation
Bext  Ba cost



uM (t )  SBa sin t   n n ' sin nt   n " cosnt 
n1


empty coil
fundamental susceptibility
U1S
U1S
'
1 
1
SBa
UN
 U1C  U1C
"

SBa
UN
sample magnetization
higher harmonic susceptibilities
U nS
n ' 
nU N
U nC
n"
nU N
Real magnetization loss measurement:
Pick-up coil
Calibration necessary
M  C  udt
sample
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:
LOCK-IN
Rogowski
coil
LN2
channel A
channel B
generator
transformer
Im
AMPLIFIER
power supply
Psample  I mU B
sample
Loss measurement from the side of AC power supply:
YI hysteresis loop registration for superconducting magnet (Wilson 1969)
SC magnet
 Udt
Y
RI
I
y
x
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