1. Superconductor dynamics
Fedor Gömöry
Institute of Electrical Engineering
Slovak Academy of Sciences
Dubravska cesta 9, Bratislava, Slovakia

2. magnetic moment of a current loop [Am2]
Some useful formulas:
magnetic moment of a current loop
[Am2]
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] m I S Y
average of
magnetic field ui
linked magnetic flux
number of turns
area of single turn

3. Outline:
1. Hard superconductor in varying magnetic field
Magnetization currents: Flux pinning
Coupling currents
3. Possibilities for reduction of magnetization currents
4. Methods to measure magnetization and AC loss

4. Outline:
1. Hard superconductor in varying magnetic field
Magnetization currents: Flux pinning
Coupling currents
3. Possibilities for reduction of magnetization currents
4. Methods to measure magnetization and AC loss

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

6. 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
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

7. (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]
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, jc , elsewhere
in the simplest version (first approximation) jc =const.

8. Outline:
1. Hard superconductor in varying magnetic field
Magnetization currents: Flux pinning
Coupling currents
3. Possibilities for reduction of magnetization currents
4. Methods to measure magnetization and AC loss

9. Transport of electrical current e.g. the critical current measurement
0 A A A
80 A A A

10. Transport of electrical current e.g. the critical current measurement
0 A A A
j =+ jc
j =0
80 A A A

11. Transport of electrical current e.g. the critical current measurement
0 A A A
j =+ jc
j =0
80 A A A

12. Transport of electrical current e.g. the critical current measurement
0 A A A
j =+ jc
j =0
80 A A A
j =- jc

13. Transport of electrical current e.g. the critical current measurement
0 A A A
j =+ jc
j =0
80 A A A
j =- jc

14. Transport of electrical current e.g. the critical current measurement
0 A A A
j =+ jc
j =0
80 A A A
j =- jc
persistent magnetization current

15. Transport of electrical current
AC cycle with Ia less than Ic : neutral zone
persistent magnetization current
    A

16. AC transport in hard superconductor is not dissipation-less (AC loss)
neutral zone:
j =0, E = 0  U
check for hysteresis in I vs.  plot

17. AC transport loss in hard superconductor
hysteresis  dissipation  AC loss

18. Hard superconductor in changing magnetic field
   mT
     mT

19. Hard superconductor in changing magnetic field
   mT
     mT

20. Hard superconductor in changing magnetic field
   mT
     mT

21. Hard superconductor in changing magnetic field
   mT
     mT

22. Hard superconductor in changing magnetic field
   mT
     mT

23. Hard superconductor in changing magnetic field
   mT
     mT

24. Hard superconductor in changing magnetic field
   mT
     mT

25. Hard superconductor in changing magnetic field
dissipation because of flux pinning Ba x y
volume loss density Q [J/m3]
magnetization:
(2D geometry)

26. Round wire from hard superconductor in changing magnetic fieldMs Bp
Ms saturation magnetization, Bp penetration field

27. Round wire from hard superconductor in changing magnetic field
estimation of AC loss at Ba >> Bp

28. for Ba<Bp for Ba>Bp
(infinite) slab in parallel magnetic field – analytical solution B
penetration field w
for Ba<Bp j
for Ba>Bp

29. Slab in parallel magnetic field – analytical solution3

30. Outline:
1. Hard superconductor in varying magnetic field
Magnetization currents: Flux pinning
Coupling currents
3. Possibilities for reduction of magnetization currents
4. Methods to measure magnetization and AC loss

31. Two parallel superconducting wires in metallic matrixBa
coupling currents
in the case of a perfect coupling:
   mT

32. Magnetization of two parallel wires
uncoupled:
coupled:

33. Magnetization of two parallel wires
uncoupled:
coupled:
how to reduce the coupling currents ?

34. Composite wires – twisted filamentslp
j
good interfaces
bad interfaces

35. Composite wires – twisted filaments
coupling currents (partially) screen the applied field
Bext Bi t
 - time constant of the magnetic flux diffusion
A.Campbell (1982) Cryogenics 22 3
K. Kwasnitza, S. Clerc (1994) Physica C
K. Kwasnitza, S. Clerc, R. Flukiger, Y. Huang (1999)
Cryogenics
shape factor
(~ aspect ratio)
in AC excitation
round wire

36. Outline:
1. Hard superconductor in varying magnetic field
Magnetization currents: Flux pinning
Coupling currents
3. Possibilities for reduction of magnetization currents
4. Methods to measure magnetization and AC loss

37. at large fields proportional to Bp ~ jc w
Persistent currents:
at large fields proportional to Bp ~ jc w
= 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
width of superconductor (perpendicular to the applied magnetic field)

38. effect of the field orientation
perpendicular field
parallel field

39. Magnetization loss in strip with aspect ratio 1:1000B B
Ba/m0 [A/m]

40. = reduction of the width e.g. striation of CC tapes
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

41. but in operation the filaments are connected at magnet terminations
striation of CC tapes
but in operation the filaments are connected at magnet terminations B B
coupling currents will appear
=> transposition necessary

42. = filaments (in single tape) or strands (in a cable)
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

43. Outline:
1. Hard superconductor in varying magnetic field
Magnetization currents: Flux pinning
Coupling currents
3. Possibilities for reduction of magnetization currents
4. Methods to measure magnetization and AC loss

44. 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

45. ideal magnetization loss measurement:

46. pick-up coil wrapped around the sample
induced voltage um(t) in one turn: um
Area S
pick-up coil voltage processed by integration either numerical or by an electronic integrator:

47. Method 1: double pick-up coil system with an electronic integrator :
measuring coil, compensating coil um uM uc M B M
 = 0
 = 
 = /2
 = 3/2
AC loss in one magnetization cycle [J/m3]:

48. Harmonic AC excitation – use of complex susceptibilities
Temperature dependence:
fundamental component n =1 T ” ’ -1
AC loss per cycle
energy of magnetic shielding

49. Method 2: Lock-in amplifier
– phase sensitive analysis of voltage signal spectrum
in-phase and out-of-phase signals
reference signal necessary to set the
frequency
phase
taken from the current energizing the
AC field coil
reference um uM
UnS
Lock-in amplifier
UnC uc

50. Method 2: Lock-in amplifier – only at harmonic AC excitation
empty coil sample magnetization
fundamental susceptibility
higher harmonic susceptibilities

51. Real magnetization loss measurement:
Pick-up coil
sample
Calibration necessary
by means of:
measurement on a sample
with known properrties
calibration coil
numerical calculation …

52. AC loss can be determined from the balance of energy flows
AC power flow
AC loss in SC object
AC power supply
AC power flow
AC power flow

53. Solution 1- detection of power flow to the sample
AC power flow
AC loss in SC object
AC power supply
AC power flow
AC power flow

54. Solution 2- elimination of parasitic power flows
AC loss in SC object
AC power supply
AC power flow

55. Loss measurement from the side of AC power supply:
sample
power supply

56. Loss measurement from the side of AC power supply:
Y(I) hysteresis loop registration for superconducting magnet (Wilson 1969)
SC magnet y x

57. Conclusions:
Hard superconductors in dynamic regime produce heat because of magnetic flux pinning -> transient loss, AC loss
Extent of dissipation is proportional to macroscopic magnetic moments of currents induced because of the magnetic field change
Hysteresis loss (current loop entirely within the superconductor) reduced by the reduction of superconductor dimension (width)
Coupling loss (currents connecting parallel superconductors) reduced by the transposition (twisting) and the control of transverse resistance
Minimization of loss often in conflict with other requirements
Basic principles are known, particular cases require clever approach and innovative solutions