1. Characterization of the local critical current fluctuation along the length in industrially produced CC tapes Fedor Gömöry, Miro Adámek, Asef Ghabeli, Lubmír Frolek, Ján Šouc Institute of Electrical Engineering, Slovak Academy of Sciences, Bratislava, Slovakia
Acknowledgement for providing the Ic(x) data:
Markus Bauer (THEVA)
Hanwoo Cho (Sunam)

2. Outline Introduction Statistical description (Gauss, Weibull)
“Overall” critical current and prediction of I-V curve
Tape in self-field conditions
Cable with perfect/no current sharing
Pancake coil
Coil from CORT cable, case of perfect/no current sharing
Chances for a current sharing
4) Weak points and appearance of hot spot
5) Conclusions

3. Introduction
starting point for the design of a superconducting device:
Ic(B,q) of the tape
data obtained on a short sample:

4. Introduction data provided by the manufacturer:
design and computation assume a constant Ic: which value should be used?

5. Statistical description
L: tape length N: number of data points
space resolution
Gauss:
mean IG , standard deviation G , probability density
Cumulative distribution probability:
Weibull:
scale IW , shape kW , probability density
Cumulative distribution probability:

6. Statistical description
example:
Gauss:
mean 647 A, standard deviation 17 A
statistical analysis:
Weibull:
scale 654 A, shape 48

7. Statistical description
comparison of two statistical descriptions (data from 6 tapes):
Gauss: better known in the community
Weibull: higher significance of the low performance portion

8. Prediction of I-V curve – tape in self field
Tapestar data: 1 2 3 4 5

9. Prediction of I-V curve – tape in self field
Ic1, Ic2, Ic3, ………………Ici,……………………………………………………... IcN I
average critical current:
assuming the local I-V dependence as power law, with constant n: E I Ec
overall electric field:
overall critical current:
Yinshung Wang et al., Cryogenics

10. Prediction of I-V curve – tape in self field
meaning of the overall critical current
single number sufficient to predict the I-V curve
total energy dissipation
depends only on statistical parameters of the distribution (the same result for reordered values of local Ic,i)
could be directly derived from the parameters of statistical distribution?

11. Prediction of I-V curve – tape in self field
ideal case would be
computing exercise based on real tape data (L = 53 m, x = 1 cm)
Gauss:
mean A
standard deviation 3.5 A
Weibull:
scale A
shape 71

12. Prediction of I-V curve – tape in self field
computing exercise with Gaussian distribution:
artificial data sets generated by modification of  , keeping
tested quantity:

13. Prediction of I-V curve – tape in self field
empirical fit found from this computing exercise:
I-V curve of a single tape is predictable from its Ic statistics,
i.e. the mean value and the standard deviation
then can be determined

14. Prediction of I-V curve – cable from transposed tapes
analysis neglecting the influence of magnetic field created by the current itself
Ic11, Ic12, Ic13, ……………….. Ic1i,…………………………………………………………... Ic1N I
Ic21, Ic22, Ic23, ……………….. Ic2i,…………………………………………………………... Ic2N
IcM1, IcM2, IcM3, ……………….. IcMi,…………………………………………………………... IcMN

15. Prediction of I-V curve – cable from transposed tapes
a) Perfect terminations, no current sharing between j = 1,2,….M tapes along the length
Ic11, Ic12, Ic13, ……………….. Ic1i,…………………………………………………………... Ic1N I
Ic21, Ic22, Ic23, ……………….. Ic2i,…………………………………………………………... Ic2N
IcM1, IcM2, IcM3, ……………….. IcMi,…………………………………………………………... IcMN I1 I2 Im V
at zero termination resistance: 

16. Prediction of I-V curve – cable from transposed tapes
a) Perfect terminations, no current sharing between j = 1,2,….M tapes along the length
Ic11, Ic12, Ic13, ……………….. Ic1i,…………………………………………………………... Ic1N I1
Ic21, Ic22, Ic23, ……………….. Ic2i,…………………………………………………………... Ic2N I2
Icable
IcM1, IcM2, IcM3, ……………….. IcMi,…………………………………………………………... IcMN Im V

17. Prediction of I-V curve – cable from transposed tapes
a) Perfect terminations, no current sharing between j = 1,2,….M tapes along the length
at L >> x
I-V curve of a cable is predictable from the Ic statistics of single tape
overall critical current of the cable is due to Ic fluctuations reduced by (1-tape)

18. Prediction of I-V curve – cable from transposed tapes
b) Perfect terminations, possible current sharing between tapes along the length
Ic11, Ic12, Ic13, ……………….. Ic1i,…………………………………………………………... Ic1N
Icable
Ic21, Ic22, Ic23, ……………….. Ic2i,…………………………………………………………... Ic2N
IcM1, IcM2, IcM3, ……………….. IcMi,…………………………………………………………... IcMN
V(x)
I1(x)
I2(x)
Im(x)

19. Prediction of I-V curve – cable from transposed tapes
b) Perfect terminations, possible current sharing between tapes along the length
Ic11, Ic12, Ic13, ……………….. Ic1i,…………………………………………………………... Ic1N
Icable
Ic21, Ic22, Ic23, ……………….. Ic2i,…………………………………………………………... Ic2N
IcM1, IcM2, IcM3, ……………….. IcMi,…………………………………………………………... IcMN
E(x)
xi=ix
xi-1=(i-1)x

20. Prediction of I-V curve – cable from transposed tapes
b) Perfect terminations, possible current sharing between tapes along the length
at L >> x and M >>1:
in the cable with perfect current sharing the effect of Ic fluctuations is reduced and the mean value of Ic is relevant

21. Prediction of I-V curve – pancake coil
M turns, average radius Rc M
Ic1, Ic2, Ic3, ………………Ici,……………………………………………………... IcN
single piece of tape used for the coil winding
Turn 1
Turn 2
Turn M
in difference to the self-field case, now the turns experience various magnetic fields

22. Prediction of I-V curve – pancake coil
Step 1: estimation of the critical currents of turns made from perfectly uniform tape
Ic calculated using a uniform Jcoil approximation
Ic(B,q) of the tape
Turn 1,2,……j......M
uniform Jcoil -> B(r,z) -> Jc(r,z)
Icoil in each turn

23. Prediction of I-V curve – pancake coil
Step 1: estimation of the critical currents of turns made from perfectly uniform tape
Turn 1,2,……

24. Prediction of I-V curve – pancake coil
Step 2: introduction of fluctuations
assumptions:
2Rc >> x
standard deviation remains the same also in magnetic field
 replacing for each turn the computed Ic, turn, j by the correction derived from the overall computed in self-field conditions
Method 1: using the whole set of Ic(x) data

25. Prediction of I-V curve – pancake coil
Step 2: introduction of fluctuations
Method 2: provided the standard deviation is known, simplified estimation can be done utilizing the relation between  and 
e.g. at  = 4% and n = 20
tape =
and ~ 1.75% reduction of Ic

26. Prediction of I-V curve – coil from CORT cable
10 cm
33 cm
Ján Šouc et al., CCA 2019 Poster

27. Prediction of I-V curve – coil from CORT cable
Step 0:
short sample data on Jc(B, )
Step 1:
local critical currents of a uniform tape
2.41010
Jc [A/m2]
1.51010
uniform Jcoil approximation
0.1
B [T]
coil made of T turns (index t)
cable made of Ntapes = 8 tapes, index k
(2 layers x 4 tapes)

28. Prediction of I-V curve – coil from CORT cable
Alternative 1: no current sharing between parallel tapes
electric field on turn t:
because of tape’s transposition, all the tapes go through all the positions, experience approximately the same magnetic field, and
voltage on the terminations of coil with Nturns, total cable length L:

29. Prediction of I-V curve – coil from CORT cable
Alternative 1: no current sharing between parallel tapes
introduction of critical current fluctuations ( )
voltage on the coil terminations:
e.g. at  = 5% and n = 30
tape =
and ~ 4% reduction of Ic

30. Prediction of I-V curve – coil from CORT cable
Alternative 2: perfect current sharing between parallel tapes
critical current of cable turn t:
because of tape’s transposition, this does not change along the coil turn
voltage on the coil terminations in case of perfect current sharing:

31. Prediction of I-V curve – coil from CORT cable
Alternative 2: perfect current sharing between parallel tapes
at 2R >> x and Ntapes >>1:
current sharing compensates the fluctuations :

32. Prediction of I-V curve – coil from CORT cable
Comparison with experiment: SuperPower tape SCS4050-AP,
IG = 111 A,  =1.58%
cable made of 2 layers x 4 tapes, voltage signals detected on the outer layer

33. Chance for a current sharing
numerical simulation: current transfer to metallic stabilizer (Ag 3 µm)
50% reduction of Ic on 4 mm
current in SC layer
current in stabilizer
4 mm
3 mm
current transfer length ~ 3 mm

34. Appearance of a hot spot
Gauss:
mean : IG = 511 A
standard deviation: 43 A
at n =30

35. Appearance of a hot spot
analytic electro-thermal model for checking the long-term stability
Marc Dhalle, Anne Bergen et al., MT-25 in Amsterdam, talk Or09-03 I

36. Appearance of a hot spot
analytic electro-thermal model for checking the long-term stability
dimensionless time:
dimensionless temperature:
Marc Dhalle, Anne Bergen et al., MT-25 in Amsterdam, talk Or09-03

37. Appearance of a hot spot
Computation using the properties of a typical FCL tape
Ic = 500 A
Tc = 87 K
k(T0) = 45 W/(K.m)
D = 140 W/(K2m)
Cp = 15 J/(K.m)

38. Conclusions
Tape characterization by the mean value and standard deviation of Ic allows to predict the I-V curve for a tape with Ic fluctuations
I-V curve is controlled by the “overall critical current” always lower than the mean
Current sharing among parallel tapes in cabled conductors reduces the effect of Ic fluctuations and the overall critical current could be identical to the Ic mean value
For a complete tape characterization it is necessary to know the minimum value of Ic in order to check the probability of hot spot creation by an analytical model