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)

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

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

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

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:  

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

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

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

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 43 71-77 920030

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?  

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 187.5 A standard deviation 3.5 A Weibull: scale 188.5 A shape 71

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

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

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

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:   

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    

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)

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)

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        

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

Prediction of I-V curve – pancake coil M turns, average radius Rc 1 2................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

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    

Prediction of I-V curve – pancake coil Step 1: estimation of the critical currents of turns made from perfectly uniform tape Turn 1,2,……......10

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

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 = 0.0173   and ~ 1.75% reduction of Ic

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

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)  

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:  

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 = 0.0423   and ~ 4% reduction of Ic

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:  

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 :    

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

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

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

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          

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

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)

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