Estimation of performance of Nb3Sn CICC with thermal strain distribution

In the last few years, the critical temperature (T c) of several ITER cable-in-conduit conductors (CICC) was determined by magnetization measurements at zero current. The distribution of the critical temperatures (T c), caused by variation of strain in the Nb3Sn strands, was found to vary with the number of load cycles. A comparison with mechanical modelling requires the strain distribution, while the T c distribution is sufficient to determine the CICC performance. The current sharing temperature (T cs) is calculated supposing that the measured distribution of T c is representative of its variation along a single strand and that the current is uniformly distributed among the strands. The T cs values, found for background and peak magnetic field, are compared with the results of the DC test in the SULTAN facility. The presented estimation techniques lead to an overestimation of T cs. The differences of the estimated and measured T cs values are discussed including the effect of different current-voltage characteristics of single strands and CIC conductors.


Introduction
For the design of cable-in-conduit conductors (CICCs), it would be desirable to know the relation between the strand and the cable properties. The uncertainty in the prediction of the performance of large Nb3Sn CICCs made it necessary to qualify each of the ITER (International Experimental Thermonuclear Reactor) conductors by tests in the SULTAN (SUpraLeiterTestANlage German acronym for superconductor test facility) facility. In contrast to Nb-Ti, the critical current density of Nb3Sn depends not only on field and temperature but also on the intrinsic strain experienced by the superconducting filaments. Furthermore, Nb3Sn is a brittle material susceptible to filament breakages, when the tensile strain exceeds a certain limit (irr).
It is always possible to explain the measured initial cable performance by a supposed value of the thermal strain and a load dependent extra strain. However, the test of large Nb3Sn CICCs indicated that cyclic loading, performed by ramping up and down the conductor current in the high background field of the SULTAN facility, led in some of the conductors to a decrease of the current sharing temperature (Tcs). The conductor performance was also found to depend on the number of warm-up and cool-down (wucd) cycles. Furthermore, the width of the transition in Nb3Sn cables, described by the power law E = Ec (I/Ic) n (E electric field, Ec = 0.1 µV/cm, I operation current, Ic critical current) defining the n factor, was found to be broadened (i.e. reduced n factor) in comparison with the behavior of single strands. It should be noted that the n factor of Nb-Ti CICC is comparable to that of the strands [1]. The prediction of the CICC performance suffers from the not exactly known thermal strain, strain distribution and the difficulty to distinguish between reversible strain effects and an irreversible degradation.
Mechanical modelling provided first evidence that there exists a distribution of thermal strains in CICCs [2]. Measurements of the cable critical temperature (Tc), performed at the Swiss Plasma Center (SPC) [3], [4], [5], [6], are in line with the existence of a Tc distribution caused by a distribution of the thermal strain. Details of the susceptibility measurements and the procedure to find the Tc distribution are described in [3], [4]. The determination of the thermal strain distribution requires the knowledge of the scaling of the critical current density of the Nb3Sn strands with temperature, field and strain.

Scaling relation for the critical current of Nb3Sn
The critical current density of Nb3Sn strands depends on temperature, magnetic field and the intrinsic strain experienced by the superconducting filaments. One of the most important results is the fact that the dependence of the normalized pinning force Fp = jc  B = C g() h(t) fp(b) can be represented in a form in which the dependencies on temperature, field and strain are separable [7]. Here, jc is the critical current density, B the magnetic field and C a constant. The function g depends only on the intrinsic strain , h only on the reduced temperature t and fp only on the reduced field b. Different parametrizations, proposed by Ekin [8], Durham University [9] and Twente University [10], [11], are compared in [12]. In the ITER project, a modified version of the Twente scaling is used to describe jc(T,B,) [12]. The scaling relations for the critical surface are where t = T/Tc() is the reduced temperature, b = B/Bc2(T,) the reduced magnetic field, s() the strain function, C1 a constant and p and q the low and high field exponent of the pinning force. Equation (2) provides the dependence of the upper critical field (Bc2) on strain and temperature while equation (3) indicates the strain dependence of Tc.
where Bc2m(0,0) is the maximum value of Bc2 at T = 0 and  = 0, while Tcm(0) is the maximum value of Tc at zero strain and the strain function can be expressed as Here, Ca1 and Ca2 are strain fitting constants, 0a is the residual strain component and m the tensile strain at which the maximum critical current is reached. Susceptibility measurements of several cable-in-conduit conductors provided the critical temperature and the broadening of the transition caused by a distribution of the thermal strain. The The strain dependencies of the upper critical field and the critical current can be expressed implicitly using the ratio of Tc() to Tcm(0).
The strand scaling parameters of all studied CICCs are listed in table 1. The value of the constant C1 has been adapted to the witness strand results assuming an intrinsic strain of -0.15% for the measurements on ITER barrels. The left leg of CSJA5 L is made of Hitachi strands, while the right leg CSJA5 R uses Furukawa strands. The strands of TFCN4, TFKO4 and TFRF4 were manufactured by Western Superconducting Technologies (WST), Korea Advanced Technology (KAT) and Bochvar, respectively. In the nomenclature, CS indicates a conductor for the central solenoid, while TF represents a conductor for the toroidal field coil. The conductors were supplied by the Chinese (CN), Japanese (JA), Korean (KO) and Russian (RF) Domestic Agencies.

Measured Tc distributions
For the CSJA5 SULTAN sample, the critical temperature was measured for the initial state and after 3950 load cycles, 5970 load cycles plus one warm-up and cool-down (wucd) and finally after 7920 load cycles and two wucds. The Tc distributions for the two conductor legs, made of different strands, which have been extracted from the measurements [3] are shown in figure 1. The mean values of Tc, defined as Tc,mean = f(Tc)Tc (here f(Tc) is the probability to find Tc in the measured cable crosssection), are listed in table 2. In both conductor legs, cyclic loading leads to an increased mean value of Tc, while the width of the distribution is only weakly affected.
The Tc distributions found for the two legs of TFCN4 are presented in figure 2. For the left leg, the Tc distribution after 1000 load cycles and an additional warm-up and cool-down is significantly broadened, while the effect is much less pronounced for the right leg. In the left leg the mean Tc is reduced from 17.28 to 17.14 K, while it is slightly enhanced from 17.14 K to 17.19 K in TFCN4 R.
The Tc distributions found for the right legs of TFKO4 and TFRF4 are shown in figures 3 and 4, respectively. For TFKO4, the Tc distribution is shifted to lower values and slightly more broadened after 1000 load cycles. The mean Tc values found before and after cyclic loading are 17.30 and 17.19 K, respectively. The wucd leads to a pronounced further broadening of the Tc distribution, while Tc,mean of 17.22 K is not much changed. The initial Tc distribution of TFRF4 is narrower than that after 1000 load cycles and an additional wucd. However, the mean Tc increases from 17.21 to 17.45 K due to the load and thermal cycles.      filaments, which is considered as an estimate of Tcm(0), provides s() = (Tc,mean/Tcm(0)) 3 . Using equations (7) and (8) and the scaling parameters C1, Bc2m(0,0) and Tcm(0), provided in table 1, the critical current per strand can be estimated.
Variation of T until Ic equals the average strand current in the Tcs measurement provides an estimation of the current sharing temperature of the cable. The results are presented in Section 4. This estimation ignores the effects of the broadness of the transition characterized by the n factor in an Ic measurement and the m factor in a Tcs measurement (E = Ec (T/Tcs) m ).
3.3. Estimation of Tcs using the measured Tc distribution Next, we take into account both, the Tc distribution and the width of the transition. In an Ic measurement the evolution of the electric field close to the transition is well described by the power law E = Ec (I/Ic) n , where Ec is the criterion used to define Ic. Empirically it was found that the n factor is mainly a function of the critical current. This means that the n factor is the same for different combinations of T, B and  leading to the same value of Ic [13]. Typically, the dependence of n on Ic can be well represented by the scaling relation n = 1 + r Ic k . In figure 5, the relation between n and Ic is shown for the Hitachi witness strands of CSJA5 L. The values of r and k of the considered strands are provided in table 3. The measured n factors of the CICC in question are also listed in table 3. The measurement of Tcs is used to characterize the performance of the SULTAN samples. In figure 6, the electric field versus temperature data of CSJA5 are presented. The measured data can be well described by the power law E = Ec (T/Tcs) m . In both conductor legs, the m factor decreases with increasing number of load and thermal cycles. As an example the estimation of the initial performance of the left leg of CSJA5 is described. The estimation assumes insulated strands and a uniform current distribution among the strands. Furthermore, it is supposed that the probability to find a value of Tc() along the length of each strand is the same as the measured probability f(Tc) measured in a single cross-section. In the estimation of the CICC performance, we are facing the problem that the parameters of the Twente scaling relation are found from transport measurements, whereas the Tc distribution is obtained from magnetization measurements. The two different types of measurements may lead to systematic differences in the Tc, and hence Tc values, obtained from different types of measurements, should not be used in one Tc ratio. In our analysis, the magnetization measurements of Tc of the cable and the free-standing strands are only used to determine the values of the strain function s() = (Tc()/Tc(filaments)) 3 , whereas the values of the reduced temperature in equations (7) and (8) are based on the scaling parameter Tcm(0) quoted in table 1.
The Tc distributions, presented in figures 1 to 4, do not include a possible contribution of the Lorentz force to the strain distribution, and hence to the Tc distribution. Moreover, the entanglement of field and strain distributions is not known, which may be different at the low-field, high-load side and the high-field, low-load side. For simplicity, Tcs is estimated supposing a constant magnetic field in the      Figure 9. Comparison of the initial electric field versus temperature characteristics of CSJA5 found from a simulation with ncable = 0.4 nstrand(Ic) and a calculation using ncable = 12.2 independent of Ic.
The Tcs values estimated for the initial performance of CSJA5 L are shown in figure 8. The presented data indicate that the value of the estimated Tcs depends weakly on the ratio ncable/nstrand. The estimated Tcs values for Bb and Bp are 7.87 and 7.24 K, respectively, as compared to 6.87 K (blue line) obtained from the SULTAN test.
In a second estimation, the n factor of 12.2, found from the SULTAN test, is used to estimate Tcs. For simplicity, the Ic dependence of n is omitted, i.e. n = 12.2 independent of Ic(Tc()). The results of the two different estimations for Bp are compared in figure 9. The use of the measured cable n factor leads to a slightly lower Tcs of 7.16 K than the use of 0.4 nstrands. The m factor of 51.8, obtained from an estimation using the measured cable n factor, is significantly larger than the measured m factor of 34.7.

Results and discussion
In figure 10, the results of different estimations of Tcs using the peak field are presented for the two legs of CSJA5. In both legs, strain relaxation leads to an increase of the measured Tcs for increasing number of load cycles. All estimations suggest that the Tcs after cyclic loading is increased, which is in line with the trend found from the SULTAN test. However, even for the peak magnetic field the estimated Tcs values are higher than the measured values. The highest values are provided by an estimation based on the mean Tc value. For the left leg, the Tcs measured after 7920 cycles and two wucds is 7.06 K, while an estimation based on the mean Tc leads to a value of 7.92 K. The lowest Tcs value of 7.55 K is obtained from the use of the Tc distribution and the measured cable n factor. The estimated Tcs exceeds the measured one by around 0.5 K. An estimation based on the background field and the measured cable n factor provides a Tcs of 8.12 K, more than 1 K higher than the measured value (see table 4). For the right leg, the measured Tcs after cyclic loading is 7.4 K, while the estimation using Bp and the measured cable n factor leads to 7.96 K, which is 0.56 K higher than the measured value. For Bb the estimation provides a Tcs as high as 8.51 K (see table 4). The gap between measured and estimated Tcs values is in both legs comparable.
In figure 11, the results for both legs of TFCN4 are presented. The initial Tcs values of 6.55 K for the left and 6.58 K for the right leg are nearly identical. The estimation, based on Bp, the cable n factor and the Tc distribution, provides Tcs values of 8.07 K (left leg) and 7.59 K (right leg). The difference of estimated and measured Tcs is as large as 1.52 K for the left leg, while it is 1.01 K for the right leg. For the left leg, the estimated Tcs after cyclic loading is reduced to 7.12 K, while the measured value is 6.35 K. In line with the experimental data, Tcs decreases with cyclic loading, however, the effect is much more pronounced in the simulation. The measured Tcs of the right leg decreases by 0.31 K, whereas the estimated value based on Bp and the cable n factor, increases by 0.1 K, i.e. the difference of the measured and estimated Tcs after cyclic loading is 1.42 K. The results for the right leg of TFKO4 are shown in figure 12. The initial Tcs of 6.89 K drops to 6.6 K after 1000 load cycles plus wucd. The estimation, based on Bp and the cable n factor provides 7.68 K before cyclic loading and 7.39 K after cycling. The corresponding differences of measured and estimated Tcs values are 0.79 K for both, the initial and the final performance. This means that measured and simulated changes in Tcs are identical.
The current sharing temperatures of the right leg of TFRF4 are provided by figure 13. Due to cycling the measured Tcs increases from 5.92 K to 6 K, while the estimation, based on Bp and the cable n factor, provides an increase of Tcs from 7.43 to 7.7 K. Again the estimated Tcs value is significantly larger than the measured one, leading to a difference of 1.7 K after cycling. The estimated increase of Tcs due to strain relaxation is 0.29 K larger than the measured one.   In general, the estimations of Tcs presented in this work overestimate the performance of the considered CIC conductors. An estimation of Tcs based on the mean value of Tc leads to the highest overestimation of the current sharing temperature. The use of the measured cable n factor and the observed Tc distribution leads to the smallest difference of estimated and measured Tcs values. However, this procedure leads to an overestimation of the cable m factor. A simulation using a ratio of ncable to nstrand including variation of nstrand with Ic, which reproduces the measured m factor, leads to Tcs values between the two extremes.
A further piece of information is the variation of Tcs with cycling. Except for the right leg of TFCN4, the estimated Tcs shows the same trend (increase or decrease) as the measured Tcs. A comparison of the changes of the measured initial and final Tcs with the estimation based on the cable n factor, the Tc distribution and the background field (see table 4) indicates that for CSJA5 L the measured value increases by 0.19 K, while the estimation provides an enhancement of 0.37 K. For CSJA5 R, the measured change in Tcs of 0.19 K is even closer to the Tcs of 0.25 K found from the estimation. In case of TFKO4 both, the measured and the estimated change in Tcs, is -0.29 K. The estimated increase of the Tcs of TFRF4 due to strain relaxation is 0.25 K as compared to the measured change of 0.08 K. A considerable discrepancy of measured and estimated changes in Tcs has been found for TFCN4. The test provides a reduction of Tcs after cycling of -0.2 and -0.31 K for the left and right leg, respectively. The estimated change of Tcs is -0.9 K for the left leg and +0.1 K for the right leg.
A possible explanation for the overestimated performance could be an additional strain contribution caused by the Lorentz force, which is always present in the SULTAN Tcs measurement and absent in the measurement of Tc distribution used in the Tcs estimation. The variation of the magnetic field within the conductor cross-section has a significant impact on the estimated Tcs values as can be seen from the differences found for calculations using the background or the peak field. In addition, the Tc distribution may vary along the length of the conductor in the high field zone. A main problem in the estimation of the current sharing temperature is the broadened transition in the cable, i.e. reduced n  11 and m factors. The reduction of the cable n factor can be caused by redistribution of current among the strands and/or irreversible degradation of the strands accompanied by reduced n factors. For CSJA5, showing an improvement of Tcs with cyclic loading, the probability for strand degradation is expected to be much smaller than in conductors showing cyclic load degradation. The difference of estimated and measured Tcs is smallest for this conductor. For all TF conductors, the difference of estimated and measured Tcs is significantly larger than for CSJA5 (see figure 14). A strand degradation cannot be excluded for the TF conductors.

Conclusion
The presented estimations overestimate the performance of the considered CIC conductors. The trends for the changes in Tcs are typically in line with the measured data. However, the estimated changes in Tcs are typically larger than the measured ones. The results suggest that the knowledge of the Tc distribution, caused by thermal strain distribution, provides only an incomplete characterization of Nb3Sn cable-in-conduit conductors. The overestimation of the conductor performance may be partly caused by the unknown contribution of the Lorentz force to the strain experienced by the Nb3Sn strands during the Tcs measurement. Moreover, the transition in cable-in-conduit conductors is broadened, reflected by a reduced n or m factor, which needs to be taken into consideration in an estimate of the cable performance. Finally, the Tc of the Nb3Sn strands is not only broadened because of the strain distribution but also susceptible to variations of the Sn concentration in the Nb3Sn filaments [14] possibly affecting the estimation of the strain function.