2.1. AC susceptibility

The complex AC susceptibility $\chi ={\chi }^{\prime }-j{\chi }^{″}$ (${\chi }^{\prime }$ and ${\chi }^{″}$ are real and imaginary parts of $\chi$) of an infinitely long superconducting strand with a round cross-section of radius a was numerically calculated with a uniform transverse ac field based on the critical-state model with a constant J_c before. D-X Chen and E. Pardo made such calculations based on the critical-state model using a magnetic energy minimization (MEM) method, and the results were presented as tables and figures in Ref.[16,17].

Considering a superconducting strand placed in a uniform applied ac field $H_{m}sin\omega t$ in transverse direction, we calculate its complex AC susceptibility $\chi ={\chi }^{\prime }-j{\chi }^{″}$ based on the critical-state model with a constant J_c. For a superconducting strand magnetized by a transverse applied field H and starting from the zero-field cooled state along the initial magnetization curve Mini(H) with increasing H from 0 to H_m, the screening currents profiles are calculated using the procedure detailed in^[18]. Finally, ${\chi }^{\prime }$ and ${\chi }^{″}$ as functions of H_m are obtained from Fourier analysis in the following^[19]:

where m(t) is the magnetic moment for an ac magnetic field. Hence, AC susceptibility ${\chi }^{\prime }+i{{\chi }^{″}}_{}$ is calculated as:

As we can see in Eq. (1) and (2), ${\chi }^{\prime }$=${\chi }^{″}$=0 for even n. The AC susceptibility for odd n is given by:

According to the above equations, ${\chi }^{\prime }$ and ${\chi }^{″}$ versus H_m could be obtained using Matlab software. It is interesting that ${\chi }^{\prime }$ and ${\chi }^{″}$ versus H_m/H_p (H_p is the full penetration field) functions for a long cylinder in longitudinal and transverse fields are very similar after we use MEM method, so we may use the exact analytical result of the longitudinal cylinder to approximate that of the transverse cylinder strand. The expressions of ${\chi }^{\prime }$ and ${\chi }^{″}$ are as follows^[17]:

when $x=H_m/H_p\le 1,$ and

when $x\ge 1.$ where the demagnetizing factor-based parameter called shape susceptibility ${\chi }_0=2$ and $H_p=2J_{c}a/\pi$ for round cross-section of strand and J_c is the critical current density.

2.2. Hysteresis loss

The losses in high field superconductors exposed to changing magnetic fields arise from the irreversible nature of flux pinning. As the field rises (or falls), flux moves into (or out of) the superconductor which means the individual vortices must break free of their pinning centers and move through the material. This movement is an irreversible dissipative process which generates heat. Here we follow an approximate approach, based on the following assumptions:

● the critical current density is uniform in the filament and strand cross section;

● the n-value of CICC strands is relatively so large that the critical current density can be taken to be constant for the AC susceptibility calculation;

● transport current effects are neglected and the cable is not saturated.

It gives rise to an irreversible magnetization curve like the innermost loop and the energy dissipation Q (J/m³) around a cycle is given by^[4]:

where M is the magnetization, defined as the magnetic moment per unit volume, produced by the screening currents caused by magnetic field, ${\mu }_0$ is permeability of vacuum and H is the external field, if we take the AC susceptibility into consideration for the calculation of area of magnetization curve, hysteresis loss Q_hys can be also determined from:

where H_m is the amplitude of the applied AC magnetic field, ${\chi }^{″}$ is the imaginary part of the AC susceptibility $\chi$ which have discussed in section 2.1.

The biggest difference between this method for hysteresis loss calculation and the classical method described in^[4] is that AC susceptibility which could be obtained by experiment or parameterized function from Eq. (4)-(7) easily. Therefore, this method is considered to be an effective way to calculate hysteresis loss for CICCs.

2.3. Coupling loss

The coupling loss is originated from the inductive loops whose resistive couplings are distributed among the strands in cable or within the strands. And this loss depends on the characteristics of the cable such as twist pitch, matrix resistivity of conductor, presence of wraps, coating or solder. It can be estimated by the well-known formulas in the following^[4]:

where P is the loss power per unit volume, n $\tau$ is coupling time constant of the conductor and B_i is the rate of change of field inside the conductor. For steady-state regimes, we may assume that $\frac{\partial B_i}{\partial t}=\frac{\partial B_e}{\partial t}$, where B_e is the external changing field. But for small ripples where the induced currents screen the inside of the strand, a more detailed equation is expressed by:

The general solution for B_i is:

For applied field $B_e=\Delta B\sin \omega t$,

The time constant $\tau$ is the decay time constant of the induced currents. For monolithic conductors the time constant can be written as^[5]:

where L_p is the twist pitch (or the characteristic length of a coupling current path) and ${\rho }_{\perp }$ is the effective electrical resistivity in the transverse direction which can be obtained from experiment.

Assume one only dominant coupling loss time constant is present (inter-filament coupling loss), then the coupling loss over an extended frequency range Q_cpl is given by^[4]:

where B_a is the amplitude of the applied AC magnetic field and n is the shape factor and equal to 2 for round strand. Inter-strand coupling loss is generally treated analogously to inter-filament coupling loss. Therefore, the equations presented above are also used for a cable by adopting an appropriate effective time constant ${\tau }_{eff}$, which is the sum over the N stages composing the cable of the multiple time constants n_i, which based on the assumption that the coupling currents in a given stage do not interfere with the coupling currents of the other stages, detailed in^[4,27].

The method enables an easy treatment of any number of cabling stages. Consequently, this is the most frequently used model for the calculation of inter-strand coupling loss in CIC conductors.

3.1 Sultan samples

The selected conductor called China’s forth toroidal field (TF) Conductor (TFCN4) is based on composite Nb₃Sn internal tin strands manufactured by Western Superconducting Technologies Company (WST). Cabling and jacketing operations are performed in collaboration between Institute of Plasma Physics Chinese Academy of Sciences (ASIPP) and Baosheng Company in China according to a layout determined by ITER TF conductor specification^[20].The cable layout was the same for all conductors. Every cable contains 900 Nb₃Sn strands with 522 copper strands cabled together around a central cooling spiral. The jacket for conductor is compacted using the compacting machine with a set of rollers. The compaction was done in one single step by a set of rollers. The after compaction conductor’s outer diameter is 43.75 mm, inner diameter is 39.75 mm, so the jacket tube thickness is 2 mm. The entire sample was heat-treated according to the following temperature schedule at Centre De Recherché En Physique Des Plasmas (CRPP) Villigen, Switzerland: 210 ℃ for 50 hours, 340 ℃ for 25 hours, 450 ℃ for 25 hours, 575 ℃ for 100 hours and 650℃ for 100 hours. The rate of temperature change is 5 ℃/h.

Typical cross sections of the CNTF4 strand and conductor are provided in Fig. 1. The properties of the strands and conductor used for the manufacturing of the tested Sultan samples are summarized in Table 1 and Table 2.

Fig. 1.Full-screen View

ITER-type TF Nb₃Sn internal tin strand (0.82 mm in diameter) cross section (left) and cross section of TF CICC with diameter of 43.7 mm (right)

3.2. Sultan test

A hairpin type TF sample tested in the SULTAN facility is made of two 3.5 m parallel conductor sections (legs) which are connected with an overlapping joint^[21]. The two legs are identified by the positions. Once a sample is completely done, it is inserted into the SULTAN facility where it is cooled down to 4.2 K. The SULTAN magnet generates a high magnet field of almost 11 T which covers 450 mm of sample length^[22]. The sample condition is monitored by means of 24 voltage taps and 16 thermometers distributed on both legs.

In general, the measurement of AC losses is based on the traditional approach, the so-called calorimetric measurement in SULTAN which is primarily devoted to qualification tests of full-size CICCs for fusion magnets. The boundary conditions of CICC sample in SULTAN test is schematically illustrated in Fig. 2^[12]. Calorimetric method is the most direct methods and offer from medium to high sensitivity and good precision for both small samples. The AC loss, E, is measured by gas flow calorimetric method and is evaluated from the increase of the coolant temperature ΔT (ΔT=T2-T1, is shown in Fig. 2), the coolant specific heat C_p, the coolant mass flow rate dm/dt, AC field frequency f and the cable volume V. The AC loss per cm³ and per cycle of CICC (E) can be calculated by^[12]:

Fig. 2.Full-screen View

CICC sample boundary conditions in Sultan

The SULTAN test consists of measurements in AC and DC conditions. Both types of measurements are performed on the sample at initial condition after it is inserted into the SULTAN testing well and cooled down to 4.2 K. The total loss of the conductor can be measured from the AC test, while the transition index n and the current sharing temperature T_cs are derived from I_c measurement during DC test. Then a certain number of electromagnetic sinusoidal cyclic loadings are performed with ramping-up and down operating current of 68 kA in a background field of 10.78 T.

4.1. AC susceptibility

According to the calculation procedures described in section 2.1, ${\chi }^{\prime }$ and ${\chi }^{″}$ as function of critical current density J_c at different amplitude of applied field $\Delta B$ are plotted in Fig. 3. As the Nb₃Sn strands are very sensitive to the strain and experience severe loads by electromagnetic force and thermal contraction in the TF conductor, J_c was obtained from test result of Pacman facility at the University of Twente^[22]. The strain description model (DEV-model) is used to calculate the critical current of the Nb₃Sn composites. The parameters for ITER scaling law are listed in Table 3^[24,25]. The result of the I_c calculation of CNTF4 strand is summarized in Fig. 4 as a function of strain, temperature and magnetic field. And the critical current density J_c of the strand is 5.68×10⁹ A/m² (Cu: non-Cu = 1, critical current I_c is about 1500 A at DC field B_dc=2 T, T=4.2 K in SULTAN test). And for the TFCN4 strands with applied magnetic field $B_{ac}=0.3\sin \omega t$ (T) in Sultan, hence ${\chi }^{\prime }$ and ${\chi }^{″}$ are approximately -1.6942 and 0.1257.

Fig. 3.Full-screen View

$-{\chi }^{\prime }$ and ${\chi }^{″}$ as function of critical current density J_c with different amplitude of applied field $\Delta B$=$\pm$0.3 T, $\pm$ 0.2 T, $\pm$ 0.1 T, respectively. The left of y-axis represents $-{\chi }^{\prime }$ and the right of y-axis represents ${\chi }^{″}$.

Fig. 4.Full-screen View

Critical current (I_c) of CNTF4 strand as a function of applied strain using DEV-model. I_c with square-solid symbol was calculated at magnetic field B= 14 T, 12 T, 10 T, 8 T, 6 T, respectively when temperature T= 4.2 K which is shown on the left of y-axis, also, I_c with triangle-dash dot symbol was calculated at temperature T=4.2 K, 6 K, 8 K, 10 K, 12 K, respectively when magnetic field B= 2 T which is shown on the left of y-axis.

It is worth explaining that AC susceptibility of small samples with about 1 mm diameter can be obtained by test under the comparatively high frequency condition, but the CIC conductors with large diameter cannot be tested so far. But fortunately, AC susceptibility only depends on the characteristics such as J_c and the conductor diameter, so we can calculate AC susceptibility of the size of conductors (not only TFCN4 conductor) using the equations in section 2.1.

4.2. AC loss

For the sake of providing the result of Sultan test, we calculate AC loss under the same conditions as Sultan, i.e. the background magnetic field B_dc=2 (T), the alternating field $B_{ac}=0.3\sin \omega t$ (T) and the transport current I=0 (A). So from the above analysis, main results of the calculation of hysteresis loss as function of critical current density at different $\Delta B$ based on ac susceptibility, coupling loss and total loss as function of frequency (the coupling loss constant $n\tau$ is about 122.78 ms before cycling from the well-known equations in section 2.3 and^[4,27].) are presented in Fig. 5 and 6, respectively. From Fig. 3 and 5, the hysteresis loss is considered to be independent of frequency due to the assumptions in section 2.2. From Fig. 5, it is obviously seen that the hysteresis loss increase firstly and then decrease with the increase of critical current density. Essentially, the conductor will be fully penetrated due to the low critical current density and then the hysteresis loss is proportional to J_c in low critical current density region. On the contrary, the conductor is not penetrated when the critical current density is high and then hysteresis loss is inversely proportional to J_c in high critical current density region. As to the Sultan sample, the background magnetic field is small (about 2 T) and then critical current density is high according to scaling law in table.3, in this case, hysteresis loss is very small compared to coupling loss for most of Sultan conductors. But in the real operation environment of ITER (at above 10 T background and fast transport current), the conductor may be fully penetrated and the hysteresis loss would be high.

Fig. 5.Full-screen View

Hysteresis loss as function of critical current density at different $\Delta B$

Fig. 6Full-screen View

The numerical results of coupling loss and total AC loss of CNTF4 as function of frequency with different amplitude of applied field $\Delta$B=±0.1T, ±0.2T, ±0.3T, respectively. The symbol with solid line represents total AC loss while the symbol with dash line represents coupling loss. Here the definition of volume is all the strands volume in conductor.

Meanwhile, AC loss of TFCN4 is measured in the transverse field of ±0.3 T alternating with frequencies from 0.1 Hz to 1 Hz in a background field of 2 T before and after the current load cycling. AC test result per cycle for TFCN4 before and after the cycling for the whole range of frequencies is given in Fig. 7^[28]. After cycling, the AC loss decreased significantly, which is due to the significant increase of the contact resistance in conductors with Cr coated Nb₃Sn strands. In Cr coated strands, the contact resistance was observed to decrease during the heat treatment, due to the oxygen diffusion, and increase again when transverse forces are applied during the load cycling. The contact resistance remains high when the cable is unloaded^[29]. Therefore, before electromagnetic load cycling is applied, AC loss of the CIC conductors was larger.

Fig. 7.Full-screen View

Results of TFCN4 sample ac losses investigation in SULTAN test facility

It should be pointed that the coupling loss per unit volume of the conductor per cycle increases linearly with frequency and with the square of amplitude of magnetic field, which can deduce the effective coupling current time constant $n\tau$^[4] from the slope of the results at the low frequency range shown in Fig. 7:

For applied field $B=\Delta B\sin \omega t$,

where $\alpha$ is the slope of the results at the low frequency range. From Eq. (13) and (14), the $n\tau$ obtained from the initial slope of the losses curve (f: 0.1 Hz-0.2 Hz) is about 137.21 ms. The comparison between numerical method and measurement results of AC loss for TFCN4 is shown in Fig. 8 and the results show a relatively good agreement (at B_dc = 2 T, T = 4.2 K, $B_{ac}=0.3\sin \omega t$ (T)). As we can see in Fig. 8, experimental results are found to be deviating from the numerical results. On the one hand, AC susceptibility is complicated at very low frequency because the hysteretic losses due to pinning and the eddy-current losses are not simply additive which was described in paper “Low-frequency AC susceptibility due to the Interplay of Loss Mechanisms”, actually, AC susceptibility at low frequency is a little larger than that of the numerical results from the Eq. (4)-(7), which leads to the experimental results is a little larger than that of numerical at low frequency; on the other hand, coupling loss saturates and subsequently decreases with frequency due to shielding of the interior of the conductor at high frequency which was also described in^[5]. However, the saturation of conductor was not taken into account when we used the classical method of coupling loss calculation because the maximum frequency of Sultan test of CNTF4 is 1 Hz, which leads to the experimental results is a little smaller than that of numerical at high frequency.

Fig. 8.Full-screen View

Comparison of AC loss versus frequency between numerical method and experimental method. The red curve with round symbol was calculated using $n\tau$=122.78 which is obtained by numerical analysis and the blue curve with triangle symbol was calculated using $n\tau$=137.21 which is obtained by the initial slope of Sultan test.

4.3. DC results

The DC test is the most time consuming activity in the TF sample test program which is aimed to measure the main parameters of the conductor, such as the current sharing temperature T_cs and the critical current I_c (n-value). And strand witness samples for left and right conductors are wound each onto “ITER barrels”, and are heat treated together with the CICC sections in order to verify the heat treatment by measuring the strand I_c. The test of the strands for TFCN4 is carried out at CRPP, immersed in liquid helium and at applied fields between 9 and 15 T. Fig. 9 illustrates the results of I_c measurement and n-value versus applied field and it is obvious that Fig. 9 shows a good agreement with Fig. 4 which was obtained by scaling law.

Fig. 9.Full-screen View

(a) critical current vs. applied field for witness strands at 4.2 K, 10 μV/m criterion and (b) n-value vs. applied field for witness strands at 4.2 K, 10 μV/m criterion

The index of transition or n-value data is characterized using an empirical modified power law ^[25]:

As previously observed in^[30], $s(T,\epsilon )$ is approximately a constant for all temperatures and applied strains and $r(T,\epsilon )$ only merely depends on strain properties^[31]. From Fig. 9 we find that $r(T,\epsilon )$ and $s(T,\epsilon )$ are 3.235 and 0.4122, respectively, which are similar to those of other advanced internal-tin strands^[32]. As given in Fig. 10, n-value versus critical current correspondingly to the specimens #1 and #2 of TFCN4 was presented and the parameterized function curve was also plotted. Obviously, it shows a very nice agreement between DC test result of Sultan and the result of parameterized function curve for CNTF4 strand, and this curve would be also available for other internal-tin strands.

Fig. 10.Full-screen View

n-value vs. critical current for witness strands at 4.2 K, 10 μV/m criterion

One of the n-value applications is that AC loss of CIC conductors with transport current can be calculated. In transport current–voltage (I-V) measurements of most CICC’s strands, I changes with V in the full penetration regime following roughly a power law (PL): $V\propto I^n$ ^[33], so that I_c has to be defined as that when electric field E=V/l=E_c, where l is the distance between both voltage taps and criterion E_c=10 $\text{μV/m}$ (at B=4-11 T, T=4-10 K) is routinely defined. The I-V curve comes from a PL relates to E(J), which is a characteristic of collective flux creep and expressed as:

Based on (16), the transport loss Q is calculated by:

As we know, logarithm is introduced to extract the index-n in Eq. (20):

Then,

where E and I are obtained from the measurement results based on four probe method, and n-value of internal-tin strands can also be obtained by the above parameterized function curve.

The transport current loss of low temperature superconductor (LTS) with power-law is described in Eq. (21), on the one hand, this method is further work because lots of work need to be solved such as the inhomogeneous distribution of critical current density based on Norris model for a round superconducting strand, on the other hand, it is also a worthwhile work because the existed numerical formulas for calculating hysteresis loss and coupling loss of LTS are complicated and time consuming, so it would be convenient if this method is also available for calculation of AC loss of LTS.

As for the CIC conductors, the result of the above equations for transport current loss calculation is part of the total loss. In consideration of the coupling current existing filaments and Cu-matrix, it is supposed that the result of Eq. (17) is including hysteresis loss and part of coupling loss of CICC and further relative research will be done in near future.