Shielding current in a copper-plated multifilament coated conductor wound into a single pancake coil and exposed to a normal magnetic field

A single pancake coil wound with a copper-plated multifilament coated conductor, with four filaments, was put in a cusp magnetic field, and the magnetic field was measured near the coil at 30 K. A similar experiment was performed by using another reference single pancake coil wound with a monofilament coated conductor. Numerical electromagnetic field analyses of these coils were carried out, and the calculated shielding current-induced fields (SCIFs) were compared with the measured ones in both coils. The temporal behaviour of the calculated SCIF in the coil wound with the four-filament coated conductor was also compared with a series of exponential components, in which a coupling time constant extrapolated from short sample experiments was used as the time constant of the primary component. Current distributions in the coated conductors wound into the pancake coils were visualised. In particular, the temporal behaviours of the current distributions in the four-filament coated conductor and their influence on the SCIF were discussed.

Furukawa Electric Co., Ltd and SuperPower Inc. together developed multifilament coated conductors whose superconductor layers were divided into four filaments by laser striation and subsequently were plated with copper using the usual plating process of SuperPower Inc. [24]. In the copperplated four-filament coated conductor, the plated copper makes current sharing possible between the superconductor filaments. The coupling losses of the stacked short samples were measured to determine their coupling time constants, and it was found that the conductivity of plated copper dominates their transverse conductance [24]. The determined coupling time constants of the stacked short samples were also extrapolated to obtain a scaling law for the coupling time constant against the length of the coated conductor wound into a pancake coil. However, the temporal behaviour of the shielding current in a long copper-plated four-filament coated conductor wound into a pancake coil as well as the SCIF in the coil has not been clarified.
The objective of this study was to clarify the temporal behaviour of the coupling current and the SCIF in a pancake coil wound with the copper-plated four-filament coated conductor. To achieve this objective, a single pancake coil wound with the copper-plated four-filament coated conductor was prepared, as well as a reference single pancake coil wound with a monofilament coated conductor. A magnetic field was applied normal to the coated conductor wound into a coil by putting the coil in a cusp magnetic field. With this configuration, it was possible to study the electromagnetic coupling among filaments in a long multifilament coated conductor wound into a pancake coil by using only one coil. The measured and calculated SCIFs in these two coils were compared, and the temporal behaviour of the calculated coupling current was examined.
This paper is organised as follows. In section 2, the two single pancake coils, each of which was put in a cusp magnetic field, and the experimental setup are described. In section 3, the model for numerical electromagnetic field analyses is given. Section 4 presents results and discussion. Finally, section 5 provides the conclusions.

Single pancake coils and experimental setup for applying normal magnetic fields
Two single pancake coils were prepared for the experiments: one wound with a copper-plated four-filament coated conductor, which was striated from one end to the other, and a reference one wound with a monofilament coated conductor. In both coils, the conductor length was 20 m. The specifications of the coated conductors as well as the single pancake coils are listed in tables 1 and 2, respectively. Each coil was put in a cusp magnetic field generated by a pair of double  The critical current measurements were done as routine measurements in the manufacturing process, whose error is up to ∼5 A. b These values for 20 m coated conductors with which pancake coils were wound were determined based on the critical currents measured at B=0.1−0.5 T using a short sample taken from the same lot as the 20 m monofilament coated conductor. It was assumed that B Km of both 20 m coated conductors was same as that of the short sample. J c0 of each 20 m coated conductor was given as (I c,20 /I c,ss )J c0,ss where I c,20 , I c,ss , and J c0,ss are the critical current of the 20 m coated conductor measured in liquid nitrogen at self-field, that of the short sample, and J c0 of the short sample, respectively. pancake coils as shown in figures 2(a) and (b). The single pancake coil was placed between two double pancake coils, and the iron rings were placed inside and outside the single pancake coil for field shaping. The single pancake coil was conduction-cooled, but no current was supplied to it. Approximately 0.5 T of normal magnetic field could be applied to the coated conductor of the single pancake coil by providing 43 A of current to the double pancake coils, which were cooled independently of the single pancake coil. With this experimental setup, it was possible to study the electromagnetic behaviour of a long coated conductor wound into a pancake coil in the normal magnetic field rather than a short sample in the normal magnetic field. In other words, it was possible to simulate the electromagnetic condition of a pancake coil in a stack of pancake coils-a top or bottom one, for example-by using only one single pancake coil. It was not possible to study the effect of a multifilament coated conductor by using a single pancake coil to which a transport current is supplied but no external magnetic field is applied. A top view of a single pancake coil is shown in figure 3. A Hall sensor was installed in the coil, with which the radial component of the magnetic field at r HS =49 mm, where r is the distance from the centre of the single pancake coil, could be measured.

Model for electromagnetic field analyses [24, 30]
The current vector potential T defined using current density J as is used for the formulation in the model for electromagnetic field analyses. By using the thin-strip approximation, where only the current density component tangential to the analysed layer is considered [30], the governing equation derived from Faraday's law and Biot-Savart's law is given as follows: In this equation, T and T′ are the current vector potential components normal to the analysed layer at the field point and source point, respectively; n and n′ are the normal vectors of the analysed layer at the field point and source point, respectively; r is the vector from the source point to the field point; t s is the thickness of the analysed layer; σ is the conductivity of the analysed layer; S′ is the entire area of the analysed layer. S′ consists of triangular elements and is curved in a coated conductor wound into a coil. The spiral geometry of a single pancake coil was considered, because it was necessary to obtain the coupling current distribution along the coated conductor essentially. The analysed layer consisted of superconducting   filaments, whose width and equivalent conductivity were w s and σ s , respectively, and narrow resistive strips between superconductor filaments, whose width and conductivity were w n and σ n , respectively [24]. The analyses were made for the upper half of the coil because of its symmetry. The superconducting property was given by the power law electric field E-current density J characteristic [31,32]: where E 0 is 10 −4 V m −1 , and J c is the critical current density, which is solely determined by the magnetic field component normal to the superconductor layer B ⊥ by Kim's model [33] as where J c0 is the critical current density at zero magnetic field, and B Km is a constant. Subsequently, the equivalent conductivity of the superconductor σ s (nT) is derived as The transverse conductance between superconductor filaments G t , which is the value per unit length along the coated conductor, is represented by narrow resistive strips between superconductor filaments in the model. The conductivity of the resistive strip σ n is related to the transverse conductance G t as Algebraic multigrid preconditioning, hierarchical matrices, and parallel computation techniques were used to enhance the performance of computation [34].

Measured magnetic fields
The single pancake coil wound with the monofilament coated conductor was cooled at 30 and 110 K, the magnetic field was applied, and the radial magnetic field was measured. Figure 4(a) shows the temporal evolutions of the measured magnetic fields B m (t) as well as the currents supplied to the double pancake coils I dp (t) to generate a magnetic field. Figure 4(b) is the similar plots for the coil wound with the copper-plated four-filament coated conductor. In each figure, B m (t) at 110 K does not contain the SCIF of the single pancake coil itself, which was not superconducting. In other words, it is the magnetic field generated by the pair of double pancake coils. The double pancake coils were superconducting even when the single pancake coil was at 110 K. Therefore, B m (t) at 110 K might not be completely proportional to the current supplied to the double pancake coils because of the SCIF of the double pancake coils themselves. The following equation was fit to B m (t) at 110 K for each coil , 7 dp,e,ee dp0 dp1 dp1 dp2 dp2 where B dp0 , B dp1 , B dp2 , τ dp1 , and τ dp2 are fitting parameters.
Using fitted values for each coil listed in table 3, B dp,e,ee of equation (7) can be fitted well to B m (t) at 110 K of each coil. It is assumed that the magnetic field generated by the double pancake coils is reproducible, and, then, B dp,e,ee (t) using parameters listed in table 3 is subtracted from B m (t) at 30 K to obtain the experimentally-obtained SCIF B sc,e (t) for each coil.
The obtained B sc,e (t) of the coil wound with the monofilament coated conductor and that of the coil wound with the copperplated four-filament coated conductor are compared in figure 5. B sc,e (t) in the coil wound with the four-filament coated conductor decays much more rapidly than B sc,e (t) in another. Even at the beginning of the flat top of the applied magnetic field (t=0), B sc,e (t) in the coil wound with fourfilament coated conductor is smaller than B sc,e (t) in another.
The reason for this may be the decay of the coupling current during the field ramp-up.

Applied magnetic field in electromagnetic field analyses
In the electromagnetic field analyses, it was assumed that the applied magnetic field was uniform. The slight temporal change in the applied magnetic field caused by the decay of SICF of the double pancake coils was also neglected. Then, the applied magnetic field in the electromagnetic field analyses B a,c (t) was given by the following equations: where I dp (t) in the measurements at 30 K shown in figure 4 was used. B a,c,ft was set at 0.5 T, which was a rounded value of B m (t) measured at 110 K, and I dp,ft was set at 43.1 A, which was the averaged values at the flat tops in the measurements at 30 K. Using B a,c (t) given by equations (8) and (9), the electromagnetic field analyses were carried out for each coil, and, then, the calculated magnetic fields were subtracted from B a,c (t) to obtain the calculated values of SCIF B sc,c (t).

Comparison between calculated and measured SCIFs in coil wound with monofilament coated conductor to determine n value
First, the temporal changes of the calculated SCIF B sc,c (t) were compared with the experimentally-obtained SCIF B sc,e (t) in the coil wound with the monofilament coated conductor. The n value of equation (5) and the offset to the measured magnetic field B os were used as fitting parameters. n was varied from 35 to 50. When n=41 and B os =8.10 mT, the temporal change of the calculated SCIF B sc,c (t) agreed well with the measured one B sc,e (t) as shown in figure 6. These n and B os are used in the analyses for all results shown in the latter part of this work. The determined n value reflects E-J characteristic at a low E range, which could indeed govern the temporal behaviour of the shielding current in a long time scale [35]. If the n value had been determined by the usual transport technique, it would have reflected the E-J characteristic at a relatively high E range (around 10 −4 V m −1 ) and might have been inappropriate for examining the temporal behaviour of the shielding current in a long time scale.

Comparison between calculated and measured SCIFs in coil wound with four-filament coated conductor
Next, the temporal changes of the calculated and measured SCIFs in the coil wound with the copper-plated four-filament coated conductor were compared. It was assumed that n of the power law E-J characteristic of the copper-plated four-filament coated conductor is 41 as well. For electromagnetic field analyses, it was necessary to set the conductivity of the narrow resistive strip σ n , i.e. the transverse conductance between filaments. In previous work, the authors obtained the transverse conductance between filaments G t,ss , namely, the conductivity of the resistive strip in the model σ n,ss , at 77 K from the measurements and calculations of coupling losses of short samples of the copper-plated four-filament coated conductor as follows [24]  Here, the subscripts 'ss' mean that the value was obtained from short sample data. This previous work also suggested that the conductivity of plated copper dominates G t in the copper-plated four-filament coated conductor [24]. If it is assumed that G t,ss , namely, σ n,ss , is determined only by the conductivity of plated copper, σ n,ss at temperature T is given as   If it is further assumed that RRR of the plated copper is 45 [36], Cu Cu [37], and, then, one obtains We varied σ n around this value, 5×10 9 Ω −1 m −1 through 2×10 10 Ω −1 m −1 , including 5.7×10 9 Ω −1 m −1 , and carried out the electromagnetic field analyses. B os as well as σ n was used as the fitting parameter when comparing the calculated SCIF B sc,c (t) with the experimentally-obtained SCIF B sc,e (t). When σ n =1×10 10 Ω −1 m −1 and B os =10.1 mT, B sc,c (t) can be fitted well to B sc,e (t) except at the beginning of the flat top of the applied magnetic field (t=0-2000 s) as shown in figure 7. These σ n and B os as well as n of 41 are used in the analyses for all results shown in the latter part of this work.

Decay time constants of SCIF in coil wound with four-filament coated conductor
In a previous study, it was pointed out that the coupling time constant τ c corresponding to the entire length l of a coated conductor wound into a pancake coil is given as follows [24]: The constant a, which is proportional to the transverse conductance between filaments, was approximately 2 s m −2 at 77 K for the copper-plated four-filament coated conductor [24]. Considering σ n,ss (T=77 K)=1.02×10 9 Ω −1 m −1 as shown in equation (11) and σ n (T=30 K)=1×10 10 Ω −1 m −1 , with which B sc,c (t) can be fitted well to B sc,e (t), one can reasonably assume that the constant a might be around 20 s m −2 at 30 K: 20 at 30 K. 15 1 2 Because the length of the four-filament coated conductor wound into the pancake coil in this work was 20 m, τ 1 becomes 8000 s. First, the following equation was fit to B sc,c (t) in the period between t=2000 and 10 000 s, where B sc,c (t) agrees well with B sc,e (t), using B 0 and B 1 as fitting parameters: ; 8000 s. 16 sc,ee1 0 1 1 1 Using B 0 and B 1 as listed in table 4, B sc,ee1 (t) can be fitted well to B sc,c (t) in the period.
In the earlier period between t=0 and 2000 s, B sc,c (t) contains the following residual, which decays rapidly: Two exponential components were fit with shorter time constants to this residual. Consequently, a series of exponential components B sc,ee3 (t) given as equation (18) can be fitted well to B sc,c (t) in the entire period between t=0 and 10 000 s using B 0 through B 3 and τ 1 through τ 3 as listed in table 4 B sc,c (t) and B sc,ee3 (t) are compared in figure 8. In this figure, each term in equation (18) is also plotted. The first term in equation (18) represents the field induced by persistent shielding current in superconductor filaments, and the second term represents the field induced by the coupling current flowing in the entire length of the coated conductor. The origin of the third and fourth terms with shorter time constants is discussed in the next section. Table 3. Constants in equation (7), exp , dp,e,ee dp0 dp1 dp1 dp2 dp2 with which B dp,e,ee can be fitted well to B m (t) at 110 K of each coil.

Four-filament coated conductor
Monofilament coated conductor   3 3 with which B sc,ee3 (t) can be fitted well to B sc,c (t) of pancake coil wound with four-filament coated conductor in the entire period between t=0 and 10 000 s. At t=0 in each figure, one can observe multiple loops of shielding current whose spatial characteristic lengths are different: some expand to the entire length of the coated conductor, and some are localised near the ends of the coated conductor. If a uniform magnetic field is applied to a straight coated conductor that was not wound into a pancake coil, the loops of the shield current could have a single characteristic length, expanding to the entire length of the coated conductor. A coated conductor wound into a pancake coil exposed to a cusp magnetic field is analogous to the stacked short pieces of a coated conductor exposed to a normal magnetic field. Inner or outer turns of the pancake coil behave similarly to coated conductors near the top or the bottom of the stack. If the applied magnetic field is not large enough to penetrate to the lateral centre of the coated conductor in the middle of the pancake coil or the stack, more shielding current could be induced in the inner or outer turns of the pancake coil or the coated conductors near the top or the bottom of the stack. This might be the mechanism of the multiple loops of shielding current with different characteristic lengths.
Whereas the loops of current are rather persistent in the monofilament coated conductor as shown in figure 9, the loops of current in the four-filament coated conductor vary temporally as shown in figure 10, because they contain coupling current, which flows through the transverse conductance (resistance) and is not persistent. In particular, coupling current with a short characteristic length decays rapidly, because the inductance is proportional to the length and the transverse resistance is inversely proportional to the length. Such a rapidly decaying coupling current with a short characteristic length could be the origin of the components with short time constants in the SCIF, such as the third and the fourth terms in equation (18).
Next, lateral current distributions across each coated conductor were examined at various longitudinal positions as listed in table 5. Because the magnetic flux penetrates in almost the same way from the inner end and the outer end of a coated conductor wound into a pancake coil, locations between the inner end and the longitudinal centre of a coated conductor were chosen. Figure 11 shows the lateral current distributions across the monofilament coated conductor at t=0 and 10 000 s. The magnetic flux hardly penetrates to the lateral centre of the coated conductor-in particular, at the longitudinal centre of the coated conductor (position D). Because the shielding current is rather persistent, the temporal changes in the current distributions are not remarkable. A finite E-J characteristic of the superconductor, i.e. a finite nonlinear resistivity of the superconductor, caused slight temporal changes in the current distributions [8,35], which led to the gradual decay of the SCIF shown in figure 6. Figure 12 shows the lateral current distributions across the four-filament coated conductor at t=0 and 10 000 s. Near the end of the coated conductor (positions A and B), filaments are decoupled even at t=0. At position C, which is 3.53 m from the inner end of the coated conductor, filaments Figure 8. Comparison between calculated SCIF B sc,c (t) of single pancake coil wound with a copper-plated four-filament coated conductor and a series of exponential components: 3 where the constants are listed in table 4.  are coupled at t=0 but are decoupled at t=10 000 s. Such a decoupling process of filaments leads to the rapid decay of the SCIF in the coil wound with the four-filament coated conductor as shown in figure 7. At the longitudinal centre of the coated conductor (position D), which is 9.65 m from the inner end of the coated conductor, filaments are almost coupled even at t=10 000 s. As discussed in section 4.5, the time constant of the dominant coupling current, which flows in the entire length of the coated conductor, is 8000 s. This is consistent with the coupled filaments at the longitudinal centre of the coated conductor at t=10 000 s, because a time around the coupling time constant is not long enough to decouple filaments at the longitudinal centre of multifilament coated conductor [24].

Conclusion
We measured and calculated the SCIF in a pancake coil wound with a 20 m long copper-plated four-filament coated conductor and in another pancake coil wound with a monofilament coated conductor with the same length, both of which were exposed to 0.5 T of the magnetic field which was normal to their tape faces. The measured and calculated temporal behaviours of the SCIF agreed well with each other: the SCIF decayed more rapidly in the pancake coil wound with the copper-plated four-filament coated conductor than in the pancake coil wound with a monofilament coated   conductor. Electromagnetic field analysis shows that the decay of the SCIF in the pancake coil wound with the copperplated four-filament coated conductor was dominated by the coupling current flowing in the entire length of the coated conductor and that coupling currents localised near the ends of the coated conductor generated additional minor components with short time constants.