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

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

$$\begin{eqnarray}&&\begin{array}{l}{B}_{{\rm{dp}},{\rm{e}},{\rm{ee}}}\left(t\right)={B}_{{\rm{dp}}0}+{B}_{{\rm{dp}}1}\exp \left(-t/{\tau }_{{\rm{dp}}1}\right)\\ \,\,\,\,\,+\,{B}_{{\rm{dp}}2}\exp \left(-t/{\tau }_{{\rm{dp}}2}\right),\end{array}\end{eqnarray} \tag{ 7 }$$

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 copper-plated 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 four-filament 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.

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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:

$$\begin{eqnarray}&&{B}_{{\rm{a}},{\rm{c}}}\left(t\right)={B}_{{\rm{a}},{\rm{c}},{\rm{ft}}}:{\rm{at}}\,{\rm{flat}}\,{\rm{top}},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{B}_{{\rm{a}},{\rm{c}}}\left(t\right)=\displaystyle \frac{{I}_{{\rm{d}}{\rm{p}}}\left(t\right)}{{I}_{{\rm{d}}{\rm{p}},{\rm{f}}{\rm{t}}}}{B}_{{\rm{a}},{\rm{c}},{\rm{f}}{\rm{t}}}:{\rm{d}}{\rm{u}}{\rm{r}}{\rm{i}}{\rm{n}}{\rm{g}}\,{\textstyle \text{ramp-up}},\end{eqnarray} \tag{ 9 }$$

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).

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.

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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]

$$\begin{eqnarray}&&{G}_{{\rm{t}},{\rm{ss}}}\left(T=77\,{\rm{K}}\right)=3.47\times {10}^{7}{{\rm{\Omega }}}^{-1}{{\rm{m}}}^{-1},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\sigma }_{{\rm{n}},{\rm{ss}}}\left(T=77\,{\rm{K}}\right)=\displaystyle \frac{{w}_{{\rm{n}}}}{{t}_{{\rm{s}}}}{G}_{{\rm{t}}}=1.02\times {10}^{9}{{\rm{\Omega }}}^{-1}\,{{\rm{m}}}^{-1}.\end{eqnarray} \tag{ 11 }$$

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

$$\begin{eqnarray}&&{\sigma }_{{\rm{n}},{\rm{ss}}}\left(T\right)=\displaystyle \frac{{\sigma }_{{\rm{Cu}}}\left(T\right)}{{\sigma }_{{\rm{Cu}}}\left(T=77\,{\rm{K}}\right)}{\sigma }_{{\rm{n}},{\rm{ss}}}\left(T=77\,{\rm{K}}\right).\end{eqnarray} \tag{ 12 }$$

If it is further assumed that RRR of the plated copper is 45 [36], ${\sigma }_{{\rm{Cu}}}\left(T=30\,{\rm{K}}\right)/{\sigma }_{{\rm{Cu}}}\left(T=77\,{\rm{K}}\right)=\,5.58$ [37], and, then, one obtains

$$\begin{eqnarray}&&{\sigma }_{{\rm{n}},{\rm{ss}}}\left(T=30\,{\rm{K}}\right)=5.69\times {10}^{9}{{\rm{\Omega }}}^{-1}{{\rm{m}}}^{-1}\cong 5.7\times {10}^{9}{{\rm{\Omega }}}^{-1}{{\rm{m}}}^{-1}.\end{eqnarray} \tag{ 13 }$$

We varied σ_n around this value, 5 × 10⁹ Ω⁻¹m⁻¹ through 2 × 10¹⁰ Ω⁻¹ m⁻¹, including 5.7 × 10⁹ 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 × 10¹⁰ 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.

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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]:

$$\begin{eqnarray}&&{\tau }_{1}=a{l}^{2}.\end{eqnarray} \tag{ 14 }$$

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 × 10⁹ Ω⁻¹ m⁻¹ as shown in equation (11) and σ_n(T = 30 K) = 1 × 10¹⁰ 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:

$$\begin{eqnarray}&&{\tau }_{1}=20{l}^{2}\,\mathrm{at}{\rm{}}\,30\,{\rm{K}}.\end{eqnarray} \tag{ 15 }$$

Because the length of the four-filament coated conductor wound into the pancake coil in this work was 20 m, τ₁ 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 B₀ and B₁ as fitting parameters:

$$\begin{eqnarray}&&{B}_{{\rm{sc}},{\rm{ee}}1}\left(t\right)={B}_{0}+{B}_{1}\exp \left(-t/{\tau }_{1}\right);{\tau }_{1}=8000\,{\rm{s}}.\end{eqnarray} \tag{ 16 }$$

Using B₀ and B₁ 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:

$$\begin{eqnarray}&&{B}_{{\rm{sc}},{\rm{c}}}\left(t\right)-{B}_{{\rm{sc}},{\rm{ee}}1}\left(t\right).\end{eqnarray} \tag{ 17 }$$

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 B₀ through B₃ and τ₁ through τ₃ as listed in table 4

$$\begin{eqnarray}&&\begin{array}{l}{B}_{{\rm{sc}},{\rm{c}}}\left(t\right)\cong {B}_{{\rm{sc}},{\rm{ee}}3}\left(t\right)={B}_{0}+{B}_{1}\exp \left(-t/{\tau }_{1}\right)\\ \,\,\,\,+\,{B}_{2}\exp \left(-t/{\tau }_{2}\right)+{B}_{3}\exp \left(-t/{\tau }_{3}\right).\end{array}\end{eqnarray} \tag{ 18 }$$

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.

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Figures 9 and 10 show the current lines in the monofilament coated conductor wound into a coil and in the four-filament coated conductor wound into a coil, respectively. It should be noted that the figures are expanded laterally for visibility. The loops of currents in both figures are shielding currents induced by electromagnetic induction during the ramp-up of the cusp magnetic field, which is normal to the wide face of the coated conductor. 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.

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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.

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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 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].

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