Coupling time constants of striated and copper-plated coated conductors and the potential of striation to reduce shielding-current-induced fields in pancake coils

The shielding-current-induced field is a serious concern for the applications of coated conductors to magnets. The striation of the coated conductor is one of the countermeasures, but it is effective only after the decay of the coupling current, which is characterised with the coupling time constant. In a non-twisted striated coated conductor, the coupling time constant is determined primarily by its length and the transverse resistance between superconductor filaments, because the coupling current could flow along its entire length. We measured and numerically calculated the frequency dependences of magnetisation losses in striated and copper-plated coated conductors with various lengths and their stacks at 77 K and determined their coupling time constants. Stacked conductors simulate the turns of a conductor wound into a pancake coil. Coupling time constants are proportional to the square of the conductor length. Stacking striated coated conductors increases the coupling time constants because the coupling currents in stacked conductors are coupled to one another magnetically to increase the mutual inductances for the coupling current paths. We carried out the numerical electromagnetic field analysis of conductors wound into pancake coils and determined their coupling time constants. They can be explained by the length dependence and mutual coupling effect observed in stacked straight conductors. Even in pancake coils with practical numbers of turns, i.e. conductor lengths, the striation is effective to reduce the shielding-current-induced fields for some dc applications.


Introduction
The large magnetisation caused by shielding (screening) currents in coated conductors is a serious concern for their applications to magnets for MRI, NMR, and particle accelerators, because it deteriorates their field qualities (shielding-current-induced field or SCIF) [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The striation of a coated conductor to divide its superconductor layer into filaments is one of the approaches to reduce the magnetisation of a coated conductor . The striation is effective to reduce the magnetisation, i.e. SCIF, only after the decay of the coupling current flowing through the transverse resistance [39,45]. Even if a striated coated conductor is not twisted, the coupling current could decay at a certain time constant (coupling time constant) owing to the finite length of the conductor and its finite transverse resistance [24,25,28]. Although the coupling time constants of non-twisted striated coated conductors wound into coils may be much larger than those of twisted multifilamentary conductors, we can wait until the coupling currents sufficiently decay after the excitation of magnets in some dc applications.
From the viewpoint of the rapid decay of coupling current, larger transverse resistance between superconductor filaments is preferable in a striated coated conductor [41,44]. However, if it is too large or infinite, the static and transient current sharing between superconductor filaments might be prevented, and, subsequently, the tolerance against local defects of superconductor filaments and the stability of the coated conductor against time-dependent disturbance might be affected. It should be noted that the required transverse conductance for current sharing could depend on the spatial scale of local defect, spatial and time scales of disturbance, and the characteristic time of current and/or magnetic field change.
Multifilamentary coated conductors whose superconductor layers were striated using a laser and subsequently plated with copper were developed by Furukawa Electric Co., Ltd and SuperPower Inc. In these conductors, the copper plating allows current sharing between the superconductor filaments. Moreover, it covers and protects the striated edges of superconductor layers against chemical degradation. However, their small transverse resistances owing to the plated copper result in an increase in the coupling time constants.
The objective of this study is to evaluate the coupling time constants of non-twisted striated and copper-plated coated conductors of various forms: single conductors with various lengths; stacked conductors with various separations between conductors, numbers of conductors, and lengths; conductors wound into pancake coils. Using these characterisations, we aim to clarify the potential of striated and copper-plated coated conductors to reduce the SCIF in pancake coils.
The paper is organised as follows. Section 2 describes the concept of coupling currents in non-twisted striated coated conductors. In this section, first, we explain the paths of coupling currents in non-twisted striated coated conductors and the factors that determine the coupling time constants. Subsequently, we provide the Debye function for the frequency dependence of coupling loss, which is useful to obtain the coupling time constant. In section 3, we explain the method of measurement of magnetisation loss and the model for numerical electromagnetic field analysis to determine the coupling time constants. In section 4, we present the experimental results to evaluate the coupling time constants of straight striated coated conductors and their stacks and compare them with the results of numerical electromagnetic field analyses. The discussion in this section focuses on the influences of stacking and conductor length. In section 5, we estimate the coupling time constants in the striated coated conductors wound into pancake coils and discuss the potential of striation to reduce the SCIF in pancake coils. Finally, section 6 presents our conclusions.
2. Coupling currents in non-twisted striated coated conductors 2.1. Paths of coupling currents in non-twisted striated coated conductors and coupling time constants First, we consider a straight striated coated conductor. A temporally-changing external magnetic field with a component normal to its superconductor layer can induce a coupling current, which is the shielding current flowing across the transverse resistances between superconductor filaments as shown in figure 1. The coupling time constant, which is the decay time constant of a coupling current, is determined by the self-inductance of the path of the coupling current and the resistance along the path. The resistance along the path is mainly the transverse resistance between the superconductor filaments [40,43]. The self-inductance of the path is determined primarily by the conductor length.
Next, we consider a striated coated conductor wound into a pancake coil. When such pancake coils are stacked together, the striated coated conductor may be exposed to the magnetic field with a normal component; subsequently, this normal field component could induce a coupling current. In the case of such a striated coated conductor wound into a pancake coil, the mutual coupling between the sections of the path of the coupling current in different turns of the pancake coil might influence the inductance as shown in figure 2.  In general, the frequency dependence of the coupling loss of a multifilamentary superconductor exposed to a transverse magnetic field that changes sinusoidally is given by a Debye function as where Q c is the coupling loss per unit length per cycle, A * is a factor determined by the geometry of the conductor, H m is the amplitude of the applied magnetic field, f is the frequency of the applied magnetic field, and τ c is the coupling time constant [46,47]. Now, we define a characteristic frequency f c as and plot the coupling loss Q c against frequency f schematically in figure 3: Q c is almost proportional to f when f = f c ; Q c reaches its peak when f = f c ; Q c is almost proportional to 1/f when f ? f c . This type of curve is called a Debye curve, which is useful to determine the coupling time constant τ c from the magnetisation losses measured or calculated at various f. By fitting a Debye curve to the measured or calculated magnetisation losses which are dominated by coupling losses using the method of least-squares, we can determine the coupling time constant τ c and the geometry factor A * in equation (1).

Method of measurement of magnetisation loss
The hysteresis loss component of magnetisation loss is proportional to H m 3 when the magnetic field H m is small, whereas the coupling loss component is proportional to H . m 2 Therefore, coupling losses dominate magnetisation losses when the magnetic field is small, and the frequency is near f c . Consequently, if we apply a small magnetic field to a striated coated conductor and measure its magnetisation losses at various frequencies, we can determine τ c by fitting equation (1) to the measured magnetisation losses. We measured the magnetisation losses of short pieces of striated coated conductors in liquid nitrogen by using our ac loss measurement system [22,25,48]. Figure 4(a) shows the schematic view of the system [48,49]. Transverse ac magnetic fields with sinusoidal waveforms were applied to a sample using a copper magnet, and the magnetisation losses were measured using a linked pick-up coil (LPC) installed around the sample conductor as shown in figure 4(b) [48]. The magnetisation loss per unit length per cycle Q m is given as follows:   where C pc is the calibration factor of the LPC, h pc is its height, N pc is its number of turn, L pc is the length of a turn of the LPC along the sample conductor. The calibration factor C pc is determined by the cross-sectional shape of the LPC and is 2.042 for the LPC used in this study. The number of turn N pc is 996, and other parameters are shown in figure 4(b). H rms is the rootmean-square value of H m (√2 H m ), and V m,rms is the loss component of the LPC output voltage (root-mean-square value) measured using a lock-in amplifier. In all the experiments reported in this paper, μ 0 H m was 1 mT. As the sensitivity of the pick-up coil depends on the sample length, the LPC was calibrated. We measured the eddy current losses of copper rods of various lengths in liquid nitrogen. The ratio between the measured loss of the copper rod of length l (mm) and that of the copper rod of length 100 mm c l given as follows was used as the calibration factor, where Q Cu,100 and Q Cu,l are the measured losses of copper rods of lengths 100 mm and l (mm), respectively. We multiplied Q m calculated using equation (3) by c l to obtain the magnetisation loss of the sample superconductor of length l (mm), because 100 mm is the standard sample length in our experimental system. The calibration factors c l used in this study are listed in table 1.

3.2.
Model for numerical electromagnetic field analysis of striated coated conductors [50,51] The numerical electromagnetic field analyses were carried out for a layer consisting of superconductor filaments, whose width is w s , and narrow resistive strips between the superconductor filaments, whose width is w n , as shown in figure 5.
We refer to this layer as the analysed layer. We assumed that the thickness of the resistive strips is the same as that of the superconductor filaments t s . Narrow resistive strips represented the transverse resistances between the superconductor filaments, although the components that contribute the transverse resistance and their geometries must be complicated in real striated coated conductors. The governing equation for the numerical electromagnetic field analysis was derived from Faraday's law and Biot-Savart's law. We also applied the thin-strip approximation, where only the current density component tangential to the analysed layer is considered, and its normal component is neglected [52]. The current vector potential T is used for the formulation: The derived governing equation is as follows: In this equation, T and T′ are the current vector potential components normal to the analysed layer at the field point (the point where the field (potential) is calculated) and source point (the point where a current flows to generate the magnetic field at the field 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; B ext is the applied external magnetic field; σ is the (equivalent) conductivity; S′ is the entire area (wide face) of the analysed layer. S′ consists of triangular elements, and it is flat or curved, respectively, in a straight coated conductor and a coated conductor wound into a pancake coil. Further, n and n′ are identical and constant in a non-twisted straight coated conductor, but they are different and vary in a coated conductor wound into a pancake coil. Equation (6) was discretized, and the derived system of equations were solved to obtain the time-dependent spatial distribution of T. A typical number of unknown valuables T is around 30 000. The details of the model are given in [50,51]. For the sake of simplicity, we assumed that the magnetic field dependence of the critical current density J c was solely determined by the magnetic field component normal to the superconductor layer B ⊥ . The superconducting property is Figure 5. Thin analysed layer consisting of superconductor filaments, whose width is w s , and narrow resistive strips between superconductor filaments, whose width is w n , for numerical electromagnetic analyses. The thickness of the layer is t s . The conductivity of the resistive strips is denoted with σ n . The transverse conductance between the superconductor filaments per unit length along the coated conductor is denoted with G t .   Width of groove between superconductor filaments (w g ) c 57 μm -Width of resistive strip between superconductor filaments in numerical electromagnetic field analyses (w n ) d 50 μm 5 0 μm Transverse conductance between superconductor filaments per unit length along the coated conductor (G t,e ) e 3.47 × 10 7 Ω -1 m -1 3.47 × 10 7 Ω -1 m -1 Conductivity of narrow resistive strip in the model for numerical electromagnetic field analyses (σ n ) e 1.02 × 10 9 Ω -1 m -1 1.02 × 10 9 Ω -1 m -1 a Equation (7):   [53,54]: Subsequently, the equivalent conductivity of the superconductor σ s (nT) was derived as In order to reduce the computation time required for iterative calculations, we set the maximum equivalent conductivity as 10 300 Ω −1 m −1 . The magnetic field dependence of the local critical current density is given by Kim's model as where J c0 is the critical current density at zero magnetic field and B 0 is a constant [55].
The conductivity of the resistive strips σ n representing the transverse resistances between the superconductor filaments was set as a constant value for an analysis.
After calculating T, we can obtain J, i.e. the current distribution, by using equation (5). Using the following expression, we can calculate the magnetisation loss per unit length per cycle: where l is the conductor length. In all the numerical electromagnetic field analyses reported in this paper, sinusoidal transverse magnetic fields with μ 0 H m of 1 mT were applied to the conductors.

4 mm wide, four-filament coated conductor
The specifications of the striated coated conductor used in the experiments and numerical electromagnetic field analyses are listed in table 2, and its schematic cross section is shown in figure 6, respectively. Its superconductor layer with silver protective layer was cut into four filaments using a laser, covered with additional silver layer and subsequently plated with copper. Grooves between superconductor filaments cut by laser might be filled with residual materials melted by laser, silver, and copper. Eventually, their superconductor filaments were connected electrically mainly through the plated copper. From the difference between the critical currents before and after the striation (laser cutting), we simply assumed that 4.27% of the superconductor layer was lost; the estimated width of each striated groove between the superconductor filaments w g is 57 μm. In the numerical electromagnetic field analyses, the width of resistive strip w n was set as the rounded value of 50 μm. The value of conductivity of the resistive strip σ n in table 2 will be determined in section 4.2. Measurements of magnetisation loss to determine the coupling time constants and numerical electromagnetic field analyses were conducted using single straight conductors and stacked straight conductors.

Coupling time constants of single striated coated conductors of various lengths, transverse conductance between superconductor filaments, and current distributions
We measured the magnetisation losses of single straight striated coated conductors. In figure 7, the measured magnetisation losses Q m of single conductors of various lengths l are plotted against frequency f with closed symbols. The broken line denotes the Debye curve with τ c = 8.66 ms fitted to the measured magnetisation losses for l = 100 mm using the method of least-squares. The frequency f c , which provides the peak of the Q m -f plot, decreases with an increase in conductor length. By fitting equation (1) to the measured Q m -f plots, the coupling time constants τ c for various conductor lengths are determined and plotted against conductor length with closed symbols in figure 8. These experimentally determined coupling time constants τ c are almost proportional to the square of conductor length l, and the following  For the single conductor with l = 100 mm, we varied conductivity of the resistive strip σ n and carried out the numerical electromagnetic field analyses in order to determine the value of σ n with which the measured Q m -f plot can be reproduced. The value of σ n obtained is 1.02 × 10 9 Ω -1 m -1 , and according to this value, τ c = 8.66 ms ( f c = 18.4 Hz). This value of σ n is used in the numerical electromagnetic field analyses in the following part of this paper. The transverse conductance between the superconductor filaments per unit length along the coated conductor G t shown in figure 5 can be calculated as follows: When σ n is 1.02 × 10 9 Ω -1 m -1 , G t is 3.47 × 10 7 Ω -1 m -1 . This G t determined by the experiments and numerical electromagnetic field analyses is denoted as G t,e in the following discussion. The transverse conductance between the superconductor filaments could be the parallel circuits consisting of the 20 μm thick copper layer plated over the superconductor layer, a much thinner additional silver layer, and materials in the 1.7 μm thick groove (copper, silver, and melted residuals) as shown in figure 9. It is reasonable to assume that the copper layer could dominate the transverse conductance, because it is much thicker and is more conductive than others. We estimate the transverse conductance based on this assumption. We assume that the current flows uniformly along the copper layer whose thickness and effective length that contribute the transverse conductance (resistance) are t Cu and l Cu , respectively, as shown in figure 9. The transverse conductance between the superconductor filaments per unit length along the coated conductor G t,Cu is given as follows: where σ Cu is the conductivity of copper, which is 5 × 10 8 Ω -1 m -1 at 77 K and t Cu is 20 μm. In figure 10, the calculated G t,Cu is plotted against effective copper length l Cu and is compared with G t,e . When the effective copper length is w g (57 μm), G t,Cu is much larger than G t,e . This is expected, because w g should be the minimum path of the transverse current flowing through the copper layer. If the effective copper length is assumed to be approximately 300 μm, G t,Cu becomes equal to G t,e . There are advantages and a drawback in the transverse resistance (conductance) dominated by the plated copper. The thickness of the plated copper can be uniform along a conductor and controllable. The consequent uniformity and controllability of the transverse resistance could be advantages. We can increase the transverse resistance, i.e. decrease the coupling time constant, by decreasing the copper thickness or by using a copper alloy instead of copper. A drawback is decreasing transverse resistance with  Model of transverse conductance (resistance) between superconductor filaments dominated by the copper layer plated over the superconductor layer. We assume that the current flows uniformly along the copper layer whose thickness and effective length contributing the transverse conductance (resistance) are t Cu and l Cu , respectively. The conductivity of the copper is denoted with σ Cu . The transverse conductance between the superconductor filaments per unit length along the coated conductor is denoted with G t,Cu .
decreasing temperature. Consequently, the coupling time constant could be inversely proportional to the temperaturedependent resistivity of copper. It should be noted that the interlayer resistances between copper, silver, and superconductor layers can reduce the transverse conductance.
We carried out the numerical electromagnetic field analyses at various frequencies for single conductors of various conductor lengths. In figure 7, the calculated magnetisation losses are plotted with open symbols. In this figure, the calculated losses are reasonably consistent with the measured ones. In the electromagnetic field analyses, the coupling currents are confined in the layer whose thickness is t s shown in figure 5, whereas the coupling currents in the experiments expand in the copper layer on the superconductor layer as shown in figure 9. The difference in the coupling current paths may cause the difference between the measured and calculated losses. By fitting equation (1)

Influence of stacking striated coated conductors on coupling time constants
We measured the magnetisation losses of the stacks of straight striated coated conductors. Stacked conductors simulate the turns of a conductor wound into a pancake coil. We varied the number of stacked conductors n c and the separation between conductors d, whereas conductor length l was fixed at 50 mm. When we stacked the conductors, polyimide sheets and/or GFRP sheets were placed to maintain the separation between conductors, and a GFRP sample holder was used to reduce the lateral misalignments of stacked conductors in most of the experiments.
In figure 13, the measured magnetisation losses of stacked conductors with various numbers of stacked conductors n c and the separation between conductors d = 0.1 mm are plotted against frequency f with closed symbols. The characteristic frequency f c , which gives the peak of a Q m -f plot, decreases with increasing number of stacked conductors. In figure 14, the measured magnetisation losses of stacked conductors with various separations between conductors d and number of stacked conductor n c = 8 are plotted against frequency f with closed symbols. The characteristic frequency increases with increasing separation between conductors. In these figures, we can observe the influence of stacking on the Q m -f plots. Further, Q m -f measurements were carried out for various combinations of n c and d. We determine the values of coupling time constant for various combinations of n c and d by fitting equation (1) to the measured Q m -f plots. In figure 15, the experimentally determined coupling time constants are plotted against number of stacked conductors with closed symbols where d = 0.05-1 mm. The coupling time  constants increase with increasing number of stacked conductors. Figure 16 schematically shows the coupling currents in stacked conductors. The mutual coupling of the coupling currents in different conductors could increase coupling time constants. In figure 17, the experimentally determined coupling time constants are plotted against separation between conductors with closed symbols where n c = 1-8. The    coupling time constants decrease with increasing separation between conductors. When d = 4.1 mm, the coupling time constant for n c = 2 falls to 2 ms, which is almost equal to the coupling time constant of a single conductor. With increasing separation between conductors, mutual coupling between the coupling currents in different conductors could be reduced. This could be the reason for the decrease in coupling time constant with increasing separation between conductors. In figure 15, the increasing coupling time constant saturates at larger number of stacked conductors. This saturation is more remarkable for larger separation between conductors. As shown in figure 17, the coupling currents in distant conductors are less coupled. This could be the reason for the saturation, because some of the other conductors are located more distantly in a stack with large number of stacked conductors and/or separation between conductors.
We carried out the numerical electromagnetic field analyses at various frequencies for stacked conductors with various n c and d. In figures 13 and 14, the calculated magnetisation losses are plotted with open symbols. We also determine coupling time constants for various combinations of number of stacked conductors and separation between conductors by fitting equation (1) to the calculated Q m -f plots. These theoretically determined coupling time constants are plotted with open symbols in figures 15 and 17. In figure 15, the theoretically determined coupling time constants and the experimentally determined coupling time constants saturate at larger number of stacked conductors although the former is larger than the latter. As mentioned in section 4.2, the difference between the coupling current paths may cause the difference between the experimentally determined and theoretically determined coupling time constants.

Coupling time constants of stacked striated coated conductors of various lengths
In figure 18, the experimentally determined and theoretically determined coupling time constants are plotted against conductor length with closed circles and open circles, respectively, for n c = 4 and d = 0.1 mm. The corresponding values   for single conductors (n c = 1) shown in figure 8 are replotted with triangles. For n c = 16 and d = 0.1 mm, the theoretically determined coupling time constants are plotted with open squares. The experimentally determined or theoretically determined coupling time constants of stacked striated coated conductors for each n c are proportional to l 2 . Equation (11) was fitted to them by using the method of least-squares in order to determine the factor of proportionality a, and the determined values of a are shown in the figure. The fitted curves for the experimentally determined and theoretically determined coupling time constants are shown with the solid and broken lines, respectively.
Assuming that any value of coupling time constant is proportional to the square of conductor length, the factor of proportionality a of equation (11) can be determined directly from the coupling time constant for a single conductor length. In figure 15, the value of a was calculated directly from each coupling time constant by using equation (11); each point in the figure represents coupling time constant in the left vertical axis and the factor of proportionality in the right vertical axis. Similar to coupling time constant, the factor of proportionality saturates at larger number of stacked conductors.
The results in section 4 can be summarised as follows.
(1) The coupling time constant of stacked conductors and that of a single conductor are proportional to the square of the conductor length. We studied the coupling time constants of the striated coated conductors wound into pancake coils using numerical electromagnetic field analyses. In order to reduce the computation loads, we used a 2 mm wide, two-filament coated conductor. Comprehensive analyses were performed for the followings: (1) stacked straight conductors with various numbers of stacked conductors; (2) conductors wound into pancake coils with various numbers of turns n t ; (3) single straight conductors whose lengths l are identical to those of (2).
The detailed specifications of the 2 mm wide, two-filament coated conductor are listed in table 2, and those of the pancake coils are listed in table 3. In the model, the actual spiral geometry of the conductor in a pancake coil as shown in figure 19 was considered. A sinusoidal magnetic field whose amplitude was 1 mT was applied normally and uniformly to the striated coated conductors with σ n = 1.02 × 10 9 Ω -1 m -1 and w n = 50 μm.
In preparation for the analyses of pancake coils, we investigate the stacked 2 mm wide, two-filament coated conductors. In figure 20, coupling time constant τ c as well as the factor of proportionality a of equation (11), which were calculated directly from each coupling time constant, are plotted against number of stacked conductors where d = 0.1 mm and l = 50 mm. Because of mutual coupling, the factor of proportionality a of equation (11) increases with increasing number of stacked conductors n c , but it saturates around n c = 16. For this 2 mm wide, two-filament coated conductor, a = 1.42 s m −2 when n c = 1, and 2.75 s m −2 when n c = 16. We will use these two values in the discussion below.
Then, we compare the coupling time constants τ c of the conductors wound into pancake coils and those of the reference single straight conductors whose lengths are identical to the conductors wound into pancake coils. In figure 21, the calculated magnetisation loss Q m of conductors wound into pancake coils with various numbers of turns, i.e. conductor lengths, and those of single straight conductors with the corresponding conductor lengths are plotted against frequency with closed symbols and open symbols, respectively. We determined coupling time constants by fitting equation (1)  Figure 19. Spiral geometry of coated conductor wound into pancake coil. The inner and outer radii of a pancake coil are R in and R out , respectively. The separation between superconductor layers is t cc + d, where t cc and d are the thickness of coated conductor and the separation between coated conductors, respectively.
to the calculated Q m -f plots. In figure 22, the determined coupling time constants of the conductors wound into pancake coils and those of straight conductors are plotted against conductor length, i.e. number of turns, with closed triangles and open reverse triangles, respectively. The coupling time constants of the conductors wound into pancake coils are larger than those of the straight conductors. The former increases more sharply with increasing length than the latter. In this figure, the solid line shows coupling time constant calculated by using equation (11)  . This suggests that the mutual coupling between the sections of the coupling current path in different turns could be the reason for larger coupling time constant and its sharp increase with increasing length in pancake coils. In other words, equation (11) with a of 2.75 s m −2 , which is derived for a 16-stacked straight conductor but not for pancake coils, is reasonably applicable to estimate the coupling time constants of the conductors wound into pancake coils.

Analytical estimation of coupling time constant of 4 mm wide, four-filament coated conductor wound into pancake coil
Similar to the discussion in the last part of section 5.1, using equation (11) with a of 1.60 s m −2 , which is the experimentally determined value for the eight-stacked straight conductors shown in figure 15, we can estimate the coupling time constant of the 4 mm wide, four-filament coated conductor wound into a pancake coil. Here, we consider a pancake coil whose inner radius is 60 mm, d is 0.1 mm, and n t is 200, i.e. l is 100.3 m. The estimated coupling time constant is 1.61 × 10 4 s (4.5 h), which is not a small value, but we can wait until the decay of the coupling current in some applications.    figure 19, which is the value for the single straight conductor.

Conclusion
The coupling time constants of striated and copper-plated coated conductors are proportional to the square of the conductor length and increase when they are stacked because of the mutual coupling between coupling currents in stacked conductors. These observations are useful for estimating the coupling time constants of the striated coated conductors wound into pancake coils.
In the striated and copper-plated coated conductors, the plated copper layers dominate the transverse resistances between superconductor filaments. Consequently, the transverse resistance can be uniform along a conductor, and we can increase the transverse resistance, i.e. decrease the coupling time constant, by decreasing the copper thickness or by using a copper alloy. However, the transverse resistance could decrease with decreasing temperature: the coupling time constant could be inversely proportional to the temperaturedependent resistivity of copper.
The striation of coated conductors is effective for reducing the shielding-current-induced field in pancake coils even if they are plated with copper after striation in order to enhance the current sharing. Although the striated coated conductor in a pancake coil is not twisted, the coupling current decays gradually owing to the finite length of the conductor. The increasing number of turns should naturally increase the coupling time constant owing to the increasing conductor length, i.e. increasing self-inductance and decreasing entire transverse resistance of the coupling current path. Moreover, the mutual coupling between the sections of the coupling current path in different turns results in an increase in the coupling time constants. Even if we consider these factors, the coupling time constant of a pancake coil wound with a 100-m-long conductor is 4.5 h, which is not short but might be tolerable in some dc applications.