Effects of Crack Size Distribution and Voltage Probe Spacing on Variation of Critical Current and n-Value along the Longitudinal Direction in Superconducting Tape with Cracks

Abstract

The effects of crack size distribution and voltage probe spacing on the variation of critical current and n-value along the longitudinal direction in heterogeneously cracked superconducting tape were studied using a Monte Carlo simulation combined with a model of crack-induced current shunting. The simulation results were as follows. The variation of the critical current along the longitudinal direction of the tape increases with increasing crack size distribution width, whereas it decreases when the voltage probe spacing is larger. The variation of n-value along the longitudinal direction is larger than that of critical current. The largest crack contributes most to the synthesis of the voltage-current curve, and this contribution increases with the crack size distribution width. Next, a model to predict the upper and lower bounds of distributed critical current and n-values was applied to the simulation results. It was confirmed that the critical current and n-values are within the upper and lower bounds in any crack size distribution width. In addition, it was revealed that the critical current value shifts from the lower to upper bound and the n-value shifts from the upper to lower bound with the increase in the crack size distribution width. Furthermore, an equivalent crack-current shunting model was applied to the simulation results. The multiple n-values for a critical current value and the correlation diagram between the n-value and critical current could be described with this model.

1. Introduction

Two types of superconducting tape have been developed: coated tape, in which a substrate is coated with a superconducting phase RE(Y, Sm, Dy, Gd, …)Ba₂Cu₃O_7−δ (REBCO), and filamentary tape, in which filaments of the superconducting phase, such as Bi₂Sr₂Ca₂Cu₃O_10+x, Nb₃Sn, Nb₃Al, and MgB₂, are embedded in metals. During fabrication and operation, both types are subjected to thermal, mechanical, and electromagnetic stresses and strains. At high stresses and strains, the superconducting phase cracks, reducing critical current I_c and n-value in both coated^1–9) and filamentary^10–21) tapes.

Cracking of the superconducting layer or filaments usually occurs heterogeneously; thus, the I_c and n-values differ among specimens^{5,7–9,11,12,16)} and vary along the specimen length.^7,11,12) We have studied the I_c and n-value of both coated and filamentary tapes^2,3,7,16) experimentally and analytically by applying a current shunting model at cracks.¹⁰⁾ We have been developing a Monte Carlo simulation method^8,9) combined with the current shunting model at cracks to systematically obtain a wide range of data about the effects of crack size distribution and specimen length on I_c and n-values. We have shown that this simulation method^8,9) reproduces the experimental observation that the local information about critical current values in a specimen is diluted in longer specimens.²²⁾

In this work, we used the Monte Carlo method combined with the current shunting model to determine the effects of the crack size distribution and voltage probe spacing on the variation of critical current and n-value along the tape length. The simulation results were analyzed by a model that obtains the upper and lower bounds of I_c and n-values for large voltage probe spacing from the statistical data for small voltage probe spacing. The simulation results were analyzed also by an equivalent crack-current shunting model where multiple cracks are replaced by an equivalent crack. These approaches provided important information on the superconducting properties of cracked tape.

2. Simulation Model and Procedure

2.1 Model tape consisting of sections and regions

Representative REBCO-coated tape was used for modeling. The configuration of the model tape employed in the present work is shown in Fig. 1(a). The tape was 180 cm long and consisted of a series of 120 sections with length L₀ = 1.5 cm. Each section had one stress-induced crack of a different size. The voltage probes were attached in steps of L = L₀ = 1.5 cm and L = 5L₀ = 7.5 cm. The critical current, I_c, and n-value for the sections with a length of 1.5 cm and for the regions that consisted of five sections and were 7.5 cm long were measured. There were 24 regions with length L = 7.5 cm in the model tape.

Fig. 1

Model used in the present work. (a) Model tape consisting of sections with length L = L₀ = 1.5 cm and regions with length L = 5L₀ = 7.5 cm. (b) Configuration of the region between the voltage probes, consisting of a series circuit of five sections, where each section has a crack with a different size. (c) Case A (extreme case of the crack array in (b)), in which the largest crack is in all sections. (d) Case B (extreme case of the crack array in (b)), in which the largest crack is in one section and other cracks in other sections are negligible compared with the largest crack. Case A in (c) and Case B in (d) are used in Subsection 2.4 to obtain the upper and lower bounds of the critical current and n-values of the region from the V–I curve of the largest crack section.

The voltage–current (V–I) curves, I_c values, and n-values of the 120 sections for a wide variety of crack size distributions were determined by the Monte Carlo method combined with a model of crack-induced current shunting, the procedure for which is shown in Subsection 2.2. The V–I curves of the 7.5 cm regions were synthesized from the V–I curves of the sections that constitute the region by the procedure shown in Subsection 2.3.

2.2 Calculating V–I curves of sections with the Monte Carlo method combined with a model of crack-induced current shunting

To calculate the V–I curves of the cracked sections, we used a modified form^{2,3,7–9,15,16)} of the model of Fang et al.¹⁰⁾ to describe the experimental critical current and n-value of filamentary and coated superconductors with stress-induced cracks. In this work, the modified form was used. Details have been reported in our previous work^{2,3,7–9,15,16)} and we provide a brief outline as follows.

Figure 2 shows a schematic representation of the current path in a section with a crack.⁸⁾ In the transverse cross section of the superconducting REBCO layer that contains a crack, the cracked part and the ligament form a parallel electric circuit. We define the ratio of the cross-sectional area of the cracked part to the total cross-sectional area of the REBCO layer as f. The REBCO layer in the ligament with an area ratio 1 − f transports current I_RE. At the cracked part with area ratio f, current I_s (= I − I_RE) shunts into a stabilizer, such as Ag and Cu. The electrical resistance of the shunting circuit in the section is R_t,section, and the voltage that develops at the ligament that transports current I_RE is V_RE. The voltage that develops at the cracked part by shunting current I_s, V_s = I_sR_t,section, is equal to V_RE because the cracked part and ligament form a parallel circuit. We define the current transfer length as s (≪ L₀), I_c and n-value of the sections in the non-cracked state as I_c0 and n₀, respectively, and the critical electric field for determining the critical current as E_c (= 1 µV/cm in this work). We express the V–I curve of the cracked section (L₀ = 1.5 cm) as^{2,3,8,9,15,16)}   

\begin{equation} V = E_{\text{c}}L_{0}(I/I_{\text{c0}})^{n_{0}} + V_{\text{RE}} \end{equation} (1)

  

\begin{align} I &= I_{\text{RE}} + I_{\text{s}} \\ &= I_{\text{c0}}L_{\text{p,section}} [V_{\text{RE}}/(E_{\text{c}}L_{\text{0}})]^{1/n_{\text{0}}} + V_{\text{RE}}/R_{\text{t,section}} \end{align} (2)

The term L_p,section in eq. (2) is called the ligament parameter of the sections. We derived this term ($L_{\text{p,section}} = (1 - f)(L_{0}/s)^{1/n_{0}})$)^2,3,8,9) by modifying the formulations of Fang et al.¹⁰⁾

Fig. 2

Schematic representation of a section and the current path at a crack.⁸⁾

Because a wide distribution of ligament parameter L_p,section corresponds to a wide distribution of f (crack size) and the standard deviation of 1 − f is the same as that of f, the standard deviation of L_p,section, ΔL_p,section, was used to monitor the crack size distribution width; the larger ΔL_p,section, the wider the crack size distribution. To formulate the distribution of L_p,section of the cracked sections with length L₀, the normal distribution function was used, as in previous work.^8,9) For the average of L_p,section, L_{p,section,ave}, the cumulative probability, F(L_p,section), was expressed by   

\begin{equation} F(L_{\text{P,section}}) = \frac{1}{2}\left\{1 + \text{erf}\left(\frac{L_{\text{p,section}} - L_{\text{p,section,ave}}}{\sqrt{2}\Delta L_{\text{p,section}}}\right)\right\} \end{equation} (3)

L_{p,section,ave} was taken to be 0.670 to describe a representative situation, where the cracks reduce the critical current of sections by approximately 1/3 on average compared with the non-cracked state. Six cases with ΔL_p,section = 0.01, 0.025, 0.05, 0.1, 0.15, and 0.2 were calculated to reveal the effect of the L_p,section distribution (corresponding to the crack size distribution) on I_c and n-value of 1.5 cm sections and 7.5 cm regions containing five sections. The L_p,section value for each cracked section was obtained using eq. (3) and the following Monte Carlo method. We generated a random value, RND, in the range of 0–1 for each section. Then, setting F(L_p,section) = RND and substituting the L_{p,section,ave} and ΔL_p,section values mentioned above in eq. (3), we derived the L_p,section value that satisfies eq. (3). This procedure was repeated 120 times for each ΔL_p,section value, and the L_p,section values were obtained for 120 sections. Because the random value, RND, is different from section to section, the L_p,section values (and hence the crack sizes) are also different.

The V–I curve of each cracked section with length L₀ = 1.5 cm was calculated by substituting the L_p,section value obtained by the procedure above, and the values R_t,section = 2 µΩ, I_c0 = 200 A, and n₀ = 40 into eqs. (1) and (2). The values for R_t,section, I_c0, and n₀ were taken from the experimental average values for cracked and non-cracked copper-stabilized 2.5-µm-thick DyBCO-coated tape, the V–I curves of which were measured at 77 K in a self-magnetic field for a voltage probe spacing of 1.5 cm.⁵⁾

This simulation procedure based on the Monte Carlo method was repeated 120 times and 120 sets of (V–I curve, I_c value, n-value) for 120 sections were obtained for each of ΔL_p,section = 0.01, 0.025, 0.05, 0.1, 0.15, and 0.2.

2.3 Calculating the V–I curve of the regions, and estimating the critical current and n-value

Because the 7.5 cm region consists of a series electric circuit of five 1.5 cm sections (Fig. 1(a)), the current in the region is the same as that in all sections, and the region’s voltage is the sum of the voltages of all sections,   

\begin{equation} I = I_{\text{S}i}(i = \text{1 to 5}) \end{equation} (4)

  

\begin{equation} V = \sum_{i = 1}^{5}V_{\text{S}i} \end{equation} (5)

Using the V–I curves of sections obtained by eqs. (1) and (2), we obtained the V–I curves of the regions by using eqs. (4) and (5).

From the V–I curves of the sections and regions, the critical currents of the sections, I_c,section, and regions, I_c,region, were obtained with the electric field criterion of E_c = 1 µV/cm, corresponding to the critical voltages V_c = 1.5 (sections) and 7.5 µV (regions). The n-values of the sections, n_section, and regions, n_region, were obtained by fitting the E–I curves to the form of E ∝ Iⁿ in the electric field range of E = 0.1–10 µV/cm, namely by fitting the V–I curves to the form of V ∝ Iⁿ in the voltage range of V = 0.15–15 µV (sections) and V = 0.75–75 µV (regions).

2.4 Upper and lower bounds of I_c and n-value of the region consisting of five sections

In this work, the subscripts “section”, “region”, “upper”, “lower”, and “ave” for I_c, n, and V indicate the condition under which the values were obtained. For instance, I_c,section and I_c,region refer to the I_c values of a section and region, respectively, and I_{c,section,ave} and I_c,region.ave refer to the average values for I_c,section and I_c,region, respectively. I_{c,region,upper} and I_{c,region,lower} refer to the upper and lower bounds for I_c,region, and I_{c,region,upper,ave} and I_{c,region,lower,ave} refer to the average values for I_{c,region,upper} and I_{c,region,lower}, respectively. The subscripts for the n-value and V are used in the same way.

In the 7.5 cm region consisting of a series of 1.5 cm sections, there are five cracks of different sizes (Fig. 1(b)). The superconductivity is lost first in the section with the largest crack. Accordingly, the voltage that develops at the largest crack section is the highest, and hence it contributes most to the region’s voltage. In the present work, by using the V–I curve of the largest crack section in the extreme Cases A and B (Fig. 1(c) and (d)), we obtain the upper and lower bounds of the critical current and n-value of each region as follows.

In Case A, the crack size is the same as the largest crack in all sections (Fig. 1(c)). In this case, the V–I curve is the same in all sections. Accordingly, the region’s voltage, given by the sum of the voltage of all sections in the region, corresponds to the upper bound of the region’s voltage, V_region,upper. The n-value of the region, defined as the index in the form of V ∝ Iⁿ, corresponds to the upper bound, n_region,upper, because V is given by the upper bound (V_region,upper). On the contrary, as V_region,upper reaches V_c (7.5 µV) at the lowest I for the given L_p (namely for the given crack size), the I_c value corresponds to the lower bound, I_{c,region,lower}.

In Case B, the crack size of one section is far larger than that of the other sections. In this case, the region’s voltage is the same as that of the largest crack section, where the voltages developed in the other sections are too low and do not contribute to the region’s voltage. The model of the region in this case is an assembly of one severely cracked section with the largest crack and non-cracked sections (Fig. 1(d)). This case gives the lower bound for the region’s voltage, V_region,lower. As V_region,lower reaches V_c at the highest I for the given L_p (crack size), Case B gives I_{c,region,upper}. This, in turn, means that Case B gives a lower bound for the n-value, n_region,lower.

Case A gives the V_region,upper–I curve, I_{c,region,lower}, and n_region,upper, and Case B gives the V_region,lower–I curve, I_{c,region,upper}, and n_region,lower. The V_region,upper–I and V_region,lower–I curves were obtained with eqs. (1), (2), (4), and (5) by finding the largest crack section with the smallest L_p,section value. From these curves, the I_{c,region,upper}, I_{c,region,lower}, n_region,upper, and n_region,lower values were obtained for each region and were compared with the I_c and n-values from the simulation.

3. Results and Discussion

3.1 Variation of I_c and n-value of sections and regions with position along the longitudinal direction of a superconducting tape

Figure 3 shows the variation of the critical current of sections, I_c,section (a, b, c), and the n-value of sections, n_section (a′, b′, c′), with position in REBCO-coated tape, obtained by a Monte Carlo simulation at L_{p,section,ave} = 0.67 and ΔL_p,section = 0.025 (a, a′), 0.1 (b, b′), and 0.2 (c, c′). The average critical current, I_{c,section,ave}, and average n-value, n_section,ave, of the sections were ∼136 A and ∼28 for any value of ΔL_p,section. This result is obtained with L_{p,section,ave} = 0.67, which was set to be the same for any ΔL_p,section. While the I_{c,section,ave} and n_section,ave values were the same for any value of ΔL_p,section, both the I_c,section and n_section values varied greatly with position along the longitudinal direction when the value of ΔL_p.section increased, that is, when the difference in crack size increased. The increased difference in crack size among the sections produced big differences in the I_c and n-value of the regions, as shown below.

Fig. 3

Variation of (a), (b), (c) I_c and (a′), (b′), (c′) n-values of 1.5 cm sections with position along the longitudinal direction in REBCO-coated tape obtained by a Monte Carlo method combined with the model of crack-induced current shunting for L_{p,section,ave} = 0.67 and ΔL_p,section = (a), (a′) 0.025, (b), (b′) 0.1, and (c), (c′) 0.2.

Figure 4 shows the variation of critical current I_c,region, its upper bound, I_{c,region,upper}, and lower bound, I_{c,region,lower} (a–f), and n-value n_region, its upper bound n_region,upper, and lower bound n_region,lower (a′–f′) of the 7.5 cm region with position along the longitudinal direction of the tape. The values were obtained by the simulation with L_{p.section,ave} = 0.67 and ΔL_p.section = 0.01 (a, a′), 0.025 (b, b′), 0.05 (c, c′), 0.10 (d, d′), 0.15 (e, e′), and 0.20 (f, f′). ● shows I_c,region and n_region, and Δ and ∇ show the upper (I_{c,region,upper}, n_region,upper) and lower (I_{c,region,lower}, n_region,lower) bounds, respectively. Figure 5 shows the variation of I_c,region, I_{c,region,upper}, and I_{c,region,lower} with position, where representative data for ΔL_p,section = 0.01 (a), 0.05 (b), and 0.15 (c) were taken from Fig. 4(a), (c), and (e), respectively, and were plotted at high magnification to show the details of the relationships among I_c,region, I_{c,region,lower}, and I_{c,region,upper}.

Fig. 4

Variation of the values of (a)–(f) I_c,region (●), I_{c,region,upper} (△), and I_{c,region,lower} (▽), and (a′)–(f′) n_region (●), n_region,upper (△), and n_region,lower (▽) of the 7.5 cm region with position along the longitudinal direction of the tape, obtained by the simulation for L_{p.section,ave} = 0.67 and ΔL_p.section = (a), (a′) 0.01, (b), (b′) 0.025, (c), (c′) 0.05, (d), (d′) 0.10, (e), (e′) 0.15, and (f), (f′) 0.20.

Fig. 5

Variation of I_c,region, I_{c,region,upper}, and I_{c,region,lower} with position along the longitudinal direction, where the data for ΔL_p,section = (a) 0.01, (b) 0.05, and (c) 0.15, taken from Fig. 4(a), (c), and (e), respectively, are drawn at high magnification to show the details of the relationship among the values of I_c,region, I_{c,region,lower}, and I_{c,region,upper}.

The average values of I_c,region, I_{c,region,lower}, and I_{c,region,upper}, (hereafter I_c,region,ave, I_{c,region,lower,ave}, and I_{c,region,upper,ave}, respectively) were obtained for each ΔL_p,section value (0.01–0.20) from Fig. 4(a)–(f). The average values of n_region, n_region,lower, and n_region,upper, (hereafter n_region,ave, n_{region,lower,ave}, and n_{region,upper,ave}, respectively) were obtained for each ΔL_p,section value from Fig. 4(a′)–(f′). Figure 6 shows the variation of I_c,region,ave, I_{c,region,lower,ave}, and I_{c,region,upper,ave} (a), and n_region,ave, n_{region,lower,ave}, and n_{region,upper,ave} (b) with increasing ΔL_p,section. The following results were obtained from Figs. 4–6.

Fig. 6

Variation of (a) I_c,region,ave, I_{c,region,lower,ave}, and I_{c,region,upper,ave}, and variation of (b) n_region,ave, n_{region,lower,ave}, and n_{region,upper,ave} with ΔL_p.section.

(1) Increasing ΔL_p,section, namely, the difference in crack size among the sections, increases the variation of I_c,region and n_region of the regions with position (Figs. 4 and 5).

(2) When L_{p,section,ave} was 0.67, namely the average crack size was the same, the average of the critical current and n-value of sections, I_{c,section,ave} and n_section,ave, were ∼136 A and ∼28, respectively, for any value of ΔL_p,section (Fig. 3). However, because the region consists of sections, the largest crack reduces the critical current and n-value of the region.^7–10,16) The size of the largest crack increases with increasing ΔL_p,section; thus, I_c,region,ave and n_region,ave decreased with increasing ΔL_p,section (Fig. 6).

(3) It has been shown experimentally that the variation of I_c value along the tape length depends strongly on the voltage probe spacing L; larger probe spacing gives the appearance of higher uniformity of the I_c values.²²⁾ In the present work, the standard deviations of the I_c values of sections (L = 1.5 cm), ΔI_c,section, were 11.4, 22.6, and 43.1 A, and those of regions (L = 7.5 cm), ΔI_c,region, were 3.3, 16.3, and 32.9 A, for ΔL_p,section = 0.05, 0.1, and 0.2, respectively. These results show that ΔI_c,region was lower than that of ΔI_c,section. Consequently, the experimental behavior was reproduced by the present simulation.

(4) The difference between I_{c,region,upper} and I_{c,region,lower} was around 10 A in this work. Thus, in the vertical axis scale (0 to I_c0 (= 200 A)) in Fig. 4, the difference among I_c,region, I_{c,region,upper}, and I_{c,region,lower} cannot be distinguished clearly. When the vertical axis is magnified, the difference is visible (Fig. 5). On the contrary, the difference among n_region, n_region,upper, and n_region,lower is clear on the vertical axis scale (0 to n₀ (= 40)) in Fig. 4. This result shows that the n-value of the region is more sensitive to the difference in crack size among the sections than the I_c value.

(5) The I_c,region values for a small difference in crack size among the sections (Fig. 5(a)) are near to the I_{c,region,lower} values, given by Case A (Subsection 2.4). With increasing difference in crack size among the sections (Fig. 5(b), (c)), I_c,region shifts toward I_{c,region,upper}, given by Case B (Subsection 2.4).

(6) The values of n_region for a small difference in crack size among the sections (Fig. 4(a′)) are near to the values of n_region,upper, given by Case A. With increasing difference in crack size among sections (Fig. 4(b′)–(f′)), n_region shifts toward n_region,lower, given by Case B.

(7) The behavior described in (5) and (6), namely the shift of the critical current from the lower to upper bound and the shift of n-value from the upper to lower bound with increasing crack size distribution width, is shown clearly by the average critical current and n-value (Fig. 6(a) and (b)).

3.2 Analysis of the V–I curves of regions with an equivalent crack-current shunting model and the behavior of ligament parameter L_p,region and electrical resistance in the shunting circuit R_t,region

The experimental results for multiple cracks between the voltage probes in cracked tapes are described well by replacing multiple cracks with an equivalent crack in the crack-current shunting model because the shunting mechanism for the cracks is the same.^2,7,9,15,16) In this subsection, the V–I curves of the regions obtained by the simulation are analyzed with an equivalent crack-current shunting model to reveal the importance and utility of ligament parameter L_p,region and electrical resistance in the shunting circuit R_t,region of the region in determining I_c,region and n_region. In using an equivalent crack-current shunting model for the V–I curves of the regions, L₀ = 1.5 cm was replaced with L = 7.5 cm, and L_p,section and R_t,section were replaced with L_p,region and R_t,region in eqs. (1) and (2), respectively. Modified eqs. (1) and (2) were fitted to the V–I curves of the regions, from which L_p.region and R_t,region were obtained.

Figure 7 shows three examples (Exs. 1, 2, and 3) of the simulated V–I curves of the 7.5 cm regions and five 1.5 cm sections that constitute the region (a–c), comparison of the simulated V–I curves of the 7.5 cm regions with the V_region,upper–I and V_region,lower–I curves (a′–c′), and the analysis results of the simulated V–I curves of the 7.5 cm regions with an equivalent crack-current shunting model (a′′–c′′). Examples 1, 2, and 3 in Fig. 7 had the following statistical features. The average ligament parameter values of the sections (L_{p,section,ave}) were mainly in the range of 0.68–0.70, and the average critical current values (I_{c,section,ave}) and average n-values (n_section,ave) of the sections were generally in the ranges of 136–140 A and 27–30, respectively. Thus, the sections in these examples had similar average crack sizes, average critical current values, and average n-values, whereas the crack size distribution width was different.

Fig. 7

Examples (Exs. 1, 2, and 3) of (a)–(c) V–I curves of the 1.5 cm sections and 7.5 cm regions, obtained by the simulation, (a′)–(c′) V–I curves of the regions compared with the V_region,upper–I and V_region,lower–I curves, and (a′′)–(c′′) the results of using an equivalent crack-current shunting model for the V–I curves of the regions, where the solid and dotted curves show the V–I curves of the regions and the V–I curves reproduced with the obtained L_p,region and R_t,region values, respectively.

The V–I curves of the regions and sections in Fig. 7(a)–(c) show the following features. The V–I curve of the region is near that of the lowest I_c section that has the largest crack. Thus, the lowest I_c section contributes most to the synthesis of the V–I curve of the region. The sections with the next largest cracks also contribute to the synthesis of the V–I curve of the region when the interspacing among the V–I curves of the sections is small, and the difference in crack size is small. In contrast, the sections with the next largest cracks do not contribute when the interspacing among the V–I curves of the sections is large, and the difference in crack sizes is large.

The V_region–I curves of the region in Fig. 7(a)–(c) are compared with the V_region,upper–I and V_region,lower–I curves in Fig. 7(a′)–(c′). The V_region–I curve is in between the V_region,upper–I and V_region,lower–I curves. The V_region–I curve is near the V_region,upper–I curve when the V–I curves of the sections are close together (when the difference in crack size among sections is small) as in Fig. 7(a′). The V_region–I curve shifts to the V_region,lower–I curve with the increase in interspacing among the V–I curves of the sections (with the increase in difference in crack size among sections) as in Fig. 7(b′), (c′). The V_region–I curve shift causes the shift in I_c,region from I_{c,region,lower} to I_{c,region,upper} and the shift in n_region from n_region,upper to n_region,lower with increasing ΔL_p,section (Fig. 6).

Figure 7(a′′)–(c′′) show the results of using an equivalent crack-current shunting model for the V–I curves of the regions. The values of L_p,region and R_t,region were 0.673 and 8.7 µΩ for Ex. 1, 0.624 and 4.5 µΩ for Ex. 2, and 0.386 and 2.3 µΩ for Ex. 3 (Fig. 7(a′′)–(c′′)). The solid curves in Fig. 7(a′′)–(c′′) are the V–I curves obtained from the simulation and the dotted curves are the V–I curves back-calculated by substituting the calculated values of L_p,region and R_t,region into eqs. (1) and (2). The V–I curves, and the critical current and n-value of the regions are reproduced well by the equivalent crack-current shunting model (Fig. 7(a′′)–(c′′)).

Figure 8(a) shows the variation of the estimated values of the ligament parameter of the regions (L_p,region) with position. The variation of L_p,region with position increases with increasing ΔL_p,section, that is, with the increasing crack size distribution width. Comparing the results in Fig. 8(a) with those in Fig. 4, the variation of L_p,region with position is similar to that of I_c,region. Then, using the set values (L_p,region, I_c,region), the value of I_c,region was plotted against L_p,region (Fig. 8(b)). L_p,region has a nearly linear relationship with I_c,region, indicating that L_p,region plays a role in determining the critical current as a first approximation. L_p,region reflects the crack size of the lowest I_c section that contributes most to the synthesis of the V–I curve of the region (Fig. 7). The nearly linear relation between L_p,region and I_c,region has been observed in the analyzed results of the experimentally measured V–I curves of cracked tapes with the equivalent crack-current shunting model.^2,3,7,16) These results indicate that the size of the largest crack in the region is the primary factor in determining the critical current.

Fig. 8

(a) Variation of the estimated values of L_p,region with position for ΔL_p,section = 0.01, 0.05, and 0.15 and (b) plot of I_c,region against L_p,region from the obtained sets of (L_p,region, I_c,region) values for ΔL_p,section = 0.01 to 0.2.

Figure 9(a) shows the variation of electrical resistance of the shunting circuit (R_t,region) in the regions with position along the tape length. Figure 9(b) shows the variation of the average values of R_t,region (R_t,region,ave) with ΔL_p,section and the average values of the upper (R_{t,region,upper,ave}) and lower (R_{t,region,lower,ave}) bounds for reference. The variation of R_t,region with position increases greatly with increasing ΔL_p,section (increasing crack size distribution width) (Fig. 9(a)) similar to other property values (I_c (Fig. 5), n-value (Fig. 4(a′)–(f′)), and L_p,region (Fig. 8(a))). The average electrical resistance of the shunting circuit (R_t,region,ave) decreases with increasing ΔL_p,section, and shifts from the upper bound to the lower bound with increasing crack size distribution width (Fig. 9(b)). This shift reflects the change in interspacing among the V–I curves of the sections, from narrow interspacing that gives a large R_t,region value, to wide interspacing that gives a low R_t,region value.

Fig. 9

Variation of the estimated values of (a) R_t,region with position and (b) R_t,region,ave with ΔL_p,section.

The correlation between I_c,region and n_region values is not determined uniquely because there can be multiple n-values for one I_c value, depending on the positional relation among the V–I curves of sections, as shown by Exs. 4 and 5 in Fig. 10(a) and (b). The (I_c, n) values of the regions of Exs. 4 and 5 were (133 A, 21.0) and (133 A, 13.2), respectively. The I_c values were the same, but the n-values were different. Thus, there can be multiple n-values for one I_c value.

Fig. 10

Importance of R_t,region values for the correlation diagram between n_region and I_c,region. Examples 4 and 5 have the same I_c,region value of 133 A and different n_region values of (a) 21.0 and (b) 13.2 due to the difference in positional relationship among the V–I curves of the sections, which is reflected in the different R_t,region values of 6.13 (Ex. 4) and 2.43 µΩ (Ex. 5). (c) n_region–I_c,region correlation diagram, which can be described by using multiple R_t,region values for the average, upper bound, and lower bound.

The R_t,region value is large when the V–I curves of the sections are near the V–I curve of the largest crack section (difference in crack size is small), but it decreases when the spacing among the V–I curves becomes large (difference in crack size is large) (Fig. 7). Thus, R_t,region reflects the positional relationship among the V–I curves of the sections, and consequently, it reflects the difference in crack size among the sections. Similarly, Exs. 4 (Fig. 10(a)) and 5 (Fig. 10(b)) reflect the positional relationship among the V–I curves: R_t,region is as high as 6.13 µΩ in Ex. 4 because the V–I curves are close together, but it is as low as 2.43 µΩ in Ex. 5 because the V–I curves are widely separated. These results indicate that using multiple R_t,region values is suitable for constructing diagrams of the correlation between I_c and n-values. This is because the correlation between I_c and n-values is not determined uniquely due to the dispersed positional relationship among the V–I curves of the sections.

Figure 10(c) shows an example of the correlation diagram between n_region and I_c,region, obtained by the present simulation method for L_{p,section,ave} = 0.40, 0.67, and 0.94, L = 7.5 cm, and ΔL_p,section = 0.15. When the spacing among the V–I curves of the sections is small (difference in crack size among sections is small), many sections contribute to the voltage of the region and R_t,region becomes high, which acts to raise the n-value. When the spacing among the V–I curves of the sections is large (difference in crack size among sections is large), fewer sections contribute to the region’s voltage, and R_t,region becomes low, which acts to reduce the n-value. Therefore, in heterogeneously damaged superconducting tape, the upper bound, lower bound, and average in the n_region–I_c,region diagram of regions, arising from the difference in crack size among sections, can be expressed by using multiple R_t,region values (Fig. 10(c)).

4. Conclusions

1. (1)    The variation of critical current along the longitudinal direction of the tapes increases with increasing crack size distribution width for voltage probe spacings of 1.5 and 7.5 cm. The extent of the variation decreases for larger voltage probe spacing, as observed experimentally, indicating that local information about critical current values is diluted when the voltage probe-distance is large.
2. (2)    Compared with the critical current, the n-value exhibits larger variation with position along the longitudinal direction.
3. (3)    Among the V–I curves of the sections, the V–I curve of the section with the largest crack contributes most to the synthesis of the V–I curve of the region consisting of the sections. The contribution of the largest crack section increases with increasing difference in crack size among the sections.
4. (4)    In modeling, Case A, an extreme case where all of the 1.5 cm sections that constitute the 7.5 cm region have cracks with the same size as the largest crack, was used to give a lower bound for the critical current and an upper bound for the n-value of the region. Case B, an extreme case where one section has a far larger crack than other sections, was used to give an upper bound for the critical current and lower bound for the n-value. The critical current and n-values of the regions are within the upper and lower bounds for any crack size distribution width. The difference in critical current value between the upper and lower bounds was relatively small compared with the difference in n-value.
5. (5)    When the crack size distribution width is small, the critical current values measured for regions are near the lower bound given by Case A. With increasing crack size distribution width, the critical current shifts from the lower bound given by Case A to the upper bound given by Case B. In contrast, when the crack size distribution width is small, the n-value measured for regions is near the upper bound given by Case A, but it shifts from the upper bound given by Case A to the lower bound given by Case B with increasing crack size distribution width.
6. (6)    Analyzing the simulation results for the regions with an equivalent crack-current shunting model showed that the ligament parameter of the region has a nearly linear relationship with the critical current and the size of the largest crack is the primary factor in determining critical current value. The electrical resistance of the current shunting circuit in the region reflects the positional relation among the V–I curves of the sections. Hence, the multiple n-values observed for a critical current value and the correlation diagram between the n-value and critical current can be described by using this feature.

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