2020 Volume 61 Issue 4 Pages 766-775
The effects of the size of the largest crack and size difference among cracks on critical current of superconducting tape with multiple cracks of different sizes in the superconducting layer were investigated by a model analysis and a Monte Carlo simulation, using the specimens consisting of a series circuit of local sections where each section has one crack of different size from each other. It was revealed that, with increasing distribution-width of crack size, the increase in the size of the largest crack acts to reduce the critical current and the increase in the crack size-difference among the sections acts to raise the critical current, and these conflicting two effects are summed up and determine the critical current value. To describe this feature quantitatively, we expressed the statistics of the size of the largest crack in specimens with the Gumbel’s extreme value distribution function. Also we monitored the effect of the difference in crack size among the sections on specimen’s critical current, using the number of the sections equivalent to the largest crack-section at the critical voltage for determination of specimen’s critical current. In this monitoring, small number of the sections equivalent to the largest crack-section corresponds to large difference in crack size among the sections. With the present approach, the effect of the increase in size of the largest crack, which acts to reduce critical current, and the effect of the increase in difference in crack size among the sections, which acts to raise critical current, with increase in distribution-width of crack size, could be estimated separately.
With model analysis and Monte Carlo simulation, it was clarified and quantified that the critical current Ic of a superconducting tape with multiple cracks of different sizes is given by the sum of the lower bound of the critical current Ic,lower determined by the size of the largest crack and the increment of the critical current ΔIc resulting from the size difference among the cracks.
During fabrication and operation, superconducting tapes are subjected to thermal, mechanical, and electromagnetic stresses/strains. At high stresses/strains, the superconducting phase is cracked, which reduces critical current Ic and n-value in both coated1–12) and filamentary13–21) tapes. Cracking of the superconducting layer/filament occurs heterogeneously, due to which the Ic- and n-values differ from specimen to specimen6–12) and vary along the specimen length.7–12,14,15) The same situation takes place under the existence of the heterogeneously distributed Ic-reducing defects, introduced during the tape-fabrication process.22,23) It is required to have full knowledge of the relation of heterogeneously distributed defects/cracks to superconducting property for safety design.
We have been developing a Monte Carlo simulation method7–12) combined with the current shunting model at cracks13) to detect the effects of crack size distribution on Ic- and n-values. We have shown that, among the voltage V–current I curves of the local sections that constitute the specimen, the V–I curve of the section with the largest crack contributes most to the synthesis of the V–I curve of the specimen.9–12) Accordingly, the size of the largest crack is a primary factor in determination of Ic-value of specimen. However, not only the size of the largest crack but also the difference in crack size among the sections affects the Ic-value of specimen,9–12) since the latter changes the interspacing among the V–I curves of the local sections and hence the V–I curve of the specimen.9,12)
In the present work, we took up the stress-induced cracks as the representative of the defects, and attempted to describe the Ic values from the viewpoint of the roles of the size of the largest crack and the size difference among the cracks in determination of the specimen’s Ic values. As the tools, we used the Monte Carlo simulation method mentioned above, Gumbel’s extreme value distribution24) for calculation of the size-distribution of the largest crack among the specimens and a new method that monitors the effect of the size difference among the cracks on the specimen’s Ic values. Details of the procedure of the simulation and analysis, and the results, are reported in this paper.
RE(Y, Sm, Dy, Gd, …)Ba2Cu3O7−δ layer-coated superconducting tape (hereafter noted as REBCO tape) was used for the model specimen whose configuration is shown in Fig. 1(a) and (b). The specimens were 4.5, 15 and 30 cm long, consisting of 3, 10 and 20 local sections with a length L0 = 1.5 cm. Each section had one crack with a different size from each other. Current path in a cracked section is shown schematically in Fig. 2, in which one section in a REBCO tape specimen is representatively drawn. In the transverse cross-section in which a crack exists, the cracked part and the ligament part constitute a parallel electric circuit.
Modeling for analysis. (a) The model specimen consisting of N local sections with a length L0 = 1.5 cm, having a crack in each section. (b) Configuration of multiple cracks of different sizes in specimen. (c) An extreme Case A (N sections have the same size-crack). (d) Another extreme Case B (one section has a crack and other sections have no crack). (e) Schematic drawing of the number of sections equivalent to the largest crack-section Keq (1 ≤ Keq ≤ N), which is used to monitor the effect of the difference in crack size among the sections shown in (b) on critical current of the specimen. The case of Keq = 3 is drawn as an example.
Schematic representation of the current path in a section with a crack.7)
Details of the simulation procedure have been reported in our preceding works.7–12) Here, we show an outline. The following technical terms were used.
The cracked- and ligament-parts constitute a parallel electric circuit in the cracked section.13) Accordingly, the voltage Vs developed at the cracked part is equal to the voltage VRE developed at the ligament part (Vs = VRE). In this work, we use VRE as a representative of VRE and Vs. The transport current I is the sum of the REBCO-transported current IRE in the ligament part and the shunting current Is in the cracked part (I = IRE + Is).
The V–I curve of the cracked section is expressed as,2,3,9–12)
| \begin{equation} V = E_{\text{c}}L_{0}\left(\frac{I}{I_{\text{c0}}}\right)^{n_{0}}{} + V_{\text{RE}} \end{equation} | (1) |
| \begin{equation} I = I_{\text{RE}} + I_{\text{s}} = I_{\text{c0}}L_{\text{p}}\left[\frac{V_{\text{RE}}}{E_{\text{c}}L_{0}}\right]^{1/n_{0}} {}+ \frac{V_{\text{RE}}}{R_{\text{t}}} \end{equation} | (2) |
The distribution of Lp was formulated using the normal distribution function, as in our former works.9–12) The cumulative probability F(Lp) and density probability f(Lp) for the average of Lp, Lp,ave, are expressed as
| \begin{equation} F(L_{\text{p}}) = \frac{1}{2}\left\{1 + \text{erf}\left(\frac{L_{\text{p}} - L_{\text{p,$\,$ave}}}{\sqrt{2}\Delta L_{\text{p}}}\right)\right\} \end{equation} | (3) |
| \begin{equation} f(L_{\text{p}}) = \frac{1}{\sqrt{2\pi}\Delta L_{\text{p}}}\exp\left\{-\frac{(L_{\text{p}} - L_{\text{p,$\,$ave}})^{2}}{2(\Delta L_{\text{p}})^{2}}\right\} \end{equation} | (4) |
The V–I curve of each cracked section was calculated, by substituting the Lp-value obtained by the Monte Carlo method, and the values of Ic0 = 200 A, n0 = 40 and Rt = 2 µΩ, taken from our former experimental work on REBCO tape,5,9–12) into eqs. (1) and (2).
As the specimen consists of a series electric circuit of N sections (N = 3, 10 and 20 for L = 4.5, 15 and 30 cm, respectively) (Fig. 1(a)), the current in the specimen is the same as that in all sections, and the specimen’s voltage is the sum of the voltages of all sections,
| \begin{equation} I_{\text{specimen}} = I_{\text{section(${i}$)}}\quad (i = \text{1 to $N$}) \end{equation} | (5) |
| \begin{equation} V_{\text{specimen}} = \sum_{i = 1}^{N} V_{\text{section(${i}$)}} \end{equation} | (6) |
From the V–I curves of the sections and specimens, the Ic-values of the sections and specimens were obtained with the electric field criterion of Ec = 1 µV/cm, corresponding to the critical voltages Vc = EcL0 for sections and Vc = EcL = EcNL0 for specimens.
2.3 Model to obtain the upper and lower bounds of critical current of specimens9–12)The present model specimen consists of a series of N sections and contains N cracks of different sizes (Fig. 1(b)). The superconductivity is lost first in the section with the largest crack. Therefore, the voltage developed at the largest crack-section is the highest among all sections, and it contributes most to the specimen’s voltage. We can obtain the upper and lower bounds of Ic of specimen using the V–I curve of the largest crack-section among all sections, as follows.9–12) Hereafter the ligament parameter of the largest crack-section (= smallest ligament-section) among all sections is expressed as Lp,smallest.
Case A (Fig. 1(c)) corresponds to the extreme case where the crack size is the same as the largest crack in all sections. Thus, the V–I curves of all sections are the same. Accordingly, the specimen’s voltage, given by the sum of the voltage of all sections, corresponds to the upper bound of the specimen’s voltage, Vupper, under the given size of the largest crack. As the Vupper reaches the critical voltage Vc at lowest I, the Ic value in Case A corresponds to the lower bound, Ic,lower.
Case B (Fig. 1(d)) corresponds to the extreme case where the specimen is an assembly of one severely cracked section and non-cracked sections. This case gives the lower bound Vlower for the specimen’s voltage under the given size of the largest crack. As Vlower reaches Vc at highest I, the Ic value in Case B corresponds to the upper bound Ic,upper.
Case A gives the Vupper–I curve and Ic,lower, and Case B gives the Vlower–I curve and Ic,upper for specimen under the given size of the largest crack (= smallest ligament). The Vupper–I and Vlower–I curves of specimens were calculated with eqs. (1), (2), (5) and (6) by setting Lp = Lp,smallest that refers to the smallest ligament (= largest crack)-section. From the calculated curves, the Ic,upper and Ic,lower values were obtained with the criterion of Ec = 1 µV/cm.
2.4 Monitoring of the contribution of the difference in crack size among sections to the critical current of specimenIn Case A, number of N sections have the largest crack (Fig. 1(c)), and, in Case B, one section has the largest crack (Fig. 1(d)). In practical situation as shown in Fig. 1(b), not only the largest crack-section but also the other sections contribute to the voltage of the specimen. In this study, for the quantitative estimation of the effect of the positional relation of the V–I curves of the sections, induced by the difference in crack size among the sections, on the critical current of specimens, the sum of the voltages of sections (= the voltage of the specimen) was replaced by the sum of the voltages of a number of Keq sections equivalent to the largest crack-section and the voltages of N–Keq sections without cracks. As an example of this replacement, the case of Keq = 3 is shown schematically in Fig. 1(e) where the voltage of the specimen at V = Vc, which is the sum of the voltages of the sections containing the cracks of different sizes in Fig. 1(b), is expressed as the sum of the voltages of Keq (= 3) sections equivalent to the largest crack-section and voltages of N–Keq non-cracked sections at V = Vc. In this approach, the V–I curves of the largest crack-section could be expressed by eqs. (1) and (2) as before, and the V–I curve of the specimen can be expressed by.
| \begin{equation} V = E_{\text{c}}L\left(\frac{I}{I_{\text{c0}}}\right)^{n_{0}} {}+ K_{\text{eq}}V_{\text{RE}} \end{equation} | (7) |
Figure 3 shows the difference in Ic value due to the difference in Keq value under the condition of the smallest ligament parameter Lp,smallest = 0.5 and specimen length L = 15 cm, as an example. The V–I curves of the specimen for Keq = 1 to 10 are shown in Fig. 3(a), in which Keq = 10 and 1 corresponds to Cases A and B, respectively. The Keq value exists in between 1 and N (= L/L0), as schematically shown in Fig. 1(c–e). The Ic value of the 15 cm-specimen is equal to Ic,lower when Keq = N = 10 (Case A), it increases with decreasing Keq and becomes the upper bound value Ic,upper when Keq = 1 (Case B), as shown in Fig. 3(b).
An example showing the effects of Keq value on critical current. (a) V–I curves of the specimen for Keq = 1 to 10, in which Keq = 10 and 1 corresponds to Cases A and B, respectively and (b) change in critical current of the specimen with varying Keq value, calculated for the smallest ligament parameter Lp,smallest = 0.5 and specimen length L = 15 cm.
The results in Fig. 3(b) show evidently that the Ic value increases with decreasing Keq; namely with increasing difference in crack size among the sections under the given size of the largest crack (= under the given size of the smallest ligament, which is given by Lp,smallest = 0.5 in this example). In this way, the Ic value is affected not only by the size of the largest crack that is monitored by Lp,smallest but also by the difference in crack size among the sections that is monitored by Keq. It is noted that, while the Ic value is dependent both on Lp,smallest- and Keq-values, the Ic,lower value (Case A) is determined solely by the Lp,smallest value, as will be shown in detail in subsection 3.1. The difference ΔIc between the Ic and Ic,lower, ΔIc = Ic − Ic,lower, is attributed to the effect of the difference in crack size among the sections. As an example, the case of Keq = 3 is picked up and the ΔIc and its relation to Ic and Ic,lower are drawn in Fig. 3(b). In this way, Ic is given by the sum of the effect of the largest crack size (Ic,lower) and that of the difference in crack size among the sections (ΔIc). This result will be used in subsections 3.1 and 3.2 for separate estimation of the effects of the size of the largest crack and the difference in crack size among sections on Ic value.
Figure 4 shows the examples showing the effect of difference in crack size among the sections in 15 cm-specimen constituted of 10 sections (a, a′) on the positional relation of the V–I curves among the sections and between the specimen and the sections, (b, b′) on the positional relation among the V–I, Vupper–I and Vlower–I curves of specimen, and (c, c′) on the positional relation of the V–I curve of the specimen obtained by simulation and the V–I curves of the specimen calculated with eqs. (1), (2) and (7) for Keq = 1 to 10, where Keq = 1 and 10 refer to Case B and Case A, respectively. The results of the examples Ex. 1 in Fig. 4(a–c) and Ex. 2 in Fig. 4(a′–c′), taken up from the simulation results, correspond to the cases where the difference in ligament parameter (Lp) value (= difference in crack size) among the sections is small and large, respectively. In these examples, the average of the ligament parameter, Lp,ave, (namely average of crack size) was common (Lp,ave = 0.67) and the distribution-width of crack size was different.
Effect of difference in crack size among the sections on the V–I curve and critical current of specimen. Ex. 1 and Ex. 2 are examples taken from the simulation results, where the difference in crack size among the sections is small in Ex. 1 but large in Ex. 2, while the average crack size is common. (a, a′) show the positional relation of the V–I curves among the sections and between the specimen and the sections, (b, b′) comparison of the simulated V–I curves of specimens with the calculated Vupper–I and Vlower–I curves, and (c, c′) comparison of the simulated V–I curves of specimens with the calculated V–I curves for Keq = 1 to 10.
When the crack sizes of the sections are close to each other as in Ex. 1, the V–I curves of all sections exist near to the V–I curve of the largest crack-section. Therefore, the voltages of many sections contribute to the rise of the voltage of the specimen. On the other hand, when the crack sizes of the sections are different as in Ex. 2, the interspacing among the V–I curves of the sections is large. Therefore, the voltage only of the largest crack-section or a few sections with the relatively large cracks contributes to synthesize the V–I curve of the specimen. The critical currents of the extreme Cases A and B are obtained from the Vupper–I and Vlower–I curves, respectively. In actual specimens, the situation is in between Case A and Case B, and accordingly the V–I curves exist in between the Vupper–I and Vlower–I curves, and the critical current values exist in between Case A and Case B (Fig. 3(a, b)).
The Ic value of specimen is given as the current at V = Vc = EcL. The relation of Vc to Ic, Keq and VRE and the relation of Ic to Lp,smallest and VRE in the smallest ligament-section are expressed by eqs. (8) and (9) from eqs. (7) and (2), respectively:
| \begin{equation} V_{\text{c}} = E_{\text{c}}L = E_{\text{c}}L\left(\frac{I_{\text{c}}}{I_{\text{c0}}}\right)^{n_{0}}{} + K_{\text{eq}}V_{\text{RE}} \end{equation} | (8) |
| \begin{equation} I_{\text{c}} = I_{\text{RE}} + I_{\text{s}} = I_{\text{c0}}L_{\text{p,$\,$smallest}}\left[\frac{V_{\text{RE}}}{E_{\text{c}}L_{0}}\right]^{1/n_{0}}{} + \frac{V_{\text{RE}}}{R_{\text{t}}} \end{equation} | (9) |
When the distribution-width of the crack size is small as in Ex. 1, Keq value is high (4.61). With increasing distribution-width of crack size, Keq value decreases (1.55 in Ex. 2) and asymptotically approaches unity at large difference in crack size among the sections (Fig. 3(b)). With the present approach, Keq values for wide range of ΔLp and L can be obtained.
Figure 5 shows the changes of (a–c) the smallest ligament parameter Lp,smallest values and (a′–c′) critical current Ic values of specimens with increase in standard deviation of the ligament parameter ΔLp, obtained by simulation for specimen length L = (a, a′) 4.5 cm, (b, b′) 15 cm and (c, c′) 30 cm. The circle (○) indicates the value of Lp,smallest and Ic of each specimen, and the square (□) indicates the average value (Lp,smallest,ave and Ic,ave). Evidently, the Lp,smallest,ave and Ic,ave of specimens decreased with increasing ΔLp. Also the extent of their decrease with increasing ΔLp was enhanced in longer specimen. These results suggest that, on average, the size of the smallest ligament in specimen decreases (= size of the largest crack in specimen increases) with increasing ΔLp and L, which acts to reduce Ic.
Changes of (a–c) the smallest ligament parameter Lp,smallest and (a′–c′) critical current Ic of the specimens with increasing standard deviation of the ligament parameter ΔLp, obtained by simulation for specimen length L = 4.5, 15 and 30 cm. The circle symbol indicates the value of each specimen and the square symbol indicates the average value.
Figure 6 shows the plot of (a–c) Ic value of each specimen against the Lp,smallest value, and plot of (a′–c′) Ic,ave value of specimens against the Lp,smallest,ave value, where the Ic and Lp,smallest values were averaged for each set of L- and ΔLp-values. The lower bound Ic,lower and upper bound Ic, upper of critical current were obtained as a function of Lp,smallest through the calculation of the V–I curves, by setting Lp = Lp,smallest in eq. (2) and Keq = N (Case A) and 1 (Case B) in eq. (7).
Plot of (a–c) critical current Ic against the smallest ligament parameter Lp,smallest, and (a′–c′) average critical current Ic,ave against the average smallest ligament parameter Lp,smallest,ave. (a, a′), (b, b′) and (c, c′) refer to the simulation results for L = 4.5, 15 and 30 cm, respectively. For comparison, the calculated relations of Ic,upper to Lp,smallest and Ic,lower to Lp,smallest are presented with the solid and broken lines, respectively.
The results in Fig. 6 show that, when the specimen is short (4.5 cm), the difference between Ic,upper and Ic,lower is small. This feature suggests that the smallest ligament (= largest crack)-section plays a significant role in determination of Ic-value especially in short specimens, as has been shown in our preceding work.12) When the Ic,upper–Ic,lower is small as in Fig. 6(a, a′), the Ic is approximately expressed as12)
| \begin{equation} I_{\text{c}} \cong I_{\text{c,$\,$lower}} \end{equation} | (10) |
When the Lp,smallest value is known, Ic,lower can be calculated under the condition of Keq = N. As shown in our preceding work,12) the Lp,smallest-value and its distribution can be obtained by using the Gumbel’s extreme value distribution function.24) The average of Lp,smallest values of the specimens, Lp,smallest,ave, for each set of ΔLp- and L-values is obtained by24)
| \begin{equation} L_{\text{p,smallest,ave}}= \lambda-\alpha\gamma \end{equation} | (11) |
| \begin{equation} F(\lambda)=1/N \end{equation} | (12) |
| \begin{equation} \alpha=1/\{Nf(\lambda)\} \end{equation} | (13) |
Plots of (a) average smallest ligament parameter, Lp,smallest,ave, and (b) average lower bound of critical current Ic,lower,ave, of the specimens with length L = 4.5, 15 and 30 cm, against the standard deviation of ligament parameter ΔLp. (c) Plots of Ic,lower,ave against Lp,smallest,ave. Open and closed symbols refer to the values obtained by simulation and calculation, respectively.
It is important that the largest crack, monitored by Lp,smallest, plays a role to give the Ic,lower, as shown in Fig. 7(c), where the Ic,lower,ave values in Fig. 7(b) are plotted against the corresponding Lp,smallest,ave values in Fig. 7(a). The Ic,lower,ave has one to one relation to the Lp,smallest,ave regardless the values of the specimen length L and distribution-width of crack size ΔLp. The Ic,lower,ave arises in Case A where the crack size is the same in all sections in specimen, and hence the Ic,lower,ave is not dependent on specimen length. This result shows that the Ic,lower,ave value is determined solely by the size of the largest crack. The decrease in critical current with increasing ΔLp and L is caused by the increase in the size of the largest crack. It is also important that, while the critical current is almost the same as the lower bound when ΔLp and L are small, the difference between Ic and Ic,lower, ΔIc = Ic − Ic,lower, increases with increasing ΔLp and L (Fig. 6). As indicated in Figs. 3 and 4, the wider distribution of crack size, which corresponds to smaller Keq value, leads to wider positional spacing among the V–I curves of sections, which acts to raise Ic value under the given size of the largest crack. Namely, Ic is given by the sum of the Ic,lower that is determined by the size of the largest crack and ΔIc that is determined by the difference in crack size among the sections, as has been shown in Fig. 3(b). In the next subsection, the effect of the difference in crack size among the sections on ΔIc is evaluated by using the Keq-value.
3.2 Role of the difference in crack size among the sections in determination of critical current of specimenFigure 8 shows the changes in number of sections equivalent to the largest crack-section, Keq, and the difference between the critical current and its lower bound (ΔIc = Ic − Ic,lower) with increase in distribution-width of crack size (ΔLp). The values of Keq were estimated by the procedure stated in subsection 2.4. The variations of Keq and Keq,ave (average of Keq-values at each set of ΔLp- and L-values) with increasing ΔLp for L = 4.5, 15 and 30 cm are shown in Fig. 8(a–c). The Keq decreases with increasing ΔLp. This decrease reflects the shift of interspacing among the V–I curves of the sections, from narrow interspacing that gives a large Keq value, to wide interspacing that gives a small Keq value.
Plot of (a–c) number of sections equivalent to the smallest ligament (largest crack)-section, Keq, and (a′–c′) difference between the critical current and its lower bound ΔIc (= Ic − Ic,lower), against the standard deviation of ligament parameter ΔLp. (a, a′), (b, b′) and (c, c′) refer to the result for L = 4.5, 15 and 30 cm, respectively.
The values of ΔIc (= Ic − Ic,lower) were taken from the simulation results for each ΔLp value in Fig. 6. The increments of ΔIc value with increasing ΔLp for L = 4.5, 15 and 30 cm are shown in Fig. 8(a′–c′). The results show that the ΔIc increases with increasing ΔLp; the larger the difference in crack size among sections, the larger becomes the increment of ΔIc.
Figure 9 shows the plot of the average values of ΔIc, ΔIc,ave, at ΔLp = 0.01∼0.15 against the average of the equivalent number of the largest crack-section, Keq,ave, for L = 4.5, 15 and 30 cm. The feature that the decrease in Keq contribute to raise ΔIc for a given Lp,smallest value (Figs. 3 and 4) is reproduced well. This result shows that Keq value is useful as a tool to estimate the contribution of the difference in crack size among the sections in determination of ΔIc.
Average of difference between the critical current and its lower bound, ΔIc,ave = Ic,ave − Ic,lower,ave, plotted against the average number of sections equivalent to the smallest ligament (largest crack)-section, Keq,ave.
As has been shown in subsections 3.1 and 3.2, the smallest ligament parameter Lp,smallest referring to the largest crack-section and the number of the sections equivalent to the largest crack-section Keq are useful tools to estimate the lower bound of the critical current Ic,lower and the contribution of difference in crack size among the sections to critical current ΔIc, respectively. When the Keq- and Lp,smallest-values are known by simulation or calculation, the critical current Ic is calculated by Ic = Ic,lower + ΔIc. In this subsection, the simulation results of the changes in Ic,ave-, Ic,lower,ave- and ΔIc,ave-values as a function of the distribution-width of ligament size (= distribution-width of crack size) ΔLp for each specimen length L (4.5, 15 and 30 cm) will be reproduced.
For reproduction of the simulation results, the following calculations were carried out. The Lp,smallest,ave values for ΔLp = 0.01∼0.15 under the specimen length of L = 4.5, 15 and 30 cm were calculated using eqs. (3), (4), (11), (12) and (13), as has been shown in Fig. 7. The Ic,lower,ave value for each set of ΔLp- and L-values was obtained by calculation of the V–I curve through substituting the Lp,smallest,ave and Keq = N into eqs. (2) and (7) and application of the criterion of Ec = 1 µV/cm. In the same way, the Ic,ave value was obtained by substituting Lp,smallest = Lp,smallest,ave and Keq = Keq,ave into eqs. (2) and (7). The ΔIc,ave was calculated by ΔIc,ave = Ic,ave − Ic,lower,ave.
Figure 10 shows the changes of the Ic,ave values obtained by simulation (○) and calculation (●), the Ic,lower,ave values obtained by calculation (◇), and the ΔIc,ave values obtained by calculation (□) with increasing ΔLp for L = 4.5, 15 cm and 30 cm. As shown in Fig. 10, the simulation results of Ic,ave values are described well by the calculation of Ic,ave = Ic,lower,ave + ΔIc,ave where the Ic,lower,ave-value is determined solely by the Lp,smallest,ave value and the ΔIc,ave value is determined by the difference in crack size among the sections under the given value of Lp,smallest,ave. In this way, the role of the largest crack size and that of the difference in crack size among the sections in determination of critical current are separately estimated by the present approach.
Effects of the standard deviation of the ligament parameter ΔLp on average critical current Ic,ave obtained by simulation (○) and calculation (●), average lower bound of critical current Ic,lower,ave obtained by calculation (◇), and average increment of critical current from the lower bound, ΔIc,ave (= Ic,ave − Ic,lower,ave), obtained by calculation (□), for specimen length L = (a) 4.5 cm, (b) 15 cm and (c) 30 cm.
With increase in distribution-width of crack size, the increase in size of the largest crack acts to reduce the critical current, while the increase in the difference in crack size among the sections acts to raise the critical current. The sum of these conflicting effects determines the critical current. The present approach, using the ligament parameter to monitor the crack size and the number of sections equivalent to the largest crack-section to monitor the effect of the difference in crack size among the sections on critical current, can be a useful tool for estimation of critical current of the tape with heterogeneous cracks in the superconducting layer.
3.4 Analysis of the increment of critical current induced by the difference in crack size among the sections in the largest crack-sectionIn this subsection, the increment of the critical current induced by the difference in crack size among the sections, ΔIc,ave, will be analyzed from the viewpoint of the contribution of the REBCO layer-transported current ΔIRE,ave at the ligament part and shunting current ΔIs,ave at the cracked part in the largest crack-section.
The contributions of ΔIRE,ave and ΔIs,ave to the ΔIc,ave as a function of ΔLp for L = 4.5, 15 and 30 cm were calculated in the following procedure. Substituting the values of Keq,ave obtained by simulation, Ic,ave obtained by the calculation procedure stated in the subsection 3.3, specimen length L and V = Vc = EcL into eq. (8), we obtained the voltage, VRE,ave in the largest crack-section at the critical voltage Vc of the specimen. Then, substituting the VRE = VRE,ave and Lp,smallest = Lp,smallest,ave calculated by the procedure stated in subsection 3.1 into eq. (9), the IRE,ave given by $I_{\text{c0}}L_{\text{p,smallest,ave}}\{ (V_{\text{RE,ave}}/E_{\text{c}}L_{0})^{1/n_{0}}\} $ and the Is,ave given by VRE,ave/Rt were calculated. In the same way, substituting Keq = Keq,ave = N, Ic = Ic,lower,ave, Lp,smallest = Lp,smallest,ave and corresponding L-value into eqs. (8) and (9), we calculated the IRE,ave and Is,ave for the lower bound Ic,lower of critical current. Noting the IRE,ave and Is,ave for the lower bound as IRE,lower,ave and Is,lower,ave, respectively, and setting ΔIRE,ave = IRE,ave − IRE,lower,ave and ΔIs,ave = Is,ave − Is,lower,ave, we had
| \begin{align} \Delta I_{\text{c,ave}}& = I_{\text{c,ave}} - I_{\text{c,lower,ave}} = I_{\text{RE,ave}}-I_{\text{RE,lower,ave}}\\ &\quad +I_{\text{s,ave}}-I_{\text{s,lower,ave}} = \Delta I_{\text{RE,ave}}+\Delta I_{\text{s,ave}} \end{align} | (14) |
Figure 11 shows the increment of critical current from the lower bound, ΔIc,ave (= Ic,ave − Ic,lower,ave), the increment of the shunting current at the cracked part, ΔIs,ave, and the increment of the REBCO-layer transported current at the ligament part, ΔIRE,ave, in the largest crack-section with increasing ΔLp for L = 4.5, 15 and 30 cm. The following features are read. (i) The phenomenon “the increment of the critical current ΔIc,ave due to the difference in crack size among the sections becomes large as Keq,ave decreases (Fig. 9)” is attributed to the increase in the shunting current at the cracked part and the current transported by the REBCO layer at the ligament part in the largest crack-section. This is because, with decrease in Keq,ave value, the number of the sections that contribute to the voltage of the specimen decreases and hence the largest crack-section generates a higher voltage at the critical voltage Vc of the specimen at which the critical current is estimated. (ii) This phenomenon becomes more prominent with increasing ΔLp due to the increase in interspacing among the V–I curves of the sections, which reduces Keq,ave-value. Also this phenomenon becomes more prominent for longer specimens due to the increase in Vc (= EcL) which raises Is,ave and IRE,ave.
Average increment of the critical current of specimen ΔIc,ave, and the shunting current ΔIs,ave and REBCO-layer transported current ΔIRE,ave in the largest crack-section at V = Vc = EcL, with increasing standard deviation of crack size ΔLp and specimen length L.
