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The influence of tensile strain to critical current of Bi2223 composite tape
Physica C 468 (2008) 1801–1804
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Physica C journal homepage: www.elsevier.com/locate/physc
The influence of tensile strain to critical current of Bi2223 composite tape Y. Mukai a, J.K. Shin a, S. Ochiai a,*, H. Okuda a, M. Sugano a, K. Osamura b a b
Department of Materials Science and Engineering, Graduate School of Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan Research Institute for Applied Sciences, Sakyo-ku, Kyoto 606-8202, Japan
a r t i c l e
i n f o
Article history: Available online 27 May 2008 PACS: 74.25.Sv 74.25.Ld Keywords: Bi2223 Composite tape Tensile strain Local critical current distribution
a b s t r a c t As the stress-induced damage evolution is different from position to position in the sample, the local critical current is scattered in a sample, affecting on the overall current. The present work aimed to describe the distribution of local critical current and its relation to overall critical current under tensile stress for Bi2223/Ag superconducting composite tape. In the experiment, seven voltage probes were attached in a step of 10 mm. The local critical current and n-value at 77 K under various applied stress levels were measured for a voltage probe distance 10 mm and the overall ones for a probe distance 60 mm. Main results are summarized as follows. The overall critical current and n-value were described well by using the voltage summation model in which the sample was regarded as a one dimensional series circuit. For the low applied stress, the distribution of local critical current was described with the three parameter Weibull distribution function. Using the measured distribution of the local critical current, an experimental relation of critical current to n-value and the voltage summation model, and applying the Monte Carlo method, the overall critical current was predicted, which was in good agreement with the experimental results. Based on these results, the sample length dependence of critical current of the sample damaged by tensile stress was discussed. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction
2. Experimental procedure
High temperature superconductor is expected to be utilized in many applications such as electric power cable, magnetic levitation train and MRI scanner. It is capable to manufacture long size composite tape enough for actual use [21]. The fracture strain of (Bi, Pb)2Sr2Ca2Cu3O10 filament that transports the superconducting current, is, however, around 0.1%,being quite lower than that of other traditional superconductors (Nb3Sn:0.5–1.2% [15,16], Nb3Al:0.6–1.0% [17], Nb–Ti:2% [18]). During fabrication/winding and operation, mechanical and electromagnetic stresses are subjected to the Bi2223/Ag composite superconductors, which change the superconducting property [1–13]. Under applied stress, there co-exist less and more damaged parts within a sample [6,7,13]. Thus the local critical current (and also n-value) is different from part to part. The aim of the present work is to describe the critical current distribution and its relation to overall critical current under applied stress.
The following two types of multifilamentary Bi2223 composite tapes were used.
* Corresponding author. Tel.: +81 75 753 4834; fax: +81 75 753 4841. E-mail address: [email protected] (S. Ochiai). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.05.088
(1) High Ic tape constituting of Bi2223 filaments, Ag and Ag–Li alloy, fabricated at American Superconductor Corporation. The width and thickness of the tape were 4.1 and 0.22 mm, respectively. (2) High strength tape constituting of the high Ic tape mentioned above and laminated reinforcement of stainless steel sheet, fabricated at American Superconductor Corporation. The width and thickness of the tape were 4.1 and 0.31 mm, respectively. Fig. 1 shows the configuration of the test piece. Seven voltage probes were attached in a step of 10 mm with solder. The overall critical current of the samples with a length of 60 mm was measured from the voltage between the probes 1 and 7. The critical current Ic of local elements (E1 to E6 in Fig. 1) with a 10 mm length was measured from the voltage between i and i + 1 probes (i = 1 to 6) of each element (E1 to E6) and that of overall sample measured by the four probe method. The sample tape was cooled down to
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Fig. 1. The configuration of local elements (E1–E6) and overall sample for test.
77 K, the voltage (V)current (I) curve was measured in a self magnetic field. The strain was measured by a couple of Nyilas type strain gauge [19]. From the result using such a test piece (Fig. 4) for both high Ic and high strength tapes, the distribution of Ic of local elements and it is relation to the overall Ic were obtained. 3. Results and discussion 3.1. Change of critical current and n-value of local elements and overall ones with increasing applied tensile stress in high Ic and high strength tapes Fig. 2 shows the changes of Ic and n-value with increasing applied tensile stress r for local elements and overall sample in (a) high Ic and (b) high strength tapes. Following features are found. (1) The overall Ic and n-value decreased slightly with increasing applied stress r up to around r = 75 MPa and 370 MPa in high Ic and high strength tapes, respectively, due to the strain effect [5]. Beyond such stresses, the overall Ic values decreased seriously with increasing stress, due to the damage evolution [5,6]. (2) The Ic and n-value of the elements also decreased with increasing stress, showing the similar tendency to those of the overall sample. (3) The Ic and n-value were different among the local elements at any applied stress. (4) The overall Ic and n-value were lower than the average values of the local elements but nearly equal to or higher than the lowest Ic and n-value among the elements. (5) There is a tendency that the element and overall sample with high Ic have high n-value. 3.2. Relation of critical current and n-value of elements with those of overall sample In present work, the 1 lV/cm criterion was used for definition of the critical current. Accordingly, the local and overall critical cur-
Fig. 2. Measured and calculated variations of overall (L = 60 mm) and local (10 mm) critical current Ic and n-value with increasing tensile stress in (a) high Ic and (b) high strength tapes.
rents were estimated as the currents at the generated voltage 1 and 6 lV, respectively. As the overall sample is composed of a series circuit of local elements (Fig. 1), the overall voltage near the transition from super- to normal conductivity is given the sum of the voltage of all elements (voltage summation approach [13]). Such an approach was used to describe the relation of critical current and n-value of the local elements to those of the overall sample as follows. The voltage (V(i)) and n-value (n(i)) of the i-element is expressed by VðiÞ ¼ AðiÞInðiÞ . For each element with a length 10 mm, Ai can be expressed practically by AðiÞ ¼ f1=Ic ðiÞgnðiÞ under the 1 lV/cm criterion for determination of Ic. Thus, the overall voltage V(overall) is expressed by
VðoverallÞ ¼ AðoverallÞInðoverallÞ ¼ ðI=Ic ðiÞÞnðiÞ
ð1Þ
where n(overall) is the n-value for overall sample. Setting V(overall) = 6 lV, and substituting the measured values of Ic(i) and n(i)
Y. Mukai et al. / Physica C 468 (2008) 1801–1804
of the constituting elements (i = 1 to 6) into Eq. (1), the overall Ic(overall) for 1 lV/cm criterion is calculated. Also, n(overall) is estimated by fitting the curve fI=Ic ðiÞgnðiÞ to A(overall)In(overall) in the range of V = 0.6 to 60 lV. The calculation result for overall Ic and n-value are presented in Fig. 2. The measured Ic and n-value at each stress of the overall sample were described well. This result indicates that, if the distribution of Ic and n-value of the local elements are known in advance, the overall Ic and n-value can be express by the voltage summation approach. In the following Section 3.3, the distribution function of Ic of elements under each tensile stress conditions and the relation of Ic to n will be obtained experimentally. Then applying the voltage summation approach and using a Monte Carlo simulation method, the Ic and n-value of overall sample at arbitrary sample length will be simulated. 3.3. Description of the distribution of the critical current Ic of local elements at low applied tensile stress
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3.4. Simulation of the critical current Ic and n-value of overall sample from the distribution of those of local elements The Monte Carlo simulation method using the voltage summation model, developed in our preceding work [20], was applied to the present result to simulate the Ic and n-value of the overall sample with arbitrary length from the distribution of them of the constituting elements with a length 1 cm. For simulation of critical current of samples with a q cm length, the values of Ic, (i) and n(i) of q constituting elements are needed. The Ic(i) and n(i) were obtained in the following procedure. At low stresses (r < 75 MPa and 370 MPa for high Ic and high strength tapes, respectively), the estimated Weibull distribution of Ic (Eq. (2)) was used as the FIc relation. At high stresses (r > 75 MPa and 370 MPa for high Ic and high strength tapes, respectively), as the Weibull distribution function given by Eq. (2) was not suitable for description of FIc relation, the measured FIc relation presented in Fig. 4 was used. The FIc relation in the
In order to formulate the distribution of Ic of local elements (Fig. 2), the three parameter-Weibull distribution function [14] was employed. According to this distribution function, the cumulative probability F of critical current Ic of the elements with a length 10 mm is expressed as
F ¼ 1 exp½fðIc Icmin Þ=I0 gm
ð2Þ
where Ic,min is the minimum critical current, I0 the scale parameter and m the shape parameter. Eq. (2) is rewritten in the form lnln{(1F)1} = mln(IcIc,min)mln(I0).The plot of lnln{(1F)1} against ln(IcIc,min) has been called as Weibull plot. The cumulative distribution function F was calculated by F = (j0.3)/(N+0.4), where j is element number in the ascending sequence of Ic and N is total number of tested elements (N = 18 and 12 for high Ic and high strength tapes, respectively). Fig. 3 shows an example of the Weibull plot obtained by the regression analysis for high Ic sample at r = 60 MPa. As typically shown in this example, high linearity between the lnln{(1F)1} and ln(IcIc,min) was obtained for the results at low applied stresses (r < 75 MPa for high Ic and r < 370 MP for high strength tapes). The typical estimated values of Ic,min, m, and I0 were 150 A, 0.8 A and 3 A, respectively, for high Ic tapes at r < 75 MPa, and 130 A, 0.75 A and 3 A, respectively, for high strength tapes at r < 370 MPa.
Fig. 3. An example of the weibull plot of critical current Ic. The Ic values of high Ic sample under r = 60 MPa is presented as an example.
Fig. 4. The cumulative probability F of critical current Ic of (a) high Ic and (b) high strength tapes.
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range outside the measured FIc relation was inferred by extrapolating the measured FIc relation as shown with the dotted curves in Fig. 4. The local critical current of the i element, Ic(i), was determined in the computer by generating a random value R(i) (0 1) and setting F(i) = R(i) in the FIc relation. For the Ic(i) value determined by the procedure stated above, the corresponding n-value, n(i) was obtained by the experimental result for the nIc relation, which was expressed by n = 1.55 + 0.092Ic and 1.54 + 0.073Ic for high Ic and high strength samples, respectively. The procedure stated above was repeated for i = 1 to q by generating further random value R(i) (i = 1 to q), and the values of Ic(i) and n(i) for i = 1 to q were determined. Substituting thus determined Ic(i) and n(i)-values (i = 1 to q) into Eq. (1), the Ic and n-value of one overall sample with a length q cm was calculated. By repeating such a procedure by using different random value series, the average overall Ic and n-value of the overall samples with a q cm length were obtained. Simulated average critical current of the sample was plotted against the length as shown in Fig. 5. The measured Ic value of 6 cm sample was also presented for comparison, which was well described by the present simulation method. The simulation result in Fig. 5 suggests that the critical current decreases only slightly against the length at low stress (0 60 MPa and 0 370 MPa for high Ic and high strength tapes, respectively). At higher stresses,
the Ic of high Ic sample decreases largely with increasing length in comparison with that of high strength one. This stems from the wider distribution of critical current in damaged high Ic sample than in damaged high strength one. 4. Conclusions (1) The correlation of distributed critical current (77 K, in self magnetic field) and n-value of local elements to overall critical current under applied tensile stress was described comprehensively with a voltage summation model for both high Ic and high strength tapes. (2) Within the range of tensile stress lower than about 75 MPa and 370 MPa for high Ic and high strength tapes respectively, the distribution of local critical current could be described with the Weibull distribution function. (3) A simulation method to describe the overall critical current from the local ones and the sample length dependence of critical current was presented, in which the experimentally measured cumulative probability of critical current, the empirical formula of the n-value as a function of critical current, voltage summation model and Monte Carlo method were combined. With this method, the experimentally observed correlation of distribution of local critical current to overall critical current and the sample length dependence of critical current could be described well. Also it was shown that the wide distribution of critical current of the constituting elements leads to large reduction in critical currents in long sample. Acknowledgements The authors wish to express their gratitude to The Ministry of Education, Culture, Sports, Science and Technology, Japan and the Japan Society for the Promotion of Science for Grant-in-aid (No. 18106011) and NEDO (New Energy and Industrial Technology Development Organization, Japan) for the Grant-in-aid (No. 2004EA004). References
Fig. 5. Simulated critical current – sample length relation of (a) high Ic and (b) high strength tapes, together with the measured critical current values of 6 cm samples (closed circles) for comparison.
[1] S. Ochiai, K. Hayashi, K. Osamura, Cryogenics 32 (1992) 799. [2] B. ten Haken, A. Beuink, H. ten Kate, IEEE Trans. Appl. Supercond. 7 (1999) 2043. [3] H. Kitaguchi, K. Itoh, H. Kumakura, T. Takeuchi, K. Togano, W. Wada, IEEE Trans. Appl. Supercond. 11 (2001) 3058. [4] R. Passerini, M. Dhalle’, E. Giannini, G. Witz, B. Seeber, R. Flükiger, Physica C 371 (2002) 173. [5] K. Osamura, M. Sugano, K. Matsumoto, Supercond. Sci. Technol. 16 (2003) 971. [6] S. Ochiai, T. Nagai, H. Okuda, S.S. Oh, M. Hojo, M. Tanaka, M. Sugano, K. Osamura, Supercond. Sci. Technol. 16 (2003) 988. [7] K. Katagiri, H.S. Shin, K. Kasaba, T. Tsukinokizawa, K. Hiroi, T. Kuroda, K. Itoh, H. Wada, Supercond. Sci. Technol. 16 (2003) 995. [8] H.S. Shin, K. Katagiri, Supercond. Sci. Technol. 16 (2003) 1012. [9] H.J.N. van Eck, D.C. van der Laan, M. Dhallé, B. ten Haken, H.H.J. ten Kate, Supercond. Sci. Technol. 16 (2003) 1026. [10] M. Hojo, M. Nakamura, T. Matsuoka, M. Tanaka, S. Ochiai, M. Sugano, K. Osamura, Supercond. Sci. Technol. 16 (2003) 1043. [11] S. Ochiai, N. Miyazaki, D. Doko, T. Nagai, M. Nakamura, H. Okuda, S.S. Oh, M. Hojo, M. Tanaka, K. Osamura, J. Nuclear Mater. 329–333 (2004) 1585. [12] T. Kuroda, K. Itoh, H. Kumakura, H. Wada, K. Katagiri, Supercond. Sci. Technol. 16 (2003) 1120. [13] S. Ochiai, D. Doko, H. Rokkaku, M. Fujimoto, H. Okuda, M. Hojo, M. Tanaka, M. Sugano, K. Osamura, M. Miura, Physica C 445–449 (2006) 746. [14] W. Weibull, Appl. Mech. 28 (1951) 293. [15] S. Ochiai, K. Osamura, M. Ryoji, Trans. Iron and Steel Inst. Jpn. 28 (1998) 973. [16] S. Ochiai, S. Nishino, M. Hojo, K. Watanabe, Supercond. Sci. Technol. 8 (1995) 863. [17] S. Ochiai, T. Sawada, S. Nishino, H. Sekino, M. Hojo, Y. Yamda, K. Hayashi, K. Ayai, M. Koganeya, M. Ono, J. Mater. Sci. Lett. 18 (1999) 137. [18] M. Hojo, N. Iwasaki, F. Sekino, S. Ochiai, S. Sasaki, K. Watanabe, Cryogenics 39 (1999) 627. [19] A. Nyilas, K. Osamura, M. Sugano, Supercond. Sci. Technol. 16 (2003) 1036. [20] S. Ochial, M. Fujimoto, H. Okuda, S.S. Oh, D.W. Ha, Supercond. Sci. Technol. 20 (2007) 800. [21] K. Ueda, O. Tsukamoto, S. Nagaya, H. Kimura, S. Akita, IEEE Trans. Appl. Supercond. 3 (2003) 1946.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
The influence of tensile strain to critical current of Bi2223 composite tape Y. Mukai a, J.K. Shin a, S. Ochiai a,*, H. Okuda a, M. Sugano a, K. Osamura b a b
Department of Materials Science and Engineering, Graduate School of Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan Research Institute for Applied Sciences, Sakyo-ku, Kyoto 606-8202, Japan
a r t i c l e
i n f o
Article history: Available online 27 May 2008 PACS: 74.25.Sv 74.25.Ld Keywords: Bi2223 Composite tape Tensile strain Local critical current distribution
a b s t r a c t As the stress-induced damage evolution is different from position to position in the sample, the local critical current is scattered in a sample, affecting on the overall current. The present work aimed to describe the distribution of local critical current and its relation to overall critical current under tensile stress for Bi2223/Ag superconducting composite tape. In the experiment, seven voltage probes were attached in a step of 10 mm. The local critical current and n-value at 77 K under various applied stress levels were measured for a voltage probe distance 10 mm and the overall ones for a probe distance 60 mm. Main results are summarized as follows. The overall critical current and n-value were described well by using the voltage summation model in which the sample was regarded as a one dimensional series circuit. For the low applied stress, the distribution of local critical current was described with the three parameter Weibull distribution function. Using the measured distribution of the local critical current, an experimental relation of critical current to n-value and the voltage summation model, and applying the Monte Carlo method, the overall critical current was predicted, which was in good agreement with the experimental results. Based on these results, the sample length dependence of critical current of the sample damaged by tensile stress was discussed. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction
2. Experimental procedure
High temperature superconductor is expected to be utilized in many applications such as electric power cable, magnetic levitation train and MRI scanner. It is capable to manufacture long size composite tape enough for actual use [21]. The fracture strain of (Bi, Pb)2Sr2Ca2Cu3O10 filament that transports the superconducting current, is, however, around 0.1%,being quite lower than that of other traditional superconductors (Nb3Sn:0.5–1.2% [15,16], Nb3Al:0.6–1.0% [17], Nb–Ti:2% [18]). During fabrication/winding and operation, mechanical and electromagnetic stresses are subjected to the Bi2223/Ag composite superconductors, which change the superconducting property [1–13]. Under applied stress, there co-exist less and more damaged parts within a sample [6,7,13]. Thus the local critical current (and also n-value) is different from part to part. The aim of the present work is to describe the critical current distribution and its relation to overall critical current under applied stress.
The following two types of multifilamentary Bi2223 composite tapes were used.
* Corresponding author. Tel.: +81 75 753 4834; fax: +81 75 753 4841. E-mail address: [email protected] (S. Ochiai). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.05.088
(1) High Ic tape constituting of Bi2223 filaments, Ag and Ag–Li alloy, fabricated at American Superconductor Corporation. The width and thickness of the tape were 4.1 and 0.22 mm, respectively. (2) High strength tape constituting of the high Ic tape mentioned above and laminated reinforcement of stainless steel sheet, fabricated at American Superconductor Corporation. The width and thickness of the tape were 4.1 and 0.31 mm, respectively. Fig. 1 shows the configuration of the test piece. Seven voltage probes were attached in a step of 10 mm with solder. The overall critical current of the samples with a length of 60 mm was measured from the voltage between the probes 1 and 7. The critical current Ic of local elements (E1 to E6 in Fig. 1) with a 10 mm length was measured from the voltage between i and i + 1 probes (i = 1 to 6) of each element (E1 to E6) and that of overall sample measured by the four probe method. The sample tape was cooled down to
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Fig. 1. The configuration of local elements (E1–E6) and overall sample for test.
77 K, the voltage (V)current (I) curve was measured in a self magnetic field. The strain was measured by a couple of Nyilas type strain gauge [19]. From the result using such a test piece (Fig. 4) for both high Ic and high strength tapes, the distribution of Ic of local elements and it is relation to the overall Ic were obtained. 3. Results and discussion 3.1. Change of critical current and n-value of local elements and overall ones with increasing applied tensile stress in high Ic and high strength tapes Fig. 2 shows the changes of Ic and n-value with increasing applied tensile stress r for local elements and overall sample in (a) high Ic and (b) high strength tapes. Following features are found. (1) The overall Ic and n-value decreased slightly with increasing applied stress r up to around r = 75 MPa and 370 MPa in high Ic and high strength tapes, respectively, due to the strain effect [5]. Beyond such stresses, the overall Ic values decreased seriously with increasing stress, due to the damage evolution [5,6]. (2) The Ic and n-value of the elements also decreased with increasing stress, showing the similar tendency to those of the overall sample. (3) The Ic and n-value were different among the local elements at any applied stress. (4) The overall Ic and n-value were lower than the average values of the local elements but nearly equal to or higher than the lowest Ic and n-value among the elements. (5) There is a tendency that the element and overall sample with high Ic have high n-value. 3.2. Relation of critical current and n-value of elements with those of overall sample In present work, the 1 lV/cm criterion was used for definition of the critical current. Accordingly, the local and overall critical cur-
Fig. 2. Measured and calculated variations of overall (L = 60 mm) and local (10 mm) critical current Ic and n-value with increasing tensile stress in (a) high Ic and (b) high strength tapes.
rents were estimated as the currents at the generated voltage 1 and 6 lV, respectively. As the overall sample is composed of a series circuit of local elements (Fig. 1), the overall voltage near the transition from super- to normal conductivity is given the sum of the voltage of all elements (voltage summation approach [13]). Such an approach was used to describe the relation of critical current and n-value of the local elements to those of the overall sample as follows. The voltage (V(i)) and n-value (n(i)) of the i-element is expressed by VðiÞ ¼ AðiÞInðiÞ . For each element with a length 10 mm, Ai can be expressed practically by AðiÞ ¼ f1=Ic ðiÞgnðiÞ under the 1 lV/cm criterion for determination of Ic. Thus, the overall voltage V(overall) is expressed by
VðoverallÞ ¼ AðoverallÞInðoverallÞ ¼ ðI=Ic ðiÞÞnðiÞ
ð1Þ
where n(overall) is the n-value for overall sample. Setting V(overall) = 6 lV, and substituting the measured values of Ic(i) and n(i)
Y. Mukai et al. / Physica C 468 (2008) 1801–1804
of the constituting elements (i = 1 to 6) into Eq. (1), the overall Ic(overall) for 1 lV/cm criterion is calculated. Also, n(overall) is estimated by fitting the curve fI=Ic ðiÞgnðiÞ to A(overall)In(overall) in the range of V = 0.6 to 60 lV. The calculation result for overall Ic and n-value are presented in Fig. 2. The measured Ic and n-value at each stress of the overall sample were described well. This result indicates that, if the distribution of Ic and n-value of the local elements are known in advance, the overall Ic and n-value can be express by the voltage summation approach. In the following Section 3.3, the distribution function of Ic of elements under each tensile stress conditions and the relation of Ic to n will be obtained experimentally. Then applying the voltage summation approach and using a Monte Carlo simulation method, the Ic and n-value of overall sample at arbitrary sample length will be simulated. 3.3. Description of the distribution of the critical current Ic of local elements at low applied tensile stress
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3.4. Simulation of the critical current Ic and n-value of overall sample from the distribution of those of local elements The Monte Carlo simulation method using the voltage summation model, developed in our preceding work [20], was applied to the present result to simulate the Ic and n-value of the overall sample with arbitrary length from the distribution of them of the constituting elements with a length 1 cm. For simulation of critical current of samples with a q cm length, the values of Ic, (i) and n(i) of q constituting elements are needed. The Ic(i) and n(i) were obtained in the following procedure. At low stresses (r < 75 MPa and 370 MPa for high Ic and high strength tapes, respectively), the estimated Weibull distribution of Ic (Eq. (2)) was used as the FIc relation. At high stresses (r > 75 MPa and 370 MPa for high Ic and high strength tapes, respectively), as the Weibull distribution function given by Eq. (2) was not suitable for description of FIc relation, the measured FIc relation presented in Fig. 4 was used. The FIc relation in the
In order to formulate the distribution of Ic of local elements (Fig. 2), the three parameter-Weibull distribution function [14] was employed. According to this distribution function, the cumulative probability F of critical current Ic of the elements with a length 10 mm is expressed as
F ¼ 1 exp½fðIc Icmin Þ=I0 gm
ð2Þ
where Ic,min is the minimum critical current, I0 the scale parameter and m the shape parameter. Eq. (2) is rewritten in the form lnln{(1F)1} = mln(IcIc,min)mln(I0).The plot of lnln{(1F)1} against ln(IcIc,min) has been called as Weibull plot. The cumulative distribution function F was calculated by F = (j0.3)/(N+0.4), where j is element number in the ascending sequence of Ic and N is total number of tested elements (N = 18 and 12 for high Ic and high strength tapes, respectively). Fig. 3 shows an example of the Weibull plot obtained by the regression analysis for high Ic sample at r = 60 MPa. As typically shown in this example, high linearity between the lnln{(1F)1} and ln(IcIc,min) was obtained for the results at low applied stresses (r < 75 MPa for high Ic and r < 370 MP for high strength tapes). The typical estimated values of Ic,min, m, and I0 were 150 A, 0.8 A and 3 A, respectively, for high Ic tapes at r < 75 MPa, and 130 A, 0.75 A and 3 A, respectively, for high strength tapes at r < 370 MPa.
Fig. 3. An example of the weibull plot of critical current Ic. The Ic values of high Ic sample under r = 60 MPa is presented as an example.
Fig. 4. The cumulative probability F of critical current Ic of (a) high Ic and (b) high strength tapes.
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range outside the measured FIc relation was inferred by extrapolating the measured FIc relation as shown with the dotted curves in Fig. 4. The local critical current of the i element, Ic(i), was determined in the computer by generating a random value R(i) (0 1) and setting F(i) = R(i) in the FIc relation. For the Ic(i) value determined by the procedure stated above, the corresponding n-value, n(i) was obtained by the experimental result for the nIc relation, which was expressed by n = 1.55 + 0.092Ic and 1.54 + 0.073Ic for high Ic and high strength samples, respectively. The procedure stated above was repeated for i = 1 to q by generating further random value R(i) (i = 1 to q), and the values of Ic(i) and n(i) for i = 1 to q were determined. Substituting thus determined Ic(i) and n(i)-values (i = 1 to q) into Eq. (1), the Ic and n-value of one overall sample with a length q cm was calculated. By repeating such a procedure by using different random value series, the average overall Ic and n-value of the overall samples with a q cm length were obtained. Simulated average critical current of the sample was plotted against the length as shown in Fig. 5. The measured Ic value of 6 cm sample was also presented for comparison, which was well described by the present simulation method. The simulation result in Fig. 5 suggests that the critical current decreases only slightly against the length at low stress (0 60 MPa and 0 370 MPa for high Ic and high strength tapes, respectively). At higher stresses,
the Ic of high Ic sample decreases largely with increasing length in comparison with that of high strength one. This stems from the wider distribution of critical current in damaged high Ic sample than in damaged high strength one. 4. Conclusions (1) The correlation of distributed critical current (77 K, in self magnetic field) and n-value of local elements to overall critical current under applied tensile stress was described comprehensively with a voltage summation model for both high Ic and high strength tapes. (2) Within the range of tensile stress lower than about 75 MPa and 370 MPa for high Ic and high strength tapes respectively, the distribution of local critical current could be described with the Weibull distribution function. (3) A simulation method to describe the overall critical current from the local ones and the sample length dependence of critical current was presented, in which the experimentally measured cumulative probability of critical current, the empirical formula of the n-value as a function of critical current, voltage summation model and Monte Carlo method were combined. With this method, the experimentally observed correlation of distribution of local critical current to overall critical current and the sample length dependence of critical current could be described well. Also it was shown that the wide distribution of critical current of the constituting elements leads to large reduction in critical currents in long sample. Acknowledgements The authors wish to express their gratitude to The Ministry of Education, Culture, Sports, Science and Technology, Japan and the Japan Society for the Promotion of Science for Grant-in-aid (No. 18106011) and NEDO (New Energy and Industrial Technology Development Organization, Japan) for the Grant-in-aid (No. 2004EA004). References
Fig. 5. Simulated critical current – sample length relation of (a) high Ic and (b) high strength tapes, together with the measured critical current values of 6 cm samples (closed circles) for comparison.
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