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Evaluation of AC losses in cable conductors using thin superconducting tapes with non-uniform Jc distribution
Physica C 442 (2006) 139–144 www.elsevier.com/locate/physc
Evaluation of AC losses in cable conductors using thin superconducting tapes with non-uniform Jc distribution Ryoji Inada *, Yuichi Nakamura, Akio Oota Department of Electrical and Electronic Engineering, Toyohashi University of Technology, 1-1 Tempaku-cho, Toyohashi, Aichi 441-8580, Japan Intelligent Sensing System Research Center, Toyohashi University of Technology, 1-1 Tempaku-cho, Toyohashi, Aichi 441-8580, Japan Received 1 March 2006; received in revised form 24 April 2006; accepted 26 April 2006 Available online 16 June 2006
Abstract The critical current density (Jc) in the YBCO superconducting layer in the typical coated conductor is known to be not uniform along a lateral direction of its cross section. Non-uniformity of Jc values in a superconducting layer is the dominant factors of the AC loss characteristics together with its geometrical shape and arrangement in an actual power device. In this study, we investigated the AC loss properties on the cable conductors using thin superconducting tapes with non-uniform Jc distributions through the numerical calculations. A rectangular cross sectional tape with large aspect ratio AR (>103) of its cross section and non-uniform Jc values along a tape width were used as the strand for assembling the single-layer cable conductors. Several tape strands were arranged on a cylindrical former, in a parallel way to the conductor length. Numerical calculation of magnetic field distributions and loss values under AC current transmission were performed, by taking into account the locally-varying Jc values in a tape strand and geometrical factors of conductors. The influence of non-uniformity of Jc values along a tape width on loss under AC current transmission in the cable conductors was discussed. 2006 Elsevier B.V. All rights reserved. PACS: 02.60.x; 84.70.p; 85.25.K Keywords: Superconducting tapes; Aspect ratio; Jc distribution; AC losses; Cable conductors
1. Introduction Recently, the fabrication technologies of YBCO coated conductors are growing rapidly, as long sample length of several 10–100 m class and high critical current density (Jc) of MA/cm2 class have become available [1–4]. Generally, the YBCO layer thickness in a typical coated conductor is a few lm, which leads to very large aspect ratio (AR) of its cross section (= 103–104) compared with conventional Ag/Bi2223 tapes. According to this geometrical anisotropy of superconducting layer, the electromagnetic *
Corresponding author. Address: Department of Electrical and Electronic Engineering, Toyohashi University of Technology, 1-1 Tempakucho, Toyohashi, Aichi 441-8580, Japan. Tel.: +81 532 44 6735; fax: +81 532 44 6757. E-mail address: [email protected] (R. Inada). 0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.04.099
properties in coated conductors show quite complex behaviors. In addition, it is experimentally clarified that the Jc values in the YBCO layers in coated conductors are not uniform along a lateral direction of its cross section [5,6]. Since the magnetic field penetration inside the superconductor with large AR value strongly depends on the Jc distribution along a width direction [7–9], AC loss characteristics of coated conductors are strongly influenced by the Jc distributions of them [9,10]. For the applications of coated conductors to AC power cables, they will be assembled in cylindrical shapes to obtain desired current capacity. Previously, we investigated the AC loss characteristics in cable conductors composed of superconducting tapes with large AR values (>103) and uniform Jc distribution through the numerical calculations, and examined the influence of size and arrangement of the tape strands on magnetic field distributions and loss
140
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generations inside the conductors [11]. However, the magnetic field penetration into the tape strands inside the cable conductors might be influenced not only by the geometrical factor of conductors but also by the non-uniformity of Jc values in tape strands. Therefore, it is crucial to examine the AC loss properties in cable conductors using thin superconducting tapes with non-uniform Jc distributions, for developing conductors with low loss values satisfying the practical demand. In this paper, we numerically evaluated the AC loss properties in cable conductors using thin superconducting tapes with various Jc distributions along a lateral direction of a tape. The rectangular cross sectional tapes with large AR values (>1000) and various Jc distributions along its width were used as the strand for assembling the conductor models. Several tape strands were arranged on a cylindrical former, in a parallel way to the conductor axis. Numerical calculation of magnetic field distributions and loss values under AC current transmission were performed, by taking into account the locally-varying Jc values in each tape strand, the arrangements of tape strands and geometrical factors of conductors. Based on the calculated results, the influence of non-uniformity of Jc values along a width of each tape strand on the loss in cable conductors was discussed. 2. Calculation method
losses in cable conductors. The peaking Jc profile has the distribution that Jc value shows its peak at the center part of tape section and becomes lower as the lateral position x goes from center to tape edges. On the other hand, the trapezoidal Jc profile has the distribution that Jc value decrease only near the edge part of tape section (jxj > 2.0 mm) while the other part has uniform Jc values (jxj < 2.0 mm). We also assumed that all tape strands in the conductor have same Jc distributions. 15 tape strands were arranged on a cylindrical former in a parallel way to the conductor axis, with the gap dgap of 0–1.0 mm among the adjacent tape strands. The definition of geometrical parameters in the conductors is also shown in Fig. 2. The specifications of single-layer cables are summarized in Table 1. The numerical calculations of magnetic field distributions and AC transport losses in the single-layer cable models were performed, with taking into account the actual arrangements and the lateral Jc distributions of tape strands in conductors [7,11,12] based on the Norris’s theory [13]. For the calculation, each superconductor tape inside conductor is regarded as a bundle of straight thin fibers with their cross sectional area dS. Under the condition of current amplitude I0 < Ic, a fiber transports a cur-
Gap dgap
wS
C
The rectangular cross sectional superconducting tape with width (wSC) of 5.0 mm and thickness (tSC) of 2.0 lm was used as the strand for assembling the single-layer cable models. The aspect ratio AR (=wSC/tSC) of the tape section is 2500. The critical current (Ic) of the individual tape was fixed on 100 A, which corresponds to the average critical current density Jc0 (=Ic/wSCtSC) of 1.0 · 1010 A/m2. As shown in Fig. 1, the various Jc distributions along a lateral direction of tape section (uniform, peaking, and trapezoidal Jc profiles) were assumed for the calculation of AC
m
tSC = 2 μm Tape strand
Cylindrical former
(15 pieces)
R
10
1.5x10
10
1.0x10
10
Fig. 2. A schematic diagram of cross section of the single-layer cable model composed of thin superconducting tapes as strands. The number of tape strands (N) is fixed to 15.
Uniform JC Peaking JC Trapezoidal JC
2
Critical current density JC (A/m )
2.0x10
=5m
Table 1 Characteristics of the single-layer cable model for AC loss calculation
9
5.0x10
IC = 100 A AR = 2500
x wSC = 5 mm
0.0 -2
-1
0
1
2
Lateral position x (mm) Fig. 1. Assumed Jc distributions along a lateral direction of a rectangular superconducting tape with aspect ratio AR = 2500 used as a strand of cable conductors. The critical current (Ic) of a tape strand is fixed to 100 A.
Tape strand Tape width wSC Tape thickness tSC Aspect ratio of tape section AR Critical current Ic Average critical current density Jc0
5.0 mm 2.0 lm 2500 100 A 1.0 · 1010 A/m2
Single-layer cable Number of tape strands N Critical current Ic Gap between adjacent tapes dgap Former radius R
15 1500 A 0–1.0 mm 11.8–14.1 mm
R. Inada et al. / Physica C 442 (2006) 139–144
rent fragment dI = Jc(r)dS outside the current free region, while carrying no current inside the current free region. Here, Jc(r) represents the critical current density varied with the position r in the cross section of a superconductor. The origin is taken as an arbitrary position in the current free region in a specific tape in the conductor. By regarding the assembled conductors as the bundle of straight superconductor tapes, the loss density Qd(r) per-cycle in the observation point r for superconductor region in a specific tape strand in the conductors can be expressed as [7,11,12]
Fig. 3 shows the calculation results of transport loss values Qt of the tape strand with different lateral Jc distributions in an isolated state, as a function of reduced current amplitude i = I0/Ic (lower horizontal axis) and actual transport current amplitude I0 (upper horizontal axis). In the calculations, the tape was regarded as the bundle of 80 000 fibers. The numbers of meshes along a width and a thickness direction of the tape section were fixed to 1600 and 50, respectively. The results are compared with the analytical Qt values for an elliptical and a thin strip superconductor with based on the Norris’s theory [13], which are expressed as follows:
Qt-strip ¼
l0 I 2c i2 ð1 iÞlnð1 iÞ þ i for ellipse; p 2
Transport loss Qt (J/m/cycle)
-5
10
-6
10
IC = 100 A AR = 2500
0.1
1
Reduced current amplitude i = I0 / IC Fig. 3. Transport losses Qt per-cycle per-unit length for an isolated rectangular tape with AR = 2500 in an isolated state plotted against the current ratio i = I0/Ic (lower axis) and actual current amplitude I0 (upper axis). Also shown are the analytical loss values in an elliptical superconductor (a dashed line) and in a thin strip superconductor (a solid line) predicted by Norris.
where i is the current amplitude I0 normalized by Ic of a superconductor. As can be seen, the Qt values in rectangular tapes with uniform Jc distributions agree well with the analytical values for a thin strip model. The deviation from the analytical values are less than 10% even at the reduced
I0 = 0.9IC
5
3
4x10
Uniform JC Peaking JC Trapezoidal JC
5
3x10
5
2x10
5
1x10
Center
0 0.0
(a)
Tape edge
0.5
1.0
1.5
2.0
2.5
Lateral position x (mm)
I0 = 0.5IC
1.0x10
5
5.0x10
4
(b)
ð5Þ
-4
10
-7
ð4Þ
l0 I 2c ½ð1 iÞ lnð1 iÞ þ ð1 þ iÞ lnð1 þ iÞ i2 for thin strip; p
10
10
Loss density Qd (J/m /cycle)
3. Results and discussion
Uniform JC Peaking JC Trapezoidal JC
-3
3
Here, l0 is the magnetic permeability in vacuum (=4p · 107 H/m), rself and rother are the integral coordinates in 1 1 the specific tape and the other tapes in the conductor, respectively. The shape of current free region in the each tape strand under I0 < Ic is determined based on the fact that the total magnetic flux Utotal(r) at peak current shows the minimum at current free region in a tape strand [7,14]. The transport loss values per-unit length of the conductor per-cycle Qt are obtained by integration of Qd(r) over the whole part of superconductor region in the conductor.
100
10
ð1Þ
Here, Utotal(r) is the total magnetic flux at peak current I0 passing through between the current-free region and observation point r, Uself(r) and Uother(r) are the magnetic flux at peak current I0 generated by the specific tape own and by the other tapes, respectively. The magnetic flux components of Uself(r) and Uother(r) with taking into account the position-dependent critical current densities Jc(r) are calculated as follows [7,11,12]: Z l0 jr rself self 1 j J c ðr1 Þ ln Uself ðrÞ ¼ ð2Þ dS self 1 ; 2p jrself 1 j Z l jr rother j 1 J c ðrother Uother ðrÞ ¼ 0 Þ ln : ð3Þ dS other 1 1 2p jrother j 1
Qt-ellipse ¼
Transport current amplitude I0 (A) 10
-2
Loss density Qd (J/m /cycle)
Qd ðrÞ ¼ 4J c ðrÞUtotal ðrÞ ¼ 4J c ðrÞ½Uself ðrÞ þ Uother ðrÞ:
141
Uniform JC Peaking JC Trapezoidal JC
Tape edge Center
0.0 0.00
1.75
2.00
2.25
2.50
Lateral position x (mm)
Fig. 4. Distributions of loss density Qd per-cycle on the broad face of an isolated rectangular tape with different Jc distribution: (a) I0 = 0.9Ic and (b) I0 = 0.5Ic.
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R. Inada et al. / Physica C 442 (2006) 139–144
current amplitude I0 = 0.2Ic. By addressing this fact, we confirmed the validity of the calculation procedure. It is also evident that the non-uniformity of Jc values along a tape width strongly influences the Qt values. The tape with peaking Jc distribution has significantly higher Qt values than the case with uniform Jc in the whole current range. On the other hand, the Qt values of the tape with trapezoidal Jc distribution almost agree with the analytical values for a thin strip where I0 > 0.7Ic. However, they deviate from the analytical values in the range of I0 < 0.6Ic and the deviation becomes more remarkable with decreasing I0. To examine the effect of non-uniformity of Jc values on the loss values in an isolated tape, the calculated results for the loss density (Qd) distributions on the broad face of a tape are shown in Fig. 4. As can be seen, higher Qd values are observed in the position near the tape edge (x = 2.5 mm) because the change of magnetic flux in ACcycle are most frequent around the tape edge. However, the distribution of Qd values along a tape width is strongly influenced by the lateral Jc distribution of the tape. In the case with peaking Jc distribution, the flux penetration region with loss density Qd > 0 extends toward the center part of tape regardless the current amplitude I0. This causes the significant increase of Qt values in a tape with peaking -1
I0 = 0.9IC
-2
dgap = 1.0 mm, I0 = 0.9IC 5
10
-3
Uniform JC Peaking JC Trapezoidal JC
10
-4
0.0
0.2
0.4
0.6
0.8
1.0
Gap dgap (mm)
(a)
Tape edge
0.5
1.0
1.5
2.0
2.5
Lateral position x (mm)
10
dgap = 0.3 mm, I0 = 0.9IC 5
Uniform JC Peaking JC Trapezoidal JC
2x10
3
Transport loss Qt (J/m/cycle)
0 0.0
I0 = 0.5IC
-4
Uniform JC Peaking JC Trapezoidal JC
-5
10
-6
2x10
Center
(a)
Loss density Qd (J/m /cycle)
-3
10
5
1x10
-3
2x10
(b)
Uniform JC Peaking JC Trapezoidal JC
2x10
3
10
Loss density Qd (J/m /cycle)
Transport loss Qt (J/m/cycle)
10
Jc profile shown in Fig. 3. On the other hand, as shown in Fig. 4(a), the Qd values at the condition of I0 = 0.9Ic are slightly decreased quite near the edge part (x > 2.3 mm) of a tape with trapezoidal Jc distribution because of the decrease of Jc only near the tape edge (see Fig. 1). In the lateral position of x < 2.2 mm, the Qd values are slightly larger than the case with uniform Jc. With decreasing I0 values, as shown in Fig. 4(b), the decrease of Qd values near the tape edge is disappeared and extension of the flux penetration region with Qd > 0 toward the center part of tape are observed, as well as the results with peaking Jc distribution. These results explain the difference of loss values Qt which depend on the non-uniformity of Jc values well therein shown in Fig. 3. Fig. 5 shows the loss values Qt in single-layer cable conductors composed of tape strands with different lateral Jc distribution, as a function of the gap dgap of adjacent tape strands. As can be seen, the Qt values in all conductors are reduced with decreasing the gap dgap. The dependence of Qt values on the dgap values becomes more significant at lower current amplitude (Fig. 5(b)). Our previous studies suggest that the reduction of the Qt values with decreasing the dgap values is caused by the suppression of magnetic flux passing through at the edge part of each tape strand [11]. In addition, it is also evident that the non-uniformity of Jc values along a lateral direction of each strand strongly influences the Qt values in cable conductors. The conductor composed of the tape strands with peaking Jc distribution has
0.0
0.2
0.4
0.6
0.8
1.0
Gap dgap (mm)
Fig. 5. Transport losses Qt per-cycle per-unit length for single-layer conductors plotted against the gap dgap of adjacent tape strands: (a) I0 = 0.9Ic and (b) I0 = 0.5Ic.
(b)
5
1x10
Tape edge Center
0 0.0
0.5
1.0
1.5
2.0
2.5
Lateral position x (mm)
Fig. 6. Distributions of loss density Qd per-cycle on the broad face of a tape strand in single-layer conductors at I0 = 0.9Ic, composed of tapes with different Jc distribution: (a) dgap = 1.0 mm and (b) dgap = 0.3 mm.
R. Inada et al. / Physica C 442 (2006) 139–144
significantly higher Qt values than the case with uniform Jc. Furthermore, at higher current amplitude near Ic, the change of Qt values due to the dgap values are suppressed compared with the results with other Jc profiles (Fig. 5(a)). On the other hand, the Qt values of the conductor using tape strands with trapezoidal Jc distribution are slightly greater than the values with the case of uniform Jc at higher current amplitude near Ic (Fig. 5(a)). As the current amplitude I0 decreases, however, the difference of the Qt values with the cases of uniform Jc and trapezoidal Jc becomes remarkable (Fig. 5(b)). The dependence of the Qt values in single-layer conductors on the lateral Jc distributions of each tape strand is quite similar with that in the tape in an isolated state shown in Fig. 3. To examine the effect of Jc distributions in each tape strand on the loss values in single-layer cable conductors, the calculated results for the loss density (Qd) distributions on the broad face of a tape strand in conductors are shown in Figs. 6 and 7, respectively. As can be seen, the transport loss in single-layer conductors is mainly generated near the edge part of each tape strand. The absolute values of Qd near the tape edge (x = 2.5 mm) are lower than those in an isolated tape (Fig. 4) and also reduced with decreasing dgap values, because of the influence for the magnetic flux generated by the adjacent tapes [11]. However, it is clearly observed that the distributions of Qd along a lateral direction of a tape strand in the conductor are quite similar with 4
dgap = 1.0 mm, I0 = 0.5IC
3
Loss density Qd (J/m /cycle)
6.0x10
4.0x10
4
2.0x10
4
Tape edge
Center
0.0 0.00
(a)
2.00
2.25
4.0x10
2.0x10
4
Uniform JC Peaking JC Trapezoidal JC
Tape edge
1.75
2.00
We investigated the AC loss properties on cable conductors using thin superconductor tapes with non-uniform Jc distributions through the numerical calculations. The calculations of loss values under AC current transmission were performed with taking into account the actual arrangements and the lateral Jc distributions of tape strands in conductors and geometrical factors of conductors. The calculated results suggest that the loss values in single-layer conductors are strongly affected not only by the arrangement of tape strands inside a conductor but also by the non-uniformity of Jc values along a lateral direction of each tape strand. As well as the case of a tape in an isolated state, the decrease of Jc values near the edge part of a tape strand leads to the significant increase of loss values under AC current transmission. This is mainly attributed to the extension of flux penetration region in going from the edge part to the center part of tape section, reflecting the lower Jc values near the tape edges. Therefore, in addition with the optimization with conductor structure, the improvement of uniformity of Jc values along a lateral direction in coated conductors is quite important for reducing the AC loss generated inside the cable conductors.
This work was supported in part by Grant-in-Aids for Scientific Research (No. 17206026) from the Japanese Society of the promotion of science, and also by that (No. 17760233) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. It was also supported in part from Research Foundation for the Electrotechnology of Chubu (No. R-17211) and by the 21st Century COE Program ‘‘Intelligent Human Sensing’’, from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
Center
0.0 0.00
4. Conclusion
Acknowledgements
dgap = 0.3 mm, I0 = 0.5IC
4
those in a tape in an isolated state. The non-uniformity of Jc values in each tape strand also affect to the loss generation in a tape strand. The decrease of Jc values near the edge part of a tape strand causes the extension of flux penetration region with loss density Qd > 0 toward the center part of tape strands. This is the reason for the increase of loss values in conductors which depend on the non-uniformity of Jc values in tape strands shown in Fig. 5. From these results, it is suggested that the improvement of uniformity of Jc values along a tape width in coated conductors has crucial importance to suppress the AC loss generation inside the cable conductors, together with the optimization of cable structures.
2.50
4
3
Loss density Qd (J/m /cycle)
1.75
Lateral position x (mm)
6.0x10
(b)
Uniform JC Peaking JC Trapezoidal JC
143
2.25
2.50
Lateral position x (mm)
Fig. 7. Distributions of loss density Qd per-cycle on the broad face of a tape strand in single-layer conductors at I0 = 0.5Ic, composed of tapes with different Jc distribution: (a) dgap = 1.0 mm and (b) dgap = 0.3 mm.
References [1] Y. Shiohara, Y. Aoki, Physica C 426–431 (2005) 1. [2] K. Kakimoto, Y. Sutoh, N. Kaneko, Y. Iijima, T. Saitoh, Physica C 426–431 (2005) 858.
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[3] T. Watanabe, R. Kuriki, H. Iwai, T. Muroga, S. Miyata, A. Ibi, Y. Yamada, Y. Shiohara, IEEE Trans. Appl. Supercond. 15 (2005) 2566. [4] R. Teranishi, T. Izumi, Y. Shiohara, Supercond. Sci. Technol. 19 (2006) S4. [5] N. Amemiya, Y. Shinkai, Y. Iijima, K. Kakimoto, K. Takeda, Supercond. Sci. Technol. 14 (2001) 611. [6] N. Amemiya, O. Maruyama, M. Mori, K. Kashima, T. Watanabe, S. Nagaya, Y. Shiohara, Physica C, submitted for publication. [7] R. Inada, A. Oota, H. Fujimoto, Physica C 378–381 (2002) 1133.
[8] K. Kajikawa, Y. Mawatari, T. Hayashi, K. Funaki, Supercond. Sci. Technol. 17 (2004) 553. [9] O. Tsukamoto, Supercond. Sci. Technol. 18 (2005) 596. [10] J. Ogawa, Y. Sawai, H. Nakayama, O. Tsukamoto, D. Miyagi, Physica C 401 (2005) 171. [11] R. Inada, Y. Nakamura, A. Oota, T. Fukunaga, Physica C 426–431 (2005) 1309. [12] R. Inada, Y. Nakamura, A. Oota, T. Fukunaga, IEEE Trans. Appl. Supercond. 15 (2005) 2871. [13] W.T. Norris, J. Phys. D: Appl. Phys. 3 (1970) 489. [14] T. Fukunaga, R. Inada, A. Oota, Appl. Phys. Lett. 72 (1998) 3362.
Evaluation of AC losses in cable conductors using thin superconducting tapes with non-uniform Jc distribution Ryoji Inada *, Yuichi Nakamura, Akio Oota Department of Electrical and Electronic Engineering, Toyohashi University of Technology, 1-1 Tempaku-cho, Toyohashi, Aichi 441-8580, Japan Intelligent Sensing System Research Center, Toyohashi University of Technology, 1-1 Tempaku-cho, Toyohashi, Aichi 441-8580, Japan Received 1 March 2006; received in revised form 24 April 2006; accepted 26 April 2006 Available online 16 June 2006
Abstract The critical current density (Jc) in the YBCO superconducting layer in the typical coated conductor is known to be not uniform along a lateral direction of its cross section. Non-uniformity of Jc values in a superconducting layer is the dominant factors of the AC loss characteristics together with its geometrical shape and arrangement in an actual power device. In this study, we investigated the AC loss properties on the cable conductors using thin superconducting tapes with non-uniform Jc distributions through the numerical calculations. A rectangular cross sectional tape with large aspect ratio AR (>103) of its cross section and non-uniform Jc values along a tape width were used as the strand for assembling the single-layer cable conductors. Several tape strands were arranged on a cylindrical former, in a parallel way to the conductor length. Numerical calculation of magnetic field distributions and loss values under AC current transmission were performed, by taking into account the locally-varying Jc values in a tape strand and geometrical factors of conductors. The influence of non-uniformity of Jc values along a tape width on loss under AC current transmission in the cable conductors was discussed. 2006 Elsevier B.V. All rights reserved. PACS: 02.60.x; 84.70.p; 85.25.K Keywords: Superconducting tapes; Aspect ratio; Jc distribution; AC losses; Cable conductors
1. Introduction Recently, the fabrication technologies of YBCO coated conductors are growing rapidly, as long sample length of several 10–100 m class and high critical current density (Jc) of MA/cm2 class have become available [1–4]. Generally, the YBCO layer thickness in a typical coated conductor is a few lm, which leads to very large aspect ratio (AR) of its cross section (= 103–104) compared with conventional Ag/Bi2223 tapes. According to this geometrical anisotropy of superconducting layer, the electromagnetic *
Corresponding author. Address: Department of Electrical and Electronic Engineering, Toyohashi University of Technology, 1-1 Tempakucho, Toyohashi, Aichi 441-8580, Japan. Tel.: +81 532 44 6735; fax: +81 532 44 6757. E-mail address: [email protected] (R. Inada). 0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.04.099
properties in coated conductors show quite complex behaviors. In addition, it is experimentally clarified that the Jc values in the YBCO layers in coated conductors are not uniform along a lateral direction of its cross section [5,6]. Since the magnetic field penetration inside the superconductor with large AR value strongly depends on the Jc distribution along a width direction [7–9], AC loss characteristics of coated conductors are strongly influenced by the Jc distributions of them [9,10]. For the applications of coated conductors to AC power cables, they will be assembled in cylindrical shapes to obtain desired current capacity. Previously, we investigated the AC loss characteristics in cable conductors composed of superconducting tapes with large AR values (>103) and uniform Jc distribution through the numerical calculations, and examined the influence of size and arrangement of the tape strands on magnetic field distributions and loss
140
R. Inada et al. / Physica C 442 (2006) 139–144
generations inside the conductors [11]. However, the magnetic field penetration into the tape strands inside the cable conductors might be influenced not only by the geometrical factor of conductors but also by the non-uniformity of Jc values in tape strands. Therefore, it is crucial to examine the AC loss properties in cable conductors using thin superconducting tapes with non-uniform Jc distributions, for developing conductors with low loss values satisfying the practical demand. In this paper, we numerically evaluated the AC loss properties in cable conductors using thin superconducting tapes with various Jc distributions along a lateral direction of a tape. The rectangular cross sectional tapes with large AR values (>1000) and various Jc distributions along its width were used as the strand for assembling the conductor models. Several tape strands were arranged on a cylindrical former, in a parallel way to the conductor axis. Numerical calculation of magnetic field distributions and loss values under AC current transmission were performed, by taking into account the locally-varying Jc values in each tape strand, the arrangements of tape strands and geometrical factors of conductors. Based on the calculated results, the influence of non-uniformity of Jc values along a width of each tape strand on the loss in cable conductors was discussed. 2. Calculation method
losses in cable conductors. The peaking Jc profile has the distribution that Jc value shows its peak at the center part of tape section and becomes lower as the lateral position x goes from center to tape edges. On the other hand, the trapezoidal Jc profile has the distribution that Jc value decrease only near the edge part of tape section (jxj > 2.0 mm) while the other part has uniform Jc values (jxj < 2.0 mm). We also assumed that all tape strands in the conductor have same Jc distributions. 15 tape strands were arranged on a cylindrical former in a parallel way to the conductor axis, with the gap dgap of 0–1.0 mm among the adjacent tape strands. The definition of geometrical parameters in the conductors is also shown in Fig. 2. The specifications of single-layer cables are summarized in Table 1. The numerical calculations of magnetic field distributions and AC transport losses in the single-layer cable models were performed, with taking into account the actual arrangements and the lateral Jc distributions of tape strands in conductors [7,11,12] based on the Norris’s theory [13]. For the calculation, each superconductor tape inside conductor is regarded as a bundle of straight thin fibers with their cross sectional area dS. Under the condition of current amplitude I0 < Ic, a fiber transports a cur-
Gap dgap
wS
C
The rectangular cross sectional superconducting tape with width (wSC) of 5.0 mm and thickness (tSC) of 2.0 lm was used as the strand for assembling the single-layer cable models. The aspect ratio AR (=wSC/tSC) of the tape section is 2500. The critical current (Ic) of the individual tape was fixed on 100 A, which corresponds to the average critical current density Jc0 (=Ic/wSCtSC) of 1.0 · 1010 A/m2. As shown in Fig. 1, the various Jc distributions along a lateral direction of tape section (uniform, peaking, and trapezoidal Jc profiles) were assumed for the calculation of AC
m
tSC = 2 μm Tape strand
Cylindrical former
(15 pieces)
R
10
1.5x10
10
1.0x10
10
Fig. 2. A schematic diagram of cross section of the single-layer cable model composed of thin superconducting tapes as strands. The number of tape strands (N) is fixed to 15.
Uniform JC Peaking JC Trapezoidal JC
2
Critical current density JC (A/m )
2.0x10
=5m
Table 1 Characteristics of the single-layer cable model for AC loss calculation
9
5.0x10
IC = 100 A AR = 2500
x wSC = 5 mm
0.0 -2
-1
0
1
2
Lateral position x (mm) Fig. 1. Assumed Jc distributions along a lateral direction of a rectangular superconducting tape with aspect ratio AR = 2500 used as a strand of cable conductors. The critical current (Ic) of a tape strand is fixed to 100 A.
Tape strand Tape width wSC Tape thickness tSC Aspect ratio of tape section AR Critical current Ic Average critical current density Jc0
5.0 mm 2.0 lm 2500 100 A 1.0 · 1010 A/m2
Single-layer cable Number of tape strands N Critical current Ic Gap between adjacent tapes dgap Former radius R
15 1500 A 0–1.0 mm 11.8–14.1 mm
R. Inada et al. / Physica C 442 (2006) 139–144
rent fragment dI = Jc(r)dS outside the current free region, while carrying no current inside the current free region. Here, Jc(r) represents the critical current density varied with the position r in the cross section of a superconductor. The origin is taken as an arbitrary position in the current free region in a specific tape in the conductor. By regarding the assembled conductors as the bundle of straight superconductor tapes, the loss density Qd(r) per-cycle in the observation point r for superconductor region in a specific tape strand in the conductors can be expressed as [7,11,12]
Fig. 3 shows the calculation results of transport loss values Qt of the tape strand with different lateral Jc distributions in an isolated state, as a function of reduced current amplitude i = I0/Ic (lower horizontal axis) and actual transport current amplitude I0 (upper horizontal axis). In the calculations, the tape was regarded as the bundle of 80 000 fibers. The numbers of meshes along a width and a thickness direction of the tape section were fixed to 1600 and 50, respectively. The results are compared with the analytical Qt values for an elliptical and a thin strip superconductor with based on the Norris’s theory [13], which are expressed as follows:
Qt-strip ¼
l0 I 2c i2 ð1 iÞlnð1 iÞ þ i for ellipse; p 2
Transport loss Qt (J/m/cycle)
-5
10
-6
10
IC = 100 A AR = 2500
0.1
1
Reduced current amplitude i = I0 / IC Fig. 3. Transport losses Qt per-cycle per-unit length for an isolated rectangular tape with AR = 2500 in an isolated state plotted against the current ratio i = I0/Ic (lower axis) and actual current amplitude I0 (upper axis). Also shown are the analytical loss values in an elliptical superconductor (a dashed line) and in a thin strip superconductor (a solid line) predicted by Norris.
where i is the current amplitude I0 normalized by Ic of a superconductor. As can be seen, the Qt values in rectangular tapes with uniform Jc distributions agree well with the analytical values for a thin strip model. The deviation from the analytical values are less than 10% even at the reduced
I0 = 0.9IC
5
3
4x10
Uniform JC Peaking JC Trapezoidal JC
5
3x10
5
2x10
5
1x10
Center
0 0.0
(a)
Tape edge
0.5
1.0
1.5
2.0
2.5
Lateral position x (mm)
I0 = 0.5IC
1.0x10
5
5.0x10
4
(b)
ð5Þ
-4
10
-7
ð4Þ
l0 I 2c ½ð1 iÞ lnð1 iÞ þ ð1 þ iÞ lnð1 þ iÞ i2 for thin strip; p
10
10
Loss density Qd (J/m /cycle)
3. Results and discussion
Uniform JC Peaking JC Trapezoidal JC
-3
3
Here, l0 is the magnetic permeability in vacuum (=4p · 107 H/m), rself and rother are the integral coordinates in 1 1 the specific tape and the other tapes in the conductor, respectively. The shape of current free region in the each tape strand under I0 < Ic is determined based on the fact that the total magnetic flux Utotal(r) at peak current shows the minimum at current free region in a tape strand [7,14]. The transport loss values per-unit length of the conductor per-cycle Qt are obtained by integration of Qd(r) over the whole part of superconductor region in the conductor.
100
10
ð1Þ
Here, Utotal(r) is the total magnetic flux at peak current I0 passing through between the current-free region and observation point r, Uself(r) and Uother(r) are the magnetic flux at peak current I0 generated by the specific tape own and by the other tapes, respectively. The magnetic flux components of Uself(r) and Uother(r) with taking into account the position-dependent critical current densities Jc(r) are calculated as follows [7,11,12]: Z l0 jr rself self 1 j J c ðr1 Þ ln Uself ðrÞ ¼ ð2Þ dS self 1 ; 2p jrself 1 j Z l jr rother j 1 J c ðrother Uother ðrÞ ¼ 0 Þ ln : ð3Þ dS other 1 1 2p jrother j 1
Qt-ellipse ¼
Transport current amplitude I0 (A) 10
-2
Loss density Qd (J/m /cycle)
Qd ðrÞ ¼ 4J c ðrÞUtotal ðrÞ ¼ 4J c ðrÞ½Uself ðrÞ þ Uother ðrÞ:
141
Uniform JC Peaking JC Trapezoidal JC
Tape edge Center
0.0 0.00
1.75
2.00
2.25
2.50
Lateral position x (mm)
Fig. 4. Distributions of loss density Qd per-cycle on the broad face of an isolated rectangular tape with different Jc distribution: (a) I0 = 0.9Ic and (b) I0 = 0.5Ic.
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current amplitude I0 = 0.2Ic. By addressing this fact, we confirmed the validity of the calculation procedure. It is also evident that the non-uniformity of Jc values along a tape width strongly influences the Qt values. The tape with peaking Jc distribution has significantly higher Qt values than the case with uniform Jc in the whole current range. On the other hand, the Qt values of the tape with trapezoidal Jc distribution almost agree with the analytical values for a thin strip where I0 > 0.7Ic. However, they deviate from the analytical values in the range of I0 < 0.6Ic and the deviation becomes more remarkable with decreasing I0. To examine the effect of non-uniformity of Jc values on the loss values in an isolated tape, the calculated results for the loss density (Qd) distributions on the broad face of a tape are shown in Fig. 4. As can be seen, higher Qd values are observed in the position near the tape edge (x = 2.5 mm) because the change of magnetic flux in ACcycle are most frequent around the tape edge. However, the distribution of Qd values along a tape width is strongly influenced by the lateral Jc distribution of the tape. In the case with peaking Jc distribution, the flux penetration region with loss density Qd > 0 extends toward the center part of tape regardless the current amplitude I0. This causes the significant increase of Qt values in a tape with peaking -1
I0 = 0.9IC
-2
dgap = 1.0 mm, I0 = 0.9IC 5
10
-3
Uniform JC Peaking JC Trapezoidal JC
10
-4
0.0
0.2
0.4
0.6
0.8
1.0
Gap dgap (mm)
(a)
Tape edge
0.5
1.0
1.5
2.0
2.5
Lateral position x (mm)
10
dgap = 0.3 mm, I0 = 0.9IC 5
Uniform JC Peaking JC Trapezoidal JC
2x10
3
Transport loss Qt (J/m/cycle)
0 0.0
I0 = 0.5IC
-4
Uniform JC Peaking JC Trapezoidal JC
-5
10
-6
2x10
Center
(a)
Loss density Qd (J/m /cycle)
-3
10
5
1x10
-3
2x10
(b)
Uniform JC Peaking JC Trapezoidal JC
2x10
3
10
Loss density Qd (J/m /cycle)
Transport loss Qt (J/m/cycle)
10
Jc profile shown in Fig. 3. On the other hand, as shown in Fig. 4(a), the Qd values at the condition of I0 = 0.9Ic are slightly decreased quite near the edge part (x > 2.3 mm) of a tape with trapezoidal Jc distribution because of the decrease of Jc only near the tape edge (see Fig. 1). In the lateral position of x < 2.2 mm, the Qd values are slightly larger than the case with uniform Jc. With decreasing I0 values, as shown in Fig. 4(b), the decrease of Qd values near the tape edge is disappeared and extension of the flux penetration region with Qd > 0 toward the center part of tape are observed, as well as the results with peaking Jc distribution. These results explain the difference of loss values Qt which depend on the non-uniformity of Jc values well therein shown in Fig. 3. Fig. 5 shows the loss values Qt in single-layer cable conductors composed of tape strands with different lateral Jc distribution, as a function of the gap dgap of adjacent tape strands. As can be seen, the Qt values in all conductors are reduced with decreasing the gap dgap. The dependence of Qt values on the dgap values becomes more significant at lower current amplitude (Fig. 5(b)). Our previous studies suggest that the reduction of the Qt values with decreasing the dgap values is caused by the suppression of magnetic flux passing through at the edge part of each tape strand [11]. In addition, it is also evident that the non-uniformity of Jc values along a lateral direction of each strand strongly influences the Qt values in cable conductors. The conductor composed of the tape strands with peaking Jc distribution has
0.0
0.2
0.4
0.6
0.8
1.0
Gap dgap (mm)
Fig. 5. Transport losses Qt per-cycle per-unit length for single-layer conductors plotted against the gap dgap of adjacent tape strands: (a) I0 = 0.9Ic and (b) I0 = 0.5Ic.
(b)
5
1x10
Tape edge Center
0 0.0
0.5
1.0
1.5
2.0
2.5
Lateral position x (mm)
Fig. 6. Distributions of loss density Qd per-cycle on the broad face of a tape strand in single-layer conductors at I0 = 0.9Ic, composed of tapes with different Jc distribution: (a) dgap = 1.0 mm and (b) dgap = 0.3 mm.
R. Inada et al. / Physica C 442 (2006) 139–144
significantly higher Qt values than the case with uniform Jc. Furthermore, at higher current amplitude near Ic, the change of Qt values due to the dgap values are suppressed compared with the results with other Jc profiles (Fig. 5(a)). On the other hand, the Qt values of the conductor using tape strands with trapezoidal Jc distribution are slightly greater than the values with the case of uniform Jc at higher current amplitude near Ic (Fig. 5(a)). As the current amplitude I0 decreases, however, the difference of the Qt values with the cases of uniform Jc and trapezoidal Jc becomes remarkable (Fig. 5(b)). The dependence of the Qt values in single-layer conductors on the lateral Jc distributions of each tape strand is quite similar with that in the tape in an isolated state shown in Fig. 3. To examine the effect of Jc distributions in each tape strand on the loss values in single-layer cable conductors, the calculated results for the loss density (Qd) distributions on the broad face of a tape strand in conductors are shown in Figs. 6 and 7, respectively. As can be seen, the transport loss in single-layer conductors is mainly generated near the edge part of each tape strand. The absolute values of Qd near the tape edge (x = 2.5 mm) are lower than those in an isolated tape (Fig. 4) and also reduced with decreasing dgap values, because of the influence for the magnetic flux generated by the adjacent tapes [11]. However, it is clearly observed that the distributions of Qd along a lateral direction of a tape strand in the conductor are quite similar with 4
dgap = 1.0 mm, I0 = 0.5IC
3
Loss density Qd (J/m /cycle)
6.0x10
4.0x10
4
2.0x10
4
Tape edge
Center
0.0 0.00
(a)
2.00
2.25
4.0x10
2.0x10
4
Uniform JC Peaking JC Trapezoidal JC
Tape edge
1.75
2.00
We investigated the AC loss properties on cable conductors using thin superconductor tapes with non-uniform Jc distributions through the numerical calculations. The calculations of loss values under AC current transmission were performed with taking into account the actual arrangements and the lateral Jc distributions of tape strands in conductors and geometrical factors of conductors. The calculated results suggest that the loss values in single-layer conductors are strongly affected not only by the arrangement of tape strands inside a conductor but also by the non-uniformity of Jc values along a lateral direction of each tape strand. As well as the case of a tape in an isolated state, the decrease of Jc values near the edge part of a tape strand leads to the significant increase of loss values under AC current transmission. This is mainly attributed to the extension of flux penetration region in going from the edge part to the center part of tape section, reflecting the lower Jc values near the tape edges. Therefore, in addition with the optimization with conductor structure, the improvement of uniformity of Jc values along a lateral direction in coated conductors is quite important for reducing the AC loss generated inside the cable conductors.
This work was supported in part by Grant-in-Aids for Scientific Research (No. 17206026) from the Japanese Society of the promotion of science, and also by that (No. 17760233) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. It was also supported in part from Research Foundation for the Electrotechnology of Chubu (No. R-17211) and by the 21st Century COE Program ‘‘Intelligent Human Sensing’’, from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
Center
0.0 0.00
4. Conclusion
Acknowledgements
dgap = 0.3 mm, I0 = 0.5IC
4
those in a tape in an isolated state. The non-uniformity of Jc values in each tape strand also affect to the loss generation in a tape strand. The decrease of Jc values near the edge part of a tape strand causes the extension of flux penetration region with loss density Qd > 0 toward the center part of tape strands. This is the reason for the increase of loss values in conductors which depend on the non-uniformity of Jc values in tape strands shown in Fig. 5. From these results, it is suggested that the improvement of uniformity of Jc values along a tape width in coated conductors has crucial importance to suppress the AC loss generation inside the cable conductors, together with the optimization of cable structures.
2.50
4
3
Loss density Qd (J/m /cycle)
1.75
Lateral position x (mm)
6.0x10
(b)
Uniform JC Peaking JC Trapezoidal JC
143
2.25
2.50
Lateral position x (mm)
Fig. 7. Distributions of loss density Qd per-cycle on the broad face of a tape strand in single-layer conductors at I0 = 0.5Ic, composed of tapes with different Jc distribution: (a) dgap = 1.0 mm and (b) dgap = 0.3 mm.
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