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Effective helical-pitch adjustment in a high-Tc superconducting cable for reducing AC losses
Physica C 412–414 (2004) 1206–1211 www.elsevier.com/locate/physc
Effective helical-pitch adjustment in a high-Tc superconducting cable for reducing AC losses H. Noji *, S. Ooyama, K. Nakajima Department of Electric Engineering, Miyakonojo National College of Technology, 473-1 Yoshio-cho, Miyakonojo, Miyazaki 885-8567, Japan Received 29 October 2003; accepted 13 January 2004 Available online 1 June 2004
Abstract We have reported that AC losses in a high-Tc superconducting transmission cable fabricated by Tokyo Electric Power Company and Sumitomo Electric Industries Ltd. are calculated correctly by using an electric-circuit model. According to the calculated results, the circumferential field losses are dominant in the total losses. The helical pitches of each layer in the cable are designed to obtain almost same layer currents, which gives the minimum self-field losses. We think that the optimum helical pitches giving the minimum total losses are different from the helical pitches designed by the companies and calculate the optimum ones in the condition of the same helical direction of each layer in the cable. As a result, for example, it is found that the AC loss of 2.1 W m1 cm3 at transporting 1 kArms can be reduced to 1.8 W m1 cm3 (about 14% reductions) after redesigning the cable with the optimum helical pitches. The optimum helical pitches are obtained for each given transport current. After redesigning, the distribution of layer currents is not uniform and the circumferential fields are reduced. Ó 2004 Elsevier B.V. All rights reserved. PACS: 84.70.+p; 85.25.Kx Keywords: High-Tc superconducting transmission cable; Electric-circuit model; AC losses; Bi-2223 tapes
1. Introduction We have reported that the electric-circuit (EC) model is useful for an analysis of AC electric properties in HTS cable [1,2]. In the previous paper, we reported that the losses due to the applied fields, in particular, circumferential fields B_ cm , are
*
Corresponding author. Tel.: +81-986-47-1198; fax: +81986-47-1208. E-mail address: [email protected] (H. Noji).
dominant in total losses of the one-phase cable. The applied fields on each layer are generated by layer currents. In general, it is believed that minimum losses are obtained by uniform layer’s current distribution [3,4]. Therefore, the TEPCO/ SEI’s cable is designed to make uniform all layer current, 4 layer currents passing through the conductor and 2 layer currents passing through the electrical shielding (see Fig. 1), by means of helical pitch adjustment. (Though, the results of our calculation for the TEPCO/SEI’s cable clarify that the shielding currents are not perfectly uniform.) It
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.140
H. Noji et al. / Physica C 412–414 (2004) 1206–1211
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conductor and the electrical shielding have four layers and two layers, respectively. Each layer is formed by a helical winding of HTS tapes. The direction of helical winding for each layer is indicated in Table 1. 2.1. The self-field losses The HTS tapes in the cable are exposed in the self-fields and the applied fields. The self-fields and the applied fields are generated by the AC transport currents passing through own tape and other tapes, respectively. We suppose that the complex self-fields, B_ selfkm and B_ self?m , and the complex applied field B_ applkm for the tape in layer m (m ¼ 1–6) are applied to the tape cross-section as shown in Fig. 1(a) and (b), respectively. The losses due to the perpendicular component of the self-fields, wself?m , dominate the self-field losses, in other words the transport losses, wself m . The wself?m are accounted for by the equation by Norris for a strip superconductor [5]:
Fig. 1. Schematic diagrams of the HTS tape’s cross-section exposed in the self-field and the applied field. (a) The HTS tape exposed in the self-field, (b) the one exposed in the applied field, (c) a concept of the cause generating the self-field loss and (d) a concept of the cause generating the applied field loss.
is clear that the uniform layer currents minimize the losses due to self-fields. However, it is uncertain that the total losses dominated by the losses due to B_ c are minimized by equalization of all layer current. We have studied the optimum helical pitches giving minimum total losses. In this paper, we show a comparison with AC electric properties of the TEPCO/SEI’s cable and the cable redesigned by helical pitch adjustment to minimize total losses for given transport currents.
wself?m ¼
Ic2 l0 f p ð1 im Þ lnð1 im Þ þ ð1 þ im Þ lnð1 þ im Þ i2m ðW m1 Þ for im < 1: ð1Þ
Here, the im is the normalized current in the tape and are described by im ¼ ^Im =ðNm Ic Þ, where ^Im is the peak layer current passing through layer m (^Im is the absolute value of complex layer current I_m ), Nm is the number of tapes and Ic is the critical current of the tape. Moreover, l0 is the magnetic permeability in vacuum and f is the frequency of
2. Calculation The one-phase cable is comprised of conductor, electrical insulation and electrical shielding. The Table 1 Main parameters of the HTS cable Layer m
Radius rm (mm)
Helical direction
Helical pitch length Pm SEI (mm)
Helical pitch length Pm minð0:5kAÞ (mm)
Helical pitch length Pm minð1kAÞ (mm)
Helical pitch length Pm minð1:5kAÞ (mm)
Number of tapes Nm
1 2 3 4 5 6
8.65 9.10 9.55 10.00 18.05 18.50
S S Z Z S S
130 305 400 115 350 530
80 255 405 120 510 1000
80 185 600 115 425 685
80 165 1000 135 485 1000
13 13 14 14 27 27
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H. Noji et al. / Physica C 412–414 (2004) 1206–1211
electric source. On the other hand, the wself m are also accounted for by the Norris equation [5]: wself m ¼
Ic2 l0 f p
im ð1 im Þ lnð1 im Þ þ ð2 im Þ 2
ðWm1 Þ
the EC model. V_con and V_sh are complex voltages per unit length generated at the conductor and the electrical shielding, respectively [1]. I_con and I_sh are complex transport currents through the conductor and the electrical shielding, respectively. Eqs. (5) show six simultaneous equations for the voltages
ð2Þ
for im < 1:
V_con ¼ fRm þ jxðLam þ Lcm ÞgI_m þ jx
4 X
ðMaml þ Mcml ÞI_l jx
and
l 6¼ m ðV m1 Þ;
V_sh ¼ fRm þ jxðLam þ Lcm ÞgI_m jx
4 X
ðMaml þ Mcml ÞI_l þ jx
and
6 X
ð5Þ ðMaml þ Mcml ÞI_l ;
l¼5
l¼1
m ¼ 5–6
ðMaml þ Mcml ÞI_l ;
l¼5
l¼1
m ¼ 1–4
6 X
l 6¼ m ðV m1 Þ;
Eqs. (1) and (2) are almost equal. Therefore, the self-field losses wself m correspond to the losses due to the B_ self?m as shown in Fig. 1(c). Calculating the self-field losses in the tape in a wide current range im > 0, we have used the approximated Norris equation as follows: w0self m ffi 0:0170 i3:371 ðW m1 Þ: m
ð3Þ
Eq. (3) is obtained by approximation of the wself m versus im property (im ¼ 0:1–0:99, Ic ¼ 50 A and f ¼ 50 Hz) by using the least-squares method. The EC model is comprised of a resistive part and an inductive part [1,2]. In this model, the layer resistance Rm is determined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2prm Þ þ Pm2 w0 self m Rm ¼ ðX m1 Þ; ð4Þ Pm Im2 Nm where pffiffiIffim is the rms value of peak layer current _ ( I m = 2). Self-inductance Lam and mutual inductance Maml (l ¼ 1–6) are related to axial fields B_ am . On the other hand, the self-inductance Lcm and mutual inductance Mcml are related to circumferential fields B_ cm . The equations for these inductances have been described in Ref. [1]. Simultaneous equations for voltages, V_con and _Vsh , and for currents, I_con and I_sh , are obtained in
where x ¼ 2pf . As indicated in Eqs. (3) and (4), the resistance of superconductor is nonlinear. Therefore, if sinusoidal current fed through the superconductor, then the voltage over the material is no more sinusoidal. However, the resistance is very small in comparing with the reactance in the cable. For example, R1 is two figures smaller than xðLa1 þ Lc1 Þ at Icon ¼ 1 kArms . Therefore, currents and voltages in this cable could approximately express with complex numbers as indicated in Eq. (5). Equations for the currents are described by ! ! 4 6 X X _Icon ¼ _Im ¼ I_sh ¼ _Im ðAÞ: ð6Þ m¼1
m¼5
As the electrical shielding might shield perfectly the magnetic fields generated by layer currents, we have assumed that the relation between I_con and I_sh is described by Eq. (6). We have solved the simultaneous equations and obtained complex layer currents I_m . The calculation was repeated until we obtained a convergence between the layer currents and the voltages, because Rm depends on Im . The calculation gives the self-field losses in the one-phase cable as follows:
H. Noji et al. / Physica C 412–414 (2004) 1206–1211
o 1 n Wself ¼ Re ðV_con þ V_sh ÞI_con ðW m1 Þ; 2
ð7Þ
is the complex conjugate of I_con . where I_con
2.2. The applied field losses We suppose that the applied fields generated by the layer currents are applied parallel to the tape face as shown in Fig. 1(b). The losses due to the applied fields B_ applkm and the parallel component of the self-fields B_ selfkm in the tape are defined as the applied field losses wBm , as shown in Fig. 1(d). The layer currents generate two kinds of applied fields on each layer as shown in Fig. 1. The equations of the complex axial field B_ am and the complex circumferential field B_ cm have been described in Refs. ^ am and B ^ cm , are obtained [1,2]. The peak values, B _ _ by absolute values of Bam and Bcm . The losses due to the combination of the transport currents and the magnetic fields applied parallel to the tape face have been calculated by Carr [6] and Magnusson [7] for a superconducting slab employing the critical state model. These losses are equal to the wBm . The equations of wBm have been described in Refs. ^ am =BP and bcm ¼ B ^ cm =BP are as[1,2]. As bam ¼ B signed to bm in these equations, the applied field ^ am and B ^ cm , wBam and wBcm , losses in the tape due to B are obtained, respectively. The equations for the applied field losses in the one-phase cable due to ^ am and B ^ cm , the axial field losses WBa and the cirB cumferential field losses WBc , have been described in Refs. [1,2]. 2.3. Total losses and minimization procedure The total losses in the one-phase cable, Wtot , are expressed as a sum of the self-field losses Wself , the axial field losses WBa and the circumferential field losses WBc . The superposition principle for Wtot is widely used for the calculation and the measurement of AC losses in HTS tapes carrying AC currents in AC magnetic fields [7–9]. The minimum total loss at given transport current was calculated as follows. Starting from the SEI/TEPCO’s helical pitches, the helical pitch is changed ±50 mm wide from the temporary fixed helical pitch to 5 mm pitch at one procedure. Here,
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the minimum helical pitches, Pm min , are described as 4rm Pm min ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmÞ:
2prm Nm wid
2
ð8Þ
1
The maximum helical-pitches, Pm max , are fixed at 1 m. The helical pitches are changed between Pm min and Pm max . Calculating the loss with changing the helical pitch, the temporary fixed helical pitch giving the minimum loss at one procedure is obtained. This procedure is done for each helical pitch from P1 to P6 . When one helical pitch is changed, the other helical pitches are fixed. The calculation is repeated until we obtained a convergence of minimum total loss. In the case that the calculation is done by the personal computer (Dimension 2400C, Intel Celeron Processor 2.60 GHz, Dell), about 10–12 h are needed to obtain the convergence for one transport current.
3. Results and discussion Fig. 2 shows the AC losses as a function of transport currents for the one-phase HTS cable. TEPCO and SEI measured the losses, Wmeas , by means of the calorimetric method. Many other research groups are measured the losses of the cable conductor by AC four-probe method [10– 12]. However, we think that the total loss of the cable could not be accurately measured by AC four-probe method, because the magnetization loss is eliminated by this method, which is inferred from the measurement technique of the losses in the HTS tape carrying AC transport currents and exposed to AC external magnetic fields [9]. In our calculation, the total losses include the transport losses and the magnetization losses. Therefore, the calculations should be compared with the losses measured by a method such as calorimetric one. The total losses, Wtot , calculated by using the EC model are almost equal to the measured losses in a wide transport current range. Therefore, our calculation method is very useful to obtain AC properties of the cable. In the total losses, the circumferential field losses, WBc , are dominant comparing with the self-field losses, Wself , and the
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H. Noji et al. / Physica C 412–414 (2004) 1206–1211
Fig. 2. The AC losses of the one-phase HTS cable as a function of transport currents. Bold line and black circles indicate the calculation and the measurements for the TEPCO/SEI’s cable, respectively. White circles indicate the minimum total losses in the cable redesigned by means of helical pitch adjustment.
axial field losses, WBa . This cable is designed to obtain uniform layer’s current distribution. In fact, the calculated layer’s current distribution shows that the layer currents are uniform in the conductor [2,3]. The uniform layer’s current distribution gives the minimum self-field losses, because the uniform layer currents restrain the drift-current phenomenon, which decreases the self-field losses in the cable [13]. However, it is not clear that the uniform layer’s current distribution gives the minimum total losses dominated by the circumferential field losses. We thought that the helical pitches giving the minimum total losses are different from the ones designed by TEPCO/SEI (see the TEPCO/SEI’s helical pitches Pm SEI in Table 1. We calculated the optimum helical pitches giving the minimum total losses in the condition of the same helical direction in the TEPCO/SEI’s cable, i.e. S, S, Z, Z, S and S direction (see Table 1). As a result, the optimum helical pitches, Pm min , and the minimum total losses, Wmin , are obtained for each given transport current. After redesigning, the total losses are reduced in all transport current range. For example, the AC loss of 0.47 W m1 cm3 can be reduced to 0.36 W m1 cm3 (about 23% reductions) at transporting 0.5 kArms ,
2.1–1.8 W m1 cm3 (about 14% reductions) at transporting 1 kArms and 5.1–4.5 W m1 cm3 (about 12% reductions) at transporting 1.5 kArms in three-phase cable after redesigning. The optimum helical pitches for each transport current are different. The optimum helical pitches at transporting 0.5, 1 and 1.5 kArms are indicated in Table 1. We also investigated the layer’s loss distribution in the cable before and after redesigning by means of helical pitch adjustment. In the losses of TEPCO/SEI’s cable WSEI , the layer losses increase gently as the top on layer 4 with increasing transport currents. In the minimum losses obtained by means of helical pitch adjustment Wmin , all layer loss without the loss on layer 4 is lower than WSEI at transport current of 0.5 kArms . Therefore, the much loss reduction is obtained at this transport current. Increasing transport current to 1 kArms , the layer loss in Wmin at layer 4 is slightly higher than the one in WSEI at same layer. Increasing transport current to 1.5 kArms , this phenomenon is prominent and the layer loss in Wmin has a big peak at layer 4. The phenomenon decreases the difference between the losses before and after redesigning with increasing transport currents. The Wself and WBa are very small and WBc is dominant in the conductor, i.e. layer number from 1 to 4. The Wself increase and WBc decrease in the electrical shielding, i.e. layer number 5 and 6. In the conductor of the TEPCO/SEI’s cable, the layer currents are uniform that leads to the minimum Wself . The layer current in the electrical shielding are not perfectly uniform. Therefore, Wself in the electrical shielding are high. In contrast with Wself , WBc are low in the electrical shielding. ^ cm are low in the elecThe circumferential fields B trical shielding, because the fields are decreased by the shielding currents passing through layer num^ cm lead to low WBc in the ber 5 and 6. The low B electrical shielding. The WBa are negligibly low, ^ am are very low compared because the axial fields B ^ cm . The Wself in the conductor increase with B slightly after redesigning, because the layer currents in the conductor are not uniform. On the other hand, the Wself in the electrical shielding decrease, because the difference of layer currents in the electrical shielding is alleviated. As a result of
H. Noji et al. / Physica C 412–414 (2004) 1206–1211
^ cm leads to the deredesigning, the decrease of B crease of WBc . 4. Conclusion We calculated the AC properties of the HTS cable by using EC model. The calculated AC losses are almost equal to the measurement obtained by TEPCO/SEI. The minimum losses in the cable are also calculated by means of helical pitch adjustment. As results, the important things indicated below are clarified. 1. The optimum helical pitches giving the minimum total loss exist for each given transport current. 2. The uniform layer’s current distribution does not give the minimum total losses. 3. The reduction of the circumferential fields leads to the minimum total losses. 4. The difference between the minimum total losses obtained by means of the optimum helical pitches and the ones obtained by the uniform layer’s current distribution decreases with increasing the transport currents. These calculations were done for the same helical direction of the TEPCO/SEI’s cable. We have continued to investigate the optimum cable
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parameters to obtain the minimum total losses in the HTS cable.
References [1] H. Noji, K. Haji, T. Hamada, Supercond. Sci. Technol. 16 (2003) 14. [2] H. Noji, K. Haji, T. Hamada, Physica C 392–396 (2003) 1134. [3] T. Masuda, T. Kato, H. Yumura, M. Hirose, S. Isojima, S. Honjo, Physica C 372–376 (2002) 1555. [4] S. Mukoyama, K. Miyoshi, H. Tsubouti, T. Yoshida, M. Mimura, N. Uno, M. Ikeda, H. Ishii, S. Honjo, Y. Iwata, IEEE Trans. Appl. Supercond. 9 (1999) 1269. [5] W.T. Norris, J. Phys. D 3 (1970) 489. [6] W.J. Carr, IEEE Trans. Magn. 15 (1979) 240. [7] N. Magnusson, Physica C 349 (2001) 225. [8] T. Yazawa, J.J. Rabbers, B. ten Haken, H.H.J. ten Kate, Y. Yamada, Physica C 310 (1998) 36. [9] N. Amemiya, K. Miyamoto, S. Murasawa, H. Mukai, K. Ohmatsu, Physica C 310 (1998) 30. [10] S. Kl€ uger Olsen, C. Træholt, A. K€ uhle, O. T ønesen, M. Da€ umling, J. Østergarrd, IEEE Trans. Appl. Supercond. 9 (1999) 833. [11] S. Mukoyama, K. Miyoshi, H. Tsubouchi, H. Tanaka, A. Takagi, K. Wada, S. Megro, K. Matsuo, S. Honjo, T. Mimura, Y. Takahashi, IEEE Trans. Appl. Supercond. 11 (2001) 2192. [12] H.M. Wen, L.Z. Lin, Y.B. Lin, Z.Y. Gao, S.T. Dai, L. Shu, Physica C 386 (2003) 52. [13] H. Noji, Supercond. Sci. Technol. 10 (1997) 552; H. Noji, T. Shibata, S. Isojima, J. Fujikami, K. Sato, H. Ishii, T. Hara, Cryogen. Eng. 33 (1998) 114 (Japanese).
Effective helical-pitch adjustment in a high-Tc superconducting cable for reducing AC losses H. Noji *, S. Ooyama, K. Nakajima Department of Electric Engineering, Miyakonojo National College of Technology, 473-1 Yoshio-cho, Miyakonojo, Miyazaki 885-8567, Japan Received 29 October 2003; accepted 13 January 2004 Available online 1 June 2004
Abstract We have reported that AC losses in a high-Tc superconducting transmission cable fabricated by Tokyo Electric Power Company and Sumitomo Electric Industries Ltd. are calculated correctly by using an electric-circuit model. According to the calculated results, the circumferential field losses are dominant in the total losses. The helical pitches of each layer in the cable are designed to obtain almost same layer currents, which gives the minimum self-field losses. We think that the optimum helical pitches giving the minimum total losses are different from the helical pitches designed by the companies and calculate the optimum ones in the condition of the same helical direction of each layer in the cable. As a result, for example, it is found that the AC loss of 2.1 W m1 cm3 at transporting 1 kArms can be reduced to 1.8 W m1 cm3 (about 14% reductions) after redesigning the cable with the optimum helical pitches. The optimum helical pitches are obtained for each given transport current. After redesigning, the distribution of layer currents is not uniform and the circumferential fields are reduced. Ó 2004 Elsevier B.V. All rights reserved. PACS: 84.70.+p; 85.25.Kx Keywords: High-Tc superconducting transmission cable; Electric-circuit model; AC losses; Bi-2223 tapes
1. Introduction We have reported that the electric-circuit (EC) model is useful for an analysis of AC electric properties in HTS cable [1,2]. In the previous paper, we reported that the losses due to the applied fields, in particular, circumferential fields B_ cm , are
*
Corresponding author. Tel.: +81-986-47-1198; fax: +81986-47-1208. E-mail address: [email protected] (H. Noji).
dominant in total losses of the one-phase cable. The applied fields on each layer are generated by layer currents. In general, it is believed that minimum losses are obtained by uniform layer’s current distribution [3,4]. Therefore, the TEPCO/ SEI’s cable is designed to make uniform all layer current, 4 layer currents passing through the conductor and 2 layer currents passing through the electrical shielding (see Fig. 1), by means of helical pitch adjustment. (Though, the results of our calculation for the TEPCO/SEI’s cable clarify that the shielding currents are not perfectly uniform.) It
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.140
H. Noji et al. / Physica C 412–414 (2004) 1206–1211
1207
conductor and the electrical shielding have four layers and two layers, respectively. Each layer is formed by a helical winding of HTS tapes. The direction of helical winding for each layer is indicated in Table 1. 2.1. The self-field losses The HTS tapes in the cable are exposed in the self-fields and the applied fields. The self-fields and the applied fields are generated by the AC transport currents passing through own tape and other tapes, respectively. We suppose that the complex self-fields, B_ selfkm and B_ self?m , and the complex applied field B_ applkm for the tape in layer m (m ¼ 1–6) are applied to the tape cross-section as shown in Fig. 1(a) and (b), respectively. The losses due to the perpendicular component of the self-fields, wself?m , dominate the self-field losses, in other words the transport losses, wself m . The wself?m are accounted for by the equation by Norris for a strip superconductor [5]:
Fig. 1. Schematic diagrams of the HTS tape’s cross-section exposed in the self-field and the applied field. (a) The HTS tape exposed in the self-field, (b) the one exposed in the applied field, (c) a concept of the cause generating the self-field loss and (d) a concept of the cause generating the applied field loss.
is clear that the uniform layer currents minimize the losses due to self-fields. However, it is uncertain that the total losses dominated by the losses due to B_ c are minimized by equalization of all layer current. We have studied the optimum helical pitches giving minimum total losses. In this paper, we show a comparison with AC electric properties of the TEPCO/SEI’s cable and the cable redesigned by helical pitch adjustment to minimize total losses for given transport currents.
wself?m ¼
Ic2 l0 f p ð1 im Þ lnð1 im Þ þ ð1 þ im Þ lnð1 þ im Þ i2m ðW m1 Þ for im < 1: ð1Þ
Here, the im is the normalized current in the tape and are described by im ¼ ^Im =ðNm Ic Þ, where ^Im is the peak layer current passing through layer m (^Im is the absolute value of complex layer current I_m ), Nm is the number of tapes and Ic is the critical current of the tape. Moreover, l0 is the magnetic permeability in vacuum and f is the frequency of
2. Calculation The one-phase cable is comprised of conductor, electrical insulation and electrical shielding. The Table 1 Main parameters of the HTS cable Layer m
Radius rm (mm)
Helical direction
Helical pitch length Pm SEI (mm)
Helical pitch length Pm minð0:5kAÞ (mm)
Helical pitch length Pm minð1kAÞ (mm)
Helical pitch length Pm minð1:5kAÞ (mm)
Number of tapes Nm
1 2 3 4 5 6
8.65 9.10 9.55 10.00 18.05 18.50
S S Z Z S S
130 305 400 115 350 530
80 255 405 120 510 1000
80 185 600 115 425 685
80 165 1000 135 485 1000
13 13 14 14 27 27
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H. Noji et al. / Physica C 412–414 (2004) 1206–1211
electric source. On the other hand, the wself m are also accounted for by the Norris equation [5]: wself m ¼
Ic2 l0 f p
im ð1 im Þ lnð1 im Þ þ ð2 im Þ 2
ðWm1 Þ
the EC model. V_con and V_sh are complex voltages per unit length generated at the conductor and the electrical shielding, respectively [1]. I_con and I_sh are complex transport currents through the conductor and the electrical shielding, respectively. Eqs. (5) show six simultaneous equations for the voltages
ð2Þ
for im < 1:
V_con ¼ fRm þ jxðLam þ Lcm ÞgI_m þ jx
4 X
ðMaml þ Mcml ÞI_l jx
and
l 6¼ m ðV m1 Þ;
V_sh ¼ fRm þ jxðLam þ Lcm ÞgI_m jx
4 X
ðMaml þ Mcml ÞI_l þ jx
and
6 X
ð5Þ ðMaml þ Mcml ÞI_l ;
l¼5
l¼1
m ¼ 5–6
ðMaml þ Mcml ÞI_l ;
l¼5
l¼1
m ¼ 1–4
6 X
l 6¼ m ðV m1 Þ;
Eqs. (1) and (2) are almost equal. Therefore, the self-field losses wself m correspond to the losses due to the B_ self?m as shown in Fig. 1(c). Calculating the self-field losses in the tape in a wide current range im > 0, we have used the approximated Norris equation as follows: w0self m ffi 0:0170 i3:371 ðW m1 Þ: m
ð3Þ
Eq. (3) is obtained by approximation of the wself m versus im property (im ¼ 0:1–0:99, Ic ¼ 50 A and f ¼ 50 Hz) by using the least-squares method. The EC model is comprised of a resistive part and an inductive part [1,2]. In this model, the layer resistance Rm is determined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð2prm Þ þ Pm2 w0 self m Rm ¼ ðX m1 Þ; ð4Þ Pm Im2 Nm where pffiffiIffim is the rms value of peak layer current _ ( I m = 2). Self-inductance Lam and mutual inductance Maml (l ¼ 1–6) are related to axial fields B_ am . On the other hand, the self-inductance Lcm and mutual inductance Mcml are related to circumferential fields B_ cm . The equations for these inductances have been described in Ref. [1]. Simultaneous equations for voltages, V_con and _Vsh , and for currents, I_con and I_sh , are obtained in
where x ¼ 2pf . As indicated in Eqs. (3) and (4), the resistance of superconductor is nonlinear. Therefore, if sinusoidal current fed through the superconductor, then the voltage over the material is no more sinusoidal. However, the resistance is very small in comparing with the reactance in the cable. For example, R1 is two figures smaller than xðLa1 þ Lc1 Þ at Icon ¼ 1 kArms . Therefore, currents and voltages in this cable could approximately express with complex numbers as indicated in Eq. (5). Equations for the currents are described by ! ! 4 6 X X _Icon ¼ _Im ¼ I_sh ¼ _Im ðAÞ: ð6Þ m¼1
m¼5
As the electrical shielding might shield perfectly the magnetic fields generated by layer currents, we have assumed that the relation between I_con and I_sh is described by Eq. (6). We have solved the simultaneous equations and obtained complex layer currents I_m . The calculation was repeated until we obtained a convergence between the layer currents and the voltages, because Rm depends on Im . The calculation gives the self-field losses in the one-phase cable as follows:
H. Noji et al. / Physica C 412–414 (2004) 1206–1211
o 1 n Wself ¼ Re ðV_con þ V_sh ÞI_con ðW m1 Þ; 2
ð7Þ
is the complex conjugate of I_con . where I_con
2.2. The applied field losses We suppose that the applied fields generated by the layer currents are applied parallel to the tape face as shown in Fig. 1(b). The losses due to the applied fields B_ applkm and the parallel component of the self-fields B_ selfkm in the tape are defined as the applied field losses wBm , as shown in Fig. 1(d). The layer currents generate two kinds of applied fields on each layer as shown in Fig. 1. The equations of the complex axial field B_ am and the complex circumferential field B_ cm have been described in Refs. ^ am and B ^ cm , are obtained [1,2]. The peak values, B _ _ by absolute values of Bam and Bcm . The losses due to the combination of the transport currents and the magnetic fields applied parallel to the tape face have been calculated by Carr [6] and Magnusson [7] for a superconducting slab employing the critical state model. These losses are equal to the wBm . The equations of wBm have been described in Refs. ^ am =BP and bcm ¼ B ^ cm =BP are as[1,2]. As bam ¼ B signed to bm in these equations, the applied field ^ am and B ^ cm , wBam and wBcm , losses in the tape due to B are obtained, respectively. The equations for the applied field losses in the one-phase cable due to ^ am and B ^ cm , the axial field losses WBa and the cirB cumferential field losses WBc , have been described in Refs. [1,2]. 2.3. Total losses and minimization procedure The total losses in the one-phase cable, Wtot , are expressed as a sum of the self-field losses Wself , the axial field losses WBa and the circumferential field losses WBc . The superposition principle for Wtot is widely used for the calculation and the measurement of AC losses in HTS tapes carrying AC currents in AC magnetic fields [7–9]. The minimum total loss at given transport current was calculated as follows. Starting from the SEI/TEPCO’s helical pitches, the helical pitch is changed ±50 mm wide from the temporary fixed helical pitch to 5 mm pitch at one procedure. Here,
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the minimum helical pitches, Pm min , are described as 4rm Pm min ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmÞ:
2prm Nm wid
2
ð8Þ
1
The maximum helical-pitches, Pm max , are fixed at 1 m. The helical pitches are changed between Pm min and Pm max . Calculating the loss with changing the helical pitch, the temporary fixed helical pitch giving the minimum loss at one procedure is obtained. This procedure is done for each helical pitch from P1 to P6 . When one helical pitch is changed, the other helical pitches are fixed. The calculation is repeated until we obtained a convergence of minimum total loss. In the case that the calculation is done by the personal computer (Dimension 2400C, Intel Celeron Processor 2.60 GHz, Dell), about 10–12 h are needed to obtain the convergence for one transport current.
3. Results and discussion Fig. 2 shows the AC losses as a function of transport currents for the one-phase HTS cable. TEPCO and SEI measured the losses, Wmeas , by means of the calorimetric method. Many other research groups are measured the losses of the cable conductor by AC four-probe method [10– 12]. However, we think that the total loss of the cable could not be accurately measured by AC four-probe method, because the magnetization loss is eliminated by this method, which is inferred from the measurement technique of the losses in the HTS tape carrying AC transport currents and exposed to AC external magnetic fields [9]. In our calculation, the total losses include the transport losses and the magnetization losses. Therefore, the calculations should be compared with the losses measured by a method such as calorimetric one. The total losses, Wtot , calculated by using the EC model are almost equal to the measured losses in a wide transport current range. Therefore, our calculation method is very useful to obtain AC properties of the cable. In the total losses, the circumferential field losses, WBc , are dominant comparing with the self-field losses, Wself , and the
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Fig. 2. The AC losses of the one-phase HTS cable as a function of transport currents. Bold line and black circles indicate the calculation and the measurements for the TEPCO/SEI’s cable, respectively. White circles indicate the minimum total losses in the cable redesigned by means of helical pitch adjustment.
axial field losses, WBa . This cable is designed to obtain uniform layer’s current distribution. In fact, the calculated layer’s current distribution shows that the layer currents are uniform in the conductor [2,3]. The uniform layer’s current distribution gives the minimum self-field losses, because the uniform layer currents restrain the drift-current phenomenon, which decreases the self-field losses in the cable [13]. However, it is not clear that the uniform layer’s current distribution gives the minimum total losses dominated by the circumferential field losses. We thought that the helical pitches giving the minimum total losses are different from the ones designed by TEPCO/SEI (see the TEPCO/SEI’s helical pitches Pm SEI in Table 1. We calculated the optimum helical pitches giving the minimum total losses in the condition of the same helical direction in the TEPCO/SEI’s cable, i.e. S, S, Z, Z, S and S direction (see Table 1). As a result, the optimum helical pitches, Pm min , and the minimum total losses, Wmin , are obtained for each given transport current. After redesigning, the total losses are reduced in all transport current range. For example, the AC loss of 0.47 W m1 cm3 can be reduced to 0.36 W m1 cm3 (about 23% reductions) at transporting 0.5 kArms ,
2.1–1.8 W m1 cm3 (about 14% reductions) at transporting 1 kArms and 5.1–4.5 W m1 cm3 (about 12% reductions) at transporting 1.5 kArms in three-phase cable after redesigning. The optimum helical pitches for each transport current are different. The optimum helical pitches at transporting 0.5, 1 and 1.5 kArms are indicated in Table 1. We also investigated the layer’s loss distribution in the cable before and after redesigning by means of helical pitch adjustment. In the losses of TEPCO/SEI’s cable WSEI , the layer losses increase gently as the top on layer 4 with increasing transport currents. In the minimum losses obtained by means of helical pitch adjustment Wmin , all layer loss without the loss on layer 4 is lower than WSEI at transport current of 0.5 kArms . Therefore, the much loss reduction is obtained at this transport current. Increasing transport current to 1 kArms , the layer loss in Wmin at layer 4 is slightly higher than the one in WSEI at same layer. Increasing transport current to 1.5 kArms , this phenomenon is prominent and the layer loss in Wmin has a big peak at layer 4. The phenomenon decreases the difference between the losses before and after redesigning with increasing transport currents. The Wself and WBa are very small and WBc is dominant in the conductor, i.e. layer number from 1 to 4. The Wself increase and WBc decrease in the electrical shielding, i.e. layer number 5 and 6. In the conductor of the TEPCO/SEI’s cable, the layer currents are uniform that leads to the minimum Wself . The layer current in the electrical shielding are not perfectly uniform. Therefore, Wself in the electrical shielding are high. In contrast with Wself , WBc are low in the electrical shielding. ^ cm are low in the elecThe circumferential fields B trical shielding, because the fields are decreased by the shielding currents passing through layer num^ cm lead to low WBc in the ber 5 and 6. The low B electrical shielding. The WBa are negligibly low, ^ am are very low compared because the axial fields B ^ cm . The Wself in the conductor increase with B slightly after redesigning, because the layer currents in the conductor are not uniform. On the other hand, the Wself in the electrical shielding decrease, because the difference of layer currents in the electrical shielding is alleviated. As a result of
H. Noji et al. / Physica C 412–414 (2004) 1206–1211
^ cm leads to the deredesigning, the decrease of B crease of WBc . 4. Conclusion We calculated the AC properties of the HTS cable by using EC model. The calculated AC losses are almost equal to the measurement obtained by TEPCO/SEI. The minimum losses in the cable are also calculated by means of helical pitch adjustment. As results, the important things indicated below are clarified. 1. The optimum helical pitches giving the minimum total loss exist for each given transport current. 2. The uniform layer’s current distribution does not give the minimum total losses. 3. The reduction of the circumferential fields leads to the minimum total losses. 4. The difference between the minimum total losses obtained by means of the optimum helical pitches and the ones obtained by the uniform layer’s current distribution decreases with increasing the transport currents. These calculations were done for the same helical direction of the TEPCO/SEI’s cable. We have continued to investigate the optimum cable
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parameters to obtain the minimum total losses in the HTS cable.
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