- Email: [email protected]
Design parameters influencing non-uniform current distributions in superconducting multi-stage stranded cables
Physica C 310 Ž1998. 358–366
Design parameters influencing non-uniform current distributions in superconducting multi-stage stranded cables Kazutaka Seo ) , Mitsuru Hasegawa, Masao Morita, Hideto Yoshimura Electromechanical System Department, AdÕanced Technology R & D Center, Mitsubishi Electric Corporation, 1-1 Tsukaguchi-honmachi 8-chome, Amagasaki, Hyogo, 661 Japan
Abstract We have suggested that non-uniform currents occur due to certain design parameters, for example a combination of cabling pitches, in a previous study. In the present study, we evaluate the occurrences of non-uniform current distributions in terms of design parameters, i.e., the combination of cabling pitches, winding methods Žsolenoidal and pancake winding. and cross-sectional shape of the cable by numerical analyses. Finally, some basic design concepts to solve this problem are proposed. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Current distribution; Superconducting cable; Design parameters
1. Introduction A non-uniform current distribution is a troublesome phenomenon of superconducting stranded cables. This means that when a larger current than the cable’s average flows through a strand, it results in the whole cable being quenched. It has been reported that in a superconducting pulse coil, the current flows in a strand could be seven times larger than the cable’s mean w1x. It has also been reported that a remarkable instability due to non-uniform current appears when the current increases rapidly w2x. This phenomenon is known as ramp-ratelimitation ŽRRL. w1,2x, which suggests that this problem is related to the inductive factor. On the other hand, in a multi-stage stranded Žsuch as 3 = 3 = 3. cable in which each strand is transposed, it is thought that a non-uniform current is not induced, unlike seven-strand cables. Usually a strand’s surface is not insulated and the strands are in contact with each other and the current can be distributed freely between the strands. In other words, even if non-uniform current occurs and the normal conducting region appears in one strand, the current re-distribution occurs immediately and the cable is not quenched. However, for a CICC ŽCable In Conduit Conductor., the heat dissipation during the current re-distribution must make the temperature rise, thus it is important to reduce the current re-distribution by establishing a uniform current distribution. It has been reported that the cause of the non-uniform current distribution was circulation currents, which were superposed on excitation currents w1x. The circulation currents were thought to be induced by both a self )
Corresponding author. Tel.: q81-6-4977127; Fax: q81-6-4977288; E-mail: [email protected]
0921-4534r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 4 9 2 - 4
K. Seo et al.r Physica C 310 (1998) 358–366
359
magnetic field and the magnetic field produced by the other turns when the cables were wound into the coil. However in multi-stage stranded superconducting cables, a non-uniform current is ideally not expected to occur by neither self fields nor external fields Žof course, when there is conspicuous non-uniform cabling in such cables, a non-uniform current distribution appears.. In this study, the mechanism of non-uniform current distribution was examined by using numerical analysis from the viewpoint of the hypotheses that the phenomenon is exaggerated in the multi-stage stranded cable by having the cable wound into the coil and by choosing certain design parameters.
2. Analysis We have previously examined the mechanism of non-uniform current distribution on the basis of the non-uniformity of induced voltages w3x. All calculations were carried out by Mathematica w4x. This software enables this sort of problem to be calculated with easy visual images. Fig. 1 shows a schematic illustration of the CICC superconducting coil used in the analysis. The coil is illustrated as being wound in two different ways, double pancake ŽDP. and solenoidal ŽSOL.. DP winding
Fig. 1. Schematic illustration of model coil and conductor. The cable was stranded 3 = 3 = 3 = 3 = 6, with a final 5th cabling pitch of 480 mm and with the 4th being the subject of the analyses. The coil consists of four DP coil elements or eight elemental SOL ones. Calculations were made on the second DP coil element from the top and for the most inner SOL coil element.
360
K. Seo et al.r Physica C 310 (1998) 358–366
employs 4 DP coil elements and SOL windings 8 SOL elements. All coil elements were connected in series and each element is 486 parallel circuits with insulated strands. These model coils can be described by the electrical circuit shown in Fig. 2. Generally, non-uniform currents can be discussed according to each coil element, because the coil elements are not joined strand to strand along their length, but are all soldered together at their ends. In the analysis, the non-uniformity of the induced voltages, which are induced by the other windings, causes a non-uniform current between the sub-cables forming one coil element. The non-uniform current distributions in the 3rd stage sub-cables were examined considering the following three assumptions. Ž1. The currents are distributed equally in the six 4th stage sub-cables constituting the final cable, because they are stranded into a concentric circle and the symmetry is good. Furthermore the inductance of the circulation current through these sub-cables is rather large. Ž2. In a coil element, the dispersion of the mutual inductances between the 18 3rd stage sub-cables is not considered to simplify the calculation. If the influence of the non-uniformity of the induced voltages is a dominant cause of non-uniform current distribution, this supposition is reasonable. Ž3. The current distribution between the 27 Ž3 = 3 = 3. strands which make up the 3rd stage sub-cable is not discussed. Of course, it is possible to calculate the distribution in the other sub-cables and strands by the same analytical methods. The circuit equations for the electrical circuits shown in Fig. 2 are given in Eq. Ž1.. This equation describes the current distributions between the six 4th stage sub-cables in the final cable. The first subsections indicate the inductance matrixes of the coil element analyzed for non-uniform current distribution. The second subsections are the mutual inductances between the 4th stage sub-cables in the coil element and the other windings, and the third subsections are the self-inductances of the other windings. Using the assumptions mentioned above, Eq. Ž1. becomes Eq. Ž2. and describes the current distribution between the three 3rd stage sub-cables, where Vex is the excitation voltage and Vind is the inductive voltage. The inductance of the whole coil becomes 14.35 mH. The method of obtaining the mutual inductance between the 3rd stage sub-cables making one 4th stage sub-cable is as follows. This is calculated being considered as two coils which are close to each other. When one turn of radius a approaches another at a distance d , the mutual inductance is given by Eq. Ž3. w5x. The self-inductance of the 3rd stage sub-cables and the mutual inductance can be calculated from the results of Eq. Ž3.. When the distance d was assumed to be 1.6 mm, the inductances of the circulation current Ž L mm q L nn y
Fig. 2. Electrical circuit. Relation between Lmm q Lnn y 2 Mmn and Vind determines non-uniform current distribution.
K. Seo et al.r Physica C 310 (1998) 358–366
361
2 Mmn . between two adjacent 3rd stage sub-cables are ; 28.44 mH for the DP coil element and ; 10.07 mH for the SOL element. Here, m and n indicate the numbers of the 3rd stage sub-cables. As mentioned below, the inductance of the circulation current controls the magnitude of the non-uniform current. d i 01 dt d i 02 L01 M021 M031 M041 M051 M061
M012 L02 M032 M042 M052 M062
M013 M023 L03 M043 M053 M063
M014 M024 M034 L04 M054 M064
M015 M025 M035 M045 L05 M065
M016 M026 M036 M046 M056 L06
dt d i 03 dt d i 04
q 2 M0
d i0 dt
q L0
d i0 dt
s Vex ,
Ž 1.
dt d i 05 dt d i 06 dt
d i1 L11 M21 M31
M12 L 22 M32
dt d i2
M13 M23 L33
dt d i3
Vind1 q M0 q L0 s Vex q Vind2 , dt dt Vind3 d i0
d i0
Ž 2.
dt
ž
M s m 0 log
8a
d
/
y2 .
Ž 3.
Non-uniform currents are not thought to occur between the 4th stage sub-cables, but between the three 3rd stage sub-cables in this calculation. Thus, Eq. Ž2. can be represented as Eq. Ž4.. In Eq. Ž4., the magnitude of the non-uniform current is decided by both the induced voltage Vind m , which is caused by the difference between one 3rd stage sub-cable and the other windings, and L mm y Mmn , which is equivalent to a half of the inductance of the circulation current circuit.
Ž L11 y M21 . d i1rdt y Vind1 s Ž L22 y M12 . d i 2rd t y Vind2 , Ž L22 y M32 . d i 2rdt y Vind2 s Ž L33 y M23 . d i 3rd t y Vind3 , 6 Ž d i 1rdt q d i 2rd t q d i 3rdt . s d i 0rdt.
Ž 4.
The geometrical traces of the 3rd stage sub-cables Žtotal number 3 = 6 s 18. are calculated. By integral calculus of magnetic vector potential Au along the trace shown in Eq. Ž5., which is produced due to the unit
Fig. 3. Magnetic vector potential Ž rAu . in the winding and integral paths Žblack arrow for the DP winding, gray one for the SOL..
K. Seo et al.r Physica C 310 (1998) 358–366
362 Table 1 Calculation models Cabling pitch 5th stage Žmm.
4th stage Žmm.
Ža. 480 Žb. 480 Žc. 480 Žd. 480 Že. 480 Žf. 480
240 240=7r9 240 240=7r9 240 240
Winding methods
Deformation
Double pancake Double pancake Double pancake Double pancake Solenoid Solenoid
Non Non Deformed Deformed Non Deformed
current Ž1.0 A. energization of the other windings, the induced voltages can be calculated. Similar approaches have been found successful in Refs. w6,7x. Vind s
d i0 dt
HAu d l.
Ž 5.
The cable cross section supposed in the analysis is shown in Fig. 1. Here r5 is the cabling radius of the final 5th stage and r4 is the radius of the 4th stage. For the final stage, it was supposed that the elliptical orbit and the ratio of the major and the minor axes is 7:6. The traces of the sub-cables are determined by the superposition of the sinusoidal functions. The distribution of magnetic vector potential Au in position in the elemental coil winding is shown in Fig. 3. As shown in Eq. Ž5., the induced voltage is obtained by integrating the magnetic vector potential of the unit current energization along the geometric traces of the three 3rd stage sub-cables numerically.
Fig. 4. Traces of the three 3rd stage sub-cables. Ža. shows cases a and e, the geometrical trace of each sub-cable differs from the others. Žb. shows cases b and d.
K. Seo et al.r Physica C 310 (1998) 358–366
363
The models used in the calculations are shown in Table 1. The parameters are combination of the cabling pitches, winding method and deformation of the strand bundle. The traces of the three 3rd stage sub-cables with different combinations of cabling pitches are shown in Fig. 4. In the cases of a and e, because the 5th stage cabling pitch is double that of the 4th one, the 3rd stage sub-cables go around the same position periodically. Notice the offset of these three traces in the figure. The integral paths of the different winding methods are shown in Fig. 5. They show 16 turns for the DP winding and eight turns for the SOL winding. Compared with Fig. 3, when the sub-cables go along the SOL
Fig. 5. Traces of the three 3rd stage sub-cables, Ža. shows cases a to d and Žb. shows cases e and f.
364
K. Seo et al.r Physica C 310 (1998) 358–366
Fig. 6. Schematic illustration of strand bundle deformation and weight function. In case d, all sub-cables are pushed in the transverse direction equally.
winding, they are in a magnetic field with the same direction. On the contrary, for the DP winding, the direction changes between the inner and outer circumferences. The deformation of the cable was modeled as shown in Fig. 6. For general CICCs, they are formed into rectangular shapes after being stranded into final cables whose original cross-sectional shapes were round. As in Fig. 6, the weight function demonstrating the transverse compression was considered in the transverse direction. Electromagnetic forces also play a role in deforming the bundle of strands when the void fraction is large. If the geometrical center of the trace of the sub-cable has an offset in the transverse direction Žsee Fig. 4a., it drifts to the bundle’s center when the conductor was deformed into a rectangle.
Fig. 7. Induced voltages in 3 = 6 3rd stage sub-cables. DPs use the left scale and SOLs the right. Non-uniformity occurs when the strand bundle is deformed Žcases c and f. and the differences are less than 0.1%.
K. Seo et al.r Physica C 310 (1998) 358–366
365
Fig. 8. Current distribution in the three 3rd stage sub-cables. A non-uniform current occurs when the bundle is deformed and is solenoid-wound Žcase f..
In Fig. 4a, for the 3rd stage sub-cable 1, the geometric trace that inclines to the coil’s outer circumference is pushed to the inside periodically. For the 3rd stage sub-cables 2 and 3, whose traces incline to the inner circumference, they are pushed outwards.
3. Results and considerations The calculated induced voltages are shown in Fig. 7. The induced voltages for the DP winding are about 3.2 mV and those for the SOL about 1.0 mV. The largest dispersions are about 0.5 mV in both cases. The current distribution was calculated for an excitation rate of 1 Ars by Eq. Ž4.. Note that the non-uniform current behaviour has no relation to the current ramp rate at all, because this circuit model is a linear system and does not contain any resistance. The ratios of the currents of each of the 3rd stage sub-cables were calculated as shown in Fig. 8. The largest current is 60% larger than the cable average in case f. The effect of the combination of the cabling pitch was evaluated by comparing case c and d. When the sub-cable goes through the same trace periodically, a large non-uniform current occurs. Regarding the winding methods, if we compare cases c and f, non-uniformity is remarkably enlarged in the case of the SOL winding Žcase f.. In a pancake coil, the direction of the magnetic field which the conductor experiences is reversed between the inner and outer circumferences of the coil Žcf. Fig. 3.. Therefore in this case it is thought that not much dispersion of the inductance is caused, even if the geometrical trace of the sub-cable is offset from the center of the cable periodically. Regarding the deformation of the cable, comparing cases a and c and cases e and f, more non-uniformity occurs when the cable is deformed. When the 5th stage cabling pitch is not double the 4th one, the non-uniform current is not exaggerated in spite of the deformation Žsee case d.. The current distributions in the lower stage sub-cables also can be evaluated by doing similar analysis. For the lower stage sub-cables, the dispersion of the induced voltage by the other windings is small. However, this needs to be examined because the inductance of the circulation current circuit between such sub-cables is small too.
4. Conclusion When a circulation current is induced by magnetic flux between sub-cables, a non-uniform current occurs in multi-stage stranded cables. The most important consideration is to select a cabling pitch which prevents strands
K. Seo et al.r Physica C 310 (1998) 358–366
366
andror sub-cables periodically going around the same traces. When there is a trace offset, it is preferable not to deform the cross-sectional shape of the strand bundle and pancake winding is preferable to solenoid winding. For the essential prevention of non-uniform current distributions, these parameters need to be optimized.
References w1x w2x w3x w4x w5x w6x w7x
N. Koizumi, K. Okuno, Y. Takahashi, H. Tsuji, S. Shimamoto, Cryogenics 36 Ž6. Ž1996. 409. V. Vysotsky, M. Takayasu, M. Ferri, J.V. Minervini, IEEE Trans. On Applied Supercond. Ž1995. 580. K. Seo, M. Morita, M. Hasegawa, H. Yoshimura, Proceedings of MT-15, PF2-04 Ž1997. in China. S. Wolfram, Mathematica, Wolfram Research Ž1992.. K. Goto, S. Yamazaki, Electromagnetism practice, Ž1970. 284. C. Sihler, R. Heller, W. Maurer, A. Ulbricht, F. Weuchner, Applied Superconductivity, Elsevier, Vol. 4, No. 1r2, 1996, 95–110. M. Shimada, N. Mitchell, IEEE Trans. On Applied Supercond. Ž1997. 759.
Design parameters influencing non-uniform current distributions in superconducting multi-stage stranded cables Kazutaka Seo ) , Mitsuru Hasegawa, Masao Morita, Hideto Yoshimura Electromechanical System Department, AdÕanced Technology R & D Center, Mitsubishi Electric Corporation, 1-1 Tsukaguchi-honmachi 8-chome, Amagasaki, Hyogo, 661 Japan
Abstract We have suggested that non-uniform currents occur due to certain design parameters, for example a combination of cabling pitches, in a previous study. In the present study, we evaluate the occurrences of non-uniform current distributions in terms of design parameters, i.e., the combination of cabling pitches, winding methods Žsolenoidal and pancake winding. and cross-sectional shape of the cable by numerical analyses. Finally, some basic design concepts to solve this problem are proposed. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Current distribution; Superconducting cable; Design parameters
1. Introduction A non-uniform current distribution is a troublesome phenomenon of superconducting stranded cables. This means that when a larger current than the cable’s average flows through a strand, it results in the whole cable being quenched. It has been reported that in a superconducting pulse coil, the current flows in a strand could be seven times larger than the cable’s mean w1x. It has also been reported that a remarkable instability due to non-uniform current appears when the current increases rapidly w2x. This phenomenon is known as ramp-ratelimitation ŽRRL. w1,2x, which suggests that this problem is related to the inductive factor. On the other hand, in a multi-stage stranded Žsuch as 3 = 3 = 3. cable in which each strand is transposed, it is thought that a non-uniform current is not induced, unlike seven-strand cables. Usually a strand’s surface is not insulated and the strands are in contact with each other and the current can be distributed freely between the strands. In other words, even if non-uniform current occurs and the normal conducting region appears in one strand, the current re-distribution occurs immediately and the cable is not quenched. However, for a CICC ŽCable In Conduit Conductor., the heat dissipation during the current re-distribution must make the temperature rise, thus it is important to reduce the current re-distribution by establishing a uniform current distribution. It has been reported that the cause of the non-uniform current distribution was circulation currents, which were superposed on excitation currents w1x. The circulation currents were thought to be induced by both a self )
Corresponding author. Tel.: q81-6-4977127; Fax: q81-6-4977288; E-mail: [email protected]
0921-4534r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 4 9 2 - 4
K. Seo et al.r Physica C 310 (1998) 358–366
359
magnetic field and the magnetic field produced by the other turns when the cables were wound into the coil. However in multi-stage stranded superconducting cables, a non-uniform current is ideally not expected to occur by neither self fields nor external fields Žof course, when there is conspicuous non-uniform cabling in such cables, a non-uniform current distribution appears.. In this study, the mechanism of non-uniform current distribution was examined by using numerical analysis from the viewpoint of the hypotheses that the phenomenon is exaggerated in the multi-stage stranded cable by having the cable wound into the coil and by choosing certain design parameters.
2. Analysis We have previously examined the mechanism of non-uniform current distribution on the basis of the non-uniformity of induced voltages w3x. All calculations were carried out by Mathematica w4x. This software enables this sort of problem to be calculated with easy visual images. Fig. 1 shows a schematic illustration of the CICC superconducting coil used in the analysis. The coil is illustrated as being wound in two different ways, double pancake ŽDP. and solenoidal ŽSOL.. DP winding
Fig. 1. Schematic illustration of model coil and conductor. The cable was stranded 3 = 3 = 3 = 3 = 6, with a final 5th cabling pitch of 480 mm and with the 4th being the subject of the analyses. The coil consists of four DP coil elements or eight elemental SOL ones. Calculations were made on the second DP coil element from the top and for the most inner SOL coil element.
360
K. Seo et al.r Physica C 310 (1998) 358–366
employs 4 DP coil elements and SOL windings 8 SOL elements. All coil elements were connected in series and each element is 486 parallel circuits with insulated strands. These model coils can be described by the electrical circuit shown in Fig. 2. Generally, non-uniform currents can be discussed according to each coil element, because the coil elements are not joined strand to strand along their length, but are all soldered together at their ends. In the analysis, the non-uniformity of the induced voltages, which are induced by the other windings, causes a non-uniform current between the sub-cables forming one coil element. The non-uniform current distributions in the 3rd stage sub-cables were examined considering the following three assumptions. Ž1. The currents are distributed equally in the six 4th stage sub-cables constituting the final cable, because they are stranded into a concentric circle and the symmetry is good. Furthermore the inductance of the circulation current through these sub-cables is rather large. Ž2. In a coil element, the dispersion of the mutual inductances between the 18 3rd stage sub-cables is not considered to simplify the calculation. If the influence of the non-uniformity of the induced voltages is a dominant cause of non-uniform current distribution, this supposition is reasonable. Ž3. The current distribution between the 27 Ž3 = 3 = 3. strands which make up the 3rd stage sub-cable is not discussed. Of course, it is possible to calculate the distribution in the other sub-cables and strands by the same analytical methods. The circuit equations for the electrical circuits shown in Fig. 2 are given in Eq. Ž1.. This equation describes the current distributions between the six 4th stage sub-cables in the final cable. The first subsections indicate the inductance matrixes of the coil element analyzed for non-uniform current distribution. The second subsections are the mutual inductances between the 4th stage sub-cables in the coil element and the other windings, and the third subsections are the self-inductances of the other windings. Using the assumptions mentioned above, Eq. Ž1. becomes Eq. Ž2. and describes the current distribution between the three 3rd stage sub-cables, where Vex is the excitation voltage and Vind is the inductive voltage. The inductance of the whole coil becomes 14.35 mH. The method of obtaining the mutual inductance between the 3rd stage sub-cables making one 4th stage sub-cable is as follows. This is calculated being considered as two coils which are close to each other. When one turn of radius a approaches another at a distance d , the mutual inductance is given by Eq. Ž3. w5x. The self-inductance of the 3rd stage sub-cables and the mutual inductance can be calculated from the results of Eq. Ž3.. When the distance d was assumed to be 1.6 mm, the inductances of the circulation current Ž L mm q L nn y
Fig. 2. Electrical circuit. Relation between Lmm q Lnn y 2 Mmn and Vind determines non-uniform current distribution.
K. Seo et al.r Physica C 310 (1998) 358–366
361
2 Mmn . between two adjacent 3rd stage sub-cables are ; 28.44 mH for the DP coil element and ; 10.07 mH for the SOL element. Here, m and n indicate the numbers of the 3rd stage sub-cables. As mentioned below, the inductance of the circulation current controls the magnitude of the non-uniform current. d i 01 dt d i 02 L01 M021 M031 M041 M051 M061
M012 L02 M032 M042 M052 M062
M013 M023 L03 M043 M053 M063
M014 M024 M034 L04 M054 M064
M015 M025 M035 M045 L05 M065
M016 M026 M036 M046 M056 L06
dt d i 03 dt d i 04
q 2 M0
d i0 dt
q L0
d i0 dt
s Vex ,
Ž 1.
dt d i 05 dt d i 06 dt
d i1 L11 M21 M31
M12 L 22 M32
dt d i2
M13 M23 L33
dt d i3
Vind1 q M0 q L0 s Vex q Vind2 , dt dt Vind3 d i0
d i0
Ž 2.
dt
ž
M s m 0 log
8a
d
/
y2 .
Ž 3.
Non-uniform currents are not thought to occur between the 4th stage sub-cables, but between the three 3rd stage sub-cables in this calculation. Thus, Eq. Ž2. can be represented as Eq. Ž4.. In Eq. Ž4., the magnitude of the non-uniform current is decided by both the induced voltage Vind m , which is caused by the difference between one 3rd stage sub-cable and the other windings, and L mm y Mmn , which is equivalent to a half of the inductance of the circulation current circuit.
Ž L11 y M21 . d i1rdt y Vind1 s Ž L22 y M12 . d i 2rd t y Vind2 , Ž L22 y M32 . d i 2rdt y Vind2 s Ž L33 y M23 . d i 3rd t y Vind3 , 6 Ž d i 1rdt q d i 2rd t q d i 3rdt . s d i 0rdt.
Ž 4.
The geometrical traces of the 3rd stage sub-cables Žtotal number 3 = 6 s 18. are calculated. By integral calculus of magnetic vector potential Au along the trace shown in Eq. Ž5., which is produced due to the unit
Fig. 3. Magnetic vector potential Ž rAu . in the winding and integral paths Žblack arrow for the DP winding, gray one for the SOL..
K. Seo et al.r Physica C 310 (1998) 358–366
362 Table 1 Calculation models Cabling pitch 5th stage Žmm.
4th stage Žmm.
Ža. 480 Žb. 480 Žc. 480 Žd. 480 Že. 480 Žf. 480
240 240=7r9 240 240=7r9 240 240
Winding methods
Deformation
Double pancake Double pancake Double pancake Double pancake Solenoid Solenoid
Non Non Deformed Deformed Non Deformed
current Ž1.0 A. energization of the other windings, the induced voltages can be calculated. Similar approaches have been found successful in Refs. w6,7x. Vind s
d i0 dt
HAu d l.
Ž 5.
The cable cross section supposed in the analysis is shown in Fig. 1. Here r5 is the cabling radius of the final 5th stage and r4 is the radius of the 4th stage. For the final stage, it was supposed that the elliptical orbit and the ratio of the major and the minor axes is 7:6. The traces of the sub-cables are determined by the superposition of the sinusoidal functions. The distribution of magnetic vector potential Au in position in the elemental coil winding is shown in Fig. 3. As shown in Eq. Ž5., the induced voltage is obtained by integrating the magnetic vector potential of the unit current energization along the geometric traces of the three 3rd stage sub-cables numerically.
Fig. 4. Traces of the three 3rd stage sub-cables. Ža. shows cases a and e, the geometrical trace of each sub-cable differs from the others. Žb. shows cases b and d.
K. Seo et al.r Physica C 310 (1998) 358–366
363
The models used in the calculations are shown in Table 1. The parameters are combination of the cabling pitches, winding method and deformation of the strand bundle. The traces of the three 3rd stage sub-cables with different combinations of cabling pitches are shown in Fig. 4. In the cases of a and e, because the 5th stage cabling pitch is double that of the 4th one, the 3rd stage sub-cables go around the same position periodically. Notice the offset of these three traces in the figure. The integral paths of the different winding methods are shown in Fig. 5. They show 16 turns for the DP winding and eight turns for the SOL winding. Compared with Fig. 3, when the sub-cables go along the SOL
Fig. 5. Traces of the three 3rd stage sub-cables, Ža. shows cases a to d and Žb. shows cases e and f.
364
K. Seo et al.r Physica C 310 (1998) 358–366
Fig. 6. Schematic illustration of strand bundle deformation and weight function. In case d, all sub-cables are pushed in the transverse direction equally.
winding, they are in a magnetic field with the same direction. On the contrary, for the DP winding, the direction changes between the inner and outer circumferences. The deformation of the cable was modeled as shown in Fig. 6. For general CICCs, they are formed into rectangular shapes after being stranded into final cables whose original cross-sectional shapes were round. As in Fig. 6, the weight function demonstrating the transverse compression was considered in the transverse direction. Electromagnetic forces also play a role in deforming the bundle of strands when the void fraction is large. If the geometrical center of the trace of the sub-cable has an offset in the transverse direction Žsee Fig. 4a., it drifts to the bundle’s center when the conductor was deformed into a rectangle.
Fig. 7. Induced voltages in 3 = 6 3rd stage sub-cables. DPs use the left scale and SOLs the right. Non-uniformity occurs when the strand bundle is deformed Žcases c and f. and the differences are less than 0.1%.
K. Seo et al.r Physica C 310 (1998) 358–366
365
Fig. 8. Current distribution in the three 3rd stage sub-cables. A non-uniform current occurs when the bundle is deformed and is solenoid-wound Žcase f..
In Fig. 4a, for the 3rd stage sub-cable 1, the geometric trace that inclines to the coil’s outer circumference is pushed to the inside periodically. For the 3rd stage sub-cables 2 and 3, whose traces incline to the inner circumference, they are pushed outwards.
3. Results and considerations The calculated induced voltages are shown in Fig. 7. The induced voltages for the DP winding are about 3.2 mV and those for the SOL about 1.0 mV. The largest dispersions are about 0.5 mV in both cases. The current distribution was calculated for an excitation rate of 1 Ars by Eq. Ž4.. Note that the non-uniform current behaviour has no relation to the current ramp rate at all, because this circuit model is a linear system and does not contain any resistance. The ratios of the currents of each of the 3rd stage sub-cables were calculated as shown in Fig. 8. The largest current is 60% larger than the cable average in case f. The effect of the combination of the cabling pitch was evaluated by comparing case c and d. When the sub-cable goes through the same trace periodically, a large non-uniform current occurs. Regarding the winding methods, if we compare cases c and f, non-uniformity is remarkably enlarged in the case of the SOL winding Žcase f.. In a pancake coil, the direction of the magnetic field which the conductor experiences is reversed between the inner and outer circumferences of the coil Žcf. Fig. 3.. Therefore in this case it is thought that not much dispersion of the inductance is caused, even if the geometrical trace of the sub-cable is offset from the center of the cable periodically. Regarding the deformation of the cable, comparing cases a and c and cases e and f, more non-uniformity occurs when the cable is deformed. When the 5th stage cabling pitch is not double the 4th one, the non-uniform current is not exaggerated in spite of the deformation Žsee case d.. The current distributions in the lower stage sub-cables also can be evaluated by doing similar analysis. For the lower stage sub-cables, the dispersion of the induced voltage by the other windings is small. However, this needs to be examined because the inductance of the circulation current circuit between such sub-cables is small too.
4. Conclusion When a circulation current is induced by magnetic flux between sub-cables, a non-uniform current occurs in multi-stage stranded cables. The most important consideration is to select a cabling pitch which prevents strands
K. Seo et al.r Physica C 310 (1998) 358–366
366
andror sub-cables periodically going around the same traces. When there is a trace offset, it is preferable not to deform the cross-sectional shape of the strand bundle and pancake winding is preferable to solenoid winding. For the essential prevention of non-uniform current distributions, these parameters need to be optimized.
References w1x w2x w3x w4x w5x w6x w7x
N. Koizumi, K. Okuno, Y. Takahashi, H. Tsuji, S. Shimamoto, Cryogenics 36 Ž6. Ž1996. 409. V. Vysotsky, M. Takayasu, M. Ferri, J.V. Minervini, IEEE Trans. On Applied Supercond. Ž1995. 580. K. Seo, M. Morita, M. Hasegawa, H. Yoshimura, Proceedings of MT-15, PF2-04 Ž1997. in China. S. Wolfram, Mathematica, Wolfram Research Ž1992.. K. Goto, S. Yamazaki, Electromagnetism practice, Ž1970. 284. C. Sihler, R. Heller, W. Maurer, A. Ulbricht, F. Weuchner, Applied Superconductivity, Elsevier, Vol. 4, No. 1r2, 1996, 95–110. M. Shimada, N. Mitchell, IEEE Trans. On Applied Supercond. Ž1997. 759.