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Relation between different critical current criteria and quench current in HTS magnets
Physica C 372–376 (2002) 1360–1363 www.elsevier.com/locate/physc
Relation between different critical current criteria and quench current in HTS magnets A. Korpela a
a,* ,
T. Kalliohaka a, J. Lehtonen a, R. Mikkonen a, J. Pitel b, P. Kov ac
b
Laboratory of Electromagnetics, Tampere University of Technology, P.O. Box 692, FIN-33101 Tampere, Finland b Institute of Electrical Engineering, Slovak Academy of Sciences, Dubravska 9, 842 39 Bratislava, Slovakia
Abstract In practice, the critical current of an HTS magnet is the current at which a thermal runaway (quench) occurs. The stability analysis required to determine the quench current, Iq is often a time consuming numerical problem. Usually the short sample critical current, Ic is measured by using some electric field criterion. In trained LTS magnets Iq can be estimated from Ic due to a steep electric field (E)–current density (J) characteristic. For HTS magnets the situation is more complicated due to the anisotropy and slanted EðJ Þ-characteristics. In this paper the theoretical maximum of Iq , Iq max of conduction cooled HTS magnets is computationally compared with different critical current criteria at 20 K. Computations are based on the Ic data measured with a Bi-2223/Ag tape. Two electric field criteria, 1 and 0.1 lV/cm, are applied to the magnets by investigating both the maximum electric field in a single turn of the coil and the voltage between the magnet poles. The critical currents obtained by these criteria are compared with Iq max in several coil geometries of a solenoidal conduction cooled HTS magnet having the wire length of 2, 5 and 10 km. Iq max is determined from the balance between the available cooling power and the total loss power generated inside the magnet. The objective is to enable a magnet designer to determine a safe operation current for an HTS magnet without performing a detailed stability analysis. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Critical current; HTS magnet
1. Introduction A thermal runaway current, Iq of a trained solenoidal LTS coil is typically around 80–90% of the short sample critical current, Ic [1]. This quite reliable estimation is possible due to a steep electric field (E)–current density (J) characteristic of
*
Corresponding author. Tel.: +358-3-365-2014; fax: +358-3365-2160. E-mail address: aki.korpela@tut.fi (A. Korpela).
an LTS wire. The situation is more complicated in HTS magnets. Due to a slanted EðJ Þ-characteristic and the material anisotropy, Iq of an HTS magnet is difficult to estimate from the short sample measurements. The computation of Iq for an HTS magnet requires a detailed stability analysis [2]. In order to have a quicker way to determine a safe operation current, Isafe the relation between the theoretical maximum of Iq , Iq max and different critical current criteria is investigated in this paper for several solenoidal conduction cooled HTS magnets
0921-4534/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 2 ) 0 1 0 2 9 - 8
A. Korpela et al. / Physica C 372–376 (2002) 1360–1363
at 20 K. The chosen temperature, T, provides an optimum for HTS magnets in the ratio of the stored energy and the power required for refrigeration [3].
2. Computational model The computations are based on the short sample data of the Bi-2223/Ag tape manufactured by American Superconductor in 1997. The power law is used to describe the EðJ Þ-characteristics of the tape: nðBÞ J E ¼ Ec ; ð1Þ Jc ðBÞ where n is the index number of the superconductor and (Jc ; Ec ) a point on the EðJ Þ-characteristics. The tape dimensions are 2:6 0:18 mm2 and 31.79% of the coil volume consists of epoxy resin [4]. The short sample data are presented in Fig. 1.
Fig. 1. Values of Ic and n for Bi-2223/Ag tape as a function of the magnitude and orientation of B at 20 K. The uppermost and undermost curves have been measured and they correspond to the parallel (0°) and perpendicular (90°) orientation of B to the broad surface of the tape, respectively. The rest of the data, from 10° to 80° with the interval of 10°, has been extrapolated according to the measurements at 4.2 K. The Ec criterion of 1 lV/cm was used in the measurements.
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Computationally investigated coils had a wire length, l, of 2, 5 and 10 km. In order to have an extensive scale of coil geometries, the number of layers (pancakes) in the coil, Nlr was varied from 2, 4, 6,. . . to 100 for each l. As the inner radius of the coils is fixed to 100 mm, the effect of the bending tensile strain on Jc can be neglected [5]. Two critical current criteria, minimum critical current, Ic min , and average critical current, Icave , are applied to the coils. Ic min is given by the lowest operation current, Iop at which the maximum electric field in the coil, Emax equals Ec [6]. Icave is determined by the voltage between the magnet poles, Vm . Icave is reached when PN Vm Ei li Eave ¼ ¼ i¼1 ¼ Ec ; ð2Þ l l where Eave is the average electric field, Ei and li are the electric field and the length of the turn i, respectively, and N is the number of turns in the coil. When Bi is known, Ei can be calculated from Eq. (1). Finite Element Method was used to calculate magnetic field distributions in the coil. Ic min and Icave are applied to the magnets using two Ec : 0.1 and 1 lV/cm. Thus, different Ic criteria are denoted as Ic min 01 , Ic min 1 , Icave 01 and Icave 1 . Two identical HTS coils with different geometries of the thermal interface may have a noteworthy difference in Iq . Therefore Iq max which is determined from the balance between the cooling power of the cryocooler, Pc and the total loss power inside the coil, Q, is used in this paper. Thus, Iop ¼ Iq max when Pc ¼ Q ¼ Vm Iop . According to Ref. [2], Iq corresponds to Q ¼ 0:68Pc for the solenoidal HTS coil with an inner radius of 50 mm, outer radius of 75 mm, axial length of 100 mm and a single thermal interface on the top surface of the magnet. The influence of the cooling conditions on the thermal runaway in HTS magnets is investigated, for example, in Ref. [7]. Pc used in this work is 30 W [8].
3. Results and discussion Fig. 2 presents Ic ðNlr Þ and Iq max ðNlr Þ in 2, 5 and 10 km coils. The perpendicular component of B to the broad surface of the tape, corresponding to the
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A. Korpela et al. / Physica C 372–376 (2002) 1360–1363
Fig. 2. Values of Ic and Iq max as a function Nlr for the investigated coil geometries at 20 K: (}), Ic min 01 ; (), Ic min 1 ; (þ), Icave 01 ; (), Icave 1 and ðOÞ; Iq max .
radial component in the coil, Br is the most significant factor when determining Ic and Iq max [6]. Emax is located near the coil flanges in the vicinity of the maximum Br , Br max . All Ic criteria are quite independent of l, if Nlr ¼ 2, because Br max ðlÞ is almost constant (0.160 T). Br dominates the values of Icave 01 and Icave 1 , because the majority of E is generated in the vicinity of Br max . Even though the average of B is increased for Iop ¼ Icave 1 from 0.293 T in 2 km coil to 0.324 T in 10 km coil, the change in distribution of E is barely noticeable. On the other hand, Iq max ðlÞ P is clearly decreasing when Nlr ¼ 2, because Vm ¼ Ei li . Ic ðlÞ-dependence becomes distinct with increasing Nlr . Furthermore, Ic ðNlr Þ and Iq max ðNlr Þ have flat minima due to changes in Br max . How-
Fig. 3. Total loss power, Q, generated inside the coil for different Ic criteria: (a) Iop ¼ Icave 1 , (b) Iop ¼ Icave 01 , (c) Iop ¼ Ic min 1 and (d) Iop ¼ Ic min 01 . The annotation is as follows: (}), l 2 km; (), l 5 km and (O), l 10 km. The dashed line corresponds to Q ¼ Pc ¼ 30 W.
ever, despite the considerable variation of Br max , Ic ðlÞ and Iq max ðlÞ dependencies remain moderate owing to small changes in Ic ðBÞ and nðBÞ above 0.5 T. Fig. 3 presents the steady-state values of Qðl; Nlr Þ for different Ic . As can be seen, Ic min 01 is too pessimistic and Icave 1 too optimistic in estimating Iq max corresponding to Q ¼ 30 W. Keeping in mind that Iq max is the theoretical maximum of Iq , Icave 01 seems to be a suitable choice for Isafe . The maximum of Q at Iop ¼ Icave 01 corresponds to 0:42Pc . Also Ic min 1 is a proper choice for Isafe , if Nlr P 10.
A. Korpela et al. / Physica C 372–376 (2002) 1360–1363
4. Conclusions In order to quickly estimate Isafe of an HTS magnet, a computational study of the relation between Iq max and different critical current criteria in conduction cooled HTS magnets was performed. In practice, Icave 01 and Icave 1 are directly obtained from the voltage–current measurement, if l is known. On the other hand, Ic min 01 and Ic min 1 are difficult to measure, but can be computationally determined from the short sample data and the distribution of B inside the coil. According to the results, Icave 01 is the most practical choice for Isafe at 20 K. Ic min 1 fits also when Nlr P 10, but is more complicated to determine. Furthermore, Ic
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criteria have flat minima with respect to the number of layers in the coil. References [1] M.N. Wilson, Superconducting Magnets, Oxford University Press, New York, 1983. [2] J. Lehtonen et al., Appl. Supercond. 1 (1999) 227–230. [3] R. Mikkonen et al., Appl. Supercond. 11 (2001) 1757–1760. [4] J. Paasi et al., Supercond. Sci. Technol. 13 (2000) 949–954. [5] P. Fabbricatore et al., Supercond. Sci. Technol. 11 (1998) 304–310. [6] J. Pitel et al., Supercond. Sci. Technol. 10 (1997) 847– 852. [7] Y. Lvovsky et al., Appl. Supercond. 11 (2001) 1840–1843. [8] Cryomech Inc., 113 Falso Drive, Syracuse, NY 13211, USA.
Relation between different critical current criteria and quench current in HTS magnets A. Korpela a
a,* ,
T. Kalliohaka a, J. Lehtonen a, R. Mikkonen a, J. Pitel b, P. Kov ac
b
Laboratory of Electromagnetics, Tampere University of Technology, P.O. Box 692, FIN-33101 Tampere, Finland b Institute of Electrical Engineering, Slovak Academy of Sciences, Dubravska 9, 842 39 Bratislava, Slovakia
Abstract In practice, the critical current of an HTS magnet is the current at which a thermal runaway (quench) occurs. The stability analysis required to determine the quench current, Iq is often a time consuming numerical problem. Usually the short sample critical current, Ic is measured by using some electric field criterion. In trained LTS magnets Iq can be estimated from Ic due to a steep electric field (E)–current density (J) characteristic. For HTS magnets the situation is more complicated due to the anisotropy and slanted EðJ Þ-characteristics. In this paper the theoretical maximum of Iq , Iq max of conduction cooled HTS magnets is computationally compared with different critical current criteria at 20 K. Computations are based on the Ic data measured with a Bi-2223/Ag tape. Two electric field criteria, 1 and 0.1 lV/cm, are applied to the magnets by investigating both the maximum electric field in a single turn of the coil and the voltage between the magnet poles. The critical currents obtained by these criteria are compared with Iq max in several coil geometries of a solenoidal conduction cooled HTS magnet having the wire length of 2, 5 and 10 km. Iq max is determined from the balance between the available cooling power and the total loss power generated inside the magnet. The objective is to enable a magnet designer to determine a safe operation current for an HTS magnet without performing a detailed stability analysis. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Critical current; HTS magnet
1. Introduction A thermal runaway current, Iq of a trained solenoidal LTS coil is typically around 80–90% of the short sample critical current, Ic [1]. This quite reliable estimation is possible due to a steep electric field (E)–current density (J) characteristic of
*
Corresponding author. Tel.: +358-3-365-2014; fax: +358-3365-2160. E-mail address: aki.korpela@tut.fi (A. Korpela).
an LTS wire. The situation is more complicated in HTS magnets. Due to a slanted EðJ Þ-characteristic and the material anisotropy, Iq of an HTS magnet is difficult to estimate from the short sample measurements. The computation of Iq for an HTS magnet requires a detailed stability analysis [2]. In order to have a quicker way to determine a safe operation current, Isafe the relation between the theoretical maximum of Iq , Iq max and different critical current criteria is investigated in this paper for several solenoidal conduction cooled HTS magnets
0921-4534/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 2 ) 0 1 0 2 9 - 8
A. Korpela et al. / Physica C 372–376 (2002) 1360–1363
at 20 K. The chosen temperature, T, provides an optimum for HTS magnets in the ratio of the stored energy and the power required for refrigeration [3].
2. Computational model The computations are based on the short sample data of the Bi-2223/Ag tape manufactured by American Superconductor in 1997. The power law is used to describe the EðJ Þ-characteristics of the tape: nðBÞ J E ¼ Ec ; ð1Þ Jc ðBÞ where n is the index number of the superconductor and (Jc ; Ec ) a point on the EðJ Þ-characteristics. The tape dimensions are 2:6 0:18 mm2 and 31.79% of the coil volume consists of epoxy resin [4]. The short sample data are presented in Fig. 1.
Fig. 1. Values of Ic and n for Bi-2223/Ag tape as a function of the magnitude and orientation of B at 20 K. The uppermost and undermost curves have been measured and they correspond to the parallel (0°) and perpendicular (90°) orientation of B to the broad surface of the tape, respectively. The rest of the data, from 10° to 80° with the interval of 10°, has been extrapolated according to the measurements at 4.2 K. The Ec criterion of 1 lV/cm was used in the measurements.
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Computationally investigated coils had a wire length, l, of 2, 5 and 10 km. In order to have an extensive scale of coil geometries, the number of layers (pancakes) in the coil, Nlr was varied from 2, 4, 6,. . . to 100 for each l. As the inner radius of the coils is fixed to 100 mm, the effect of the bending tensile strain on Jc can be neglected [5]. Two critical current criteria, minimum critical current, Ic min , and average critical current, Icave , are applied to the coils. Ic min is given by the lowest operation current, Iop at which the maximum electric field in the coil, Emax equals Ec [6]. Icave is determined by the voltage between the magnet poles, Vm . Icave is reached when PN Vm Ei li Eave ¼ ¼ i¼1 ¼ Ec ; ð2Þ l l where Eave is the average electric field, Ei and li are the electric field and the length of the turn i, respectively, and N is the number of turns in the coil. When Bi is known, Ei can be calculated from Eq. (1). Finite Element Method was used to calculate magnetic field distributions in the coil. Ic min and Icave are applied to the magnets using two Ec : 0.1 and 1 lV/cm. Thus, different Ic criteria are denoted as Ic min 01 , Ic min 1 , Icave 01 and Icave 1 . Two identical HTS coils with different geometries of the thermal interface may have a noteworthy difference in Iq . Therefore Iq max which is determined from the balance between the cooling power of the cryocooler, Pc and the total loss power inside the coil, Q, is used in this paper. Thus, Iop ¼ Iq max when Pc ¼ Q ¼ Vm Iop . According to Ref. [2], Iq corresponds to Q ¼ 0:68Pc for the solenoidal HTS coil with an inner radius of 50 mm, outer radius of 75 mm, axial length of 100 mm and a single thermal interface on the top surface of the magnet. The influence of the cooling conditions on the thermal runaway in HTS magnets is investigated, for example, in Ref. [7]. Pc used in this work is 30 W [8].
3. Results and discussion Fig. 2 presents Ic ðNlr Þ and Iq max ðNlr Þ in 2, 5 and 10 km coils. The perpendicular component of B to the broad surface of the tape, corresponding to the
1362
A. Korpela et al. / Physica C 372–376 (2002) 1360–1363
Fig. 2. Values of Ic and Iq max as a function Nlr for the investigated coil geometries at 20 K: (}), Ic min 01 ; (), Ic min 1 ; (þ), Icave 01 ; (), Icave 1 and ðOÞ; Iq max .
radial component in the coil, Br is the most significant factor when determining Ic and Iq max [6]. Emax is located near the coil flanges in the vicinity of the maximum Br , Br max . All Ic criteria are quite independent of l, if Nlr ¼ 2, because Br max ðlÞ is almost constant (0.160 T). Br dominates the values of Icave 01 and Icave 1 , because the majority of E is generated in the vicinity of Br max . Even though the average of B is increased for Iop ¼ Icave 1 from 0.293 T in 2 km coil to 0.324 T in 10 km coil, the change in distribution of E is barely noticeable. On the other hand, Iq max ðlÞ P is clearly decreasing when Nlr ¼ 2, because Vm ¼ Ei li . Ic ðlÞ-dependence becomes distinct with increasing Nlr . Furthermore, Ic ðNlr Þ and Iq max ðNlr Þ have flat minima due to changes in Br max . How-
Fig. 3. Total loss power, Q, generated inside the coil for different Ic criteria: (a) Iop ¼ Icave 1 , (b) Iop ¼ Icave 01 , (c) Iop ¼ Ic min 1 and (d) Iop ¼ Ic min 01 . The annotation is as follows: (}), l 2 km; (), l 5 km and (O), l 10 km. The dashed line corresponds to Q ¼ Pc ¼ 30 W.
ever, despite the considerable variation of Br max , Ic ðlÞ and Iq max ðlÞ dependencies remain moderate owing to small changes in Ic ðBÞ and nðBÞ above 0.5 T. Fig. 3 presents the steady-state values of Qðl; Nlr Þ for different Ic . As can be seen, Ic min 01 is too pessimistic and Icave 1 too optimistic in estimating Iq max corresponding to Q ¼ 30 W. Keeping in mind that Iq max is the theoretical maximum of Iq , Icave 01 seems to be a suitable choice for Isafe . The maximum of Q at Iop ¼ Icave 01 corresponds to 0:42Pc . Also Ic min 1 is a proper choice for Isafe , if Nlr P 10.
A. Korpela et al. / Physica C 372–376 (2002) 1360–1363
4. Conclusions In order to quickly estimate Isafe of an HTS magnet, a computational study of the relation between Iq max and different critical current criteria in conduction cooled HTS magnets was performed. In practice, Icave 01 and Icave 1 are directly obtained from the voltage–current measurement, if l is known. On the other hand, Ic min 01 and Ic min 1 are difficult to measure, but can be computationally determined from the short sample data and the distribution of B inside the coil. According to the results, Icave 01 is the most practical choice for Isafe at 20 K. Ic min 1 fits also when Nlr P 10, but is more complicated to determine. Furthermore, Ic
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criteria have flat minima with respect to the number of layers in the coil. References [1] M.N. Wilson, Superconducting Magnets, Oxford University Press, New York, 1983. [2] J. Lehtonen et al., Appl. Supercond. 1 (1999) 227–230. [3] R. Mikkonen et al., Appl. Supercond. 11 (2001) 1757–1760. [4] J. Paasi et al., Supercond. Sci. Technol. 13 (2000) 949–954. [5] P. Fabbricatore et al., Supercond. Sci. Technol. 11 (1998) 304–310. [6] J. Pitel et al., Supercond. Sci. Technol. 10 (1997) 847– 852. [7] Y. Lvovsky et al., Appl. Supercond. 11 (2001) 1840–1843. [8] Cryomech Inc., 113 Falso Drive, Syracuse, NY 13211, USA.