- Email: [email protected]
Feasibility of iron-sheathed MgB2 wires for magnet applications
Physica C 400 (2004) 89–96 www.elsevier.com/locate/physc
Feasibility of iron-sheathed MgB2 wires for magnet applications M. Ahoranta b
a,*
, J. Lehtonen a, P. Kov ac
b
a Institute of Electromagnetics, Tampere University of Technology, P.O. Box 692, 33 101 Tampere, Finland Institute of Electrical Engineering, Slovak Academy of Sciences, D ubravska cesta 9, 842 39 Bratislava, Slovakia
Received 6 March 2003; received in revised form 14 October 2003; accepted 17 October 2003
Abstract Although the development of MgB2 wires has been very intensive there is only a little information about their performance in magnet applications. In this study the possibility to use an iron-sheathed MgB2 wire in magnet applications is studied computationally. The calculations are based on the short sample critical current–magnetic flux density characteristics measured for a four-filament MgB2 /Fe wire. Maximum obtainable magnetic field, field homogeneity and stored energy are determined as a function of the coil dimensions. Magnetic field in coils is calculated with finite element method. The influence of ferromagnetic sheath material on magnet performance is clarified by examining coils wound of conductors with non-magnetic sheath. For comparison, the study is repeated with coils wound of NbTi wires. The first quantitative analysis about the performance of state-of-the-art MgB2 wires in magnet applications is presented and particulars for MgB2 magnet design are discussed. 2003 Elsevier B.V. All rights reserved. PACS: 85.25.Am; 85.25.Ly; 74.70.)b Keywords: MgB2 ; Magnet applications; Iron sheath
1. Introduction Since the discovery of the superconductivity of MgB2 at 39 K [1] lots of attention has been paid on the development of MgB2 wires and tapes suitable for magnet applications. The goal is to make conduction-cooled systems that operate at temperatures above 10 K and have low material costs. So far the self-field critical current density, Jc in the
*
Corresponding author. Tel.: +358-3-3115-2009; fax: +3583-3115-2160. E-mail address: maria.ahoranta@tut.fi (M. Ahoranta).
tapes has been 100–104 A/mm2 at 4.2 K [2]. The irreversibility field, Birr of pure MgB2 is as low as 7 T at 4.2 K but it can be increased significantly by doping. The most promising results have been obtained with thin films where alloying with oxygen increased the irreversibility field to 14 T at 4.2 K [3]. Also tape unit lengths up to 70 m have already been successfully fabricated [4]. There have been some studies on the performance of MgB2 coils. Tanaka et al. [5] and Grasso et al. [6] constructed solenoids from Ni-sheathed tape fabricated by powder in tube method. Soltanian et al. [7] manufactured small one layer solenoid coils of Cu-sheathed tape with wind and react
0921-4534/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.10.010
90
M. Ahoranta et al. / Physica C 400 (2004) 89–96
method. They noticed that Jc of the tape was almost unaffected by winding. At the moment, the best matrix material for MgB2 is considered to be iron [8–10]. The magnetization of the ferromagnetic material in MgB2 composite wire influences the measured critical current, Ic [11,12]. For example, Ic can depend on the orientation of the external magnetic field and on the magnetic history of the sample. MgB2 /Cu tapes without ferromagnetic sheath are found to be isotropic [13]. Furthermore, ferromagnetic sheath may increase hysteretic losses in AC applications. The design of MgB2 /Fe coils is more complicated compared to that of traditional LTS coils also because both the load line and the field shape are affected by the magnetization of the ferromagnetic winding. In this study the potential of the MgB2 /Fe ex situ wire material [14] for magnet applications is rated. The maximum obtainable central field, field homogeneity and energy stored in the magnetic field at operation temperatures of 4.2 and 10 K are examined as a function of the solenoidal coil dimensions. Results are compared with those obtained at 4.2 K with NbTi, which is currently the most widely used LTS material in low field magnet applications. Because rapid development is expected in the MgB2 technology the sensitivity of the magnet performance to improvements in irreversibility field and self-field critical current is also considered. Computations are made with finite element method (FEM). Only DC coils are studied here and therefore AC losses are not discussed. In actual design work stability and mechanical durability of the magnet must be considered. Regardless of their importance these factors are only briefly mentioned here.
2. Computational model The coil had a composite structure consisting of superconductor, sheath, insulation and impregnation material. The composite structure did not cause any problems in NbTi coils because all constituent materials had permeability, l, equal to the vacuum permeability, l0 . However, in MgB2 coils the iron sheath got strongly magnetized [11,13] and an accurate modeling of this composite
Fig. 1. Model used in estimation of engineering permeability of magnet material.
structure was impossible in the magnetic field computations. Therefore, the coil material was described as a homogeneous material with the engineering permeability leng ðBÞ that was calculated from the unit cell model shown in Fig. 1. The cell consisted of cross-section of one wire, insulation and epoxy. l ¼ l0 was assumed for superconductor, insulation and impregnation material. The permeability of iron was obtained from the measured virgin curve shown in the inset of Fig. 2 [12]. In computations the current density of the conductor was zero. The magnetic flux per unit length, / ¼ BLy , where Ly is the height of the unit cell, was set to flow through the cell in order to define the value of external magnetic flux density B. Owing to symmetry reasons there was no magnetic flux through edges AB and CD. On edges BC and DA, the magnetic flux was normal to the edge. In terms of the z-component of the vector potential, Az these boundary conditions were given as shown in Fig. 1. After the magnetic field inside the unit cell was solved leng was obtained from R Ly 1 Bx dy Ly 0 ; leng ¼ R Lx 1 Hx dy Lx 0 where Lx is the length of unit cell and Bx and Hx the x-components of magnetic flux density and magnetic field, respectively. The leng ðBÞ was computed at 1 mT 6 B 6 4 T and then extrapolated to 0 T
M. Ahoranta et al. / Physica C 400 (2004) 89–96
91
by the maximum magnetic flux density, Bmax ¼ maxðBðr; zÞÞ, in the coil because MgB2 material has typically large n-values [2]. Then, the error, e ¼ Jguess Jc ðBmax Þ, was calculated and Jguess was updated as iþ1 i ¼ Jguess e=4; Jguess
where i is the iteration index. The procedure was repeated until jej 6 0:025 Jguess . Calculations were performed with Matlab and FEMLAB softwares. The homogeneity of magnetic field inside the magnet bore, d ¼ maxðjB Bave j=Bave Þ;
Fig. 2. Relative permeability of magnet material as obtained from unit cell model. Inset shows relative permeability for iron.
was calculated in the spheres with center in the point (0,0) and with the radius, R, ranging from 2.5 mm to the inner radius of the coil, ri . Here, Bave is the average magnetic flux density inside the sphere. The energy stored in the magnetic field of the coil was computed from Z E¼p rA J ds; Sc
where A is the vector potential and Sc the coil cross-section.
3. Results and discussion
Fig. 3. Model used in magnet calculations. The air region is suppressed for illustrative reasons.
and to infinity where leng ð1Þ ! l0 as shown in Fig. 2. Axisymmetric 2d model for the magnetic field computations in the coil is shown in Fig. 3. In order to obtain the critical current density of the magnet the homogeneous current density Jguess was set to the coil and the resulting Bðr; zÞ was calculated with FEM. The finite element mesh consisted of 2600–3900 linear triangular elements. The critical current density of the magnet was determined
Here magnets wound of three kinds of conductor were studied. First, the performance of coils wound of iron-sheathed four-filament MgB2 tape with dimensions of 1.17 mm · 1.17 mm was studied. The properties of the MgB2 tape corresponded to a sample described in [14,15]. The measured short sample critical current is shown in Fig. 4. Because the measured data for the MgB2 tape was limited in a small range of B an extrapolation was made. It was assumed that log10 ðIc Þ ¼ aB2 þ bB þ c;
ð1Þ
where the constants a, b and c were determined with the least squares method. The shape of the extrapolation curve at 4.2 K is in good accordance with measurements but it can yield pessimistic values at 10 K [4,16,17]. Second, conductor with non-magnetic sheath was examined in order to
92
M. Ahoranta et al. / Physica C 400 (2004) 89–96
Fig. 4. Short sample critical current of iron-sheathed MgB2 tape. Symbols show measured data and solid lines correspond to extrapolation according to Eq. (1).
clarify the effect of ferromagnetic sheath material on magnet performance. Critical current was similar to the iron-sheathed MgB2 tape. In practice, the iron sheath shields the superconductor from the external field. Therefore, an MgB2 tape with non-magnetic sheath would have lower critical current than an iron sheathed one even if the properties of superconductor were equal. Finally, the study was repeated with NbTi coils. In the following these three types of coils are referred as ‘‘ferromagnetic MgB2 ’’, ‘‘non-magnetic MgB2 ’’ and ‘‘NbTi’’ coils, respectively. It was estimated that 90% of the coil consisted of the insulated wire. The filling factor corresponds to the 100 lm thick insulation which can be used in MgB2 coils fabricated with wind and react method. Then, the engineering critical current density was obtained as Jc ¼ 0:9Ic =Aw where Aw ¼ 1:88 mm2 was the cross-sectional area of the insulated wire. Data of £ 0.648 mm wire produced by Outokumpu Advanced Superconductors Ltd. were used for NbTi coil. The critical current as a function of the external magnetic flux density was well characterized with Ic ¼ 4:72B2 133B þ 901A. The 10 lm thick insulation on a round wire
leads to the filling factor of 78.5%. Then, the engineering critical current density was obtained as Jc ¼ 0:785Ic =Aw where Aw ¼ 0:35 mm2 . Critical current density, central magnetic flux density, homogeneity and energy were calculated using several dimensions of solenoid coils. For small bore coils (ri ¼ 25 mm) the outer radius, ro ranged from 35 to 125 mm and for wide bore coils (ri ¼ 50 mm) ro was between 70 and 160 mm. In each case, coils with axial lengths of h ¼ 60, 120 and 240 mm were studied. Engineering critical current densities of the ferromagnetic MgB2 magnets varied from 29 to 150 A/mm2 at 4.2 K and from 22 to 68 A/mm2 at 10 K. As expected, the critical current density decreased with decreasing ri and increasing ro and h. The central magnetic flux density, B0 as a function of coil outer radius at both operation temperatures is shown in Fig. 5. The highest achieved central field value was 3.8 T at 4.2 K with the coil dimensions ri ¼ 25 mm, ro ¼ 125 mm and h ¼ 24 mm. The result was in a good agreement with the previously presented overall critical current density limit of 100 A/mm2 [14]. The NbTi coil with same dimensions would reach B0 ¼ 10 T at 4.2 K. However, the mechanical stresses due to Lorentz forces would be unbearable in such a magnet. In general, 2.6–3.5 times higher central field values were achieved with NbTi than with MgB2 at 4.2 K. Thus, MgB2 wire was not competitive when high fields are needed. When the operation temperature increased the performance of MgB2 tape deteriorated further. The maximum achieved B0 was below 3 T at 10 K. The magnetization of ferromagnetic windings increases the maximum field in the magnet and therefore decreases Jc . On the other hand, it also helps to reach higher B0 values with given current density. Fig. 6 shows the J ðBmax Þ- and J ðB0 Þ-lines for ferromagnetic and non-magnetic MgB2 coil, with dimensions ri ¼ 25 mm, ro ¼ 85 mm, h ¼ 120 mm and the engineering Jc ðBÞ curves at different operation temperatures. For example, at 4.2 K the critical current density of the ferromagnetic MgB2 coil was 47 A/mm2 with Bmax ¼ 3:9 T and B0 ¼ 3:2 T. In the non-magnetic MgB2 coil the critical current density rose to 61 A/mm2 . However, the increase in B0 was only 0.2 T. When the same current density was applied for both coils the fer-
M. Ahoranta et al. / Physica C 400 (2004) 89–96
93
Fig. 6. Comparison between performances of ferromagnetic (stars) and non-magnetic (circles) coils. Solid lines present engineering critical current density at different temperatures. Dashed lines show Jc ðBmax Þ- (open symbols) and Jc ðB0 Þ-curves (filled symbols). Arrows show central magnetic flux densities that correspond to critical current densities at 4.2 and 10 K.
Fig. 5. Central magnetic flux density as a function of coil outer radius at operation temperatures: (a) T ¼ 4:2 K and (b) T ¼ 10 K. Solid line corresponds to the coil with ri ¼ 25 mm and dashed line to the coil with ri ¼ 50 mm. Axial lengths of the coils are 6 cm (star), 12 cm (circle) and 24 cm (diamond).
romagnetic one produced a higher central field. Magnetic field distributions in both coils operating at their Jc are shown in Fig. 7. The high field values are visible in the ferromagnetic winding. However, difference between the magnetic flux densities inside the bores was not so large. It has to be remembered that the Ic data used was measured with the iron-sheathed wire and was not a property of pure MgB2 material. The magnetic shielding increases Ic especially at low magnetic fields [13].
Thus, the removal of the ferromagnetic sheath modifies the Ic ðBÞ-curve even if the material properties of superconductor remain the same. As a whole, the effect of the magnetized sheath on B0 was quite small and it could even be positive. Because at the moment MgB2 wires are under rapid development attention must be also paid on the sensitivity of computed results to changes in Jc ðBÞ. The magnet performance with different Ic ðBÞ-data can also be illustrated with Fig. 6. Two general observations can be made. First, in nonmagnetic coils the magnet load lines were steeper compared to ferromagnetic coils. Therefore, B0 can become higher in the ferromagnetic coil than in the non-magnetic one if dIc =dB is decreased. Second, the J ðBmax Þ- and J ðB0 Þ-curves were almost linear at B > 2 T but the slope of J ðBmax Þ-line was slightly smaller than that of J ðB0 Þ-line. Thus, the difference between Bmax and B0 increases when the current density in the magnet rises. Finally, changes in self-field critical current and irreversibility field were simulated. Because at Birr Eq. (1) was not valid changes in the irreversibility field were
94
M. Ahoranta et al. / Physica C 400 (2004) 89–96 Table 1 Effect of changes in Ic ðBÞ of the MgB2 wire on the critical current density and central magnetic field of the magnet, ri ¼ 25 mm, ro ¼ 85 mm, h ¼ 120 mm, at 4.2 K Increase in Ic (0) (%)
Increase in B0 (%)
10 None 10
None 10 10
Increase in Jc (%) 2.4 5.8 8.6
Increase in B0 (%) 2.2 5.0 7.5
The stored energy in a coil can be expressed as Z E ¼ 1=2 B H dv ¼ 1=2LI 2 ; V
where V is the volume of the whole space, L the inductance and I the operation current of the coil. In ferromagnetic coils the magnetization increased the stored energy due to increased B. Inductance of magnetic coils was also dependent on the operation current as shown in Fig. 8. The inductance
Fig. 7. Magnetic flux density distributions in: (a) ferromagnetic and (b) non-magnetic MgB2 coils working at their critical current.
described with changes in parameter B1A defined as aB21A þ bB1A þ c ¼ 0 i.e. Ic ðB1A Þ ¼ 1A. Then, a and c in Eq. (1) were modified in order to simulate the changes in Ic (0 T) and B1A , b was kept constant. Table 1 lists the effects of 10% increase in Ic (0 T) and B1A on Jc and B0 of the ferromagnetic coil at 4.2 K. The increase in B0 was clearly more effective than that of Ic (0 T).
Fig. 8. Normalized magnet inductance, L (ferromagnetic)/L (non-magnetic), as a function of operation current for certain coils. Dimensions of the coils are: (1) h ¼ 6 cm, ri ¼ 50 mm, ro ¼ 160 mm (circle), (2) h ¼ 6 cm, ri ¼ 25 mm, ro ¼ 125 mm (star), (3) h ¼ 24 cm, ri ¼ 50 mm, ro ¼ 70 mm (square) and (4) h ¼ 24 cm, ri ¼ 25 mm, ro ¼ 35 mm (diamond). The inductance for corresponding non-magnetic coils are (1) 1.5 H, (2) 0.75 H, (3) 0.23 H and (4) 0.16 H.
M. Ahoranta et al. / Physica C 400 (2004) 89–96
was largest on small currents but decreased as the winding got saturated. Energies of ferromagnetic
Fig. 9. Stored energy as a function of coil dimensions for ferromagnetic MgB2 coils operating on their critical current densities. Coil dimensions are: (1) h ¼ 6 cm, ri ¼ 25 mm (stars), (2) h ¼ 24 cm, ri ¼ 25 mm (circle), (3) h ¼ 6 cm, ri ¼ 50 mm (diamond) and (4) h ¼ 24 cm, ri ¼ 50 mm (square).
Fig. 10. Homogeneity of magnetic field in a sphere of radius 1.8 cm. Filled symbols are for MgB2 coils operating on their critical current density at 4.2 K. Open symbols show the homogeneity in corresponding non-ferromagnetic coils. Coil dimensions are: (1) h ¼ 6 cm, ri ¼ 25 mm (stars), (2) h ¼ 24 cm, ri ¼ 25 mm (circle), (3) h ¼ 6 cm, ri ¼ 50 mm (diamond) and (4) h ¼ 24 cm, ri ¼ 50 mm (square).
95
MgB2 coils as a function of coil dimensions are shown in Fig. 9 for coils operating at the critical current. In ferromagnetic MgB2 coils L was 1.2–1.7 times higher than in non-magnetic coils when both magnets were working at the critical current density of the ferromagnetic magnet. However, in non-magnetic coils the critical current was larger. A high coil with wide bore was preferential for large energy. The highest obtainable energy with ferromagnetic MgB2 coils was 32 kJ at 4.2 K in the magnet with ri ¼ 50 mm, ro ¼ 160 mm and h ¼ 24 mm. In very thin coils, ro ri 6 20 mm, energy was larger in the ferromagnetic MgB2 coil at 4.2 K than in the NbTi coil owing to the strong magnetization. However, in other coils up to nine times higher energies were achieved with NbTi at 4.2 K. Fig. 10 shows the homogeneity inside the sphere with R 1:8 cm for non-magnetic and ferromagnetic MgB2 coils. The magnetized winding changed the field distribution and homogeneity inside the coil bore. The most homogeneous field was obtained with long and thin coils with small bore.
Fig. 11. Normalized field homogeneity, d (ferromagnetic coil)/d (non-magnetic coil), as a function of coil current density in a sphere of radius 1.8 cm for some magnet geometries. Dimensions of the coils are: (1) h ¼ 6 cm, ri ¼ 50 mm, ro ¼ 160 mm (circles), (2) h ¼ 6 cm, ri ¼ 25 mm, ro ¼ 125 mm (stars), (3) h ¼ 24 cm, ri ¼ 50 mm, ro ¼ 70 mm (squares) and (4) h ¼ 24 cm, ri ¼ 25 mm, ro ¼ 45 mm (diamonds). The homogeneities for corresponding non-magnetic coils are: (1) 0.50%, (2) 7.4%, (3) 0.52% and (4) 0.22%.
96
M. Ahoranta et al. / Physica C 400 (2004) 89–96
The best homogeneities in the ferromagnetic and non-magnetic MgB2 coils were 0.10%, and 0.17%, respectively. However, for most of the coils magnetization of the sheath reduced the field homogeneity. Homogeneity as a function of current density is shown in Fig. 11 for ferromagnetic and non-magnetic coils. For non-magnetic coils the field homogeneity depended only on the coil geometry and not on the current density. When ironsheathed MgB2 wire was applied the magnetic field was changed according to the saturation state of the winding. However, the dependence of the field homogeneity on the operation current was largest for short coils where the homogeneity was poor in any case.
4. Conclusions Performance of solenoidal magnets made of iron sheathed ex situ MgB2 wire was studied computationally. The maximum achieved central field with MgB2 coils was 3.8 T at 4.2 K and 3 T at 10 K. Magnets manufactured of commercial NbTi wires were able to produce 2.6–3.4 times higher central fields. Also almost one order of magnitude higher energy was reached with NbTi coils. At constant current density the magnetization of iron sheath in the tape could increase the central field in the magnet bore. Although MgB2 is cheap improvements in the current carrying capacity are still needed before it becomes attractive for conduction cooled low field applications. The magnets are more complicated to design from ironsheathed MgB2 conductors than from traditional LTS wires. The ferromagnetic sheath makes both the field homogeneity inside the bore and the coil inductance dependent on the operation current of the magnet. The magnetization of iron increases the coil inductance. On the other hand, the maximum magnetic flux density in the coil increases and thereby the critical current is reduced. Altogether, the maximum achievable central field and stored energy are almost independent of the sheath material of the MgB2 conductor.
Acknowledgements This research is supported by Ulla Tuominen Foundation, Academy of Finland project 52434 and the Grant Agency of the Slovak Academy of Sciences VEGA 2/2069/22. References [1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, Nature 410 (2001) 63. [2] R. Fl€ ukiger, H.L. Suo, N. Musolino, C. Benecude, P. Toulemonde, P. Lezza, Physica C 385 (2003) 286. [3] C.B. Eom, M.K. Lee, J.H. Choi, L.J. Belenky, X. Song, L.D. Cooley, M.T. Naus, S. Patnaik, J. Jiang, M. Rikel, A. Polyanskii, A. Gurevich, X.Y. Cai, S.D. Bu, S.E. Babcock, E.E. Hellstrom, D.C. Larbalestier, N. Rogado, K.A. Regan, M.A. Hayward, T. He, J.S. Slusky, K. Inumaru, M.K. Haas, R.J. Cava, Nature 411 (2001) 558. [4] E.W. Collings, E. Lee, M.D. Sumption, M. Tomsic, X.L. Wang, S. Soltanian, S.X. Dou, Physica C 386 (2003) 555. [5] K. Tanaka, M. Okada, H. Kumakura, H. Kitaguchi, K. Togano, Physica C 382 (2002) 203. [6] G. Grasso, A. Malagoli, M. Modica, A. Tumino, C. Ferdeghini, A.S. Siri, C. Vignola, L. Martini, V. Previtali, G. Volpini, Supercond. Sci. Technol. 16 (2003) 271. [7] S. Soltanian, J. Horvat, X.L. Wang, M. Tomsic, S.X. Dou, cond-mat/0205406. [8] H.L. Suo, C. Beneduce, M. Dhalle, N. Musolino, J.Y. Genoud, R. Fl€ ukiger, Appl. Phys. Lett. 79 (2001) 3116. [9] S. Jin, H. Mavoori, C. Bower, R.B. van Dover, Nature 411 (2001) 563. [10] Y. Feng, G. Yan, Y. Zhao, C.F. Liu, B.Q. Fu, L. Zhou, L.Z. Cao, K.Q. Ruan, X.G. Li, L. Shi, Y.H. Zhang, Physica C 386 (2003) 598. [11] P. Kovac, K. Henze, T. Melisek, I. Husek, H. Kirchmayr, Supercond. Sci. Technol. 15 (2002) 1037. [12] P. Kovac, M. Ahoranta, T. Melisek, J. Lehtonen, I. Husek, Supercond. Sci. Technol. 16 (2003) 793. [13] P. Kovac, M. Ahoranta, J. Lehtonen, I. Husek, Physica C 397 (2003) 14. [14] P. Kovac, I. Husek, T. Melisek, Supercond. Sci. Technol. 15 (2002) 1340. [15] P. Fabbricatore, M. Greco, R. Musenich, P. Kovac, I. Husek, F. G€ om€ ory, Supercond. Sci. Technol. 16 (2003) 364. [16] M. Dhalle, P. Toulemonde, C. Beneduce, N. Musolino, M. Decroux, R. Fl€ ukiger, Physica C 363 (2001) 155. [17] K.Q. Ruan, H.L. Li, Y. Yu, C.Y. Wang, L.Z. Cao, C.F. Liu, S.J. Du, G. Yan, Y. Feng, X. Wu, J.R. Wang, X.H. Liu, P.X. Zhang, X.Z. Wu, L. Zhou, Physica C 386 (2003) 578.
Feasibility of iron-sheathed MgB2 wires for magnet applications M. Ahoranta b
a,*
, J. Lehtonen a, P. Kov ac
b
a Institute of Electromagnetics, Tampere University of Technology, P.O. Box 692, 33 101 Tampere, Finland Institute of Electrical Engineering, Slovak Academy of Sciences, D ubravska cesta 9, 842 39 Bratislava, Slovakia
Received 6 March 2003; received in revised form 14 October 2003; accepted 17 October 2003
Abstract Although the development of MgB2 wires has been very intensive there is only a little information about their performance in magnet applications. In this study the possibility to use an iron-sheathed MgB2 wire in magnet applications is studied computationally. The calculations are based on the short sample critical current–magnetic flux density characteristics measured for a four-filament MgB2 /Fe wire. Maximum obtainable magnetic field, field homogeneity and stored energy are determined as a function of the coil dimensions. Magnetic field in coils is calculated with finite element method. The influence of ferromagnetic sheath material on magnet performance is clarified by examining coils wound of conductors with non-magnetic sheath. For comparison, the study is repeated with coils wound of NbTi wires. The first quantitative analysis about the performance of state-of-the-art MgB2 wires in magnet applications is presented and particulars for MgB2 magnet design are discussed. 2003 Elsevier B.V. All rights reserved. PACS: 85.25.Am; 85.25.Ly; 74.70.)b Keywords: MgB2 ; Magnet applications; Iron sheath
1. Introduction Since the discovery of the superconductivity of MgB2 at 39 K [1] lots of attention has been paid on the development of MgB2 wires and tapes suitable for magnet applications. The goal is to make conduction-cooled systems that operate at temperatures above 10 K and have low material costs. So far the self-field critical current density, Jc in the
*
Corresponding author. Tel.: +358-3-3115-2009; fax: +3583-3115-2160. E-mail address: maria.ahoranta@tut.fi (M. Ahoranta).
tapes has been 100–104 A/mm2 at 4.2 K [2]. The irreversibility field, Birr of pure MgB2 is as low as 7 T at 4.2 K but it can be increased significantly by doping. The most promising results have been obtained with thin films where alloying with oxygen increased the irreversibility field to 14 T at 4.2 K [3]. Also tape unit lengths up to 70 m have already been successfully fabricated [4]. There have been some studies on the performance of MgB2 coils. Tanaka et al. [5] and Grasso et al. [6] constructed solenoids from Ni-sheathed tape fabricated by powder in tube method. Soltanian et al. [7] manufactured small one layer solenoid coils of Cu-sheathed tape with wind and react
0921-4534/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.10.010
90
M. Ahoranta et al. / Physica C 400 (2004) 89–96
method. They noticed that Jc of the tape was almost unaffected by winding. At the moment, the best matrix material for MgB2 is considered to be iron [8–10]. The magnetization of the ferromagnetic material in MgB2 composite wire influences the measured critical current, Ic [11,12]. For example, Ic can depend on the orientation of the external magnetic field and on the magnetic history of the sample. MgB2 /Cu tapes without ferromagnetic sheath are found to be isotropic [13]. Furthermore, ferromagnetic sheath may increase hysteretic losses in AC applications. The design of MgB2 /Fe coils is more complicated compared to that of traditional LTS coils also because both the load line and the field shape are affected by the magnetization of the ferromagnetic winding. In this study the potential of the MgB2 /Fe ex situ wire material [14] for magnet applications is rated. The maximum obtainable central field, field homogeneity and energy stored in the magnetic field at operation temperatures of 4.2 and 10 K are examined as a function of the solenoidal coil dimensions. Results are compared with those obtained at 4.2 K with NbTi, which is currently the most widely used LTS material in low field magnet applications. Because rapid development is expected in the MgB2 technology the sensitivity of the magnet performance to improvements in irreversibility field and self-field critical current is also considered. Computations are made with finite element method (FEM). Only DC coils are studied here and therefore AC losses are not discussed. In actual design work stability and mechanical durability of the magnet must be considered. Regardless of their importance these factors are only briefly mentioned here.
2. Computational model The coil had a composite structure consisting of superconductor, sheath, insulation and impregnation material. The composite structure did not cause any problems in NbTi coils because all constituent materials had permeability, l, equal to the vacuum permeability, l0 . However, in MgB2 coils the iron sheath got strongly magnetized [11,13] and an accurate modeling of this composite
Fig. 1. Model used in estimation of engineering permeability of magnet material.
structure was impossible in the magnetic field computations. Therefore, the coil material was described as a homogeneous material with the engineering permeability leng ðBÞ that was calculated from the unit cell model shown in Fig. 1. The cell consisted of cross-section of one wire, insulation and epoxy. l ¼ l0 was assumed for superconductor, insulation and impregnation material. The permeability of iron was obtained from the measured virgin curve shown in the inset of Fig. 2 [12]. In computations the current density of the conductor was zero. The magnetic flux per unit length, / ¼ BLy , where Ly is the height of the unit cell, was set to flow through the cell in order to define the value of external magnetic flux density B. Owing to symmetry reasons there was no magnetic flux through edges AB and CD. On edges BC and DA, the magnetic flux was normal to the edge. In terms of the z-component of the vector potential, Az these boundary conditions were given as shown in Fig. 1. After the magnetic field inside the unit cell was solved leng was obtained from R Ly 1 Bx dy Ly 0 ; leng ¼ R Lx 1 Hx dy Lx 0 where Lx is the length of unit cell and Bx and Hx the x-components of magnetic flux density and magnetic field, respectively. The leng ðBÞ was computed at 1 mT 6 B 6 4 T and then extrapolated to 0 T
M. Ahoranta et al. / Physica C 400 (2004) 89–96
91
by the maximum magnetic flux density, Bmax ¼ maxðBðr; zÞÞ, in the coil because MgB2 material has typically large n-values [2]. Then, the error, e ¼ Jguess Jc ðBmax Þ, was calculated and Jguess was updated as iþ1 i ¼ Jguess e=4; Jguess
where i is the iteration index. The procedure was repeated until jej 6 0:025 Jguess . Calculations were performed with Matlab and FEMLAB softwares. The homogeneity of magnetic field inside the magnet bore, d ¼ maxðjB Bave j=Bave Þ;
Fig. 2. Relative permeability of magnet material as obtained from unit cell model. Inset shows relative permeability for iron.
was calculated in the spheres with center in the point (0,0) and with the radius, R, ranging from 2.5 mm to the inner radius of the coil, ri . Here, Bave is the average magnetic flux density inside the sphere. The energy stored in the magnetic field of the coil was computed from Z E¼p rA J ds; Sc
where A is the vector potential and Sc the coil cross-section.
3. Results and discussion
Fig. 3. Model used in magnet calculations. The air region is suppressed for illustrative reasons.
and to infinity where leng ð1Þ ! l0 as shown in Fig. 2. Axisymmetric 2d model for the magnetic field computations in the coil is shown in Fig. 3. In order to obtain the critical current density of the magnet the homogeneous current density Jguess was set to the coil and the resulting Bðr; zÞ was calculated with FEM. The finite element mesh consisted of 2600–3900 linear triangular elements. The critical current density of the magnet was determined
Here magnets wound of three kinds of conductor were studied. First, the performance of coils wound of iron-sheathed four-filament MgB2 tape with dimensions of 1.17 mm · 1.17 mm was studied. The properties of the MgB2 tape corresponded to a sample described in [14,15]. The measured short sample critical current is shown in Fig. 4. Because the measured data for the MgB2 tape was limited in a small range of B an extrapolation was made. It was assumed that log10 ðIc Þ ¼ aB2 þ bB þ c;
ð1Þ
where the constants a, b and c were determined with the least squares method. The shape of the extrapolation curve at 4.2 K is in good accordance with measurements but it can yield pessimistic values at 10 K [4,16,17]. Second, conductor with non-magnetic sheath was examined in order to
92
M. Ahoranta et al. / Physica C 400 (2004) 89–96
Fig. 4. Short sample critical current of iron-sheathed MgB2 tape. Symbols show measured data and solid lines correspond to extrapolation according to Eq. (1).
clarify the effect of ferromagnetic sheath material on magnet performance. Critical current was similar to the iron-sheathed MgB2 tape. In practice, the iron sheath shields the superconductor from the external field. Therefore, an MgB2 tape with non-magnetic sheath would have lower critical current than an iron sheathed one even if the properties of superconductor were equal. Finally, the study was repeated with NbTi coils. In the following these three types of coils are referred as ‘‘ferromagnetic MgB2 ’’, ‘‘non-magnetic MgB2 ’’ and ‘‘NbTi’’ coils, respectively. It was estimated that 90% of the coil consisted of the insulated wire. The filling factor corresponds to the 100 lm thick insulation which can be used in MgB2 coils fabricated with wind and react method. Then, the engineering critical current density was obtained as Jc ¼ 0:9Ic =Aw where Aw ¼ 1:88 mm2 was the cross-sectional area of the insulated wire. Data of £ 0.648 mm wire produced by Outokumpu Advanced Superconductors Ltd. were used for NbTi coil. The critical current as a function of the external magnetic flux density was well characterized with Ic ¼ 4:72B2 133B þ 901A. The 10 lm thick insulation on a round wire
leads to the filling factor of 78.5%. Then, the engineering critical current density was obtained as Jc ¼ 0:785Ic =Aw where Aw ¼ 0:35 mm2 . Critical current density, central magnetic flux density, homogeneity and energy were calculated using several dimensions of solenoid coils. For small bore coils (ri ¼ 25 mm) the outer radius, ro ranged from 35 to 125 mm and for wide bore coils (ri ¼ 50 mm) ro was between 70 and 160 mm. In each case, coils with axial lengths of h ¼ 60, 120 and 240 mm were studied. Engineering critical current densities of the ferromagnetic MgB2 magnets varied from 29 to 150 A/mm2 at 4.2 K and from 22 to 68 A/mm2 at 10 K. As expected, the critical current density decreased with decreasing ri and increasing ro and h. The central magnetic flux density, B0 as a function of coil outer radius at both operation temperatures is shown in Fig. 5. The highest achieved central field value was 3.8 T at 4.2 K with the coil dimensions ri ¼ 25 mm, ro ¼ 125 mm and h ¼ 24 mm. The result was in a good agreement with the previously presented overall critical current density limit of 100 A/mm2 [14]. The NbTi coil with same dimensions would reach B0 ¼ 10 T at 4.2 K. However, the mechanical stresses due to Lorentz forces would be unbearable in such a magnet. In general, 2.6–3.5 times higher central field values were achieved with NbTi than with MgB2 at 4.2 K. Thus, MgB2 wire was not competitive when high fields are needed. When the operation temperature increased the performance of MgB2 tape deteriorated further. The maximum achieved B0 was below 3 T at 10 K. The magnetization of ferromagnetic windings increases the maximum field in the magnet and therefore decreases Jc . On the other hand, it also helps to reach higher B0 values with given current density. Fig. 6 shows the J ðBmax Þ- and J ðB0 Þ-lines for ferromagnetic and non-magnetic MgB2 coil, with dimensions ri ¼ 25 mm, ro ¼ 85 mm, h ¼ 120 mm and the engineering Jc ðBÞ curves at different operation temperatures. For example, at 4.2 K the critical current density of the ferromagnetic MgB2 coil was 47 A/mm2 with Bmax ¼ 3:9 T and B0 ¼ 3:2 T. In the non-magnetic MgB2 coil the critical current density rose to 61 A/mm2 . However, the increase in B0 was only 0.2 T. When the same current density was applied for both coils the fer-
M. Ahoranta et al. / Physica C 400 (2004) 89–96
93
Fig. 6. Comparison between performances of ferromagnetic (stars) and non-magnetic (circles) coils. Solid lines present engineering critical current density at different temperatures. Dashed lines show Jc ðBmax Þ- (open symbols) and Jc ðB0 Þ-curves (filled symbols). Arrows show central magnetic flux densities that correspond to critical current densities at 4.2 and 10 K.
Fig. 5. Central magnetic flux density as a function of coil outer radius at operation temperatures: (a) T ¼ 4:2 K and (b) T ¼ 10 K. Solid line corresponds to the coil with ri ¼ 25 mm and dashed line to the coil with ri ¼ 50 mm. Axial lengths of the coils are 6 cm (star), 12 cm (circle) and 24 cm (diamond).
romagnetic one produced a higher central field. Magnetic field distributions in both coils operating at their Jc are shown in Fig. 7. The high field values are visible in the ferromagnetic winding. However, difference between the magnetic flux densities inside the bores was not so large. It has to be remembered that the Ic data used was measured with the iron-sheathed wire and was not a property of pure MgB2 material. The magnetic shielding increases Ic especially at low magnetic fields [13].
Thus, the removal of the ferromagnetic sheath modifies the Ic ðBÞ-curve even if the material properties of superconductor remain the same. As a whole, the effect of the magnetized sheath on B0 was quite small and it could even be positive. Because at the moment MgB2 wires are under rapid development attention must be also paid on the sensitivity of computed results to changes in Jc ðBÞ. The magnet performance with different Ic ðBÞ-data can also be illustrated with Fig. 6. Two general observations can be made. First, in nonmagnetic coils the magnet load lines were steeper compared to ferromagnetic coils. Therefore, B0 can become higher in the ferromagnetic coil than in the non-magnetic one if dIc =dB is decreased. Second, the J ðBmax Þ- and J ðB0 Þ-curves were almost linear at B > 2 T but the slope of J ðBmax Þ-line was slightly smaller than that of J ðB0 Þ-line. Thus, the difference between Bmax and B0 increases when the current density in the magnet rises. Finally, changes in self-field critical current and irreversibility field were simulated. Because at Birr Eq. (1) was not valid changes in the irreversibility field were
94
M. Ahoranta et al. / Physica C 400 (2004) 89–96 Table 1 Effect of changes in Ic ðBÞ of the MgB2 wire on the critical current density and central magnetic field of the magnet, ri ¼ 25 mm, ro ¼ 85 mm, h ¼ 120 mm, at 4.2 K Increase in Ic (0) (%)
Increase in B0 (%)
10 None 10
None 10 10
Increase in Jc (%) 2.4 5.8 8.6
Increase in B0 (%) 2.2 5.0 7.5
The stored energy in a coil can be expressed as Z E ¼ 1=2 B H dv ¼ 1=2LI 2 ; V
where V is the volume of the whole space, L the inductance and I the operation current of the coil. In ferromagnetic coils the magnetization increased the stored energy due to increased B. Inductance of magnetic coils was also dependent on the operation current as shown in Fig. 8. The inductance
Fig. 7. Magnetic flux density distributions in: (a) ferromagnetic and (b) non-magnetic MgB2 coils working at their critical current.
described with changes in parameter B1A defined as aB21A þ bB1A þ c ¼ 0 i.e. Ic ðB1A Þ ¼ 1A. Then, a and c in Eq. (1) were modified in order to simulate the changes in Ic (0 T) and B1A , b was kept constant. Table 1 lists the effects of 10% increase in Ic (0 T) and B1A on Jc and B0 of the ferromagnetic coil at 4.2 K. The increase in B0 was clearly more effective than that of Ic (0 T).
Fig. 8. Normalized magnet inductance, L (ferromagnetic)/L (non-magnetic), as a function of operation current for certain coils. Dimensions of the coils are: (1) h ¼ 6 cm, ri ¼ 50 mm, ro ¼ 160 mm (circle), (2) h ¼ 6 cm, ri ¼ 25 mm, ro ¼ 125 mm (star), (3) h ¼ 24 cm, ri ¼ 50 mm, ro ¼ 70 mm (square) and (4) h ¼ 24 cm, ri ¼ 25 mm, ro ¼ 35 mm (diamond). The inductance for corresponding non-magnetic coils are (1) 1.5 H, (2) 0.75 H, (3) 0.23 H and (4) 0.16 H.
M. Ahoranta et al. / Physica C 400 (2004) 89–96
was largest on small currents but decreased as the winding got saturated. Energies of ferromagnetic
Fig. 9. Stored energy as a function of coil dimensions for ferromagnetic MgB2 coils operating on their critical current densities. Coil dimensions are: (1) h ¼ 6 cm, ri ¼ 25 mm (stars), (2) h ¼ 24 cm, ri ¼ 25 mm (circle), (3) h ¼ 6 cm, ri ¼ 50 mm (diamond) and (4) h ¼ 24 cm, ri ¼ 50 mm (square).
Fig. 10. Homogeneity of magnetic field in a sphere of radius 1.8 cm. Filled symbols are for MgB2 coils operating on their critical current density at 4.2 K. Open symbols show the homogeneity in corresponding non-ferromagnetic coils. Coil dimensions are: (1) h ¼ 6 cm, ri ¼ 25 mm (stars), (2) h ¼ 24 cm, ri ¼ 25 mm (circle), (3) h ¼ 6 cm, ri ¼ 50 mm (diamond) and (4) h ¼ 24 cm, ri ¼ 50 mm (square).
95
MgB2 coils as a function of coil dimensions are shown in Fig. 9 for coils operating at the critical current. In ferromagnetic MgB2 coils L was 1.2–1.7 times higher than in non-magnetic coils when both magnets were working at the critical current density of the ferromagnetic magnet. However, in non-magnetic coils the critical current was larger. A high coil with wide bore was preferential for large energy. The highest obtainable energy with ferromagnetic MgB2 coils was 32 kJ at 4.2 K in the magnet with ri ¼ 50 mm, ro ¼ 160 mm and h ¼ 24 mm. In very thin coils, ro ri 6 20 mm, energy was larger in the ferromagnetic MgB2 coil at 4.2 K than in the NbTi coil owing to the strong magnetization. However, in other coils up to nine times higher energies were achieved with NbTi at 4.2 K. Fig. 10 shows the homogeneity inside the sphere with R 1:8 cm for non-magnetic and ferromagnetic MgB2 coils. The magnetized winding changed the field distribution and homogeneity inside the coil bore. The most homogeneous field was obtained with long and thin coils with small bore.
Fig. 11. Normalized field homogeneity, d (ferromagnetic coil)/d (non-magnetic coil), as a function of coil current density in a sphere of radius 1.8 cm for some magnet geometries. Dimensions of the coils are: (1) h ¼ 6 cm, ri ¼ 50 mm, ro ¼ 160 mm (circles), (2) h ¼ 6 cm, ri ¼ 25 mm, ro ¼ 125 mm (stars), (3) h ¼ 24 cm, ri ¼ 50 mm, ro ¼ 70 mm (squares) and (4) h ¼ 24 cm, ri ¼ 25 mm, ro ¼ 45 mm (diamonds). The homogeneities for corresponding non-magnetic coils are: (1) 0.50%, (2) 7.4%, (3) 0.52% and (4) 0.22%.
96
M. Ahoranta et al. / Physica C 400 (2004) 89–96
The best homogeneities in the ferromagnetic and non-magnetic MgB2 coils were 0.10%, and 0.17%, respectively. However, for most of the coils magnetization of the sheath reduced the field homogeneity. Homogeneity as a function of current density is shown in Fig. 11 for ferromagnetic and non-magnetic coils. For non-magnetic coils the field homogeneity depended only on the coil geometry and not on the current density. When ironsheathed MgB2 wire was applied the magnetic field was changed according to the saturation state of the winding. However, the dependence of the field homogeneity on the operation current was largest for short coils where the homogeneity was poor in any case.
4. Conclusions Performance of solenoidal magnets made of iron sheathed ex situ MgB2 wire was studied computationally. The maximum achieved central field with MgB2 coils was 3.8 T at 4.2 K and 3 T at 10 K. Magnets manufactured of commercial NbTi wires were able to produce 2.6–3.4 times higher central fields. Also almost one order of magnitude higher energy was reached with NbTi coils. At constant current density the magnetization of iron sheath in the tape could increase the central field in the magnet bore. Although MgB2 is cheap improvements in the current carrying capacity are still needed before it becomes attractive for conduction cooled low field applications. The magnets are more complicated to design from ironsheathed MgB2 conductors than from traditional LTS wires. The ferromagnetic sheath makes both the field homogeneity inside the bore and the coil inductance dependent on the operation current of the magnet. The magnetization of iron increases the coil inductance. On the other hand, the maximum magnetic flux density in the coil increases and thereby the critical current is reduced. Altogether, the maximum achievable central field and stored energy are almost independent of the sheath material of the MgB2 conductor.
Acknowledgements This research is supported by Ulla Tuominen Foundation, Academy of Finland project 52434 and the Grant Agency of the Slovak Academy of Sciences VEGA 2/2069/22. References [1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, Nature 410 (2001) 63. [2] R. Fl€ ukiger, H.L. Suo, N. Musolino, C. Benecude, P. Toulemonde, P. Lezza, Physica C 385 (2003) 286. [3] C.B. Eom, M.K. Lee, J.H. Choi, L.J. Belenky, X. Song, L.D. Cooley, M.T. Naus, S. Patnaik, J. Jiang, M. Rikel, A. Polyanskii, A. Gurevich, X.Y. Cai, S.D. Bu, S.E. Babcock, E.E. Hellstrom, D.C. Larbalestier, N. Rogado, K.A. Regan, M.A. Hayward, T. He, J.S. Slusky, K. Inumaru, M.K. Haas, R.J. Cava, Nature 411 (2001) 558. [4] E.W. Collings, E. Lee, M.D. Sumption, M. Tomsic, X.L. Wang, S. Soltanian, S.X. Dou, Physica C 386 (2003) 555. [5] K. Tanaka, M. Okada, H. Kumakura, H. Kitaguchi, K. Togano, Physica C 382 (2002) 203. [6] G. Grasso, A. Malagoli, M. Modica, A. Tumino, C. Ferdeghini, A.S. Siri, C. Vignola, L. Martini, V. Previtali, G. Volpini, Supercond. Sci. Technol. 16 (2003) 271. [7] S. Soltanian, J. Horvat, X.L. Wang, M. Tomsic, S.X. Dou, cond-mat/0205406. [8] H.L. Suo, C. Beneduce, M. Dhalle, N. Musolino, J.Y. Genoud, R. Fl€ ukiger, Appl. Phys. Lett. 79 (2001) 3116. [9] S. Jin, H. Mavoori, C. Bower, R.B. van Dover, Nature 411 (2001) 563. [10] Y. Feng, G. Yan, Y. Zhao, C.F. Liu, B.Q. Fu, L. Zhou, L.Z. Cao, K.Q. Ruan, X.G. Li, L. Shi, Y.H. Zhang, Physica C 386 (2003) 598. [11] P. Kovac, K. Henze, T. Melisek, I. Husek, H. Kirchmayr, Supercond. Sci. Technol. 15 (2002) 1037. [12] P. Kovac, M. Ahoranta, T. Melisek, J. Lehtonen, I. Husek, Supercond. Sci. Technol. 16 (2003) 793. [13] P. Kovac, M. Ahoranta, J. Lehtonen, I. Husek, Physica C 397 (2003) 14. [14] P. Kovac, I. Husek, T. Melisek, Supercond. Sci. Technol. 15 (2002) 1340. [15] P. Fabbricatore, M. Greco, R. Musenich, P. Kovac, I. Husek, F. G€ om€ ory, Supercond. Sci. Technol. 16 (2003) 364. [16] M. Dhalle, P. Toulemonde, C. Beneduce, N. Musolino, M. Decroux, R. Fl€ ukiger, Physica C 363 (2001) 155. [17] K.Q. Ruan, H.L. Li, Y. Yu, C.Y. Wang, L.Z. Cao, C.F. Liu, S.J. Du, G. Yan, Y. Feng, X. Wu, J.R. Wang, X.H. Liu, P.X. Zhang, X.Z. Wu, L. Zhou, Physica C 386 (2003) 578.