- Email: [email protected]
Loss contributions in superconducting magnets caused by transient magnetic fields
Fusion Engineering and Design 81 (2006) 2509–2513
Loss contributions in superconducting magnets caused by transient magnetic fields S. Tak´acs a,∗ , A. Werner b , M. Sochor b a
Institute of Electrical Engineering, Slovak Academy of Sciences, 841 04 Bratislava, Slovakia b Max-Planck-Institut f¨ ur Plasmaphysik, Greifswald, Germany Available online 5 September 2006
Abstract The mechanical relaxation processes of the support structure at energizing the W7-X stellarator with cable-in-conduit superconductor have been analyzed, employing finite element calculations for one coil with adapted boundary conditions from a global static model. In this way, the duration and the amplitude of the field change at the conductor was determined. We calculate the effect of such transient on all relevant contributions to ac losses: hysteresis losses in filaments, coupling losses in different substructures of the cable, eddy current losses in normal parts (casing, jacket). Especially, the partial screening of the cable and of the individual filaments by the normal jacket is evaluated. Then, the results are applied to the W7-X system. We emphasize that the procedure can be used in other systems consisting of more magnets, or for the mechanical movement of individual parts of the winding. The possible consequences for the heat generation and the stability of the superconducting windings are given. In addition, we show that the procedure can be used to minimize the total losses in superconducting cables. © 2006 Elsevier B.V. All rights reserved. Keywords: ac losses; Superconducting magnets; Stellarator; Stability
1. Introduction The cables used for superconducting magnets are made by multistage twisting process to reduce the coupling losses. The corresponding time constants characterize the ac losses on different twisting stages. In this sense, the coupling losses and the eddy current losses in the normal parts (casing, jacket) can be described in an analogous way. ∗ Corresponding author. Tel.: +42 1 2 5477 5823 2949; fax: +42 1 2 5477 5816. E-mail address: [email protected] (S. Tak´acs).
0920-3796/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2006.07.008
In magnetic systems consisting of more magnets, the mechanical movement of one magnet can cause field changes in the other components, leading thus to additional ac losses. We derived some working formulas [1,2] for all ac loss components at changing the magnetic field from the initial value B0 by a transient amplitude b and calculated the upper limits for the loss contributions for the system W7-X. In spite of the fact that the outer parts of the magnetic system shield the inner parts, the different loss contributions are generally assumed to act more or less independently. The shielding effects could be treated as some effective increase of the time constant of the
2510
S. Tak´acs et al. / Fusion Engineering and Design 81 (2006) 2509–2513
applied field τ 0 [1,2]. In this paper, we show that the situation can be more complicated. We show further that our procedure can be used to obtain the minimum loss of the whole structure. At first, the results of the calculations for possible mechanical movement of the magnets in the W7-X system and the corresponding field changes are given. Then, the different contributions to the ac losses are evaluated, including the partial screening effect on the individual components of the winding. The possible consequences on the total loss and the stability of the system are added in the conclusions.
2. Transient event at energizing the W7-X magnetic system The support structure of the superconducting magnets in W7-X comprises sliding bearings in order to reduce tangential forces acting on the casing of the magnets. These bearings have been carefully designed for maintaining very low friction coefficients. However, if one of those bearings looses its low friction, stick–slip events may occur leading to some transient displacement of the coil in the superconductor. For assessment of such events and whether they are potentially able to cause a quench by ac losses, the dynamics of the coil deformation has to be derived. The loads on all support elements for each magnet are available by the calculation of the global static model, which includes all magnets of one field period and the support structure inclusively the sliding bearings. These calculations have been performed by finite element (FE) calculations deploying ADINA [3,4] and more recently ANSYS [5]. In order to perform transient finite element calculations (with ANSYS), only one coil has been taken and the boundary conditions of the single coil have been derived as follows: (i) the displacement U at each FE node has directly been set using the data given by the global model. (ii) Then, the Lorentz force Fwp is applied on the winding pack. The winding pack itself is fixed by the embedding resin and reacts on pressure only. (iii) The procedure, so far, leads to reaction forces Rsup at the support elements, reaction forces Rnse at the sliding bearing and to some particular dis-
placement Uext at some artificially prolonged support elements. These artificial support elements are needed to fix the coil by displacement for avoiding so called flying structures, which would occur if the forces only determined the boundary conditions. (iv) The model of the single coil is checked by employing the boundary conditions Uext , Fwp , Rsup and Rnse . The remaining deviation between the obtained value of U and U from the global model is less than 3%. (v) Rsup and Rnse are taken time dependent for the transient analysis. Different approaches have been used with only the tangential forces of Rnse removed, or Rsup at all supports stepped to values occurring for different force equilibriums of the global model. First calculations revealed two frequency domains of the coil dynamics. A low frequency part around 4 Hz and a second one at frequencies above 20 Hz, which is caused by the spring constants of the artificial central support elements. This could be confirmed in some FE modal analysis of the coil. Therefore, the model has been expanded for taking into account one field coil and the real central support ring. Finally, the transient behavior was similar in both models. The magnetic field change has been derived applying the EFFI code [6,7] to the dynamically deformed winding pack inside the other statically deformed coils. It considers the finite conductor size by approximating its shape through cuboids. The most critical values for the perpendicular field component lead finally to a field change rate of B˙ = 0.5 T/s, the characteristic transient time τ 0 = 40 ms and field amplitude b = 20 mT. This rapid field change acts on the W7-X NbTi cable, which is a cable-in-conduit conductor of diameter d = 11.4 mm, with stabilizing square Al jacket of size w = 26 mm and resistivity ρn = 7 × 10−9 m. The number of strands is 236, the number of filaments in them 144.
3. Hysteresis losses As the hysteresis losses per cycle are independent of frequency [8], the shielding does not influence the hysteresis losses in the superconducting filaments and the results of [1] could be applied without any restriction. The highest loss density per cycle appears at changing the field in the direction of the existing field and is
S. Tak´acs et al. / Fusion Engineering and Design 81 (2006) 2509–2513
given by [1]: qh =
bBp 2μ0
(1)
where Bp is the penetration field. For W7-X with b = 20 mT, we have Bp = 21 mT, and the maximum loss generation by hysteresis losses is qh = 167 J/m3 . Multiplying with the cross-section of filaments Sf = 64 mm2 , the losses per length are then Ph = 0.0107 J/m.
4. Eddy and coupling current losses The losses in the normal parts of the superconducting system can be calculated at knowing the time constant of the system [9–13]. For a square of size w, it is given by τ1 =
μ0 πw2 16ρn
(2)
where ρn is the resistivity of the normal material. This geometry is a good assumption, as the jacket is usually square-like for easier procedure at winding the magnet. We assume the magnetic field changing with time exponentially, like Be = B0 + b[1 − exp(−t/τ 0 )]. As the induced currents causing the additional field change decrease in this way, this assumption is very close to reality. The eddy current loss density at this event is [1]: ∞
qn =
τ1 /τ0 b2 8 (−1)k 2 2μ0 π2 (2k + 1) (2k + 1)2 + τ1 /τ0 k=0
which can be approximated for τ 1 /τ 0 < 10 very precisely by qn = 0.812
τ1 /τ0 b2 2μ0 1 + 0.925τ1 /τ0
(3)
For the Al jacket of W7-X cable, we have τ 1 = 9 ms, Sn = 153.6 mm2 . The loss density is then qn = 24.2 J/m3 and the eddy current losses per length are 3.72 mJ/m. Analogously, if the time constant of the coupling currents is known, (3) can be used also for calculating the corresponding coupling losses. The time constant can be measured, or calculated from known formulas
[9–13]. For round strands or cables: μ 0 l0 2 τ= 2ρ 2π
2511
(4)
where l0 is the twist pitch and ρ the transverse resistivity between filaments or strands.
5. Eddy or coupling current losses in layer shielded by another layer For simplicity, we assume an infinitely long layer of thickness w with time constant τ shielded from both sides by layers with time constant τ 1 . This situation mimes a cylinder surrounded by a normal conducting sheath, or the strands shielded by the coupling currents in a cable. Then, the inner layer is exposed to a changing field Be = b(1 − e−t/τ0 )(1 − e−t/τ1 ). By solving the corresponding diffusion equation [14], the losses are calculated during the event like in [1]. The loss density is approximately x x(1 + u) 0.812b2 v0 g(x) + v1 g + vg q= μ0 u u (5) with x = τ/τ 0 , u = τ 1 /τ 0 , v0 = 1/2 + 1/(1 + u) − (1 + u)/(1 + 2u), v1 = 1/2 + u/(1 + u) − (1 + u)/(2 + u), v = 1/2 − 1/(2 + u) − u/(1 + 2u) and p g(p) = 1 + 0.925p This loss density can be compared with the loss density in the layer without shielding and with the “effective time constant” τ eff = τ 0 + τ 1 , which involves the effect of delaying the flux entry by the time τ 1 . There are two types of coupling currents in the cable of W7-X: between the strands with a small time constant τ c = 8 ms and within the strands with τ s = 50 ms [3]. Without considering the shielding effect, we obtain the interstrand coupling loss density qc = 10.9 J/m3 and the coupling loss density within the strands qs = 37.45 J/m3 . Taking an effective time constant τ eff = 48 and 57 ms, the corresponding loss densities would be qc = 9.16 J/m3 and qs = 31.3 J/m3 , respectively. Both parts as shielded by the normal jacket, in our case with τ 0 = 40 ms, which leads to x = 0.2, u = 0.225 for the cable and x = 1.25, u = 0.425 for the
2512
S. Tak´acs et al. / Fusion Engineering and Design 81 (2006) 2509–2513
strands, respectively. The correct results from (5) are then qc = 9.34 J/m3 , qs = 30.44 J/m3 .
6. Procedure to minimize the total losses Now, the interesting question arises whether one could minimize the total losses in shielded structures. We consider an outer region (the normal jacket) with the cross-section S0 and an inner region with S1 (the cable with the coupling currents). The eddy current and the coupling current are in the above sense considered as analogous with the time constants τ 1 and τ. The only difference is, that the time constant of the normal region is proportional to the square of the dimension perpendicular to the field (3). To obtain the minimum losses for given x, we have to minimize the function: h
0.812b2 (τ1 /τ0 )2 + q(u) 2μ0 1 + 0.925τ1 /τ0
where q(u) is given by (5) and h = S0 τ 0 /S1 τ 1 is the ratio of volumes multiplied by the corresponding time constant. The basic result is that the minimum appears only for h < 1.347. For larger values of h, the normal sheath always increases the total losses. Applying the results for W7-X with h = 5.65 for the strands, the total losses are about 2.2-times larger than would be without the aluminum.
is larger than the time constant of the changing field (τ 1 > τ 0 /2). Hence, at very fast field changes the outside normal layer does not improve the stability only, but may decrease also the total loss of the structure. Otherwise, for approximately τ 0 > τ 1 ), the existence of the shielding region increases the total losses. One has to have in mind, however, that the normal sheath is usually better cooled than the “rest” of the superconducting cable. The total loss per length Pt of the cable (sum of the hysteresis, coupling and eddy current losses) in the stellarator system W7-X are Pt = 22.65 mJ/m and the mean loss density qt in the strands (sum of the hysteresis and coupling losses) qt = 0.067 mJ/cm3 . Although the total loss density is about twice the value without the aluminum layer, it is more than two orders of magnitude below the required stability limit (17 mJ/cm3 ) [2]. All calculations in this paper were made for the transient fields. We would like to add that some of the results, mainly for the shielding, should be similar for other types of field changes. The procedure can be used also in other systems consisting of more magnets, even for the mechanical movement of individual parts of the winding.
Acknowledgement This work was partially supported by the Slovak Grant Agency for Science VEGA no. 2/6098/26.
7. Conclusions We have demonstrated how to calculate the heat generation in systems with a transient in the applied magnetic field. Useful formulas for all types of ac losses are given. The eddy or coupling current losses in a region shielded by an outer region can be generally treated by considering an increased effective time constant τ eff = τ 0 + τ 1 instead of the time constant of the applied field τ 0 , if τ 1 < τ 0 /2. The expected situation in W7-X belongs to this category, therefore we can use (5) without any restriction. The total losses can be minimized by including a shielding region of normal metal at rapid changes of the field (small h values, approximately τ 0 < τ 1 ). The coupling losses in superconducting cables can be decreased substantially if the time constant of the shielding layer
References [1] S. Tak´acs, F. G¨om¨ory, Hysteresis and coupling losses of superconducting cables at additional change of the applied magnetic field, Supercond. Sci. Technol. 18 (2005) 340–345. [2] F. G¨om¨ory, S. Tak´acs, A. Werner, M. Sochor, Theoretical estimation of electromagnetic loss at the movement of superconducting coil in the W7-X stellarator, in: Proceedings of the 16th International Conference on Magnet Technology MT-16, IEEE Trans. Appl. Supercond. 16 (2006) 123–126. [3] N. Jaksic, J. Simon-Weidner, Design analysis of the support structure stressed by large superconducting coils for a plasma fusion experiment, Comput. Struct. 81 (2003) 697– 714. [4] M. Gasparotto, F. Elio, B. Heinemann, N. Jaksic, B. Mendelevitch, J. Simon-Weidner, B. Streibl, The W7-X Team, The Wendelstein 7-X mechanical structure support elements: design and tests, in: Proceedings of the 23rd Symposium on Fusion
S. Tak´acs et al. / Fusion Engineering and Design 81 (2006) 2509–2513
[5] [6]
[7]
[8] [9]
Technology, Venice, 2004, Fusion Eng. Des. 74 (2005) 161–165. V. Bykow, Strategy of structural analysis of W7-X magnet system, in: Proceedings of the 21st SOFE, 2005. E. Harmeyer, J. Kisslinger, Improved support concept for the Helias Reactor coil system, Fusion Eng. Des. 66–68 (2003) 1025–1028. S.J. Sackett, EFFI – a code for calculating the electromagnetic field, force and inductance in coil systems for arbitrary geometry, LLNL Report UCRL-52402, 1978. C.P. Bean, Magnetization of hard superconductors, Phys. Rev. Lett. 8 (1962) 250–253. W.J. Carr Jr., ac losses in a twisted filamentary superconducting wire, J. Appl. Phys. 46 (1975) 4043–4047.
2513
[10] K.-P. J¨ungst, G. Ries, ac losses in multifilamentary superconductors at technical frequencies, IEEE Trans. Mag. MAG-13 (1977) 527–530. [11] K. Kwasnitza, Scaling law for the ac losses of multifilament superconductors, Cryogenics 17 (1977) 616–620. [12] A.M. Campbell, A general treatment of losses in multifilamentary superconductors, Cryogenics 22 (1982) 3–11. [13] S. Tak´acs, ac losses in superconducting cables and their expected values in magnetic systems. Review, Supercond. Sci. Technol. 10 (1997) 733–748. [14] N. Yanagi, S. Tak´acs, T. Mito, K. Takahata, A. Iwamoto, J. Yamamoto, Measurement of time constants for superconducting cables with Hall probes, Cryogenics 37 (1997) 783–788.
Loss contributions in superconducting magnets caused by transient magnetic fields S. Tak´acs a,∗ , A. Werner b , M. Sochor b a
Institute of Electrical Engineering, Slovak Academy of Sciences, 841 04 Bratislava, Slovakia b Max-Planck-Institut f¨ ur Plasmaphysik, Greifswald, Germany Available online 5 September 2006
Abstract The mechanical relaxation processes of the support structure at energizing the W7-X stellarator with cable-in-conduit superconductor have been analyzed, employing finite element calculations for one coil with adapted boundary conditions from a global static model. In this way, the duration and the amplitude of the field change at the conductor was determined. We calculate the effect of such transient on all relevant contributions to ac losses: hysteresis losses in filaments, coupling losses in different substructures of the cable, eddy current losses in normal parts (casing, jacket). Especially, the partial screening of the cable and of the individual filaments by the normal jacket is evaluated. Then, the results are applied to the W7-X system. We emphasize that the procedure can be used in other systems consisting of more magnets, or for the mechanical movement of individual parts of the winding. The possible consequences for the heat generation and the stability of the superconducting windings are given. In addition, we show that the procedure can be used to minimize the total losses in superconducting cables. © 2006 Elsevier B.V. All rights reserved. Keywords: ac losses; Superconducting magnets; Stellarator; Stability
1. Introduction The cables used for superconducting magnets are made by multistage twisting process to reduce the coupling losses. The corresponding time constants characterize the ac losses on different twisting stages. In this sense, the coupling losses and the eddy current losses in the normal parts (casing, jacket) can be described in an analogous way. ∗ Corresponding author. Tel.: +42 1 2 5477 5823 2949; fax: +42 1 2 5477 5816. E-mail address: [email protected] (S. Tak´acs).
0920-3796/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2006.07.008
In magnetic systems consisting of more magnets, the mechanical movement of one magnet can cause field changes in the other components, leading thus to additional ac losses. We derived some working formulas [1,2] for all ac loss components at changing the magnetic field from the initial value B0 by a transient amplitude b and calculated the upper limits for the loss contributions for the system W7-X. In spite of the fact that the outer parts of the magnetic system shield the inner parts, the different loss contributions are generally assumed to act more or less independently. The shielding effects could be treated as some effective increase of the time constant of the
2510
S. Tak´acs et al. / Fusion Engineering and Design 81 (2006) 2509–2513
applied field τ 0 [1,2]. In this paper, we show that the situation can be more complicated. We show further that our procedure can be used to obtain the minimum loss of the whole structure. At first, the results of the calculations for possible mechanical movement of the magnets in the W7-X system and the corresponding field changes are given. Then, the different contributions to the ac losses are evaluated, including the partial screening effect on the individual components of the winding. The possible consequences on the total loss and the stability of the system are added in the conclusions.
2. Transient event at energizing the W7-X magnetic system The support structure of the superconducting magnets in W7-X comprises sliding bearings in order to reduce tangential forces acting on the casing of the magnets. These bearings have been carefully designed for maintaining very low friction coefficients. However, if one of those bearings looses its low friction, stick–slip events may occur leading to some transient displacement of the coil in the superconductor. For assessment of such events and whether they are potentially able to cause a quench by ac losses, the dynamics of the coil deformation has to be derived. The loads on all support elements for each magnet are available by the calculation of the global static model, which includes all magnets of one field period and the support structure inclusively the sliding bearings. These calculations have been performed by finite element (FE) calculations deploying ADINA [3,4] and more recently ANSYS [5]. In order to perform transient finite element calculations (with ANSYS), only one coil has been taken and the boundary conditions of the single coil have been derived as follows: (i) the displacement U at each FE node has directly been set using the data given by the global model. (ii) Then, the Lorentz force Fwp is applied on the winding pack. The winding pack itself is fixed by the embedding resin and reacts on pressure only. (iii) The procedure, so far, leads to reaction forces Rsup at the support elements, reaction forces Rnse at the sliding bearing and to some particular dis-
placement Uext at some artificially prolonged support elements. These artificial support elements are needed to fix the coil by displacement for avoiding so called flying structures, which would occur if the forces only determined the boundary conditions. (iv) The model of the single coil is checked by employing the boundary conditions Uext , Fwp , Rsup and Rnse . The remaining deviation between the obtained value of U and U from the global model is less than 3%. (v) Rsup and Rnse are taken time dependent for the transient analysis. Different approaches have been used with only the tangential forces of Rnse removed, or Rsup at all supports stepped to values occurring for different force equilibriums of the global model. First calculations revealed two frequency domains of the coil dynamics. A low frequency part around 4 Hz and a second one at frequencies above 20 Hz, which is caused by the spring constants of the artificial central support elements. This could be confirmed in some FE modal analysis of the coil. Therefore, the model has been expanded for taking into account one field coil and the real central support ring. Finally, the transient behavior was similar in both models. The magnetic field change has been derived applying the EFFI code [6,7] to the dynamically deformed winding pack inside the other statically deformed coils. It considers the finite conductor size by approximating its shape through cuboids. The most critical values for the perpendicular field component lead finally to a field change rate of B˙ = 0.5 T/s, the characteristic transient time τ 0 = 40 ms and field amplitude b = 20 mT. This rapid field change acts on the W7-X NbTi cable, which is a cable-in-conduit conductor of diameter d = 11.4 mm, with stabilizing square Al jacket of size w = 26 mm and resistivity ρn = 7 × 10−9 m. The number of strands is 236, the number of filaments in them 144.
3. Hysteresis losses As the hysteresis losses per cycle are independent of frequency [8], the shielding does not influence the hysteresis losses in the superconducting filaments and the results of [1] could be applied without any restriction. The highest loss density per cycle appears at changing the field in the direction of the existing field and is
S. Tak´acs et al. / Fusion Engineering and Design 81 (2006) 2509–2513
given by [1]: qh =
bBp 2μ0
(1)
where Bp is the penetration field. For W7-X with b = 20 mT, we have Bp = 21 mT, and the maximum loss generation by hysteresis losses is qh = 167 J/m3 . Multiplying with the cross-section of filaments Sf = 64 mm2 , the losses per length are then Ph = 0.0107 J/m.
4. Eddy and coupling current losses The losses in the normal parts of the superconducting system can be calculated at knowing the time constant of the system [9–13]. For a square of size w, it is given by τ1 =
μ0 πw2 16ρn
(2)
where ρn is the resistivity of the normal material. This geometry is a good assumption, as the jacket is usually square-like for easier procedure at winding the magnet. We assume the magnetic field changing with time exponentially, like Be = B0 + b[1 − exp(−t/τ 0 )]. As the induced currents causing the additional field change decrease in this way, this assumption is very close to reality. The eddy current loss density at this event is [1]: ∞
qn =
τ1 /τ0 b2 8 (−1)k 2 2μ0 π2 (2k + 1) (2k + 1)2 + τ1 /τ0 k=0
which can be approximated for τ 1 /τ 0 < 10 very precisely by qn = 0.812
τ1 /τ0 b2 2μ0 1 + 0.925τ1 /τ0
(3)
For the Al jacket of W7-X cable, we have τ 1 = 9 ms, Sn = 153.6 mm2 . The loss density is then qn = 24.2 J/m3 and the eddy current losses per length are 3.72 mJ/m. Analogously, if the time constant of the coupling currents is known, (3) can be used also for calculating the corresponding coupling losses. The time constant can be measured, or calculated from known formulas
[9–13]. For round strands or cables: μ 0 l0 2 τ= 2ρ 2π
2511
(4)
where l0 is the twist pitch and ρ the transverse resistivity between filaments or strands.
5. Eddy or coupling current losses in layer shielded by another layer For simplicity, we assume an infinitely long layer of thickness w with time constant τ shielded from both sides by layers with time constant τ 1 . This situation mimes a cylinder surrounded by a normal conducting sheath, or the strands shielded by the coupling currents in a cable. Then, the inner layer is exposed to a changing field Be = b(1 − e−t/τ0 )(1 − e−t/τ1 ). By solving the corresponding diffusion equation [14], the losses are calculated during the event like in [1]. The loss density is approximately x x(1 + u) 0.812b2 v0 g(x) + v1 g + vg q= μ0 u u (5) with x = τ/τ 0 , u = τ 1 /τ 0 , v0 = 1/2 + 1/(1 + u) − (1 + u)/(1 + 2u), v1 = 1/2 + u/(1 + u) − (1 + u)/(2 + u), v = 1/2 − 1/(2 + u) − u/(1 + 2u) and p g(p) = 1 + 0.925p This loss density can be compared with the loss density in the layer without shielding and with the “effective time constant” τ eff = τ 0 + τ 1 , which involves the effect of delaying the flux entry by the time τ 1 . There are two types of coupling currents in the cable of W7-X: between the strands with a small time constant τ c = 8 ms and within the strands with τ s = 50 ms [3]. Without considering the shielding effect, we obtain the interstrand coupling loss density qc = 10.9 J/m3 and the coupling loss density within the strands qs = 37.45 J/m3 . Taking an effective time constant τ eff = 48 and 57 ms, the corresponding loss densities would be qc = 9.16 J/m3 and qs = 31.3 J/m3 , respectively. Both parts as shielded by the normal jacket, in our case with τ 0 = 40 ms, which leads to x = 0.2, u = 0.225 for the cable and x = 1.25, u = 0.425 for the
2512
S. Tak´acs et al. / Fusion Engineering and Design 81 (2006) 2509–2513
strands, respectively. The correct results from (5) are then qc = 9.34 J/m3 , qs = 30.44 J/m3 .
6. Procedure to minimize the total losses Now, the interesting question arises whether one could minimize the total losses in shielded structures. We consider an outer region (the normal jacket) with the cross-section S0 and an inner region with S1 (the cable with the coupling currents). The eddy current and the coupling current are in the above sense considered as analogous with the time constants τ 1 and τ. The only difference is, that the time constant of the normal region is proportional to the square of the dimension perpendicular to the field (3). To obtain the minimum losses for given x, we have to minimize the function: h
0.812b2 (τ1 /τ0 )2 + q(u) 2μ0 1 + 0.925τ1 /τ0
where q(u) is given by (5) and h = S0 τ 0 /S1 τ 1 is the ratio of volumes multiplied by the corresponding time constant. The basic result is that the minimum appears only for h < 1.347. For larger values of h, the normal sheath always increases the total losses. Applying the results for W7-X with h = 5.65 for the strands, the total losses are about 2.2-times larger than would be without the aluminum.
is larger than the time constant of the changing field (τ 1 > τ 0 /2). Hence, at very fast field changes the outside normal layer does not improve the stability only, but may decrease also the total loss of the structure. Otherwise, for approximately τ 0 > τ 1 ), the existence of the shielding region increases the total losses. One has to have in mind, however, that the normal sheath is usually better cooled than the “rest” of the superconducting cable. The total loss per length Pt of the cable (sum of the hysteresis, coupling and eddy current losses) in the stellarator system W7-X are Pt = 22.65 mJ/m and the mean loss density qt in the strands (sum of the hysteresis and coupling losses) qt = 0.067 mJ/cm3 . Although the total loss density is about twice the value without the aluminum layer, it is more than two orders of magnitude below the required stability limit (17 mJ/cm3 ) [2]. All calculations in this paper were made for the transient fields. We would like to add that some of the results, mainly for the shielding, should be similar for other types of field changes. The procedure can be used also in other systems consisting of more magnets, even for the mechanical movement of individual parts of the winding.
Acknowledgement This work was partially supported by the Slovak Grant Agency for Science VEGA no. 2/6098/26.
7. Conclusions We have demonstrated how to calculate the heat generation in systems with a transient in the applied magnetic field. Useful formulas for all types of ac losses are given. The eddy or coupling current losses in a region shielded by an outer region can be generally treated by considering an increased effective time constant τ eff = τ 0 + τ 1 instead of the time constant of the applied field τ 0 , if τ 1 < τ 0 /2. The expected situation in W7-X belongs to this category, therefore we can use (5) without any restriction. The total losses can be minimized by including a shielding region of normal metal at rapid changes of the field (small h values, approximately τ 0 < τ 1 ). The coupling losses in superconducting cables can be decreased substantially if the time constant of the shielding layer
References [1] S. Tak´acs, F. G¨om¨ory, Hysteresis and coupling losses of superconducting cables at additional change of the applied magnetic field, Supercond. Sci. Technol. 18 (2005) 340–345. [2] F. G¨om¨ory, S. Tak´acs, A. Werner, M. Sochor, Theoretical estimation of electromagnetic loss at the movement of superconducting coil in the W7-X stellarator, in: Proceedings of the 16th International Conference on Magnet Technology MT-16, IEEE Trans. Appl. Supercond. 16 (2006) 123–126. [3] N. Jaksic, J. Simon-Weidner, Design analysis of the support structure stressed by large superconducting coils for a plasma fusion experiment, Comput. Struct. 81 (2003) 697– 714. [4] M. Gasparotto, F. Elio, B. Heinemann, N. Jaksic, B. Mendelevitch, J. Simon-Weidner, B. Streibl, The W7-X Team, The Wendelstein 7-X mechanical structure support elements: design and tests, in: Proceedings of the 23rd Symposium on Fusion
S. Tak´acs et al. / Fusion Engineering and Design 81 (2006) 2509–2513
[5] [6]
[7]
[8] [9]
Technology, Venice, 2004, Fusion Eng. Des. 74 (2005) 161–165. V. Bykow, Strategy of structural analysis of W7-X magnet system, in: Proceedings of the 21st SOFE, 2005. E. Harmeyer, J. Kisslinger, Improved support concept for the Helias Reactor coil system, Fusion Eng. Des. 66–68 (2003) 1025–1028. S.J. Sackett, EFFI – a code for calculating the electromagnetic field, force and inductance in coil systems for arbitrary geometry, LLNL Report UCRL-52402, 1978. C.P. Bean, Magnetization of hard superconductors, Phys. Rev. Lett. 8 (1962) 250–253. W.J. Carr Jr., ac losses in a twisted filamentary superconducting wire, J. Appl. Phys. 46 (1975) 4043–4047.
2513
[10] K.-P. J¨ungst, G. Ries, ac losses in multifilamentary superconductors at technical frequencies, IEEE Trans. Mag. MAG-13 (1977) 527–530. [11] K. Kwasnitza, Scaling law for the ac losses of multifilament superconductors, Cryogenics 17 (1977) 616–620. [12] A.M. Campbell, A general treatment of losses in multifilamentary superconductors, Cryogenics 22 (1982) 3–11. [13] S. Tak´acs, ac losses in superconducting cables and their expected values in magnetic systems. Review, Supercond. Sci. Technol. 10 (1997) 733–748. [14] N. Yanagi, S. Tak´acs, T. Mito, K. Takahata, A. Iwamoto, J. Yamamoto, Measurement of time constants for superconducting cables with Hall probes, Cryogenics 37 (1997) 783–788.