Magnetization loss for stacks of ReBCO tapes

Size: px

Start display at page:

Download "Magnetization loss for stacks of ReBCO tapes"

Meghan Rogers
6 years ago
Views:

1 N. Bykovsky, G. De Marzi 2, D. Uglietti, P. Bruzzone, L. Muzzi 2 () (2) HTS MODELLING June, Bologna, Italy

2 Outline Introduction 2 Numerical model 3 Samples for the VSM measurements 4 Experimental results 5 Magnetization loss 6 Conclusion

3 Outline Introduction 2 Numerical model 3 Samples for the VSM measurements 4 Experimental results 5 Magnetization loss 6 Conclusion

4 HTS fusion cable concept at SPC: Strand stack of HTS tapes twisted and soldered between copper profiles: Cable copper cored Rutherford design: EDIPO sample 6 ka/2 T cable prototypes: 2 cables made of & SuperOx tapes 2 strands per cable twisted at m 6 tapes per strand twisted at 32 mm w /5

5 Work motivation! Feasibility of the proposed HTS strand and cable designs for fusion magnets was experimentally demonstrated with the DC tests performed in EDIPO Improvement of the mechanical properties of the strand against the transverse Lorentz force is the ongoing task... AC loss mechanisms of the proposed cable design: Hysteresis loss in the stacked tapes 2 Inter-tape coupling current loss 3 Inter-strand coupling current loss (dominant one in the prototypes) 4 Eddy current loss y Study of the magnetization loss in the stacked tapes is the scope of this work: influence of the number of tapes in the stack?... width of the tapes?... orientation of the applied magnetic field?... manufacturer of the HTS tape? 2/5

6 Outline Introduction 2 Numerical model 3 Samples for the VSM measurements 4 Experimental results 5 Magnetization loss 6 Conclusion

7 Description of the model Parameters n t number of tapes w width of the tape d thickness of the SC layer ( µm) D distance between the layers ( ) n j E = E c power law relation j c (B,θ) nt 2 j = Sketch of the geometry (x i, y j ) D θa # B a i = 2 Nx y 2D geometry with the uniform grid s d 3/5

8 Description of the model Parameters n t number of tapes w width of the tape d thickness of the SC layer ( µm) D distance between the layers ( ) n j E = E c power law relation j c (B,θ) nt 2 j = Sketch of the geometry (x i, y j ) D θa # B a i = 2 Nx y 2D geometry with the uniform grid s d L( I ) = N i= N j = Functional (A φ formulation) 2 C i j I j I i + A ai I i t + E c I bi n + ( I + I I b ) n+ i Minimization of L is equivalent to solving the Maxwell equations C i j matrix of mutual inductances (expressed analytically) A a vector potential of the applied magnetic field Transport current is implemented as a constraint for I 3/5

9 Procedure of the calculation I (t = ) = # B a (t + t), I tr (t + t) L minimization t = t + t # B = # B a + # B sf (I + I ) I c ( # B ) adjust I I (t + t) = I (t) + I done! 4/5

10 Procedure of the calculation I (t = ) = # B a (t + t), I tr (t + t) Magnetic moment per unit length: N m(t) = m z (t) = x k I k (t) k= t = t + t L minimization # B = # B a + # B sf (I + I ) I c ( # B ) adjust I Instantaneous power loss: N N E c P(t) = E k (t)i k (t) = k= k= I n I k (t) n+ b k Hysteresis loss: Q = P(t) dt I (t + t) = I (t) + I done! Magnetization loss: Q = m cosθ a db a 4/5

11 Benchmarking Brandt s solution Numerical model 2 j /jc s 4 s 5 s 6 s Magnetic field Ba, mt s 2 s 2 2 x, mm Time t, s Parameters: n t =, w = 4mm, N x =, n =, j c (B,θ) = 4kA/mm 2 5/5

12 Benchmarking Brandt s solution Numerical model 2 j /jc s 4 s 5 s 6 s Magnetic field Ba, mt s 2 s 2 2 x, mm Time t, s Parameters: n t =, w = 4mm, N x =, n =, j c (B,θ) = 4kA/mm 2 Verification of the n-value in the model was done by considering a saturated state of the tape (expressed analytically) 5/5

13 Benchmarking Brandt s solution Numerical model 2 j /jc s 4 s 5 s 6 s Magnetic field Ba, mt s 2 s 2 2 x, mm Time t, s Parameters: n t =, w = 4mm, N x =, n =, j c (B,θ) = 4kA/mm 2 Verification of the n-value in the model was done by considering a saturated state of the tape (expressed analytically)! Numerical model is validated 5/5

14 Stack of tapes in the critical state model Brandt s solution (n t = ): Q brandt = w j c B c q ( Ba /B c ), q (x) = 2ln(cosh x) x tanh x Bc = µ j c d π 6/5

15 Stack of tapes in the critical state model Brandt s solution (n t = ): Q brandt = w j c B c q ( Ba /B c ), q (x) = 2ln(cosh x) x tanh x Bc = µ j c d π Mawatari s solution (n t = ): Q mawatari = w j c B c q ( Ba /B c,2d/πw ) ( x q (x, a) = a 2 (x 2ξ)ln + sinh2 (/a) cosh 2 ξ D w: q (x, a ) = q (x) a 2 x 3 /3, x /a D d: q (x, a ) = q slab (x, a) = x 2/(3a), x > /a ) dξ 6/5

16 Brandt s solution (n t = ): Stack of tapes in the critical state model Q brandt = w j c B c q ( Ba /B c ), q (x) = 2ln(cosh x) x tanh x Bc = µ j c d π Mawatari s solution (n t = ): Q mawatari = w j c B c q ( Ba /B c,2d/πw ) ( x q (x, a) = a 2 (x 2ξ)ln + sinh2 (/a) cosh 2 ξ ) dξ D w: q (x, a ) = q (x). a 2 x 3 /3, x /a D d: q (x, a ) = q slab (x, a) = x 2/(3a), x > /a 2 Normalized magnetization loss n t = n t = 2 n t = 4 n t = 8 n t = 6 n t = 28 B a/b c Mawatari: n t = 6/5

17 Brandt s solution (n t = ): Stack of tapes in the critical state model Q brandt = w j c B c q ( Ba /B c ), q (x) = 2ln(cosh x) x tanh x Bc = µ j c d π Mawatari s solution (n t = ): Q mawatari = w j c B c q ( Ba /B c,2d/πw ) ( x q (x, a) = a 2 (x 2ξ)ln + sinh2 (/a) cosh 2 ξ ) dξ D w: q (x, a ) = q (x). a 2 x 3 /3, x /a D d: q (x, a ) = q slab (x, a) = x 2/(3a), x > /a 2 Normalized magnetization loss n t = n t = 2 n t = 4 n t = 8 n t = 6 n t = 28 B a/b c Mawatari: n t = n t -tape stack one can consider as the stack with infinite number of tapes but with some effective distance between the tapes D eff (n t ): ( Q nt = w j c B c q nt Ba /B c,2d/πw ).34a. q nt (x, a) q x, a + n 4.44a+.65 t 6/5

18 Brandt s solution (n t = ): Stack of tapes in the critical state model Q brandt = w j c B c q ( Ba /B c ), q (x) = 2ln(cosh x) x tanh x Bc = µ j c d π Mawatari s solution (n t = ): Q mawatari = w j c B c q ( Ba /B c,2d/πw ) ( x q (x, a) = a 2 (x 2ξ)ln + sinh2 (/a) cosh 2 ξ ) dξ D w: q (x, a ) = q (x). a 2 x 3 /3, x /a D d: q (x, a ) = q slab (x, a) = x 2/(3a), x > /a 2 Normalized magnetization loss n t = n t = 2 n t = 4 n t = 8 n t = 6 n t = 28 B a/b c Mawatari: n t = n t -tape stack one can consider as the stack with infinite number of tapes but with some effective distance between the tapes D eff (n t ): ( Q nt = w j c B c q nt Ba /B c,2d/πw ).34a. q nt (x, a) q x, a + n 4.44a+.65 t 6/5

19 Outline Introduction 2 Numerical model 3 Samples for the VSM measurements 4 Experimental results 5 Magnetization loss 6 Conclusion

Fabrication of the samples HTS tape I c, A n-value 4 mm 55 ± 2 33 3 mm 7 ± 29

K/self-field before the stacks fabrication Soldering device ensures correct tapes

20 Fabrication of the samples HTS tape I c, A n-value 4 mm 55 ± mm 7 ± 29 SuperOx 4 mm 64 ± 34 Heaters press Stack of tapes 4 mm Tapes were measured at 77 K/self-field before the stacks fabrication Soldering device ensures correct tapes stacking in the cross-section Edges of the stacks were cut by the electro-erosion process 7/5

21 Cross-section of the stacks Copper Solder Hastelloy mm Top row: 4 mm wide, µm thick tapes Bottom row: 3 mm wide, 6 µm thick tapes 3 mm wide stacks have slightly distorted geometry (parabolic shape) Thickness of the solder between tapes is negligible, i.e. D = µm/ 6µm for the modelling 8/5

22 Test program HTS tape length 4 mm length 5 mm length mm width 3 mm length 5 mm SuperOx length 5 mm Number of tapes, n t Orientation of the sample, θ a / 45 / 45 / 45 / 9 / 45 VSM system at ENEA 9/5

23 Test program HTS tape length 4 mm length 5 mm length mm width 3 mm length 5 mm SuperOx length 5 mm Number of tapes, n t Orientation of the sample, θ a / 45 / 45 / 45 / 9 / 45 Finite length effect of the samples VSM system at ENEA 9/5

24 Test program HTS tape length 4 mm length 5 mm length mm width 3 mm length 5 mm SuperOx length 5 mm Number of tapes, n t Orientation of the sample, θ a / 45 / 45 / 45 / 9 / 45 VSM system at ENEA Finite length effect of the samples Geometry aspects on the magnetization behaviour (different w, D) 9/5

25 Test program HTS tape length 4 mm length 5 mm length mm width 3 mm length 5 mm SuperOx length 5 mm Number of tapes, n t Orientation of the sample, θ a / 45 / 45 / 45 / 9 / 45 VSM system at ENEA Finite length effect of the samples Geometry aspects on the magnetization behaviour (different w, D) Shielding effect for the different number of tapes in the stack n t 9/5

26 Test program HTS tape length 4 mm length 5 mm length mm width 3 mm length 5 mm SuperOx length 5 mm Number of tapes, n t Orientation of the sample, θ a / 45 / 45 / 45 / 9 / 45 VSM system at ENEA Finite length effect of the samples Geometry aspects on the magnetization behaviour (different w, D) Shielding effect for the different number of tapes in the stack n t Effect of the orientation of the magnetic field θ a 9/5

samples Geometry aspects on the magnetization behaviour (different w, D) Shielding effect for the different number of tapes

27 Test program HTS tape length 4 mm length 5 mm length mm width 3 mm length 5 mm SuperOx length 5 mm Number of tapes, n t Orientation of the sample, θ a / 45 / 45 / 45 / 9 / 45 VSM system at ENEA Finite length effect of the samples Geometry aspects on the magnetization behaviour (different w, D) Shielding effect for the different number of tapes in the stack n t Effect of the orientation of the magnetic field θ a Magnetization loss from the area of minor magnetization loops 9/5

28 Outline Introduction 2 Numerical model 3 Samples for the VSM measurements 4 Experimental results 5 Magnetization loss 6 Conclusion

29 Finite length effect at 77 K Magnetization loops Moment per unit length m, A m l = 4mm l = 5mm l = mm.5.5 Magnetic field B a, T Saturated state of the finite length tape: m sat = ( l 4 I c w w ) ( ) /n Ḃw 2n 3l 2E c 2n + Moment is corrected by ( w/3l) Ḃ effect due to finite n-value /5

30 Magnetization loops Finite length effect at 77 K Major & minor loops SuperOx at 77 K Moment per unit length m, A m l = 4mm l = 5mm l = mm Moment per unit length m, A m major loop minor loop VSM solid model dashed.5.5 Magnetic field B a, T Saturated state of the finite length tape: m sat = ( l 4 I c w w ) ( ) /n Ḃw 2n 3l 2E c 2n + Moment is corrected by ( w/3l) Ḃ effect due to finite n-value Magnetic field B a, T major loops: minor loops: 6 T at 77 K T at 5 K T at 77 K 9 T at 5 K good agreement with the modelling, if j c (B,θ) is well defined /5

31 Shielding effect of the stacked tapes 4 mm width tapes Moment per unit length per tape m/nt, A m mm width tapes n t = n t = 8 n t = 6 n t = Magnetic field B a, T /5

32 Shielding effect of the stacked tapes 4 mm width tapes Moment per unit length per tape m/nt, A m mm width tapes n t = n t = 8 n t = 6 n t = Magnetic field B a, T Tapes were not damaged during the stacks fabrication (curves overlap in the high-field zone) /5

33 Shielding effect of the stacked tapes 4 mm width tapes Moment per unit length per tape m/nt, A m mm width tapes n t = n t = 8 n t = 6 n t = Magnetic field B a, T Tapes were not damaged during the stacks fabrication (curves overlap in the high-field zone) Smoothing of the curves is due to the self-field effect /5

34 Shielding effect of the stacked tapes 4 mm width tapes Moment per unit length per tape m/nt, A m /4 3 mm width tapes n t = n t = 8 n t = 6 n t = Magnetic field B a, T Tapes were not damaged during the stacks fabrication (curves overlap in the high-field zone) Smoothing of the curves is due to the self-field effect Tapes of different widths have comparable I c field dependence /5

35 Orientation of the magnetic field single tape at 5 K VSM sample holder Moment per unit length, A m.5.5 θ a = : B a, m z θ a = 45 : B a, m z θ a = 45 : B, m vibration # B. # B a θ a # m # B Magnetic field, T Magnetization of the tapes correspond mainly to # B component, with a slightly reduced j c due to # B component One should take into account the torque acting on the sample # τ = # m # B a in the measurements (possible mechanical issue) 2/5

36 Outline Introduction 2 Numerical model 3 Samples for the VSM measurements 4 Experimental results 5 Magnetization loss 6 Conclusion

37 VSM vs modelling.2 T = 5K, B a = ± T T = 77K, B a = 2. ±.5T Normalized magnetization loss per tape Model Model 4 mm, 4 mm, 45 3 mm, SuperOx 4 mm, Number of tapes n t Number of tapes n t 3/5

38 VSM vs modelling.2 T = 5K, B a = ± T T = 77K, B a = 2. ±.5T Normalized magnetization loss per tape Model Model 4 mm, 4 mm, 45 3 mm, SuperOx 4 mm, Number of tapes n t Number of tapes n t Numerical and experimental results are in good agreement, but I c angular dependence used in the model for the SP tapes is not well representative. 3/5

39 VSM vs modelling.2 T = 5K, B a = ± T T = 77K, B a = 2. ±.5T Normalized magnetization loss per tape Model Model 4 mm, 4 mm, 45 3 mm, SuperOx 4 mm, Number of tapes n t Number of tapes n t Numerical and experimental results are in good agreement, but I c angular dependence used in the model for the SP tapes is not well representative. 3 mm wide stacks have stronger demagnetization effect (ratio D/w is smaller) 3/5

40 VSM vs modelling.2 T = 5K, B a = ± T T = 77K, B a = 2. ±.5T Normalized magnetization loss per tape Model Model 4 mm, 4 mm, 45 3 mm, SuperOx 4 mm, Number of tapes n t Number of tapes n t Numerical and experimental results are in good agreement, but I c angular dependence used in the model for the SP tapes is not well representative. 3 mm wide stacks have stronger demagnetization effect (ratio D/w is smaller) Presumably, less regular results at 77 K is due to small absolute values of the loss ( 3 times smaller than the values at 5 K) 3/5

2Hz. Magnetization loss was extracted by the data extrapolation to Hz.

41 Comparison with the EDIPO test results Magnetization loss per strand Q [ J/(m cycle) ] for B dc = 2T and B ac =.3T at 5 K: SP SO HTS cables in the EDIPO test.2.5 Numerical model.2.22 Analytical approach.2.8 Frequency range of the applied AC field in the EDIPO test is. 2Hz. Magnetization loss was extracted by the data extrapolation to Hz. Twisting of the strand has been considered by varying θ a in the calculation Conclusions from the VSM study of the field s orientation effect on the magnetization loss were used in the analytical approach 4/5

42 Outline Introduction 2 Numerical model 3 Samples for the VSM measurements 4 Experimental results 5 Magnetization loss 6 Conclusion

43 Using the numerical modelling, analytical approach for the magnetization loss of the stack with arbitrary number of tapes is proposed. Various effects of the stack on its magnetization behavior have been figured out using the VSM measurements. Developed numerical tools are well agreed with the different experimental results and will be used further for assessment of the stack magnetization. Next modelling tasks to validate the transport current term and to improve the optimization algorithms. Acknowledgments Enric Pardo helpful advices regarding the modelling Nikolay Mineev j c (B,θ) data for the SuperOx tapes at 77 K Giordano Tomassetti advanced optimization algorithms 5/5

Practical considerations on the use of J c (B,θ) in numerical models of the electromagnetic behavior of HTS INSTITUTE OF TECHNICAL PHYS

Practical considerations on the use of J c (B,θ) in numerical models of the electromagnetic behavior of HTS INSTITUTE OF TECHNICAL PHYS Francesco Grilli and Víctor M. R. Zermeño Karlsruhe Institute of