Modeling of superconductors interacting with non-linear magnetic materials: 3D variational principles, force-free effects and applications

Transcription

1 Modeling of superconductors interacting with non-linear magnetic materials: 3D variational principles, force-free effects and applications Enric Pardo, Milan Kapolka Institute of Electrical Engineering Slovak Academy of Sciences

2 What is the optimum enery loss in superconductors? Two tapes connected at the ends For you: For superconductors: B a B a +J c +J c -J c -J c the lowest possible the highest possible! Superconductors optimize the entropy production, not the loss! Superconducors can be modelled as an optimization problem

3 Modeling of superconductors interacting with non-linear magnetic materials: 3D variational principles, force-free effects and applications Enric Pardo, Milan Kapolka Institute of Electrical Engineering Slovak Academy of Sciences

4 Modeling of superconductors interacting with non-linear magnetic materials: 3D variational principles, force-free effects and applications Enric Pardo, Milan Kapolka Institute of Electrical Engineering Slovak Academy of Sciences We acknowledge funding from: VEGA 2/0126/15 ERDF ITMS ITMS

5 Talk available at zenodo.org Citable DOI

6 General variational principle Power applications 3D modelling Non-linear magnetic materials

7 Flux-free effects cause anisotropic E(J) vortex does not create force Two critical currents: J c creates force J c Anisotropic power law: A Badia, C Lopez DOI: / /28/2/024003

8 Minimum Magnetic Entropy Production (MEMEP) Equations for given E(J) relation are the Euler-Lagrange equations of J change between two time instants A from DJ A from applied field scalar potential E Pardo et al. DOI: / /28/4/ A Bossavit DOI: / L Prigozin DOI: / A Badia, C Lopez DOI: / /28/2/024003

9 Minimum Magnetic Entropy Production (MEMEP) You find J by minimizing the functional scalar potential Cross-sectional models: if you keep the current constrains, you can ignore the scalar potential E Pardo et al. DOI: / /28/4/044003

10 General variational principle Power applications Transformers Magnets 3D modelling Non-linear magnetic materials

11 General variational principle Power applications Transformers Magnets 3D modelling Non-linear magnetic materials

12 Transformer with Roebel cable in low-voltage winding 1 MVA 11 kv/415 V 3 phase transformer Robinson Research Institute in Wellington and industrial partners Roebel cable solenoid

13 AC loss agrees with model E Pardo et al. DOI: / /28/11/ Copper current leads cause eddy current loss consistent with estimations Real large scale application ~1200 turns or strands

14 General variational principle Power applications Transformers Magnets - Real geometry - Continuous approximation - Screening current induced field 3D modelling Non-linear magnetic materials

15 General variational principle Power applications Transformers Magnets - Real geometry - Continuous approximation - Screening current induced field 3D modelling Non-linear magnetic materials

16 Example winding 26 pancakes 400 turns per pancake more than turns

17 Anisotropic field dependent J c SuperPower tape I c of 4 mm tape [A] Fit of J c from measurements at 4.2 K D K Hilton et al SuST Results useful for high-field magnets angle [degrees]

18 Anisotropic field dependent J c Power-law exponent: 30 SuperPower tape I c of 4 mm tape [A] Fit of J c from measurements at 4.2 K D K Hilton et al SuST Results useful for high-field magnets angle [degrees]

19 Important screening currents 2 A/s ramp current axial position [mm] J/J c,self-field time radius [mm] E Pardo arxiv:

20 Detailed current density at all turns J/J c,self-field axial position [mm] radius [mm] No approximation made! All turns interact with all turns Computing time around 20 days Possible to make faster calculations? E Pardo arxiv:

21 General variational principle Power applications Transformers Magnets - Real geometry - Continuous approximation - Screening current induced field 3D modelling Non-linear magnetic materials

22 Continuous approximation Pancake coil approximated by taking: Less turns No separation between turns real coil continuous approximation L Prigozhin, V Sokolovsky DOI: / /24/7/075012

23 Practically the same results but faster! Real Continuous approximation axial position [mm] J/J c,self-field radius [mm] radius [mm]

24 We computed up to turns real turns: 2.7 hours turns: 2 days fulfills requirements for high-field magnets time [s] H W Weijers et al IEEE TAS continuous S Awaji et al IEEE TAS number of turns

25 Up to elements in the superconductor Computing time scales as second power time [s] real continuous fit number of elements

26 General variational principle Power applications Transformers Magnets - Real geometry - Continuous approximation - Screening current induced field 3D modelling Non-linear magnetic materials

27 Screening currents are important E Pardo arxiv: current generated field at maximum current 15 T time 4 % of maximum generated field Important for MRI and NMR B from screening currents [T] magnetic field at bore center current [A]

28 Screening currents are important E Pardo arxiv: current Stationary state after several cycles time B from screening currents [T] magnetic field at bore center current [A]

29 The variational method is efficient for large number of elements Promising for 3D modelling

30 General variational principle Power applications 3D modelling Non-linear magnetic materials

31 General variational principle Power applications 3D modelling Novel variational principle Force-free effects in films 3D bulk Non-linear magnetic materials

32 Novel 3D variacional principle M Kapolka, E Pardo arxiv: current potential T is the minimization variable or You can forget about scalar potential! Still easy to take transport currents into account

33 Thin surface Size X B a

34 Model agrees with thin film formula Power-law exponent 1000 Tape midplane Applied field: 20 mt

35 General variational principle Power applications 3D modelling Novel variational principle Force-free effects in films 3D bulk Non-linear magnetic materials

36 Flux-free effects cause anisotropic E(J) vortex does not create force Two critical currents: J c creates force J c Anisotropic power law: A Badia, C Lopez DOI: / /28/2/024003

37 Flux-free effects in thin films There is a force-free component J c =3J c Mishev et al. DOI: / /28/10/ Force-free J c Usual depinning J c

38 General variational principle Power applications 3D modelling Novel variational principle Force-free effects in films 3D bulk Non-linear magnetic materials

39 Flux-free effect increases J c Perpendicular field component: 23 mt Film top view J/J c B a Ba B a

40 Asymmetric current saturation Perpendicular field component: 50 mt Film top view J/J c B a Ba B a

41 General variational principle Power applications 3D modelling Novel variational principle Force-free effects in films 3D bulk Non-linear magnetic materials

42 3D bulk Frequency: 50 Hz sinusoidal Power-law exponent: 100 J c = 10 8 A/m 2 applied field amplitude 200 mt

43 Good resolution for 3D more than degrees of freedom in the superconductor J y /J c no symmetries taken into account yet B a

44 3D current flow J /J c

45 Vertical component is important power law 100

46 Vertical component is important Smooth E(J) relation cannot be the cause power law 30 power law 100

47 3D variational principle for the magnetic material Reversible non-linear materials Equation B created by M B from currents non-linear relation applied B is the Euler-Lagrange equation of

48 3D variational principle for the magnetic material Problem restricted to the magnetic material volume Functionals for magnetic material and superconductor solved iteratively

49 Superconductor with non-linear magnetic substrate m 0 M [T] saturation m 0 M 300 mt initial susceptibility 500 magnetic material m 0 H [T] axis superconductor radial direction

50 Magnetic substrate saturates in part of the coil 100 turns, 50% of critical current z [mm] J/J c z [mm] M r /M s z [mm] M z /M s r [mm]

51 Conclusions

52 Cross-sectional variational method Results agree with experiments : transformer Models coils with more than turns for real cross-section turns for continuous approximation Up to half million elements in the superconductor High potential for 3D modelling

53 3D modelling Novel variational principle in T formulation Takes force-free effects into account Thin films: Tilted magnetic field changes current patterns Bulk samples: 3D current penetration Significant vertical component due to shape effects Full power of the method still not achieved!

54 Non-linear magnetic material 3D variational principle Restricts to the material volume Modelled coils with magnetic substrate for one hundred turns

55 Variational methods are suitable for power and magnet applications

56 Thank you for your attention!

57 Would you like to know more? Talk available at zenodo.org

58 AC loss in test coils agrees with experiments E Pardo et al. DOI: / /28/4/ loss [J/m] Magnetic field dependent J c Magnetic field dependent power-law exponent current amplitude [A]

59 Thin surface Frequency: 50 Hz sinusoidal Power-law exponent: 30 J c = A/m 2 Size X

60 Jc(B) dependence J c Same penetration of J as constant J c J decreased because of B (Kim model) B

61 Power loss loss [W] Highest loss at first ramp Sharp peaks at remnant state loss current [arbitrary units] time [minutes]