Perturbation of vacuum magnetic fields in W7X due to construction errors

1. Perturbation of vacuum magnetic fields in W7X due to construction errors T. Andeeva Y. Igitkhanov J. Kisslinger Outline: • Introduction concerning the generation of magnetic islands • Sensitivity of the magnetic configuration with iota=1 • Asymmetric target loads due to error fields • Impact of the perturbation fields on high and low iota case • Effect of coil shift • Effect of coil declination • Conclusion

2. Construction error possibilities • inexact coil shape • positioning errors during assembly (shift and declination) Four assembly steps • coil imbedding • half module assembly • module assembly • device integration • The construction errors produce • symmetry breaking perturbations. • introduce new island at any periodicity • modify existing islands • generate and enhance stochastic regions Perturbations due to inexact coil shape and positioning errors of single coils: Dr. Andreeva’s talk. Johann Kißlinger

3. Island geometry target x-point resonant radial field dri ~ √ (bmn/(i'*m)) Johann Kißlinger

4. Simulation of deviation with different wave length Difference of deviation aab14-aab17 nlen = 3 cross-section # cross-section # nlen = 1 nlen = 5 Johann Kißlinger

5. Sensitivity of the system Perturbation by declination of modular coils of 0.02° along a helical axis with m=1 resonant fourier component B11/Bo 1.7*10- 4 , average displacement 0.28mm, max. displacement 0.55mm j= 36° j= 0° j = 72° Johann Kißlinger

6. Perturbation with mainly B22 field component lateral and radial deviation of up to 7 mm B11/Bo 0.3*10- 4 , B22/Bo 1.9*10- 4 deviation: average 3.6mm data set: dl07 ds07 l5 s07 Johann Kißlinger

7. Statistical declination of whole modules up to 0.1° This specific distribution: B11/Bo 2.3*10- 4 , deviation: average max. 2.3 7.4mm j = 180° 30 different distributions: fourier coef. B11 B22 B33 B44 average 1.9 0.5 0.3 0.1 max. value 4.4 1.0 0.6 0.2 average dev. 3.4 mm j = 0° Johann Kißlinger

8. Statistical declination of whole modules up to 0.1° first contact with target B11/Bo 2.3*10- 4 j = 180° declination of modular coils of 0.02° along a helical axis B11/Bo 1.7*10- 4 j = 0° Johann Kißlinger

9. Footprints on targets with perturbed field, standard case each field period is statistically rotated by 0.1° (3 axis). period 1 bottom targets top target period 2 period 3 period 4 period 5 Johann Kißlinger magnetic field perturbation

10. Statistical shift of whole modules up to 3mm This specific distribution: B11/Bo 1.1*10- 4 , deviation: average 2.5 max. 3mm 10 different distributions: fourier coef. B11 B22 B33 B44 average 0.5 0.6 0.2 0.1 max. value 1.1 1.2 0.35 0.15 average dev. 1.75 mm Johann Kißlinger

11. Equal perturbation have different influences at different iota values high iota standard case low iota FP 2 FP3 Dx ≈R* Bmn /((i - ir)*m) withi - ir >>i'Dx Johann Kißlinger

12. Footprints on targets with perturbed field, high iota each field period is statistically rotated by 0.1° (3 axis). period 1 top targets bottom targets period 2 period 3 period 4 period 5 Johann Kißlinger magnetic field perturbation

13. Footprints on targets with perturbed field, low iota each field period is statistically rotated by 0.1° (3 axis). top targets period 1 bottom targets period 2 period 3 period 4 period 5 Johann Kißlinger magnetic field perturbation

14. Partly compensation of the field component B11 by use of the control coils with individual currents Field perturbation by statistical declination of 0.1° around 3 axis of whole periods, no compensation FP1 2 3 4 5 Currents in control coils top 10 -15 -18 0.0 25 kA bottom 0.0 25 10 -15 -18 kA Johann Kißlinger

15. Compensation by a constant horizontal magnetic field Field perturbation by statistical declination of 0.1° around 3 axis of whole periods. Compensation of B11 component with Bx = 12G.

16. Coordinate systemfor the coils M´

17. Coordinate system for the coils M´

19. Assumptions and scheme of modeling T. Andreeva

20. 7 real and 43 simulated coils AN11 = 1.3G; AN22 = 1.12G T. Andreeva

21. Effect of coil shifts on dB T. Andreeva

22. Effect of rotation on dB (a-varies) T. Andreeva

23. Effect of rotation on dB (b-varies) T. Andreeva

24. Effect of rotation on dB (g-varies) T. Andreeva

25. Effect of rotation on dB (a=b=g) T. Andreeva

26. Effect of shift and rotation on dB (a=b=g=0.05 degree) T. Andreeva

27. Effect of shift and rotation on dB (a=b=g=0.1 degree) T Andreeva

28. Effect of shift and rotation on dB (a=b=g=0.2 degree) T. Andreeva

29. Effect of shift and rotation on dB (a=b=g=0.3 degree) Tamara Andreeva

30. dB for an average deviation of 1mm caused by different types of coil errors Average perturbation Maximum perturbation x10-4 x10-4 T. Andreeva

31. Conclusions: • Deviations with an average value of 1.5 to 2mm with a statistical distribution may • generate effective field perturbations in the range of 2*10-4 given by the proposal. • Mainly the m=1, n=1 island appears. • The field perturbation go almost linearly with the amplitude of the deviation. • Due to the low-order islands the load on the targets is asymmetric. • In the high iota configuration the centre region is displaced while the edge region is not • so strong influenced. • The more systematic deviations due to rotation of coils and whole modules is more • effective in producing low order dB perturbations then the deviations of coil shape • and shift errors. • The small scale deviations of the manufacturing errors enhances the stochastic • structures at the edge. • The control coils are not very effective for compensating low-order error fields. Outlook: Continue the calculation in collaboration with the engineering team. Compensation of the low order error fields with the planar coils should be more effective but needs extra current feeders. Consider the possibility of evaluation of scaling law for the magnetic field perturbations.