1. 1 Perturbation of vacuum magnetic fields in W7X due to construction errors 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 T. Andeeva Y. Igitkhanov J. Kisslinger

2. 2 Johann Kißlinger inexact coil shape positioning errors during assembly (shift and declination) Four assembly steps coil imbedding half module assembly module assembly device integration Construction error possibilities 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.

3. 3 Johann Kißlinger target Island geometry x-point resonant radial field  r i ~ √ (b mn /(  ' *m))

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

5. 5 Johann Kißlinger Sensitivity of the system Perturbation by declination of modular coils of 0.02° along a helical axis with m=1 resonant fourier component B 11 /Bo 1.7*10 - 4, average displacement 0.28mm, max. displacement 0.55mm  = 0°  = 36°  = 72°

6. 6 Johann Kißlinger Perturbation with mainly B 22 field component lateral and radial deviation of up to 7 mm data set: dl07 ds07 l5 s07 B 11 /Bo 0.3*10 - 4, B 22 /Bo 1.9*10 - 4 deviation: average 3.6mm

7. 7 Johann Kißlinger Statistical declination of whole modules up to 0.1°  = 0°  = 180° This specific distribution: B 11 /Bo 2.3*10 - 4, deviation: average max. 2.3 7.4mm 30 different distributions: fourier coef. B 11 B 22 B 33 B 44 average 1.9 0.5 0.3 0.1 max. value 4.4 1.0 0.6 0.2 average dev. 3.4 mm

8. 8 Johann Kißlinger Statistical declination of whole modules up to 0.1°  = 0° B 11 /Bo 2.3*10 - 4 B 11 /Bo 1.7*10 - 4 declination of modular coils of 0.02° along a helical axis  = 180° first contact with target

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

10. 10 Johann Kißlinger Statistical shift of whole modules up to 3mm 10 different distributions: fourier coef. B 11 B 22 B 33 B 44 average 0.5 0.6 0.2 0.1 max. value 1.1 1.2 0.35 0.15 average dev. 1.75 mm This specific distribution: B 11 /Bo 1.1*10 - 4, deviation: average 2.5 max. 3mm

11. 11 Johann Kißlinger high iota standard caselow iota Equal perturbation have different influences at different iota values FP 2 FP3  x ≈ R* B mn /((  -  r )*m) with  -  r >>   '  x

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

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

14. 14 Johann Kißlinger Partly compensation of the field component B 11 by use of the control coils with individual currents 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 Field perturbation by statistical declination of 0.1° around 3 axis of whole periods, no compensation

15. 15 Compensation by a constant horizontal magnetic field Field perturbation by statistical declination of 0.1° around 3 axis of whole periods. Compensation of B 11 component with Bx = 12G.

16. 16 Coordinate system for the coils M´M´

17. 17 Coordinate system for the coils M´

18. 18

19. 19 T. Andreeva Assumptions and scheme of modeling

20. 20 7 real and 43 simulated coils AN 11 = 1.3G; AN 22 = 1.12G T. Andreeva

21. 21 Effect of coil shifts on  B T. Andreeva

22. 22 Effect of rotation on  B (  -varies) T. Andreeva

23. 23 Effect of rotation on  B (  -varies) T. Andreeva

24. 24 Effect of rotation on  B (  -varies) T. Andreeva

25. 25 Effect of rotation on  B (  =  ) T. Andreeva

26. 26 Effect of shift and rotation on  B (  degree  T. Andreeva

27. 27 Effect of shift and rotation on  B (  =0.1 degree) T Andreeva

28. 28 Effect of shift and rotation on  B (  =0.2 degree) T. Andreeva

29. 29 Tamara Andreeva Effect of shift and rotation on  B (  =0.3 degree)

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

31. 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  B 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.