Perturbation of vacuum magnetic fields in W 7

Perturbation of vacuum magnetic fields in W 7 X 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 1

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

Island geometry target res o na nt rad ial fie ld x-point dri ~ √ (bmn/(i'*m)) 3 Johann Kißlinger

Simulation of deviation with different wave length Difference of deviation aab 14 -aab 17 nlen = 3 cross-section # nlen = 1 nlen = 5 4 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. 28 mm, max. displacement 0. 55 mm j = 36° j = 0° j = 72° 5 Johann Kißlinger

Perturbation with mainly B 22 field component lateral and radial deviation of up to 7 mm B 11/Bo 0. 3*10 - 4 , B 22/Bo 1. 9*10 - 4 deviation: average 3. 6 mm data set: dl 07 ds 07 l 5 s 07 Johann Kißlinger 6

Statistical declination of whole modules up to 0. 1° This specific distribution: B 11/Bo 2. 3*10 - 4 , deviation: average max. 2. 3 7. 4 mm j = 180° j = 0° 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 7 Johann Kißlinger

Statistical declination of whole modules up to 0. 1° first contact with target B 11/Bo 2. 3*10 - 4 j = 180° declination of modular coils of 0. 02° along a helical axis B 11/Bo 1. 7*10 - 4 j = 0° 8 Johann Kißlinger

Footprints on targets with perturbed field, standard case each field period is statistically rotated by 0. 1° (3 axis). top target period 1 bottom targets period 2 period 3 period 4 period 5 9 Johann Kißlinger magnetic field perturbation

Statistical shift of whole modules up to 3 mm This specific distribution: B 11/Bo 1. 1*10 - 4 , deviation: average 2. 5 max. 3 mm 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 Johann Kißlinger 10

Equal perturbation have different influences at different iota values high iota standard case FP 2 low iota FP 3 Dx ≈ R* Bmn /((i - ir)*m) with i - ir >> i' Dx 11 Johann Kißlinger

Footprints on targets with perturbed field, high 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 12 Johann Kißlinger magnetic field perturbation

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 13 Johann Kißlinger magnetic field perturbation

Partly compensation of the field component B 11 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 FP 1 2 3 4 5 Currents in control coils top 10 -15 -18 0. 0 25 k. A bottom 0. 0 25 10 -15 -18 k. A 14 Johann Kißlinger

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 = 12 G. 15

Coordinate system for the coils M´ 16

Coordinate system for the coils M´ 17

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Assumptions and scheme of modeling 19 T. Andreeva

7 real and 43 simulated coils AN 11 = 1. 3 G; AN 22 = 1. 12 G 20 T. Andreeva

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

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

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

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

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

Effect of shift and rotation on d. B (a=b=g=0. 05 degree) 26 T. Andreeva

Effect of shift and rotation on d. B (a=b=g=0. 1 degree) 27 T Andreeva

Effect of shift and rotation on d. B (a=b=g=0. 2 degree) 28 T. Andreeva

Effect of shift and rotation on d. B (a=b=g=0. 3 degree) 29 Tamara Andreeva

d. B for an average deviation of 1 mm caused by different types of coil errors Average perturbation x 10 -4 Maximum perturbation x 10 -4 30 T. Andreeva

Conclusions: • Deviations with an average value of 1. 5 to 2 mm 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 d. 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. 31