1. 10/2004Strickler, et. al1 Reconstruction of Modular Coil Shape from Magnetic Field Measurement D. Strickler, S. Hirshman, B. Nelson, D. Williamson, L. Berry, J. Lyon Oak Ridge National Laboratory 10/2004

2. Strickler, et. al2 Near Term Goals Demonstrate a method for accurately determining the modular coil current center based on magnetic field measurement. Validate the procedure on the racetrack coil.

3. 10/2004Strickler, et. al3 The shape of (as built) modular coils can be accurately determined from magnetic measurements We have developed both linear and nonlinear methods to reconstruct the shape of an NCSX modular coil from field measurements of either │B│or [B x, B y, B z ]. It appears that a three axis hall probe, measuring components of the magnetic field in local coordinates, will work best for our application. 2000-3000 field measurements within 10-15 cm of the winding center are sufficient to reconstruct the coil centerline shape with average deviation of ≈ 0.05 mm (max. 0.25 mm) for sinusoidal distortions of magnitude 2 mm, and ≈ 0.1 mm (max. < 0.5 mm) for distortions of magnitude 5 mm, assuming –Field error in the magnetic measurements consistent with the hall probe technology, –Random error of magnitude ≤ 0.1 mm in the location of measurements.

4. 10/2004Strickler, et. al4 3-in r min Measurement locations are constrained by size and shape of the winding pack Measurement region for data collection Measurements must be taken outside coil case 3D annular region around design coil centerline used for testing reconstruction methods: r min ≤ r ≤ r max

5. 10/2004Strickler, et. al5 Data will be collected by coupling a hall probe to a multi-axis measuring device Simultaneously measure magnetic field and location of measurement. NMR probes measure │B│ very accurately, but have large sensing volume and require high field. Hall probe (www.gmw.com) measures B x, B y, B z at lower field (10 – 2000 Gauss) with Sensing volume small relative to location error (±0.1 mm)www.gmw.com

6. 10/2004Strickler, et. al6 Accuracy of hall probe Components of field error: – offset error = ±1 gauss (can be eliminated) – 0.1% of reading = 0.001B – noise = ± 0.2 gauss Measurement location error: – position error of measuring device = 0.1 mm – field sensitive volume = [0.01 mm] 3

7. 10/2004Strickler, et. al7 What are the limits on the coil for field measurement? Pulse length at ~ 300 gauss: 300 sec for 20 C temperature rise or 70 sec for 5 C rise

8. 10/2004Strickler, et. al8 Numerical methods are compared for accuracy in shape reconstruction Linear approximation appears to give best results –Solve for corrections to design coil coordinates –Use singular-value decomposition Also considered nonlinear approximation methods –Parameterization of coil centerline –Fourier or cubic spline representation –Use Levenberg-Marquardt to find coefficients

9. 10/2004Strickler, et. al9 For perturbations of prescribed phase and magnitude about the design coil shape, questions include: Number and location of measurement points? Is there sufficient accuracy in the field measurements and measurement location?

10. 10/2004Strickler, et. al10 Procedure is tested on NCSX modular coil with known shape distortion Sine, cosine coil distortions : ΔR = δr sin (mθ) Δφ = δφ sin (mθ) ΔZ = δz sin (mθ) Design coil (solid) Distorted coil (dashed) m = 5 δr = δφ = δz = 5 mm Side view Top view Random error included in: Magnetic field meaurement Location of measurements m = 5 δr = δφ = δz = 5 mm Design coil (solid) Distorted coil (dashed)

11. 10/2004Strickler, et. al11 x 0 = [x 01,y 01,z 01, …,x 0n, y 0n, z 0n ] T ideal (design) coil coordinates B 0 = B(x 0 )magnetic field of ideal coil over 3D grid x 0 + Δxcoordinates of actual coil (manufactured) B 1 = B(x 0 + Δx)magnetic field of actual coil (measured) B 1 – B 0 = Δb ≈ B  ΔxB  = dB/dx is MxN Jacobian matrix B  = U∑V T SVD Δx = ∑ σn ≠ 0 (u n T Δb)v n / σ n least-squares solution for Δx Drop ‘small’ singular values σ n to stabilize solution, making it less sensitive to data vector Δb Linear Method Solve for coordinates of coil winding center

12. 10/2004Strickler, et. al12 Distance between distorted coil and solution Distorted coil represented by N=500 segments, unit tangent vectors e d. Evaluate solution x (approximating field of distorted coil) at N points. For each point x d on distorted coil, find nearest solution point x*, and perpendicular distance from solution point to distorted coil: Δx = x* - x d, Δx ║ = [Δx∙e d ]e d,Δx┴ = Δx - Δx ║ Distance between solution and distorted coil: Δ avg = 1/N ∑ i │Δx i ┴│, Δ max = max {│Δx i ┴│}

13. 10/2004Strickler, et. al13 Solution (solid) Distorted coil (dashed) No measurement error Distribution of Δx i ┴ for winding center reconstruction Nominal error (hall probe) [m] N Δ avg = 0.02 mm Δ avg = 0.06 mm

14. 10/2004Strickler, et. al14 For linear method, Δ max occures at points near plasma

15. 10/2004Strickler, et. al15 Field match δr, δφ, δz [mm] Δ max [mm] Δ avg [mm] │B││B│2.00.6720.163 B x, B y, B z 2.00.5480.103 │B││B│5.01.5760.316 B x, B y, B z 5.01.0820.178 For similar error in data, approximating magnetic field components gives better results than matching │B│ Coil distortion ΔR = δr sin (6θ), ΔZ = … 3495 field measurements, r ≥ 20 cm I = 20 kA, δB (noise) ≤ 0.2 gauss Error in measurement location ≤ 0.1 mm

16. 10/2004Strickler, et. al16 Field error δB noise [gauss] Field error – reading [%] Measurement location error [mm] Δ max [mm] Δ avg [mm] 0000.1450.023 0.20.0010.00010.2490.055 1.20.0010.00010.2820.115 For sinusoidal coil distortions of magnitude 2 mm, solutions with Δ avg < 0.1mm are obtained for field error assumptions based on hall probe sine coil distortion with m=6, δr = δφ = δz = 2.0 mm 2250 measurements of B x, B y, B z ; r ≥ 10 cm I = 20 kA

17. 10/2004Strickler, et. al17 Number of measurements r min [m]Δ max [mm] Δ avg [mm] 22500.100.2490.055 29690.150.2380.095 34950.200.8270.121 Field measurement near the coil improves accuracy in reconstruction of winding center shape sine coil distortion with m=6, δr = δφ = δz = 2 mm field measurement error δB ≤ 0.2 gauss measurement location error ≤ 0.1 mm

18. 10/2004Strickler, et. al18 δr, δφ, δz [mm] Δ max [mm] Δ avg [mm] 20.2490.055 50.4540.097 Accurate solutions are obtained for coil distortions up to δr = δφ = δz = 5 mm sine coil distortion with m=6 field measurement error δB ≤ 0.2 gauss, r ≥ 10 cm measurement location error ≤ 0.1 mm

19. 10/2004Strickler, et. al19 x = [x(t), y(t), z(t)]coordinates of coil centerline x(t) = a x0 + ∑ a x,k cos(2πkt) + b x,k sin (2πkt), …Fourier representation c = [a x0,a x1,b x1, …,a zn, b zn ] T vector of coefficients B = B(c)magnetic field over 3D grid Minimize χ 2 = ║B(c) – B 1 ║ 2 B 1 = field measurements on 3D grid Nonlinear Method Solve for coefficients in parameterization of coil winding center Levenberg-Marquardt method to solve for coefficients Initial guess for c based on fit to design coil data Option in code for cubic spline representation: x(t) = ∑ c x,k B k (t), …

20. 10/2004Strickler, et. al20 Nonlinear shape reconstruction of distorted modular coil requires > 150 of variable coefficients sin(5θ) distortion with δr = δφ = δz = 2mm Random measurement error in field: δB/B max ≤ 1.e-5, location ≤ 0.1 mm Distorted coil (dashed) Reconstruction (solid) Field error χ = 2.323e-5 T Distorted coil (dashed) Reconstruction (solid) Field error χ = 2.076e-4 T N c = 81N c = 159

21. 10/2004Strickler, et. al21 NcNc Δ max [mm] Δ avg [mm] 1591.0150.451 3150.3090.113 Difference between solution and distorted coil is slightly larger using the nonlinear method, for similar measurement error and distortion size sine coil distortion with m=6, δr = δφ = δz = 2.0 mm 2250 measurements, 10 ≤ r ≤ 15 cm I = 20 kA, δB (noise) ≤ 0.2 gauss Error in measurement location ≤ 0.1 mm

22. 10/2004Strickler, et. al22 Summary The “as built” NCSX modular coil winding center shape can be reconstructed with average error ≈ 0.05-0.1 mm for shape distortions of magnitude 2-5 mm and measurement error within limits of the hall probe technology. 2000 - 3000 field measurements within 10-15 cm of the (design coil) winding center are sufficient to reconstruct a coil centerline represented by ~500 segments.

23. 10/2004Strickler, et. al23 To further validate method: Use multifilament current representation to approximate field ‘measurements’ computed with magfor code Test procedure on racetrack coil

24. 10/2004Strickler, et. al24 Optimization of the Vacuum Field Configuration Summary of recent STELLOPT modifications for vacuum field optimization Example

25. 10/2004Strickler, et. al25 Module (VACOPT) added to stellarator optimization code (STELLOPT) to target resonances in vacuum magnetic field Minimize size of vacuum islands resulting from winding geometry errors by varying position of modular coils in array, or displacements due to magnetic loads / joule heating by varying coil currents. Additional STELLOPT variables include rigid-body rotations, shifts about coil centroid, and vacuum field coil currents. Targets include residues of prescribed resonances, bounds on variables, and constraints on position of the island O-points. Input list now contains poloidal mode number of targeted islands, initial values of vacuum field coil currents, shifts and rotations, and initial positions of control points for each modular coil. Output includes optimal coil currents and position of control points following optimal shifts / rotations of the modular coils.

26. 10/2004Strickler, et. al26 Targeting Magnetic Islands in Vacuum Field Optimization Locate O-points of islands (order m fixed points of return map) X i = (R i,0) Following [1], compute residues of targeted islands Res i = [2 – trace(T(X i ))]/4 Optimization Vary coil currents to minimize targeted residues: min. χ 2 = Σ w i Res i 2, Subject to possible constraints, e.g.: Σ I MOD + I TF = constant R min ≤ R i ≤ R max, (in prescribed toroidal plane v = v i ) Bounds on modular coil currents [1] Cary and Hanson, Phys. Fluids 29 (8), 1986

27. 10/2004Strickler, et. al27 Example: Vacuum field optimization by shifts and rotations about the centroids of modular coils Coil currents from 1.7T, high beta scenario at t=0.1s (vacuum) Minimize m=7 residue Constrain radius of O-point by R ≤ 1.18m (Z=0) Res = 0.053 Res = 0.004

28. 10/2004Strickler, et. al28 Δx (m)Δy (m)Δz (m)θφα M10.00050.01090.00020.9866-0.94710.0031 M20.0030-0.00190.0151-0.04940.2878-0.0058 M3-0.0007-0.00090.00220.06690.07910.0056 Example (cont’d): shifts and rotations about modular coil centroids for constrained minimization of the m=7 residue α m easured with respect to unit rotation vector centered at coil centroid with sperical coordinate angles θ, φ. Maximum changes in the control points are 1.17cm (M1), 1.58cm (M2), and 0.67cm (M3).

29. 10/2004Strickler, et. al29 Summary Vacuum islands resulting from winding geometry errors may be mimized by varying position of modular coils in array Vacuum islands resulting from coil displacements due to magnetic loads / joule heating may be reduced in size by varying coil currents. Need to account for non-stellarator symmetric field errors

30. 10/2004Strickler, et. al30 Additional Slides

31. 10/2004Strickler, et. al31 Quality of linear shape reconstruction depends on number of singular values retained (N σ ) Consider sin(5θ) distortion of modular coil M1 with δr = δφ = δz = 2 mm Random measurement error in field: δB/B max ≤ 1.e-5, location ≤ 0.1 mm σ min /σ max = 0.001, N σ = 213 σ min /σ max = 0.015, N σ = 132 χ field = 2.281e-5 T Δ avg = 4.75 mm χ field = 2.310e-5 T Δ avg = 0.33 mm Distorted coil (dashed) Reconstruction (solid) Distorted coil (dashed) Reconstruction (solid)

32. 10/2004Strickler, et. al32 σ min / σ max NσNσ χ field [T] Δ avg [m] 0.0151321.697e-54.268e-4 0.0251161.702e-52.464e-4 0.050941.707e-51.503e-4 0.075821.710e-51.494e-4 0.100741.718e-51.237e-4 0.125661.732e-51.527e-4 Linear coil shape reconstruction from magnetic field │B│ sine coil distortion with m=6, δr = δφ = δz = 0.002 m 3495 measurements, 0.2 m ≤ r ≤ 0.25 m error in measurement location ≤ 1.0e-4 m error in field measurement δB/B max ≤ 2.5e-4

33. 10/2004Strickler, et. al33 Magnetic field line is described by dx/ds = B(x) In cylindrical coordinates (and assuming B φ ≠ 0), these are reduced to two field line equations: dR/dφ = RB R /B φ, dZ/dφ = RB Z /B φ Integrating the field line equations, from a given starting point (e.g. in a symmetry plane), over a toroidal field period, produces a return map X  = M(X) (X = [R,Z] t ) An order m fixed point of M is a periodic orbit: X = M m (X) The dynamics of orbits in the neighborhood of a fixed point are described by the tangent map: δX  = T(δX) (whereT 11 = ∂M R /∂R,T 12 = ∂M R /∂Z, T 21 = ∂M Z /∂R,T 22 = ∂M Z /∂Z) Field Line Equations and Tangent Map