1. MCZ 050218 1 Magnetic Allignment of Coils for NCSX M.C. Zarnstorff 11 December 2006

2. MCZ 050218 2 Goals Need to align magnetic fields to achieve target physics properties – stellarator symmetry – particularly: low-order resonant errors  flux-surface breakup Present plan is to align coils mechanically – Issue: tolerance buildup Would be better to align the coils by directly aligning the magnetic fields Today’s talk: A practical proposal for how to do this. Specific Goals: 1. Measure errors in coil placement and orientation, to give feedback for accurate positioning 2. Characterize errors in coil shapes, particularly whether all coil of the same design have the same shape 3. Provide accurate measurements of deviations from stellarator symmetry

3. MCZ 050218 3 General Approach: Mutual Inductances Use mutual inductances between coils to measure moments of B-field distribution For an set of n coils, have n(n-1)/2 linearly independent mutual inductances For each coil, there are 3 location coordinates and 3 angles of orientation For a set of n coils, there are 6(n-1) relative coordinate and orientation angle parameters For n  12, the mutual inductances contain enough information to solve for the relative positions and orientations of the coils, in principle. For n>12, have additional information, which can be used to determine the coil shapes Issue: Care about small deviations in the mutual inductances. Measuring absolute mutual inductances will be sensitive to systematic measurement errors and uncertainties. M 4,6

4. MCZ 050218 4 Better Approach: Null Symmetric Differences To increase sensitivity and reduce systematic effects, measure differences between mutual inductances that ought to be identical, due to a design symmetry Example: for a symmetric toroidal solenoid, drive two coils in series and measure induced voltage across series combination of two other coils Difference in symmetric mutual inductances should be zero. – Improves diagnostic sensitivity: measuring deviation from zero. – Deviation from zero is linear in deviation of coil positions or shapes from ideal (to 1 st order) – Less sensitive to systematics. For n even, there are n(n-2)/2 such symmetric differences.  need n  13 to be able to determine relative location and orientation parameters of coils Also can measure (n-1) differences of self-inductances, using a impedance- bridge. These differences are only sensitive to shape deviations. M 4,6 – M 12,10

5. MCZ 050218 5 Null Symmetric Differences Subtlety: Typically measuring sums of 4 mutual inductances. Sums and differences give the simple differences. E.g. M 4,6 + M 12,10 + M 4,10 + M 12,6 with overbar = coil reversal. Together these give both M 4,6 – M 12,10 M 4,10 – M 12,6 Only exception is when drive or sense coils are the the same. E.g., M 4,6 – M 8,6 = M 4,6 + M 8,6 or M 4,6 – M 4,2 = M 4,6 + M 4,2 M 4,6 – M 12,10

6. MCZ 050218 6 NCSX Final Assembly 18 = 6*3 Modular coils 18 TF coils 12 PF coils 38 = 15 + 17 + 6 Differences of self inductances 964 linearly independent, symmetric differences of mutual inductance Sufficient to determine the 270 relative location and orientation parameters In principle, should be able to measure 732 relative deviation moments of coil shape. ~15.3/coil

7. MCZ 050218 7 Diagnostic sensitivity Apply AC voltage to drive coils For field coils, L is large (> 0.001 H) and R is small (<0.01 Ohms) so the reactive term dominate for f>10 Hz So, if drive the coils with ~10V, measurements of 10  V response gives sensitivity to  M/L ~10 -6. May be able to do even better When measuring differences of mutuals involving dissimilar coils (e.g. MC to TF), driving the coils with the smallest L will give the largest sensitivity

8. MCZ 050218 8 Sensitivity  M MC1, X /L From perturbations to MC1 Location and shape (1mm amplitude) See: easily have sensitivity at this level. Probably sensitive to ~0.01 mm displacements

9. MCZ 050218 9 Sensitivity  M TF1, X /L From perturbations to TF1 Location and shape (1mm amplitude)

10. MCZ 050218 10 Sensitivity  M PF4, X /L From perturbations to PF4 Location and shape (1mm amplitude)

11. MCZ 050218 11 NCSX Half-Period Assemblies When only have two half-period assemblies, have 6 MC coils 6 linearly independent symmetric differences of mutual inductance 3 differences of self-inductance Insufficient to determine the 30 relative location and orientation parameters

12. MCZ 050218 12 Half-Period Assemblies: Introduce test jig Mount between two half-periods Arrays D and E should have ~10 coil each. Start by using mutuals between D, E, and F to self-allign.