1. Instructions for the Rod Magnet July 6, 2015Stephen Brooks, Cbeta project1 Updated to include source-only field model

2. Field of a  =1 Rod with Constant M Linear potential within rod and 1/r outside So constant B within rod and 1/r 2 outside Actually identical to cos(  ) SC dipole July 6, 2015Stephen Brooks, Cbeta project2 double potf(const double x,const double y,const double r) { double rr=(x*x+y*y)/(r*r); //if (rr>=1) return x/rr; else return x; if (rr>=1) return y/rr; else return y; } V2 fieldf(const double x,const double y,const double r) { // (d/dx,d/dy) of potf double rr=(x*x+y*y)/(r*r); if (rr>=1) return V2_new(x*x-y*y,2*x*y)/(rr*rr*r*r); else return V2_new(1,0); } These functions give the 2D potential and B field for a rod whose centre is at (x,y)=(0,0), has radius r and has magnetisation M=(1,0) T. Rotate and scale for M as needed. B r =|M| (I use 1.11T±10%) In the  =1 model these may be summed over all rods.

3. Avoiding Demagnetising Settings The definition of the H field is: – H = (1/  )B – M In our  =1 model, this reduces to H = B – M This quantity can be calculated around the perimeter of each rod and checked against the demagnetisation B-H curve of the material The Halbach arrangements should be favourable July 6, 2015Stephen Brooks, Cbeta project3

4. Theoretical Optimisation Method In my program, the response matrix dB i /d  n is first calculated using small finite differences – B i are the field components measured by probes –  n are the angles of the rods A good method is to use multi-variable Newton iteration modified to use a smoothed inverse using the SVD of the response matrix – See http://stephenbrooks.org/ap/report/2015-3/svdboundedsolve.pdf http://stephenbrooks.org/ap/report/2015-3/svdboundedsolve.pdf – You also want to use a line search + tuning of July 6, 2015Stephen Brooks, Cbeta project4

5. Fast Field Model even with Iron If  r at each point is constant (no saturated iron), the B field equations are linear in M B =  rods n B rn (B nx cos  n + B ny sin  n ) – …where B nx, B ny are pre-calculated FEM fields for having only rod n magnetised by M=(1,0)T and (0,1)T respectively, other rods M=0 – B rn is the remnant field magnitude in rod n –  n is the angle of rod n Also works for 2 magnets with interfering iron July 6, 2015Stephen Brooks, Cbeta project5

6. Required Setup Measurement: card of 10-20 Hall probes, or rotating coil aligned to magnet survey point Movement: stepper motor drivers to rotate each rod by known angles, probably requires gearing down to get mrad type accuracy – Also “brake” to clamp rods in place once done Control algorithm: computer program that can both read the field measurements and control the motors automatically July 6, 2015Stephen Brooks, Cbeta project6

7. Determining Initial Rod Angles Rotate rod n by  (to  n +  ) measure field B + Rotate back by -2  (to  n -  ) measure field B - Rotate by  to return to original position  n From the formula on the last page, B + -B - = B rn [B nx (cos(  n +  )-cos(  n -  )) + B ny (sin(  n +  )-sin(  n -  ))] = (2 sin  )[B nx (-B rn sin  n )+B ny (B rn cos  n )] Solve for -B rn sin  n and B rn cos  n, giving B rn,  n July 6, 2015Stephen Brooks, Cbeta project7

8. Use of Model for Correction Early concept was to determine local response matrix by small rod rotations each iteration – This was rather susceptible to noise as it involves differentiation between measured signals Instead, given approximate knowledge of B rn,  n the model can calculate rotations to cancel the measured error by ensuring B model,rotated - B model,now ~= B goal - B measured July 6, 2015Stephen Brooks, Cbeta project8

9. Accuracy of Model If |  n,model -  n | ≤  and |B rn,model -B rn | ≤  |B model -B| ≤  rods n (  +B rn  )(|B nx |+|B ny |) To first order B rn can be replaced by B rn,model More important is accuracy of small changes |dB model /d  n - dB/d  n | ≤ (  +B rn  )(|B nx |+|B ny |) May run into problems when d/d  is small? – Late stage optimisation often does comparatively large movements with very small effects July 6, 2015Stephen Brooks, Cbeta project9

10. Iterative Correction Convergence If the following condition is met for f<1: |  B model -  B| ≤ f|  B model | …the error should reduce by at least a factor f each time the model’s correction is applied – Excluding errors uncorrectable even in the model This isn’t true when  B model is almost zero but  B isn’t, as could happen in late optimisation – Could use new data to refit model B rn and initial  n Linear problem in [B rn cos  n,B rn sin  n ]  [B measured ] July 6, 2015Stephen Brooks, Cbeta project10