EMMA – Orbit Correction

1. EMMA – Orbit Correction David Kelliher ASTeC/STFC/RAL FFAG08, Manchester, Sept 1st - 5th , 2008

2. Contents • Orbit distortion in EMMA • Vertical orbit correction • Linear approximation method • Corrector magnet strengths • Conclusion

3. Orbit distortion in EMMA • Rapid acceleration in a non-scaling FFAG ensure that effects of integer ‘resonances’ are not seen in the orbit distortion. • Standard harmonic correction is not applicable since the phase advance between lattice elements varies with momentum.

4. Amplification factor

5. Vertical orbit correction - Introduction • It is planned to include 16 vertical corrector magnets in EMMA, one in every other cell apart from in the injection and extraction cells. Corrector magnet strengths will remain constant during acceleration. • The problem of finding the optimal corrector magnet strengths is a multidimensional minimisation. Two methods considered – brute force search and linear approximation. • PTC used to track in EMMA. Displaced quadrupoles with fringe field included. Cavities added so that the accelerated orbit distortion is calculated.

6. Vertical corrector magnet location Inside of ring D Cavity F 210mm

7. Brute force search (1) PTC tracking code used to calculate vertical orbit distortion. Corrector strengths varied, one at a time, in the range -30 to 30 mrad. Search for minimum in orbit distortion.

8. Brute force search (2) Optimise corrector magnet strengths Magnet misalignments Cancel distortion at BPMs PTC Vertical Corrector deflections

9. Brute force search (3) “Experiment” simulation Magnet misalignments BPM measurements PTC Optimise corrector magnet strengths No Magnet misalignments Cancel distortion at BPMs PTC Vertical Corrector deflections

10. Brute force search (4) • Tracking in perfectly aligned lattice with BPM distortion measurements and optimising corrector magnet deflections produces similar results to tracking in misaligned lattice.

11. Brute force search summary • Advantage: • Robust, works at all acceleration rates. • Disadvantages: • Slow – many PTC runs required in scanning through possible corrector strengths.

12. Linear Approximation method • Use differential algebra (DA) to find, to first order, dependence of orbit distortion on corrector magnet strengths. • Build up set of first order Taylor coefficients that relates the vertical position measurement made at each BPM and at each turn to each corrector magnet strength. • Linear least squares problem solved to find optimal . The target is to cancel the orbit distortion measured at each BPM, turn-by-turn.

13. Determining Taylor coefficients in simulation • Plug the magnet misalignments into PTC. The code contains a DA subroutine that allows the Taylor coefficients to be calculated. The expansion is done about zero corrector deflection. • In real experiment the magnet misalignments may be difficult to determine with accuracy.

14. Determining Taylor coefficients in practice • Find Taylor coefficients from BPM measurements by introducing a small deflection at one corrector magnet and noting the change in distortion and dividing by deflection. Repeat for each corrector magnet. • At fast acceleration rates (e.g. 120 kV per cavity), the result agrees well with DA calculation in PTC.

15. Determining Taylor coefficients in practice • For slow acceleration rates (e.g. 60 kV per cavity) the linear approximation becomes more inaccurate as the number of turns increases.

16. Compare linear approximation to PTC • Error in orbit distortion calculated by linear approximation method grows with corrector magnet deflection and with number of turns taken to complete acceleration. 60 kV per cavity, 50 micron misalignment, 10 seeds

17. Linear Approximation method summary • Advantages: • Substantially faster than brute force search. No tracking required. • Disadvantages: • More inaccurate at slower acceleration rates and at greater corrector magnet deflections.

18. Correction results 100 misalignment cases included. The mean misalignment is 50 microns. The results for 16 correctors and 4 correctors are shown. In the latter case, the 4 correctors available are the first four of the 16.

19. Corrector magnet strengths Single most effective corrector magnet out of 16 included Deflections defined at 15 MeV

20. Corrector magnet strengths Compare inclusion of single most effective corrector magnet and all 16 magnets

21. Compare correction methods Single most effective corrector magnet only is included for 20 random error seeds at 120kV per cavity.

22. Effect of changing acceleration speed • Use brute force method to calculate corrector strength at a range of voltages • Single most effective corrector magnet included • Only one error seed studied to date. Further work needed.

23. Required corrector magnet field strength Relate magnet deflection angle to corrector magnetic field times length • Rigidity (15 MeV)= 0.051 T m •  B.L = 13 G m. • B = 325 G • Requirements rule out printed circuit magnets

24. Conclusions (1) • The optimal corrector magnet deflections calculated by a relatively quick linear approximation method produced results consistent with a brute force search • However, the linear approximation method becomes more inaccurate at slower acceleration rates and at greater magnet misalignments. • It was found that it is advantageous to have 16 corrector magnets. The four most effective of these correctors account for, on average, 95% of the total correction achieved.

25. Conclusions (2) • Average corrector magnet strengths rise with the level of magnet misalignments. The maximum corrector deflection angle found is about 20mrad at 15 MeV. • Increasing the number of correctors does not reduce the maximum deflection required. • More work needs to be done to determine how the corrector magnet strengths depend on the acceleration rate.