Accelerator Physics Multipoles and Closed Orbits. G. A. Krafft Old Dominion University Jefferson Lab Lecture 13

Transcription

1 Accelerator Physics Multipoles and Closed Orbits G. A. Krafft Old Dominion University Jefferson Lab Lecture 13 Graduate Accelerator Physics Fall 015

2 Oscillators Similtaneously Excited u i t it u u Fe 1 it Fe 1 1 ut Many oscillators distributed in frequency N 1 dn Nd ui i1 U N it Fe 1 1 U d for U it Fe d Graduate Accelerator Physics Fall 015

3 Resonance Effect it Fe U i PV.. d it U Fe ip. V. d For our analytic Lorentzian it it Fe Fe U i i Energy goes in! Where does it go? Graduate Accelerator Physics Fall 015

4 Physical interpretation of Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 4

5 Linearity and superposition Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 5

6 Multipole fields Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 6

7 Multipole fields Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 7

8 Multipole fields Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 8

9 Multipole fields Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 9

10 Multipole fields Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1,

11 Multipole fields Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1,

12 Generating multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 1

13 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1,

14 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1,

15 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1,

16 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1,

17 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1,

18 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1,

19 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1,

20 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 0

21 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 1

22 Multipole fields from a current distribution Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013

23 Superconducting quadrupole - collider final focus Lecture 5 Magnetic Multipoles USPAS, Fort Collins, CO, June 10-1, 013 3

24 Kick at every turn. Solve a toy model: Dipole Error B d kb d k B x s s s il B ds Q ds i0 x s exp k s s il / Q sin k s s il ih B B i0 k k Q q k s s Q B B 11/ 4 exp B / Geometric series summed x s exp k s s x ih ih s B /Q cos kb s s kbl / k sin k L / integer resonance blows up when k L B n B B sin k s s 1 q cos k L cos k s s q sin k L B B B B 1q cos k L q B s Graduate Accelerator Physics Fall 015

25 s Closed Orbit Distortion Perform summation over all kick sources x co cos k s s k L / k i B i B sin k L / i B B bound oscillation generated by error Source (dipole powering, quad displacement, etc.) Oscillation can be observed and Using the real betatron motion corrected M s s sin s s ss, 1 the proper result is cos x s z co i i i s s i sin Graduate Accelerator Physics Fall 015

26 Beta Measurement If BPM close to steerer (there is little phase advance between them), and the tune has been measured, induce a closed orbit distortion to measure the x co s bpm i i cot Graduate Accelerator Physics Fall 015

27 Dipole Error Distribution u s s s s s s co i i i j i j i j u s s s si sin co i i 8sin i u 8sin s N cos cos s sj sin ds ds 1 Angular stuff averages assuming independence of error distributions / For quad displacements replace x / f Graduate Accelerator Physics Fall 015

28 Closed Orbit Correction Suppose orbit does not go through center of all BPMs. What do you do? (At CEBAF just steer to BPM centers!) Trim magnets added whose purpose is to bring CO as close to zero as possible. cos u s s co i i i At BPM j closed orbit reads s sin cos u u s s j co j j i i i i sj si sj si sin Measure response matrix as trim magnets (index k) varied cos u j s j k k R jkk sin i s Graduate Accelerator Physics Fall 015

29 Desire i1 1 k kj j BPM j j Correction Algorithm. If have enough trims simply update More sophisticated when less trims than BPMs, minimize N u u R u BPM 1,, N x u R trims analogous to "least squares fitting" and generally uses the same types of computer algorithms, including Singular Value Decomposition (SVD). How many BPMs/trims? Fourier Analyzing closed orbit equation Fe s il l 3/ il uco s s F l s se l l Need enough to resolve the betatron orbit and distribute uniformly in betatron phase s ds Graduate Accelerator Physics Fall 015