1. Damping ring oscillation simulation R. Apsimon

2. Assumptions Synchrotron oscillation causes revolution period to oscillate around a nominal value. –This can be converted into LO phase. Phase oscillation causes processor output amplitude to vary due to LO. –Allow LO phase to be slightly off optimal Phase oscillation also causes sampling to oscillate about peak. –Allow sampling to be slightly off peak.

3. Mathematical construct (1) S.O. causes LO phase oscillation: Φ = D s (T)Φ 0 sin(ωT) –D s (T) is the synchrotron damping term, T is the turn number LO phase causes amplitude to vary: A LO = cos(Φ + θ LO ) –θ LO is an error on the LO phasing.

4. Mathematical construct (2) Peak oscillates around sample point: A sample = A peak (3.96 – (1.4*Φ/2π + θ sample )^2)/3.96 –θ sample is the time error of the sample point from the signal peak. The total observed oscillation will be: A total = A sample *A LO

5. Damping of second harmonic

6. Effects of sampling oscillation

7. Additional effects (1) Betatron oscillation: –X-betatron frequency: ~400kHz –Y-betatron frequency: ~1.22MHz A β = D β (T)sin(ω β T) –Y-betatron ignored as above Nyquist frequency, and observed amplitude very small –Damping time very short, oscillation dies away within ~500 turns

8. Additional effects (2) S.O. is an energy oscillation –Therefore different radius in arc sections –Therefore different horizontal position in BPM This position oscillation then induces further betatron oscillations.

9. Additional effects (3) Synchrotron-betatron coupling: –Convolution between sychrotron position and betatron oscillation. AC-coupling of the ADC inputs: –The signal baseline grows as 1-e -t/t0 –t0 is the decay time, which is ~8,000 turns

10. Diff with sychrotron-betatron coupling

11. Position with S-B coupling

12. Model to data comparison Beam energy spread: –Design specification: 0.08% –Simulation result: 0.077% Synchrotron-betatron coupling –Simulation result: ~8%

13. Other effects Slow beating (~400Hz) on diff signals –I suspect this is a machine oscillation Most likely cause is the nominal orbit of the damping ring is oscillating at ~200Hz

14. Diff with S-B coupling and beating

15. Position with S-B coupling and beating

16. Coupled sum and diff (1) As previously shown: Can use to solve diff-y

17. Coupled sum and diff (2) 1 sum and 2 diffs are all that is required to completely decouple the DR BPM signals –Thanks to Glenn for spotting that!