Damping ring oscillation simulation

R. Apsimon

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.

Mathematical construct (1)

• S.O. causes LO phase oscillation:Φ = Ds(T)Φ0sin(ωT)

– Ds(T) is the synchrotron damping term, T is the turn number

• LO phase causes amplitude to vary:ALO = cos(Φ + θLO)

– θLO is an error on the LO phasing.

Mathematical construct (2)

• Peak oscillates around sample point:Asample = Apeak(3.96 – (1.4*Φ/2π + θsample)^2)/3.96

– θsample is the time error of the sample point from the signal peak.

• The total observed oscillation will be:Atotal = Asample*ALO

Damping of second harmonic

Effects of sampling oscillation

Additional effects (1)

• Betatron oscillation:– X-betatron frequency: ~400kHz– Y-betatron frequency: ~1.22MHz

Aβ = Dβ(T)sin(ωβT)

– Y-betatron ignored as above Nyquist frequency, and observed amplitude very small

– Damping time very short, oscillation dies away within ~500 turns

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.

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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

Diff with sychrotron-betatron coupling

Position with S-B coupling

Model to data comparison

• Beam energy spread:– Design specification: 0.08%– Simulation result: 0.077%

• Synchrotron-betatron coupling– Simulation result: ~8%

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

Diff with S-B coupling and beating

Position with S-B coupling and beating

Coupled sum and diff (1)

• As previously shown:

• Can use to solve diff-y

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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!