Single HOM, twopass analysis

1
Single HOM, two-pass analysis
Motivation explain the plot produced by BBU code
bi
R/Q 100 Ohm Q 10000 m12 ?106 m(c/eV) ?
2???? 2 GHz t0 (1.3 GHz)1
2
Frequency of the instability

• Dipole mode is excited by first current moment
  thru interaction with longitudinal field of the
  mode
• Infinite number of bunches with finite number of
  passes (as opposed to finite number of bunches
  with infinite number of passes for BBU in storage
  rings)
• Potentially, any frequency can be present in FT
  of the current moment for infinite delta-function
  current train
• Instability occurs with frequency close to that
  of HOM, where impedance is maximal

FT
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Getting the master equation
Sample solution
Summing geometric series
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Perturbative approach
Solve the master equation for instability
frequency treating K as small parameter
The frequency up to the first order in K
Requiring Im(?) 0 yields famous
Problem 1 Half solution is missing Problem 2
Unphysical exponent
5
Second order perturbative term
The second order term is found to be
Im(?) 0 yields quadratic equation for the
threshold current
1st order 2nd order
Observation Clearly, the other half of the
solution is not a 2nd order effect
6
Complex current approach

• Solving master equation directly for current
  gives the following

Im(I0)
solution space
Re(I0)
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Max and min currents
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Obtaining complete first order solution

• In the following limit (HOM damping is small
  of timescale of t0)
• (instability frequency shift is small
  compared to bunching frequency, or as
  seen later, equivalent to
• number of bunches in recirculating loop
  gtgt 1)
• Solving Im(I0) 0, yields the threshold and
  instability frequency

Note Its sin(?tr) not sin(?tr) Unphysical
exponent is gone.
9
Linearized solutions for instability frequency
Transcendental equation for instability frequency
can be linearized for two important cases
look at solutions closest to??
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Comparison with tracking
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Solving Im(I0)0 numerically
with increasing tr
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Comparison of tracking with numeric solution of
Im(I0)0
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Large tr (?r ltlt ?/2Q)
tr does not matter as opposed to small
accelerators case, threshold is approx. given by
Iin
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A word on quad HOM BBU
coupling term
Wake functions are identical in the form, except
for the loss factor difference
In the approximation that alignment error of
cavity transverse position dominates and causes
dipole-like BBU (b is beam pipe radius)
, i.e. 2 orders of magnitude bigger