Feedback 101

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Feedback 101

• Stuart Henderson

August 30, 2004
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Outline

• Introduction to Feedback
• Block diagram
• Uses of feedback systems (dampers, instabilities,
  longitudinal, transverse
• System requirements
• Resources
• Simplest feedback system scheme
• Ideal conditions
• Eigenvalue problem and solution
• Loop delay, delayed kick
• Closed-orbit problem
• Filtering schemes (analog/digital)
• Two turn filtering scheme
• Type of digital filters (FIR, IIR)

• Kickers
• Concepts
• Dp and dtheta calculation
• Figures of merit
• Plots of freq response, etc.
• Complete System Response
• Estimates for damping e-p

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Resources

• Several good overviews and papers on feedback
  systems and kickers
• Pickups and Kickers
• Goldberg and Lambertson, AIP Conf. Proc. 249,
  (1992) p.537
• Feedback Systems
• F. Pedersen, AIP Conf. Proc. 214 (1990) 246, or
  CERN PS/90-49 (AR)
• D. Boussard, Proc. 5th Adv. Acc. Phys. Course,
  CERN 95-06, vol. 1 (1995) p.391
• J. Rogers, in Handbook of Accelerator Physics and
  Technology, eds. Chao and Tigner, p. 494.

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Why Do We Need Feedback Systems?

• High intensity circular accelerators eventually
  encounter collective beam instabilities that
  limit their performance
• Once natural damping mechanisms (radiation
  damping for ee- machines, or Landau damping for
  hadron machines) are insufficient to maintain
  beam stability, the beam intensity can no longer
  be increased
• There are two potential solutions
• Reduce the offending impedance in the ring
• Provide active damping with a Feedback System
• A Feedback System uses a beam position monitor to
  generate an error signal that drives a kicker to
  minimize the error signal
• If the damping rate provided by the feedback
  system is larger than the growth rate of the
  instability, then the beam is stable.
• The beam intensity can be increased until the
  growth rate reaches the feedback damping rate

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Types of Feedback Systems

• Feedback Systems are used to damp instabilities
• Typical applications are bunch-by-bunch feedback
  in ee- colliders, hadron colliders to damp
  multi-bunch instabilities
• Dampers are used to damp injection transients,
  and are functionally identical to feedback
  systems
• These are common in circular hadron machines
  (Tevatron, Main Injector, RHIC, AGS, )
• Feedback systems and Dampers are used in all
  three planes
• Transverse feedback systems use BPMs and
  transverse deflectors
• Longitudinal feedback systems use summed BPM
  signals to detect beam phase, and correct with RF
  cavities, symmetrically powered striplines,

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Elements of a Feedback System

• Basic elements
• Pickup
• Signal Processing
• RF Power Amplifier
• Kicker
• Pickup is BPM for transverse, phase detector for
  longitudinal
• Processing scheme can be analog or digital,
  depending on needs
• Transverse Kicker
• Low-frequency ferrite-yoke magnet
• High-frequency stripline kicker
• Longitudinal Kicker can be RF cavity or
  symmetrically powered striplines

Signal Processing
RF amp
Kicker
Pickup
Beam
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Specifying a Feedback System

• Feedback systems are characterized by
• Bandwidth (range of relevant mode frequencies)
• Gain (factor relating a measured error signal to
  output corrective deflection)
• Damping rate
• In order to specify a feedback system for damping
  an instability, we must know
• Which plane is unstable
• Mode frequencies
• Growth rates
• RF power amplifier is chosen based on required
  bandwidth and damping rate. Typical systems use
  amplifiers with 10-100 MHz bandwidth, and
  100-1000W output power.

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Simple picture of feedback
X?

• Take simple (but not very realistic) situation
• ?-functions at pickup and kicker are equal
• 90? phase advance between kicker pickup
• Integer tune

Position measurement (coordinates x, x)
X
Kick (coordinates y, y)

• System produces a kick proportional to the
  measured displacement
• At the kicker
• At the BPM after 1 turn

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Simple picture of feedback, continued

• So x-amplitude after 1 turn has been reduced by
• Giving a rate of change in amplitude
• Giving a damping rate
• But, we made two gross simplifications
• We dont really operate with integer tune.
  Averaging over all arrival phases gives a factor
  of two reduction
• In real life, we may not be able to place the BPM
  and kicker 90 degrees apart in phase, and the
  locations will not have equal beta functions. We
  need a realistic calculation.

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Realistic damping rate calculation for simple
processing

• Follow Koscielniak and Tran
• Coordinates at pickup are (xn,x?n) on turn n
• Coordinates at kicker are (yn,y?n) on turn n
• Transport between pickup and kicker has 2x2
  matrix M1 and phase ?1
• Transport between kicker and pickup has 2x2
  matrix M2 and phase ?2
• Give a kick on turn n proportional to the
  position measured on the same turn
• Where G is the feedback gain

M2, ?2
Kicker (y,y?)
Pickup (x,x?)
M1, ?1
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Simple processing, contd

• The coordinates one turn later follow from

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More realistic damping rate calculation, contd

• After n turns the coordinates are
• This is an eigenvalue problem with solution
• The eigenvalues can be obtained from

One-turn matrix
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General solution for 2x2 real matrix

• Since we have a 2x2 real matrix, we expect two
  eigenvalues which are complex conjugate pairs.
  Writing
• Where we can identify ? as the damping rate (per
  turn), and ? as the tune, which in general will
  be modified by the feedback system
• Solution

Giving,
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Damping rate and tune shift for simple processing

• We have
• With ?p, ?p the twiss parameters at the pickup,
  ?k, ?k at the kicker, ? the tune, ?1 the phase
  advance between pickup and kicker, ?2 the phase
  advance from kicker around the ring to pickup
• Finally,

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Damping rate and tuneshift for small damping

• For weak damping,
• And
• Optimal damping rate, and no tuneshift results
  for ?190 degrees

turns-1
sec-1
radians
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Damping vs. Gain for ?190 degrees
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Tuneshift vs. Gain for ?1125 degrees
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Finite Loop Delay

• Up to this point we have ignored the fact that it
  takes time to decide on the kick strength in
  the processing electronics
• It is not necessary to kick on the same turn
• We can kick m turns later
• In this way we can wait around for the optimum
  turn to provide the optimum phase

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Closed-Orbit Problem the 2-turn filter

• Our simplification ignores another problem
• A closed orbit error in the BPM will cause the
  feedback system to try to correct this closed
  orbit error, using up the dynamic range of the
  system
• Solution
• Analog a self-balanced front-end
• Digital Filter out the closed-orbit by using an
  error signal that is the difference between
  successive turns
• 2-turn filter constructs an error signal

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2-turn filter, contd

• With
• The transfer function of the filter is
• This gives a notch filter at all the rotation
  harmonics, which are the harmonics that result
  from a closed orbit error

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2-turn Filter Frequency Response
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2-turn Filter Phase
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Kickers for Transverse Feedback Systems

• For low frequencies (lt 10 MHz), it is possible to
  use ferrite-yoke magnets, but the inductance
  limits their bandwidth
• Broadband transverse kickers usually employ
  stripline electrodes
• Stripline electrode and chamber wall form
  transmission line with characteristic impedance ZL

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Stripline Kicker Layout
VL
ZL
ZL
Beam
d
l
ZL
ZL
-VL
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Stripline Kicker Schematic Model
Zc
VK
Beam Out
Beam In
p?
p? ?p?
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Stripline Kicker Analysis

• Deflection from infinite parallel plates over
  length l, separated by distance d, at opposite DC
  voltages, /- V is
• We need to account for the finite size of the
  plates (width w, separation d). A geometry
  factor g ? 1 is introduced
• Because we want to damp instabilities that have a
  range of frequencies, we will apply a
  time-varying potential to the plates V(?).
• We need to calculate the deflection as a function
  of frequency and beam velocity.

V
-V
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Stripline g?
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Deflection by Stripline Kicker

• Stripline kicker terminated in a matched load
  produces plane wave propagating in z direction
  between the plates.
• For beam traveling in z direction
• For beam traveling in z direction
• For relativistic beams, we need the beam
  traveling opposite the wave propagation!

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Deflection by Stripline Kicker

• Where
• This can be written in phase/amplitude form as

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Powering the Stripline Kicker

• For transverse deflection, one could
• Independently power each stripline with its own
  source
• Power the pair of striplines from a single RF
  power source by splitting (e.g. with a 180 degree
  hybrid to drive electrodes differentially)
• Using a matched splitting arrangement, the
  delivered power is
• Which equals the power dissipated on the two
  stripline terminations
• So that the input voltage is

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Figures of Merit for Stripline Kickers

• One common figure of merit seen in the literature
  is the Kicker Sensitivity.
• From which we get
• Which can be written in the form
• Important points
• Deflection has a phase shift relative to the
  voltage pulse
• sin?/? shows the typical transit-time factor
  response

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Transverse Shunt Impedance

• In analogy with RF cavities, one can define an
  effective shunt impedance that relates the
  transverse voltage to the kicker power
• The frequency response has notches at
• For low frequencies, (? ?? c/l)

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• So after all this, whats the kick?
• In the low frequency limit,

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Transverse Shunt Impedance (wd, ?0.85, 50?,
d15cm)
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Transverse Shunt Impedance (wd, ?0.85, 50?,
d15cm)
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Multiple Kickers

• For N kickers, each driven with power P,
• Where PTNP is the total installed power
• To achieve the same deflection (damping rate)
  with N kickers requires only
• Example One kicker with P11000W gives same kick
  as two kickers each driven at 250 W

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Putting it all together

• The RF power amplifier puts out full strength for
  a certain maximum error signal
• The system produces the maximum deflection ??max
  for a maximum amplitude xmax
• For optimal BPM/Kicker phase, the optimal damping
  rate is
• For a Damper systems, xmax is large enough to
  accommodate the injection transient
• For a Feedback system, xmax is many times the
  noise floor

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Parameters for an e-p feedback system

• Bandwidth
• Treat longitudinal slices of the beam as
  independent bunches
• Ensure sufficient bandwidth to cover coherent
  spectrum
• Choose 200 MHz
• Damping time
• To completely damp instability, we need 200
  turns
• To influence instability, and realize some
  increase in threshold, perhaps 400 turns is
  sufficient
• Input parameters
• ?y 7 meters
• Xmax 1mm
• Stripline length 0.5 m, separation d 0.10,
  w/d 1.0
• 800 MeV

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Damping Time vs. Power at 150 MHz
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Damping Time vs. Frequency at Fixed Power