Transcript PRESENTATION TITLE

Feedback 101
Stuart Henderson
March 15-18, 2004
March 15-19, 2004
S. Henderson, IU e-p meeting
ORNL
Outline
•
•
Introduction to Feedback
– Block diagram
– Uses of feedback systems
(dampers, instabilities, longitudinal,
transverse
– System requirements
– Resources (paper)
•
–
–
–
–
•
•
•
Simplest feedback system scheme
– Ideal conditions
– Eigenvalue problem and solution
– Loop delay, delayed kick
•
Kickers
Concepts
Dp and dtheta calculation
Figures of merit
Plots of freq response, etc.
Complete System Response
Estimates for damping e-p
RF amplifiers
– Parameters, cost, etc.
•
Feedback in the ORBIT code
Closed-orbit problem
– Filtering schemes (analog/digital)
– Two turn filtering scheme
– Type of digital filters (FIR, IIR)
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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 Feedback Systems?
• High intensity circular accelerators eventually encounter
collective beam instabilities that limit their performance
• Once natural damping mechanisms (radiation damping for e+emachines, 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 e+ecolliders, 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:
Signal
Processing
RF amp
Kicker
– Low-frequency: ferrite-yoke
magnet
– High-frequency: stripline kicker
Pickup
Beam
• Longitudinal Kicker can be RF
cavity or symmetrically
powered striplines
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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:
Position
measurement
(coordinates
x, x’)
– -functions at pickup and
kicker are equal
– 90 phase advance
between kicker & pickup
– Integer tune
X
Kick (coordinates y, y’)
• System produces a kick
proportional to the measured   G x

displacement:
  x0   0 
 y0   0
 '   

   
• At the kicker:
 y0    1 /  0  0    x0 /  
x   0
• At the BPM after 1 turn:  1'   
 x  1 / 
 1 
0
  
  x0   
  


0   x0 /     
0

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Simple picture of feedback, continued
• So x-amplitude after 1 turn has been reduced by
x    Gx
• Giving a rate of change in amplitude:
dx
 Gf0 x
dt
x(t )  AeGf 0t  Aet
  1/   Gf0
• Giving a damping rate:
• But, we don’t really operate with integer tune. Averaging over all
arrival phases gives a factor of two reduction:
 opt
Gf 0

2
• 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:
y  kxn  G
'
n
xn
k  p
M2, 2
Pickup
(x,x)
Kicker
(y,y)
M1, 1
• Where G is the feedback gain
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Simple processing, cont’d
• The coordinates one turn later are given by:


y n  M 1 xn
 yn   0 
 yn 

  M 2  '   

xn 1  M 2  '
'
 yn  y 
 yn    kxn 
   0

xn 1  M 2  M 1 xn  
 k

0  

 xn   M 2 ( M 1  K ) xn
0 
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More realistic damping rate calculation,
cont’d
• After n turns the coordinates are

n
xn  [M 2 (M1  K )] x0
• This is an eigenvalue problem with solution

n
xn   e
• The eigenvalues can be obtained from
det M 2 (M1  K )  I  0
 C0
M 2 ( M 1  K )   '
 C0
det
S 0   C2

'   '
S 0   C2
S 2  0 0 


' 
S 2   k 0 
C0  kS2  
S0
C  kS
S 
'
0
'
2
One-turn matrix
'
0
0
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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
  i
 e
• 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:
a
b
det
0
c
d 
1
2
  (a  d ) 
Giving,


1/ 2
1
(a  d ) 2  4(ad  cb )  e cos   ie sin 
2
2
e
 ad  cb
(a  d )
cos  
2e
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Damping rate and tune shift for simple
processing
• We have
e2  ad  cb  (C0  kS2 )S0'  S0 (C0'  kS2' )
• 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:
e 2  (cos   p sin )(cos   p sin   k  p  k sin 2 )
  p sin (
• Finally,
1 p
p
2
sin   k  k /  p (cos2   p sin 2 ))
e2  1  k 1 sin 1  1  G sin 1
cos  
2 cos   k 1 sin 2

2 1  k 1 sin 1

1/ 2
2 cos   G sin 2

1/ 2
21  G sin 1 
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Damping rate and tuneshift for small
damping
• For weak damping,
G
sin  1
-1
turns
2
Gf0
 
sin 1 sec-1
2
 
• And
G
cos 1
2
Gf
   0 cos 1
2
  
radians
• Optimal damping rate results for 1=90 degrees
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Damping vs. Gain for 1=90 degrees
0.2
Damping Rate (turn-1)
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Gain
Exact Result
Weak Damping Approximation
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Tuneshift vs. Gain for 1=90 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:
Gf0
sin(1  2m Q)
2
Gf0
  
cos(1  2m Q)
2
 
• 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:
un  xn  xn1
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2-turn filter, cont’d
• With
xn  Ae jtn  Ae jnT0
un  xn  xn 1  Ae jnT0  Ae jnT0 e  jT0
• The transfer function of the filter is:
un
 jT0
 1 e
xn
• 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
Two-turn Filter Amplitude
2.5
2
Amplitude
1.5
1
0.5
0
0
1
2
3
4
5
6
-0.5
f/f0
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2-turn Filter Phase
2-turn filter phase
100
80
60
degrees
40
20
0
-20 0
1
2
3
4
5
6
-40
-60
-80
-100
f/f0
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Kickers for Transverse Feedback Systems
• For low frequencies (< 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
d
Beam
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 parallel plates of length l,
separated by distance d, at opposite DC
voltages, +/- V is:
2eVL l
p  Ft  eEt 
d c
• We need to account for the finite size of the
plates (width w, separation d). A geometry
factor g  1 is introduced:
p 
2eVL g l
d c
g   tanh
+V
-V
w
2d
• 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.
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Stripline g
Transverse Stripline Geometry Factor
1.2
1
G-perp
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Stripline Width/Separation
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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:
2 g VL j (t  kz)
e
d
cBy  E x
Ex 
Fx  eEx  ecBy  eEx (1   )
• For beam traveling in –z direction:
Fx  eEx  ecBy  eEx (1   )
• For relativistic beams, we need
the beam traveling opposite the
wave propagation!
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Deflection by Stripline Kicker
0
0
2 g eV
p   F (t )dt   L (1   )  e j (1  )t dt
d
l / c
l / c
p 
• Where

2 g  eV L
j 1  e  j ( k B  k L )l
d
kB 
kL 


c

c
• This can be written in phase/amplitude form as:
4 g  eV L
p 
sin e  j
d
  kB  kL l / 2
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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:
VK2
P
2Z c
• Which equals the power dissipated on the two stripline
terminations:
V2
P2
• So that the input voltage is:
L
2Z L
VK  2Z c / Z L VL
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Figures of Merit for Stripline Kickers
• One common figure of merit seen in the literature is the Kicker
Sensitivity.
p 
• From which we get:
K 
p c
eVL
eVK
K
c
ZL
4g
 
2Z C dkB
ZL
sin e  j
2Z C
• Which can be written in the form
K 
Z L 2 g l
sin   j
(1   )
e
2Z C d

• 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:
2

VK2
V2
K VK 
P


•
2Z C 2 R
2 R
2
 2g 
 sin 2 
R  Z C K   2Z L 
 dkB 
• So after all this, what’s the kick?
2
p K  eVk
 

p
E 2
e
 
2 PR
2
E
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Transverse Shunt Impedance (w=d, =0.85,
50, d=15cm)
Stripline Frequency Response
Transverse Shunt Impedance
(Ohms)
1.40E+04
1.20E+04
1.00E+04
8.00E+03
6.00E+03
4.00E+03
2.00E+03
0.00E+00
-2.00E+03
0
100
200
300
400
500
600
Frequency (MHz)
1 meter
0.5 meter
0.25 meter
0.125 meter
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Transverse Shunt Impedance (w=d, =0.85,
50, d=15cm)
Transverse Shunt Impedance (a.u.)
Stripline Frequency Response
1.20E+00
1.00E+00
8.00E-01
6.00E-01
4.00E-01
2.00E-01
0.00E+00
-2.00E-01
0
100
200
300
400
500
600
Frequency (MHz)
1 meter
0.5 meter
0.25 meter
0.125 meter
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Multiple Kickers
• For N kickers, each driven with power P,
 
e
E
2
N 2 PR 
e
E
2
2 NPT R
• Where PT=NP is the total installed power
• To achieve the same deflection (damping rate) with N kickers
requires only
P1
PT 
N
• Example: One kicker with P1=1000W 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
 opt
Gf 0 f 0  k  p  max f 0  k  p e



2
2 xmax
2 xmax E 2
2 NPT R
• 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 = 2mm
– Stripline length = 0.5 m, separation d = 0.10, w/d = 1.0
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