feedback 101
DESCRIPTION
Feedback 101. Stuart Henderson. March 15-18, 2004. Outline. Introduction to Feedback Block diagram Uses of feedback systems (dampers, instabilities, longitudinal, transverse System requirements Resources (paper) Simplest feedback system scheme Ideal conditions - PowerPoint PPT PresentationTRANSCRIPT
S. Henderson, IU e-p meeting ORNLMarch 15-19, 2004
Feedback 101
Stuart Henderson
March 15-18, 2004
S. Henderson, IU e-p meeting ORNL2
March 15-19, 2004
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
• 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
• RF amplifiers– Parameters, cost, etc.
• Feedback in the ORBIT code
S. Henderson, IU e-p meeting ORNL3
March 15-19, 2004
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.
S. Henderson, IU e-p meeting ORNL4
March 15-19, 2004
Why Feedback Systems?
• High intensity circular accelerators eventually encounter collective beam instabilities that limit their performance
• Once natural damping mechanisms (radiation damping for e+e- 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
S. Henderson, IU e-p meeting ORNL5
March 15-19, 2004
Types of Feedback Systems
• Feedback systems are used to damp instabilities– Typical applications are bunch-by-bunch feedback in e+e-
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,…
S. Henderson, IU e-p meeting ORNL6
March 15-19, 2004
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
Kicker
Pickup
RF ampSignal Processing
Beam
S. Henderson, IU e-p meeting ORNL7
March 15-19, 2004
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.
S. Henderson, IU e-p meeting ORNL8
March 15-19, 2004
Simple picture of feedback
• Take simple (but not very realistic) situation: -functions at pickup and
kicker are equal
– 90 phase advance between kicker & pickup
– Integer tune
X
X
Position measurement (coordinates x, x’)
Kick (coordinates y, y’)• System produces a kick
proportional to the measured displacement:
• At the kicker:
• At the BPM after 1 turn:
x
G
/
0
00/1
0
0
0'0
0
x
x
y
y
0/
0
0/1
0 0
0'1
1
x
xx
x
S. Henderson, IU e-p meeting ORNL9
March 15-19, 2004
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 don’t 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.
ttGf AeAetx
xGfdt
dx
0)(
0
0/1 Gf
20Gf
opt
Gxx
S. Henderson, IU e-p meeting ORNL10
March 15-19, 2004
Realistic damping rate calculation for simple processing
• Follow Koscielniak and Tran
• Coordinates at pickup are (xn,xn) on turn n
• Coordinates at kicker are (yn,yn) 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
Pickup (x,x)
Kicker (y,y)
M1, 1
M2, 2
pk
nnn
xGkxy
'
S. Henderson, IU e-p meeting ORNL11
March 15-19, 2004
Simple processing, cont’d
nnnn
nn
n
n
nn
nn
xKMMxk
xMMx
kxy
yM
yy
yMx
xMy
)(0
00
0
12121
'2''21
1
• The coordinates one turn later are given by:
S. Henderson, IU e-p meeting ORNL12
March 15-19, 2004
More realistic damping rate calculation, cont’d
• After n turns the coordinates are
• This is an eigenvalue problem with solution
• The eigenvalues can be obtained from
012 )]([ xKMMx nn
ex nn
0)(det 12 IKMM
0
00)( '
2'2
22'0
'0
0012 kSC
SC
SC
SCKMM
One-turn matrix
0det '0
'2
'0
020
SkSC
SkSC
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March 15-19, 2004
General solution for 2x2 real matrix
0det
dc
ba
sincos)(4)(2
1)(
2
1 2/12 ieecbaddada
Giving, cbade 2
e
da
2
)(cos
• 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:
ie
S. Henderson, IU e-p meeting ORNL14
March 15-19, 2004
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,
)()( '2
'00
'020
2 kSCSSkSCcbade
))sin(cos/sin1
(sin
)sinsin)(cossin(cos
22
2
22
ppkp
pp
kppp
k
ke
1112 sin1sin1 Gke
2/112
2/1
11
21
sin12
sincos2
sin12
sincos2cos
G
G
k
k
S. Henderson, IU e-p meeting ORNL15
March 15-19, 2004
Damping rate and tuneshift for small damping
• For weak damping,
• And
• Optimal damping rate results for 1=90 degrees
10
1
sin2
sin2
Gf
G
turns-1
sec-1
1cos2
G radians
10 cos2
Gf
S. Henderson, IU e-p meeting ORNL16
March 15-19, 2004
Damping vs. Gain for 1=90 degrees
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 0.05 0.1 0.15 0.2 0.25 0.3
Gain
Dam
pin
g R
ate
(tu
rn-1
)
Exact Result Weak Damping Approximation
S. Henderson, IU e-p meeting ORNL17
March 15-19, 2004
Tuneshift vs. Gain for 1=90 degrees
S. Henderson, IU e-p meeting ORNL18
March 15-19, 2004
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
)2cos(2
)2sin(2
10
10
mQGf
mQGf
S. Henderson, IU e-p meeting ORNL19
March 15-19, 2004
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:
1 nnn xxu
S. Henderson, IU e-p meeting ORNL20
March 15-19, 2004
2-turn filter, cont’d
• 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
000
0
1TjTjnTjn
nnn
Tjntjn
eAeAexxu
AeAex n
01 Tj
n
n ex
u
S. Henderson, IU e-p meeting ORNL21
March 15-19, 2004
2-turn Filter Frequency Response
Two-turn Filter Amplitude
-0.5
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6
f/f0
Am
plit
ud
e
S. Henderson, IU e-p meeting ORNL22
March 15-19, 2004
2-turn Filter Phase
2-turn filter phase
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 1 2 3 4 5 6
f/f0
degr
ees
S. Henderson, IU e-p meeting ORNL23
March 15-19, 2004
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
S. Henderson, IU e-p meeting ORNL24
March 15-19, 2004
Stripline Kicker Layout
+VLZL
ZL
-VL
ZL
ZL
Beaml
d
S. Henderson, IU e-p meeting ORNL25
March 15-19, 2004
Stripline Kicker Schematic Model
VK
Zc
Beam In
p p+ p
Beam Out
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March 15-19, 2004
Stripline Kicker Analysis
• Deflection from parallel plates of 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
-Vcl
d
eVteEtFp L2
clg
d
eVp L
2
d
wg
2tanh
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March 15-19, 2004
Stripline g
Transverse Stripline Geometry Factor
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Stripline Width/Separation
G-p
erp
S. Henderson, IU e-p meeting ORNL28
March 15-19, 2004
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!
xy
kztjLx
EcB
ed
VgE
)(2
)1( xyxx eEBeceEF
)1( xyxx eEBeceEF
S. Henderson, IU e-p meeting ORNL29
March 15-19, 2004
Deflection by Stripline Kicker
• Where
• This can be written in phase/amplitude form as:
0
/
)1(0
/
)1(2
)(
cl
tj
cl
L dted
eVgdttFp
lkkjL LBejd
eVgp )(1
2
ck
ck
L
B
jL ed
eVgp sin
4 2/lkk LB
S. Henderson, IU e-p meeting ORNL30
March 15-19, 2004
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:
c
K
Z
VP
2
2
L
L
Z
VP
22
2
LLcK VZZV /2
S. Henderson, IU e-p meeting ORNL31
March 15-19, 2004
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
Kc
eVp K
j
C
L
BC
L
L
eZ
Z
dk
g
Z
Z
eV
cpK
sin2
4
2
j
C
L ed
lg
Z
ZK
sin)1(
2
2
S. Henderson, IU e-p meeting ORNL32
March 15-19, 2004
Transverse Shunt Impedance
• In analogy with RF cavities, one can define an effective shunt impedance that relates the transverse “voltage” to the kicker power:
•
• So after all this, what’s the kick?
22
2
222
sin2
2
222
BLC
K
C
K
dk
gZKZR
R
VK
R
V
Z
VP
PRE
e
E
eVK
p
p k
22
2
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March 15-19, 2004
Transverse Shunt Impedance (w=d, =0.85, 50, d=15cm)
Stripline Frequency Response
-2.00E+03
0.00E+00
2.00E+03
4.00E+03
6.00E+03
8.00E+03
1.00E+04
1.20E+04
1.40E+04
0 100 200 300 400 500 600
Frequency (MHz)
Tra
nsve
rse
Shu
nt Im
peda
nce
(Ohm
s)
1 meter 0.5 meter 0.25 meter 0.125 meter
S. Henderson, IU e-p meeting ORNL34
March 15-19, 2004
Transverse Shunt Impedance (w=d, =0.85, 50, d=15cm)
Stripline Frequency Response
-2.00E-01
0.00E+00
2.00E-01
4.00E-01
6.00E-01
8.00E-01
1.00E+00
1.20E+00
0 100 200 300 400 500 600
Frequency (MHz)
Tra
nsve
rse
Shu
nt Im
peda
nce
(a.u
.)
1 meter 0.5 meter 0.25 meter 0.125 meter
S. Henderson, IU e-p meeting ORNL35
March 15-19, 2004
Multiple Kickers
• For N kickers, each driven with power P,
• Where PT=NP is the total installed power
• To achieve the same deflection (damping rate) with N kickers requires only
• Example: One kicker with P1=1000W gives same kick as two kickers each driven at 250 W
RNPE
ePRN
E
eT22
22
N
PPT
1
S. Henderson, IU e-p meeting ORNL36
March 15-19, 2004
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
RNPE
e
x
f
x
fGfT
pkpk
opt 2222 2
max
0
max
max00
S. Henderson, IU e-p meeting ORNL37
March 15-19, 2004
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