Transcript Observation of the electron cloud effect on pick

whofle:
W. Hofle
CERN AB/RF
Feedback systems
Introduction to feedback systems
in Accelerators

An accelerator is a complex system that requires many parameters
and sub-systems to be dynamically controlled and stabilized.

In a feedback system measured quantities are used to generate an
input to a dynamic system in order to achieve a desired output

Usually dynamic systems are described by differential equations

Frequency domain description by complex transfer function (s=s+jw)

In circular accelerators z-Transform is often used (sampled domain)

Restriction to treatment of linear time invariant systems (L T I )
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Reminder Transforms
+ oo
X(f) =
Fourier Transform

x(t) e -jwt dt

- oo
+ oo
Y(s) =
Laplace Transform

y(t) e -st dt

0
F(z) =
z-Transform
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+ oo

n=0
f(nt) z-n
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
System response in frequency domain
X(s)
Y(s)
G(s)
d2y(t)
Example harmonic oscillator:
dt2
+ w02 y(t) = x(t)
x(t)
describes an external perturbation
y(t)
describes the time evolution of the system output (response)
Transfer function:
G(s) =
1
s2 + w02
with s=jw (positive/negative w!)
G(w) =
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w02 - w2
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Feedback path
X(s)
+
-
G(s)
Y(s)
H(s)
Feedback path:
H(s)
H(s) includes transfer functions of
• sensors
• signal processing
• actuators
• and always some delay!
Output with feedback loop closed:
Y(s) = G(s) X(s) - G(s) H(s) Y(s)
Closed loop transfer function: F(s) =
Open loop transfer function:
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G(s)
1 + G(s) H(s)
G(s) H(s)
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Stability
jw
x
For stability poles of closed loop
transfer function F(s) must lie
in negative half plane
when z transform is used
poles must lie inside unit circle
s
x
Closed loop transfer function: F(s) =
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G(s)
1 + G(s) H(s)
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Example: RF cavity feedback
Ib(s)
+
Ig(s)
+
Z(s)
V(s)
Z(s)
cavity impedance
H(s) includes transfer functions of
• coupling antenna
• signal processing
• RF power amplifiers
• cavity coupling
H(s)
V
Closed loop transfer function: F(s) =
Z(s)
1 - Z(s) H(s)
Ib
Ig
R
L
C
cavity
Purpose: cancel beam induced voltage, reduce impedance seen by beam
Feedback gain limited by delay (stability), note feedback closed around cavity
RF cavity feedback absolutely essential for high beam currents, super-conducting cavities
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Beam feedback system
Feedback path closed around beam
G(s) includes characteristics of beam as dynamic system
examples are “coupled bunch feedbacks” (transverse and longitudinal plane)
with use of RF amplifiers and electromagnetic kickers, cavities,
magnetic or “electro-static” deflectors, usually “fast” (turn-by-turn)
very often wide-band these systems are wide band
other examples: tune feedback, orbit feedback
actuators are magnets, usually “slow”
For theory of coupled bunch instabilities see talk by K. Schindl
Coupled bunch feedbacks widely used in high intensity proton and lepton circular
accelerators
Purpose of coupled bunch feedbacks:
Provide stability and damp injection oscillation before filamentation occurs
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Coupled bunch feedback in circular accelerators
T0 : beam revolution time
f0 : revolution frequency
kicker
t beam
assume t signal = t beam + MT 0
M=0 -> very often not possible
(ultra-relativistic beams!)
t signal
Signal
processing
M=1: very common ->
“One -Turn-Delay” feedback
one-Turn delay corresponds to multiplication
in f-domain, (Fourier transforms) with: e-jwT0
Pick-up
in s-domain, (Laplace transforms):
e-sT0
in z-domain, (z transforms):
z-1
Longitudinal plane: ws << w0
-> many turns per oscillation period
Transverse plane: wb = (n+q)w0
->
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many oscillation periods per turn
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Transverse feedback
Consider slice of beam or bunch excersing
a coherent betatron oscillation
- pick-up measures position y
- kicker corrects angle y’
Kicker
bK
Pick-up 2
t beam
D
Pick-up 1
bPU
t signal
Signal
processing
gain g
D phase advance between pick-up 1
and pick-up 2 / kicker
signal of slice of beam
on turn n at pick-up 1 (t=nT0)
yPU (t) = (JbPU)1/2 cos(2pQt)
at kicker
yK(t’) = (JbK)1/2 cos(2pQt’+D)
t’ = t + t beam
kick by feedback system
Dy’K (t’) = g (JbPU)1/2 cos(2pQt’)
angle at kicker: y’K = (t’) = - (J/bPU)1/2 (a cos(2pQt’+D) + sin(2pQt’+D)
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Transverse feedback
x’
Bunch on
turn n=1
x’
Dy’
x
Bunch on
turn n=1
Bunch on
turn n=0
area pJ
x
Dy’
area pJ
at pick-up 1
peak kick
Bunch on
turn n=0
at kicker
Dy’ = g (JbPU)1/2
effective average kick is only half of the peak kick (averaging betatron phases)
min damping time
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t = (T0/2) g (bPU bK)1/2
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for D = ?
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
further complications
pick-up at optimum phase may not be available
- use two pick-ups in quadrature to generate a signal at the desired phase
- use a single pick-up and combine signals from M previous turns
(-> Hilbert filter, FIR), disadvantage: tune sensitive
large closed orbit variations in pick-up
- use notch filter: F notch = 1-z-1 -> gain at revolution frequency harmonics is zero:
attention signal shifted in betatron phase (half a turn!)
fast damping and tunes close to integer or half integer -> unstable
- avoid being close to half integer and integer resonance
- use two kickers p/2 apart for smooth damping close to half integer/integer
...
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Technologies used in transverse feedback systems
Pick-ups:
Couplers, buttons, electrostatic pick-ups
large bandwidth or bunch to bunch processing important
required bandwidth > 1/(2 x bunch spacing)
(LHC beam: bunch spacing = 25 ns -> bandwidth > 20 MHz)
20 MHz waveform
25 ns
Processing: often digital processing using DSPs and FPGA
parallel processing for high bunch frequency ( PEP II 2 ns)
LHC beam in SPS: 12 bit ADC @ 80MHz clocked, single FPGA used
Power amplifiers: work in baseband or at harmonics of bunch frequency
power ranges from kHz to several hundred MHz,
power amplifiers with transistors or RF tubes, 100 W - 10’s kW
Kickers:
electric, magnetic, strip-lines (electro-magnetic), cavities
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whofle:
W. Hofle
CERN SL/HRF
Scrubbing Run Results 2002
Expert Control (MMI)
Expert Control (MMI)
OP Control
OP Control
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whofle:
W. Hofle
CERN SL/HRF
Scrubbing Run Results 2002
LHC front-end electronics since 2000
electron-cloud-effect-free signal processing for LHC beam:
sharing of a new set of pick-ups with MOPOS (2.04, 2.05, 206, 207)
damper: wide-band processing at 120 MHz
choice of frequency is a compromise between cable losses, available
signal from existing pick-ups, and the wish to be interference-free
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Measurement of open loop transfer function
X(s)
+
-
G(s)
Y(s)
NWA
H(s)
NWA (Network analyzer):
Network analyzer can do a sweep
In frequency, excite beam and
measure response of open loop
Polar plot (amplitude and phase)
Feed back stable, if circles
orientated to negative real axis
Calibration, take into account
all cabling
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whofle:
W. Hofle
CERN SL/HRF
Scrubbing Run Results 2002
Adjusting the horizontal dampers at 5 MHz
Scrubbing
Run settings
2002
Improved
settings
26.06.02
(Loop delay)
H1 improved settings (-9 ns)
September 2003
H2 improved settings (+6 ns)
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whofle:
W. Hofle
CERN SL/HRF
Scrubbing Run Results 2002
Performance in the 10-20+ MHz range
Horizontal damper H1
H1 at 10 MHz OK
H1 at 20 MHz limit
H1 at 25 MHz anti-damping
V1 at 20 MHz OK
V1 at 30 MHz limit
(45 degrees, delay error!)
Vertical damper V1
V1 at 10 MHz OK
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whofle:
W. Hofle
CERN AB/RF
Feedback systems
Let’s have a look at the problems
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