Transcript Instabilities (Rings)

John Byrd
Collective Effects in Storage Rings
Lecture 1
John Byrd
Lawrence Berkeley National Laboratory
26 Sept-7 Oct 2011
MEPAS, Guanajuato, Mexico
1
John Byrd
Noticias
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Lecture Summary
John Byrd
•
•
•
Introduction to Storage Ring Collective Effects
Wakefields and Impedance
Longitudinal Multibunch collective effects and cures
–
–
–
–
–
–
•
Longitudinal coupled bunch instabilities
Measurements
Passive cures
The Robinson Instability
Harmonic RF systems
Feedback systems
Transverse multibunch collective effects and cures
–
–
–
–
–
–
Transverse coupled bunch instabilities
Measurements
Passive cures
Feedback systems
Beam-Ion instabilities
Electron cloud instabilities
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Lecture Summary
John Byrd
•
Longitudinal single bunch collective effects
–
–
–
–
–
•
Short-range longitudinal wakefields and broadband impedance
Potential well distortion
Longitudinal microwave instability
Measurements
CSR microbunching instability
Transverse single bunch collective effects
–
–
–
–
Short-range transverse wakefields and broadband impedance
Head-tail modes and chromaticity
Measurements
Damping with feedback
• Not covered here: Touschek and intrabeam scattering
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
Introduction to storage ring
collective effects
• Instabilities can significantly reduce the photon beam
brightness by increasing the average transverse beam
size, energy spread, and reducing the bunch and average
beam current of the ring.
• Bad news! Most instabilities effecting 3GLSs have been
understood and cures for most are available.
• Good news! Users want more! We need to push the
capabilities of existing and new machines.
• These lectures provide an qualitative overview of the
collective effects in 3GLSs. The emphasis is on
– minimizing their negative impact wherever possible
– experimental characterization of the effects.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
Self fields and wake fields
In a real accelerator, there is another important source of e.m. fields to
be considered, the beam itself, which circulating inside the pipe, produces
additional e.m. fields called "self-fields“:
Direct self fields
Image self fields
Wake fields
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Wake Potentials
John Byrd
F = q[Ezzˆ + ( Ex - vBy ) xˆ + ( Ey + vBx ) yˆ ] º F// + F^
there can be two effects on the test charge :
1) a longitudinal force which changes its energy,
2) a transverse force which deflects its trajectory.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
Wakes and Impedances
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
Effect of a bunch distribution
Assume that the charge q1 is continuously distributed
over z·axis according to the current distribution function
i(t) such that:
The wake function wz(t) being generated by a point
charge is a Green function (i.e. impulse response)
and allows to compute the wake potential of the
bunch distribution:
We can use this
version of Ohm’s
Law if we know the
impedance and the
current in the
frequency domain.
A convolution in the time domain is a
product in the frequency domain.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Example Wakes: Resonant Cavity
John Byrd
• The longitudinal wake of a resonant cavity is given by
where
In the frequency domain, the wake becomes an
impedance given by
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Parallel Resonant Circuit Model
John Byrd
• Treat impedance of an
isolated cavity mode as a
parallel LRC circuit
• Voltage gained is given by
Ploss =
2
V
1
2 Rs
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Resonant cavity: Quality Factor
John Byrd
Lower surface
resistance gives
higher Q. For a given
R/Q, this gives higher
R. Lower external
power is required for
a given voltage V.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Example Wakes: broadband impedance
John Byrd
• Short-range wakes are usually
defined as those where the field
damps away before the next RF
bucket.
• Many vacuum chamber
components contribute to the
short-range wakes. In the
frequency domain, these are
known as the broadband
impedance.
• The broadband impedance drives
single bunch instabilities.
• Much more on this later.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
Impedance properties
• Because the wake function is causal, the impedance has several
properties
• The impedance is a complex quantity
– Real (resistive) and Imaginary (reactive)
– Zr and Zi are even and odd functions of omega, respectively
• The real and imaginary parts are related via a Hilbert transform
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John Byrd
Analogously,
we Potential
introduce the transverse wake fun
Transverse
Wake
as a kick experienced by the trailing charge per unit of
charges:
nalogously,
introduce wake
the transverse
wake function
• The we
transverse
function
is
gously,
we introduce
transverse
wake
ick experienced
by the the
trailing
charge
per
unit function
of both
the
kick
experienced
by
the
+¥
experienced by the trailing charge per unit of both
s:
q ò ( E + v ´ B) ^ dz
trailing charge per unit of both
Dp
z
+¥
w^ (t ) = ^ = -¥
; t = 1 +t
charges:
qq1
qq1
v
q ò ( E + v ´ B) ^ dz
+¥
Dp
E + v ´ B) ^ dz ; t = z1 + t
w^ (t ) = ^ q= ò (-¥
The
Dp qq
zv transverse coupling (beam) impedance is fou
qq1
w^ (t ) = ^ =1 -¥
; thet Fourier
= 1 + ttransform of the wake function:
qq
• Theqqtransverse
coupling
(beam) v is found by
1 coupling (beam)
1
e transverse
impedance
+¥
impedance
found
by the
urier transform
of the is
wake
function:
•found
Since
Z^by
w^ (t )e - jwtdynamics
dt [W] is
(wthe
) = jtransverse
ansverse
coupling
(beam)
impedance
is
ò
Fourier transform of the wake
dominated-¥by the dipole transverse
[ W]
wedynamics
can define
the
Since the wakes,
transverse
is dominated
by the
+¥-¥
transverse
dipole
per impe
wakes,
we can define
theimpedance
transverse dipole
Z^ (w ) = j ò w^ (t )e - jwt dt [transverse
W]
per unit
nce the transverse-¥dynamics is dominated
bytransverse
the dipole
unit displacement:
transverse displacement:
r transform of the wake
function:
+¥
function:
Z ^ (w ) = j ò w^ (t )e - jwt dt
erse wakes, we can define the transverse dipole impedance
itthe
transverse
displacement:
transverse
dynamics is dominated by the dipole
wakes, we can define the transverse dipole impedance
Z (w )
ansverse displacement:
Z ^¢ = ^
[ W / m]
Z^¢ =
Z ^ (w )
r1
[ W / m]
r1
Z^¢ =
Z ^ (w )
[ W / m]
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Beam signals
John Byrd
• The response of the circuit to a current I is the convolution of the
current with impulse response of the circuit.
• Strategy: derive the beam signal in time and frequency domain
and either convolve or multiply with PU response.
a)
V(t) =
-i b
Zp
¥
–¥
dt 'i(t ')W p(t –t ')
V(w)=i(w)Z p(w)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
Single Particle Current
• Consider a point particle going around a storage with revolution
period T0 and rotation frequency f0=1/T0. The current at a fixed
point in the ring is given by
n=+ ¥
d(t–n T0) = e w0S e jnw t
S
n = –¥
n
¥
= ef0+ 2ef0 S cos(nw 0t)
n=1
i(t)= e
0
The FT of this given by
(Negative frequency components can be
folded onto positive frequency. AC
components are 2X DC component.)
I(w)=e w0S d(w–kw0)
k
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
•
Example: resonant cavity
Given a particular fill pattern or bunch spectrum, how do we calculate the signal
induced in a RF cavity or a pickup? If the cavity or pickup represent a beam
impedance Z||(w), (with a corresponding impulse response W(t)), the total signal
out is a convolution of the input with the response. In the frequency domain, this
is just a multiplication of the beam spectrum with the impedance.
impedance
beam spectrum
Frequency
V(w)=I(w)Z|| (w)
¥
V(t)= 1 S I(nw0 )Z|| (nw 0)e jnw 0t
2p n = 0
P= 1
2p
2 ¥
I(nw 0) 2Z ||(nw 0 )
S
n= 0
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Example: ALS Harmonic Cavity
John Byrd
We regularly use this
technique to tune the
fundamental and TM011
(first monopole HOM).
-20
cavity signal
cavity impedance
fres=1.5068 GHz, Q=21500
3*frf
Amplitude (dBm)
-30
-40
-50
-60
-70
-80
1.498
1.500
1.502
1.504
1.506
1.508
1.510
1.512
Frequency (GHz)
-30
fres=2.324 GHz, Q=17000
-40
Amplitude (dBm)
The high-Q cavity modes
can be tuned using the
single bunch spectrum
excited in the cavity probe.
The Q and frequency can be
found to fairly high accuracy
using this method.
-50
-60
-70
-80
2.321
2.322
2.323
2.324
Frequency (GHz)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
2.325
19
2.326
John Byrd
Spectrum of bunch distribution
• The bunch is a distribution of
particles
• For a single bunch passage, the
spectrum is just the transform of
the bunch
• For a repetitive bunch passage,
the spectrum is the series of
comb lines with an envelope
determined by the single passage
spectrum
• Assumes the bunch distribution is
stationary. More on this later.
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Example: Broadband bunch spectra
John Byrd
• Measured spectrum on a spectrum analyzer is the
product of the bunch spectrum and the pickup
impedance (button pickup, feedthru, cable)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
Single particle signal
w/synchrotron oscillations
• Now add synchrotron oscillations with an amplitude
tau
i(t)=e S
d(t–nT0+tscos(wst))
n
The FT of this given by
I(w)=e w 0S e–jnw0 (t+tscosw st)
n
jmq
e jxcosq= S
j
J
(x)e
m
m
m
–m
= e w 0S
j
Jm(wt s)S d(w+ mws–kw 0)
m
k
The sideband spectrum is very similar to that of phase (or
frequency) modulated signals
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Single particle signal
w/synchrotron oscillations
John Byrd
• The comb spectrum has added FM sidebands which are
contained within Bessel function envelopes.
Rotation harmonics follow J0, first order sidebands follow J1, etc.
1.2
m=0
Current
1.0
Note that J0~1
and J1~x for x<<1
0.8
0.6
m=1
0.4
m=2
0.2
0.0
0.0
0.2
0.4 2011, MEPAS,
0.6 Guanajuato,
0.8 Mexico 1.0
John Byrd, Oct
wt
23
Introduction to coupled bunch instabilities
John Byrd
Wake voltages which last long enough to affect a
subsequent bunch can couple the synchrotron or
betatron oscillations. Under certain
circumstances, this results in exponential growth
of the oscillations, degrading the beam quality.
In the frequency domain, the normal modes
appear as upper and lower sidebands. The
stability of a mode depends of the overlap with
impedances.
0 0
2 1
1 2
0
1
2
0 0
3
2 1
1 2
4
5
Frequency/ f0
RF cavity
0 0
2 1
1 2
6
7
8
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
0 0
9 24
John Byrd
Simple model of a LCBI
•
•
•
•
•
Consider a single bunch with
synchrotron oscillations interacting
with a high-Q resonator.
Upper and lower sidebands can be
considered a part of the synchrotron
oscillation with too little and too much
energy, respectively. (For cases above
transition)
Case a) Resonant mode tuned
symmetrically. Upper and lower
sidebands lose equal energy. No
effect.
Case b) Lower sideband (higher
energy) loses more energy to resistive
impedance. Oscillation damped.
Case c) Upper sideband (lower
energy) loses more energy. Oscillation
anti-damped.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Simple model of a LCBI: time domain
John Byrd
•
•
•
•
•
Consider a single bunch with
synchrotron oscillations interacting
with a high-Q resonator.
Early and late arrival of the
synchrotron oscillation correspond to
too little and too much energy,
respectively. (For cases above
transition)
Case a) Resonant mode tuned
symmetrically. Early and late arrival
lose equal energy. No effect.
Case b) Late arrival loses more energy
than early arrival. Oscillation damped.
Case c) Early arrival loses more
energy than late arrival. Oscillation
anti-damped.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Multibunch modes
John Byrd
• The long-range wakes couple the motion of successive bunches.
We can describe the motion of N coupled bunches and N normal
modes. Each mode, l, has a relative oscillation phase of Df= 2p l
N
For the case of 3 bunches, a snapshot of the ring for the 3
modes is shown below.
Oscillation Amplitude
a) mode l=0
1
2
3
1
b) mode l=1
c) mode l=2
0.0
0.5
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
Fractional
Distance Around Storage Ring
1.0
27
John Byrd
Multibunch spectrum
• Even though each bunch is oscillating at its natural frequency, the
multibunch modes show up as upper and lower sidebands of the rotation
harmonics.
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John Byrd
Multibunch beam signal
• Each multibunch mode can appear at a number of frequencies.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Longitudinal growth rates
John Byrd
• General growth can be
found from the fractional
1 1 1 ¶V 1 ha ¶V
=
=
energy change for a small
t 2 E ¶e 2 EQs ¶f
energy offset.
V = I0fZ | =I0 f0( rf/ rff)Z| • Sum over all impedances
w
f
This gives
f = wt = r w rf t = r f0 • Note that phase
w rf
frf
modulation for a given
time modulation is larger
at higher frequencies:
higher frequencies have
Effective impedance
larger effective
impedance.
Longitudinal growth rate
note frequency dependence of impedance
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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• Because of the periodic
nature of the beam
pulses, the long-range
wakes are aliased into
the sampling bandwidth
of the beam.
• For example, a beam
rate of 500 MHz (2
nsec bunch spacing),
all wakes are aliased
into a 250 MHz
bandwidth.
Transverse position
Multibunch aliasing
0
2
4
6
8
10
6
8
10
Turn Number
Transverse position
John Byrd
0
2
4
Turn Number
The above shows a simplified betatron oscillation around a31ring and the bar shows tha
sampled transverse position at a single point in the ring. The lower picture shows the
sampled signal (as dots) and that an infinite series of frequencies fits the sampled data
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
Example:ALS RF Cavity HOMs
John Byrd
Measured
values
10
10
10
Re(Z||) (kΩ)
#
Freq(MHz) Q
Rs(kOhm)
-------------------------------------------------------------------1
808.44
21000
1050
2
1275.1
3000
33
3
1553.55
3400
26.52
4
2853
4000
18.8
5
2353
5100
16.8
6
1807.68
2900
13.34
7
3260
1500
6
8
3260
1500
1.535
9
2124.61
1800
7.56
10
2419.3
7000
5.53
11
1309.34
810
5.508
12
3183
1500
1.86
13
1007.96
840
1.764
14
2817.4
1000
1.6
15
3149
1500
1.125
16
1846.72
2200
0.88
17
3252
1500
0.81
18
2266.6
2200
0.55
19
2625.9
1500
0.141
20
2769.1
1500
0.135
21
2968
1500
0.075
22
2979
1500
0.075
23
3243
1500
0.03
------------------------------------------------------------------1
1500
21000
1700
2
2324
17000
625.6
10
10
10
10
10
3
2
TM-011
1
0
-1
-2
-3
-4
0.5
1.0
1.5
2.0
2.5
3.0
Frequency (GHz)
Narrowband impedances for
ALS main RF cavity up to
beam pipe cutoff
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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3.5
Aliased Longitudinal Impedance
John Byrd
• Aliased longitudinal impedance with thresholds for radiation
damping and feedback (at maximum damping of 3/msec.)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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ALS Longitudinal Multibunch Spectrum
John Byrd
Measured spectrum of first-order synchrotron sidebands
at ALS with beam longitudinal unstable (LFB off)
-20
a) 20 mA
n=0
n=28
n=95
n=124
95 mA
n=0
n=28
n=95
n=124
-40
Amplitude (dBm)
-60
-80
-100
-20
-40
-60
-80
-100
499.64
499.66
542.30
542.32
644.36
644.38
688.54
Frequency (MHz)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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688.56
ALS Longitudinal Multibunch Spectrum
John Byrd
Spectrum of first-order
synchrotron sidebands at
ALS with beam longitudinally
unstable (LFB off). Compare
with the calculated Zeff from
the measured cavity modes.
This can be used to
identify the driving
HOMs.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
Coherent Frequency Shift
• In general, the interaction with the impedance, creates a
complex coherent frequency shift.
– Resistive (real) impedance gives damping or anti-damping of
beam oscillations
– Reactive (imaginary) impedance shifts the frequency of the
beam oscillations.
– Assuming the growth rate and frequency shifts are small
compared to the oscillation frequency, we can write the complex
frequency shift as
a I0
j DW =
Zeff (n w 0 + w s )
4p E e n s
1 t = Âe [ j DW]
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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Controlling narrowband impedances
John Byrd
• Damping of cavity HOMs
– heavy damping required to decrease growth rates
– decreases sensitivity to tuners, temperature
– most desirable approach if possible
• HOM tuning
– done using plunger tuners or with cavity water temperature
– requires HOM bandwidth less than revolution frequency
– difficult for many HOMs
• Vacuum chamber aperture (transverse only)
– strongly affects transverse resistive wall impedance (Z~1/b3)
– trends in insertion device design are going to small vertical
aperture (<5 mm)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
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John Byrd
HOM Tuning
100
Impedance (kΩ)
80
1.2
1.0
60
0.8
0.6
40
0.4
20
Growth rate (1/ msec)
We currently use a
combination of LFB and
tuning of HOMs to control
instabilities. Tuning of HOMs
in the main cavities is done
mainly with cavity water
temperature. Tuning is done
using 2 tuners in HCs.
1.4
main cavity TM011 mode
Q=21000, Rs=1.05 MOhm
0.2
0
0.0
0.80737
0.80889
Frequency (GHz)
2.5
harmonic cavity TM011 mode
Q=17000, Rs=0.625 MOhm
Q=16000
Q=1600
Q=800
2.0
1.5
100
1.0
50
0.5
0
0.0
2.321
2.322
2.323
2.324
2.325
2.326
Frequency (GHz)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
38
Growth rate (1/ msec)
This tuning scheme requires
that the modes are high
Q. If insufficiently damped,
thgrowth rates could
actually increase.
Impedance (kΩ)
150
Concepts for HOM-damped cavities
John Byrd
• Instability Threshold
• Reduce impedance by
• Minimize total number
of cells
– Large R/Q for
fundamental
– Large voltage/cell
• Minimize R/Q for
HOMs
– Cavity shape design
• Minimize Q for HOMs
– HOM-damping
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
39
HOM Damping
John Byrd
Modern damped normal-conducting cavity designs use waveguides coupled to
the cavity body, dissipating HOM power in loads without significantly affecting the
fundamental.
10
Shunt Impedance/ cavity (MΩ)
10
10
10
10
10
10
10
10
2
fundamental
1
0
-1
damped cavity 0-pole modes
undamped cavity 0-pole modes
equivalent impedance from
radiation damping
LER long. FB capability
for gain=600 V/ mrad
-2
-3
-4
-5
-6
0.5
1.0John Byrd, Oct 2011, MEPAS,
1.5Guanajuato, Mexico
2.0
40
2.5
John Byrd
Example: PEP-II Cavity
Modeled ringing of the PEP-II cavity.
• Cavities have large number of
resonant modes.
• Frequencies extend up to cutoff
frequency of the beam pipe (TE
or TM)
• Ringing is usually dominated by
cavity fundamental mode.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
41
John Byrd
PEP-II Impedance
• The time domain wake in frequency domain is called the
beam impedance
• Each mode can be defined by a frequency, Q, and shunt
impedance
• The effectiveness of the HOM-damping is determined by
the reduction in the shunt impedance Rs
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
42
John Byrd
PEP-II Transverse Wake
• A similar situation occurs
for dipole cavity modes
which can give transverse
beam deflections.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
43
John Byrd
HOM-damped SC cavities
• S
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
44
John Byrd
NC HOM-damped cavities
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
45
John Byrd
Bessy HOM-damped design
• Developed to be a commercially available NC HOMdamped cavity design
• Used at ALBA
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
46
John Byrd
Passive techniques
• Damp instabilities by increasing
synchrotron frequency spread within a
bunch (Landau damping)
• Decoupling the synchrotron oscillations by
varying the synchrotron frequency along a
bunch train
• Modulate synchrotron motion at multiples
of synchrotron frequency
• Difficult to predict effects in advance.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
47
Longitudinal Feedback Systems
John Byrd
Design Issues:
•Bandwidth>1/2 bunch
frequency (M*revolution freq)
to handle all possible
unstable modes
•PUs: phase or radial(
energy)
•radial:
•PUs in dispersive region
•no phase shift required and much
less signal processing.
•sensitive to DC orbit and betatron
oscillations
•insensitive to beam phase shifts
•phase:
•usually more sensitive and less noisy
than radial PU
•requires 90 degree phaseJohn
shift
Byrd,(1/4
Oct 2011, MEPAS, Guanajuato, Mexico
synchrotron period delay)
48
John Byrd
ALS LFB system
•PUs operate at 3 GHz (6*Frf); phase detection following comb filter
•synchrotron oscillations downsampled by factor 20
•QPSK modulation scheme from baseband to 1 GHz (2*Frf)
•TWT 200 W amplifier drives 4 2-element drift tubes kickers.
LFB Issues:
•drifting phase of beam
wrt. master oscillator
•variation of
synchronous
phase/synch. frequency
along bunch train.
•strong growth rates
(>1/msec)
•excitation of Robinson
mode
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
49
ALS LFB Kicker
John Byrd
The ALS uses a series drift-tube
structure as an LFB kicker. Thje drift
tubes are connected with delay lines
such that the kick is in phase.
one
two
four
Beam Shunt Impedance (Ω)
1000
100
10
1.25 GHz
1 GHz
1
0.1
0.6
0.8
1.0
1.2
1.4
Other rings have designed
a heavily loaded cavity as
a kicker. (DAFNE,BESSYII)
Frequency (GHz)
Beam impedance for 1,2, and 4 series drift-tubes.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
50
John Byrd
LFB Diagnostics
The digital LFB system
in use at ALS, PEP-II,
DAFNE, PLS, Bessy-II
has built-in diagnostic
features for recording
beam transients. This
is useful for measuring
growth rates.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
51
LFB diagnostics (cont.)
400
200
300
100
0
200
imag Z (kOhm)
The impedance of a
driving mode can be
determined by measuring
the growth rate and
frequency shift of the
beam
real Z (kOhm)
John Byrd
-100
100
-200
34
38
40
42
44
46
Cavity 1 Temperature (deg C)
300
real Z (kOhm)
600
200
100
400
0
-100
200
-200
-300
36
38
40
42
Cavity 2 Temperature (deg C)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
44
46
52
imag Z (kOhm)
Example: Recent measurements to
characterize a cavity mode showed
significantly higher impedance than
was expected. The mode was tuned
by varying the cavity water
temperature. By measuring the
frequency sensitivity to temperature,
the Q can be found.
36
Effect of FB on beam quality
John Byrd
VFB off, LFB, HFB on
The effect of the feedback systems
can be easily seen on a
synchrotron light monitor (at a
dispersive section)
Energy oscillations affect undulator harmonics
LFB off
Normalized photon flux
1.0
courtesy A. Warwick, LBL
17 mA, single
152 mA, multi
0.8
all FB on
0.6
0.4
0.2
0.0
420
425
430
435
440
445
450
Photon energy (eV)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
53
Beam loading and Robinson Instabilities
John Byrd
• Basics of beam loading
• Examples of beam loading for ALS
• Robinson’s analysis of beam cavity
interaction
• Pedersen model of BCI
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
54
John Byrd
Beam Cavity Interaction:
Equivalent Circuit
From the point of view of a rigid
beam current source, beam
loading can be represented by
excitation of of a parallel RLC
circuit. This approximation is
valid for a generator coupled to
the cavity via a circulator.
The coupling to the cavity is set
to make the beam-cavity load
appear to be a resistive load to
the external generator and
matched at some beam
current.
For zero beam, the optimal
coupling (beta) is 1. To match
at nonzero beam, the optimal
beta is 1+Pbeam/Pcavity.
The beam loading appears to the
generator as an additional resistive and
inductive load. We can adjust the cavity
coupling (beta) and the detune the cavity to
make it appear to be a perfect match.
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
55
John Byrd
Phasor representation
Voltage (MV)
-0.6
The beam and generator voltages can
represented as phasors. Common
usage has either the cavity voltage or
beam current as the reference phase.
-0.4
-0.2
0.0
0.2
0.4
0.6
1.0
-2*Ib
0.6
Itot
Vb
0.4
0.5
Current (A)
Conditions for detuning to get
minimum generator power.
Ig
Vc
0.0
-0.5
-1.0
-0.2
Ib = 0.4 A
U0 = 337.335 kV
Vcell = 0.6 MV
Rs = 4.5 MΩ
Beta=2.7
Ncell = 2
-1.0
-0.5
Vg
-0.4
-0.6
0.0
0.5
Current (A)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
0.0
56
1.0
Voltage (MV)
0.2
Phasor diagram
John Byrd
1.5
Voltage (MV)
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
1.0
0.6
-2*Ib
Itot
Vb
0.4
0.5
Voltage (MV)
1.0
0.5
0.0
-0.5
total voltage
beam voltage
generator voltage
U0
Ig
Vc
0.0
0.0
Voltage (MV)
-1.0
-1.5
0
1
2
-0.5
-1.0
Ib = 0.4 A
U0 = 337.335 kV
Vcell = 0.6 MV
Rs = 4.5 MΩ
Beta=2.7
Ncell = 2
Vg
3
4
5
-0.4
80
-0.6
60
2.0
-1.0
-0.5
6
Phase (rad)
-0.2
0.0
1.0
40
20
1.5
0
-20
1.0
-40
-60
0.5
-80
-3
-80x10
-40
-20
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
0
20
Frequency (MHz)
40
60
57
Phase (deg)
Current (A)
0.5
Impedance (MOhm)
Current (A)
0.2
Low current
John Byrd
1.5
Voltage (MV)
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
1.0
0.6
0.4
0.4
0.5
0.0
-0.5
total voltage
beam voltage
generator voltage
U0
0.2
-2*Ib
Vb
Itot
Vc
0.0
Ig
0.0
Vg
Voltage (MV)
-1.0
-1.5
0
1
2
Ib = 0.04 A
U0 = 337.335 kV
Vcell = 0.6 MV
Rs = 4.5 MΩ
Beta=2.7
Ncell = 2
-0.4
3
4
5
-0.4
80
-0.6
60
2.0
-0.6
-0.6
-0.4
6
Phase (rad)
-0.2
-0.2
-0.2
0.0
0.4
0.6
40
20
1.5
0
-20
1.0
-40
-60
0.5
-80
-3
-80x10
-40
-20
0
20
Frequency (MHz)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
40
60
58
Phase (deg)
Current (A)
0.2
Impedance (MOhm)
Current (A)
0.2
Voltage (MV)
0.6
High current limit
John Byrd
1.5
Voltage (MV)
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
1.0
-2*Ib
Itot
0.6
0.4
Vb
1
Voltage (MV)
2
Ig
0
Vc
0.0
Voltage (MV)
0.0
-0.5
total voltage
beam voltage
generator voltage
U0
-1.0
-1.5
0
1
2
6
80
-0.6
60
2.0
0
1
Current (A)
Robinson high current limit
occurs when the phase
focussing from the generator
voltage is lost. This corresponds
to the Robinson mode freq.->0.
2
40
20
1.5
0
-20
1.0
-40
-60
0.5
-80
-3
-80x10
-40
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
-20
0
20
Frequency (MHz)
40
60
59
Phase (deg)
-1
5
-0.4
-2
-2
4
Vg
Ib = 0.875 A
U0 = 337.335 kV
Vcell = 0.6 MV
Rs = 4.5 MΩ
Beta=2.7
Ncell = 2
-1
3
Phase (rad)
-0.2
Impedance (MOhm)
Current (A)
0.2
0.5
High current limit (cont.)
John Byrd
Df (kHz)
-40
-30
-20
-10
0
3.0
Vc= 1.2 MV
Rs= 4.5 MΩ, b= 2.7
2 rf cells
E= 1.9 GeV
U0= 337.335 KeV
2.5
IAC = 2*IDC (A)
2.0
1.5
1.0
300 kW
0.5
0.0
-80
-60
-40
-20
0
fz(deg)
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
60
Transients beam loading effects
John Byrd
The unequal filling of the ring (i.e. gaps) create a
transient loading of the main RF systems, causing
bunches to be at different RF phases (i.e. different
arrival times.)
I
t0

t0
time
time
For the main RF only, this effect is small (few
degrees).
John Byrd, Oct 2011, MEPAS, Guanajuato, Mexico
61