Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Accelerator PhysicsParticle Acceleration
Suba De Silva, G. A. Krafft, S. A. Bogacz,
and B. Dhital
Old Dominion University / Jefferson Lab
Lecture 13
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
RF Acceleration
• Characterizing Superconducting RF (SRF) Accelerating Structures
– Terminology
– Energy Gain, R/Q, Q0, QL and Qext
• RF Equations and Control
– Coupling Ports
– Beam Loading
• Higher Order Modes
– HOM excitation by a beam
– Wake potential
– HOM damping
• RF Focusing
• Betatron Damping and Anti-damping
2
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Terminology
• Use of multi-cell rf cavities to accelerate particles
3
1 CEBAF Cavity
1 DESY Cavity
“Cells”
5 Cell Cavity
9 Cell Cavity
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Typical cell longitudinal dimension: λRF/2
Phase shift between cells: π
Cavities usually have, in addition to the resonant structure in picture:
(1) At least 1 input coupler to feed RF into the structure
(2) Non-fundamental high order mode (HOM) damping
(3) Small output coupler for RF feedback control
Modern Jefferson Lab Cavities (1.497 GHz) are
optimized around a 7 cell design
4
Terminology
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 20205
On Axis Electric Field
2, cos 0,0, cos 2 /
2 / c.c. = 2 /
2
RF z RF
i
z RF
c z RF
E x t E x t mc e E z z dz
eE eV eE
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Some Fundamental Cavity Parameters
• Energy Gain
• For standing wave RF fields and velocity of light particles
• Normalize by the cavity length L for gradient
2
, vd mc
eE x t tdt
2, cos 0,0, cos 2 /
2 / c.c. = 2 /
2
RF z RF
i
z RF
c z RF
E x t E x t mc e E z z dz
eE eV eE
accE MV/m cV
L
6
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Shunt Impedance R/Q
• Ratio between the square of the maximum voltage delivered
by a cavity and the product of ωRF and the energy stored in a
cavity (using “accelerator” definition)
• Depends only on the cavity geometry, independent of
frequency when uniformly scale structure in 3D
• Piel’s rule: R/Q ~100 Ω/cell
CEBAF 5 Cell 480 Ω
CEBAF 7 Cell 760 Ω
DESY 9 Cell 1051 Ω
2
stored energy
c
RF
VR
Q
7
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
• As is usual in damped harmonic motion define a quality factor
by
• Unloaded Quality Factor Q0 of a cavity
• Quantifies heat flow directly into cavity walls from AC
resistance of superconductor, and wall heating from other
sources.
2 energy stored in oscillation
energy dissipated in 1 cycleQ
0
stored energy
heating power in walls
RFQ
Unloaded Quality Factor
8
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
0 10 20 30 40
Eacc [MV/m]
Q0
AC70AC72AC73AC78AC76AC71AC81Z83Z87
1011
109
1010
Q0 vs. Gradient for Several 1300 MHz
Cavities
Courtesey: Lutz Lilje
9
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Eacc vs.Time
0
5
10
15
20
25
30
35
40
45
Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06
Ea
cc[M
V/m
]
BCP
EP
10 per. Mov. Avg. (BCP)
10 per. Mov. Avg. (EP)
Courtesey: Lutz Lilje
10
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Loaded Quality Factor
• When add the input coupling port, must account for the energy
loss through the port on the oscillation
• Coupling Factor
• It’s the loaded quality factor that gives the effective resonance
width that the RF system, and its controls, seen from the
superconducting cavity
• Chosen to minimize operating RF power: current matching
(CEBAF, FEL), rf control performance and microphonics
(SNS, ERLs)
0
1 1 total power lost 1 1
stored energytot L RF extQ Q Q Q
0 01 for present day SRF cavities, 1
L
ext
Q QQ
Q
11
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Loaded Quality Factor
12
0
1 1 total power lost 1 1
stored energytot L RF extQ Q Q Q
( ),e ext inputP Q
tot diss e tP P P P
( ),t ext pickupP Q
Transmitted
power
Input power
0,dissP Q
Dissipated
power
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Loaded Quality Factor
• Introduction
• Cavity Fundamental Parameters
• RF Cavity as a Parallel LCR Circuit
• Coupling of Cavity to an rf Generator
• Equivalent Circuit for a Cavity with Beam Loading
– On Crest and on Resonance Operation
– Off Crest and off Resonance Operation
Optimum Tuning
Optimum Coupling
• RF cavity with Beam and Microphonics
• Qext Optimization under Beam Loading and Microphonics
• RF Modeling
13
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Introduction
Goal: Ability to predict rf cavity’s steady-state response and
develop a differential equation for the transient response
We will construct an equivalent circuit and analyze it
We will write the quantities that characterize an rf cavity and
relate them to the circuit parameters, for
a) a cavity
b) a cavity coupled to an rf generator
c) a cavity with beam
d) include microphonics
14
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
RF Cavity Fundamental Quantities
cV
Quality Factor Q0:
Shunt impedance Rsh (accelerator definition);
Note: Voltages and currents will be represented as complex
quantities, denoted by a tilde. For example:
where is the magnitude of and is a slowly
varying phase.
2
csh
diss
VR
P
00
Energy stored in cavity
Energy dissipated in cavity walls per radiandiss
WQ
P
c cV V
Re
i ti t
c c c cV t V t e V t V e
15
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Equivalent Circuit for an rf Cavity
Simple LC circuit representing
an accelerating resonator.
Metamorphosis of the LC circuit
into an accelerating cavity.
Chain of weakly coupled pillbox
cavities representing an accelerating
cavity.
Chain of coupled pendula as
its mechanical analogue.
16
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Equivalent Circuit for an RF Cavity An rf cavity can be represented by a parallel LCR circuit:
Impedance Z of the equivalent circuit:
Resonant frequency of the circuit:
Stored energy W:
11 1
Z iCR iL
0 1/ LC
21
2cW CV
( ) i tc cV t V e
17
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Equivalent Circuit for an RF Cavity
Average power dissipated in resistor R:
From definition of shunt impedance
Quality factor of resonator:
Note:
2
csh
diss
VR
P 2shR R
00 0
diss
WQ CR
P
1
00
0
1Z R iQ
0
1
00
0
For
1 2Z R iQ
,
Wiedemann
19.11
2
2
cdiss
VP
R
18
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cavity with External Coupling
Consider a cavity connected to an rf
source
A coaxial cable carries power from an rf
source to the cavity
The strength of the input coupler is
adjusted by changing the penetration of
the center conductor
There is a fixed output coupler, the
transmitted power probe, which picks up
power transmitted through the cavity
19
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cavity with External Coupling
Consider the rf cavity after the rf is turned off.
Stored energy W satisfies the equation:
Total power being lost, Ptot, is:
Pe is the power leaking back out the input coupler. Pt is the power
coming out the transmitted power coupler. Typically Pt is very small Ptot Pdiss + Pe
Recall
Energy in the cavity decays exponentially with time constant:
tot diss e tP P P P
0L
tot
WQ
P
/ totdW dt P
00
diss
WQ
P
0
00
L
t
Q
L
dW WW W e
dt Q
0/L LQ
20
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Decay rate equation
suggests that we can assign a quality factor to each loss mechanism, such that
where, by definition,
Typical values for CEBAF 7-cell cavities: Q0=1x1010, Qe QL=2x107.
0 0
tot diss eP P P
W W
0
1 1 1
L eQ Q Q
0e
e
WQ
P
21
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Have defined “coupling parameter”:
and therefore
It tells us how strongly the couplers interact with the
cavity. Large implies that the power leaking out of the
coupler is large compared to the power dissipated in the
cavity walls.
0
1 (1 )
LQ Q
0e
diss e
P Q
P Q
Wiedemann
19.9
22
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cavity Coupled to an RF Source
Z0
Z0
The system we want to model. A generator producing power Pg
transmits power to cavity through transmission line with characteristic impedance Z0
Between the rf generator and the cavity is an isolator – a circulator connected to a load. Circulator ensures that signals reflected from the cavity are terminated in a matched load.
23
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Transmission Lines
L
Z
L L
C C C
1nVnV
NV…2nI
NI
1nInI
1 1
1
n n n
n n n
V V i LI
I I i CV
Inductor Impedance and Current Conservation
24
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Transmission Line Equations
• Standard Difference Equation with Solution
• Continuous Limit ( N → ∞)
0 0
0 0 0 0
/2 /2
,
1 1
2sin /2
/ /
in in
n n
i i i
i i
V V e I I e
V e i LI e I e i CV
LC
V L CI e V L CI e
( ) ( ) ( )
0 0 0 0
( ) ( ) ( )
0 0 0 0
1/
, /
, /
i t kx i t kx i t kx
i t kx i t kx i t kx
LC k L C v L C
V x t V e L C I e Z I e
V x t V e L C I e Z I e
25
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cavity Coupled to an RF Source
1: k
0Z
,V I
,V I
k I Ii
Equivalent Circuit
RF Generator + Circulator Coupler Cavity
Coupling is represented by an ideal (lossless) transformer of turns ratio 1:k
26
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
From transmission line equations, forward power from RF source
Reflected power to circulator
Transformer relations
Considering zero forward power case and definition of β
Cavity Coupled to an RF Source
2
0/ 2V Z
2
0/ 2V Z
2
0
/ / 2 / /
c k
c k c
V kV k V V
i i k I I k I k V k Z
2 2 2
0 0/ 2 / / 2 /cV Z V R R k Z
27
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cavity Coupled to an RF Source
2( )
Re
g
g
i t
I t
ki t
i e
2
0
g
g
R R RZ
Z k Z
Wiedemann
19.1
Wiedemann
Fig. 19.1
0 0
1 1
2 2
c cc c
V VV kZ i V kZ i
k k
1 1 1 1 2
1
aL eff
eff g
RR R
R R Z R
Loaded cavity looks like
Effective and loaded resistance
Solving transmission line equations
28
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Powers Calculated
Forward Power
Reflected Power
Power delivered to cavity is
as it must by energy conservation!
2 22 22 220 0
0 22
0 0 0
1 111 1
8 4 1
c crefl L
a
V VP iQ Q
Z k R
2 222 2
20 002
0 0 0
111 1 1
8 4
c cg L
a
V VP iQ Q
Z k R
2 22 2
2
1 11
4 1
c cg refl diss
a a
V VP P P
R R
29
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Some Useful Expressions
Total energy W, in terms of cavity parameters
Total impedance
0
0
2 2
2 0
0
0
2
0 2 2 00
0
00 22
00
0
4 1
11
14
(1 )
4 1
(1 )1 2
diss
g diss
L
g
g
L
QP
W
P PQ
QW P
Q
QW P
Q
1
1
00
0
1 1
(1 )2
TOT
g
aTOT
ZZ Z
RZ iQ
30
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
When Cavity is Detuned
Define “Tuning angle” :
Note that:
0 00
0 0
0
2 2
0
tan 2 for
4 1=
(1+ ) 1+tan
L L
g
Q Q
QW P
2 2
4 1
(1 ) 1 tandiss gP P
Wiedemann
19.13
31
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Optimal β Without Beam
. Optimal coupling: W/Pg maximum or Pdiss = Pg
which implies for Δω = 0, β = 1
This is the case called “critical” coupling
. Reflected power is consistent with energy conservation:
. On resonance:
Dissipated and Reflected Power
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
0
2
0
2
2
4
(1 )
4
(1 )
1
1
g
diss g
refl g
QW P
P P
P P
2 2
4 11
(1 ) 1 tan
refl g diss
refl g
P P P
P P
32
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Equivalent Circuit: Cavity with Beam
Beam through the RF cavity is represented by a current generator
that interacts with the total impedance (including circulator).
Equivalent circuit:
ig the current induced by generator, ib beam current
Differential equation that describes the dynamics of the system:
, , / 2
c c LC R c
L
dV V dii C i V L
dt R dt
2
20 002 2
c c Lc g b
L L
d V dV R dV i i
dt Q dt Q dt
33
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Cavity with Beam
Kirchhoff's law:
Total current is a superposition of generator current and beam
current and beam current opposes the generator current.
Assume that voltages and currents are represented by complex
phasors
where is the generator angular frequency and
are complex quantities.
L R C g bi i i i i
Re
Re
Re
i t
c c
i t
g g
i t
b b
V t V e
i t i e
i t i e
, ,c g bV i i
34
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Voltage for a Cavity with Beam
Steady state solution
where is the tuning angle.
Generator current
For short bunches: where I0 is the average
beam current.
(1 tan ) ( )2
Lc g b
Ri V i i
0| | 2bi I
Wiedemann 19.19
0
2 22| | 2 2 4
g g g
g
a
P P Pi I
k Z R R
35
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Voltage for a Cavity with Beam
At steady-state:
are the generator and beam-loading
voltages on resonance
and are the generator and beam-loading voltages.
/ 2 / 2
(1 tan ) (1 tan )
or cos cos2 2
or cos cos
or
L Lc g b
i iL Lc g b
i i
c gr br
c g b
R RV i i
i i
R RV i e i e
V V e V e
V V V
2
2
Lgr g
Lbr b
RV i
RV i
g
b
V
V
36
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Voltage for a Cavity with Beam
Note that:
0
2| | 2 for large
1
| |
gr g L g L
br L
V P R P R
V R I
Wiedemann
19.16
Wiedemann
19.20
37
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Voltage for a Cavity with Beam
As increases, the magnitudes of both Vg and Vb
decrease while their phases rotate by .
cos i
grV e
grV
Im( )gV
Re( )gV
cos
cos
i
g gr
i
b br
V V e
V V e
38
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
cV
bI
brVbV
gV
decI
accI
b
Example of a Phasor Diagram
Wiedemann
Fig. 19.3
39
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Typically linacs operate on resonance and on crest in order
to receive maximum acceleration.
On crest and on resonance
where Vc is the accelerating voltage.
On Crest/On Resonance Operation
grVcV
brVbI
c gr brV V V
40
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
More Useful Equations
We derive expressions for W, Va, Pdiss, Prefl in terms of and the
loading parameter K, defined by: K=I0Ra/(2 Pg )
From:
0
2| |
1
| |
gr g L
br L
c gr br
V P R
V R I
V V V
0
2
2
0
2
2
0 0
0
2
0 2
21
1
41
(1 )
41
(1 )
22 1
1
( 1) 2
( 1)
c g L
g
diss g
a a diss
c
g
refl g diss a refl g
KV P R
Q KW P
KP P
I V I R P
I V KK
P
KP P P I V P P
41
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Clarifications
On Crest,
Off Crest with Detuning
000
2 2 11
1 1 2 2
aLc g L L g L
g L g
R IR IV P R R I P R K
P R P
2 22
0 0
0
2 222
0 0 0
2
=2 8
2 1 tan 2 cos sin
1 cos tan sin2
g
g
cg b b
L
c L Lg b b
L c c
Z I Z k iP
Vi i I i
R
Z k V I R I RP
R V V
42
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
More Useful Equations
For large,
For Prefl=0 (condition for matching)
2
0
2
0
1( )
4
1( )
4
g c L
L
refl c L
L
P V I RR
P V I RR
0
2
0 0
0
and
4
M
cL M
M M
c cg M M
c
VR
I
I V V IP
V I
43
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Example
For Vc=14 MV, L=0.7 m, QL=2x107 , Q0=1x1010 :
Power I0 = 0 I0 = 100 A I0 = 1 mA
Pg 3.65 kW 4.38 kW 14.033 kW
Pdiss 29 W 29 W 29 W
I0Vc 0 W 1.4 kW 14 kW
Prefl 3.62 kW 2.951 kW ~ 4.4 W
44
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Off Crest/Off Resonance Operation
Typically electron storage rings operate off crest in order
to ensure stability against phase oscillations.
As a consequence, the rf cavities must be detuned off
resonance in order to minimize the reflected power and the
required generator power.
Longitudinal gymnastics may also impose off crest
operation in recirculating linacs.
We write the beam current and the cavity voltage as
The generator power can then be expressed as:
02
and set 0
b
c
i
b
i
c c c
I I e
V V e
2 22
0 0(1 )1 cos tan sin
4
c L Lg b b
L c c
V I R I RP
R V V
Wiedemann
19.30
45
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Off Crest/Off Resonance Operation
Condition for optimum tuning:
Condition for optimum coupling:
Minimum generator power:
0tan sinLb
c
I R
V
0opt 1 cosa
b
c
I R
V
2
opt
,min
c
g
a
VP
R
Wiedemann
19.36
46
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Bettor Phasor Diagram
Off crest, synchrotron phase
gi
cos
icVe
1 tancos
i
c gc
b
VV i e ii
bi
,g opti
tan sinc b sV i
cVs
s
47
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
C75 Power Estimates
G. A. Krafft
48
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
12 GeV Project Specs
7-cell, 1500 MHz, 903 ohms
0
2
4
6
8
10
12
14
16
18
20
1 10 100
Qext (106)
P (
kW
)
21.2 MV/m, 460 uA, 25 Hz, 0 deg
21.2 MV/m, 460 uA, 20 Hz, 0 deg
21.2 MV/m, 460 uA, 15 Hz, 0 deg
21.2 MV/m, 460 uA, 10 Hz, 0 deg
21.2 MV/m, 460 uA, 0 Hz, 0 deg
13 kW
Delayen
and
Krafft
TN-07-29
460 µA*15 MV=6.8 kW49
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
7-cell, 1500 MHz, 903 ohms
0
2
4
6
8
10
12
14
16
18
20
1 10 100
Qext (106)
P (
kW
)
25.0 MV/m, 460 uA, 25 Hz, 0 deg
24.0 MV/m, 460 uA, 25 Hz, 0 deg
23.0 MV/m, 460 uA, 25 Hz, 0 deg
22.0 MV/m, 460 uA, 25 Hz, 0 deg
21.0 MV/m, 460 uA, 25 Hz, 0 deg
50
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Assumptions
• Low Loss R/Q = 903*5/7 = 645 Ω
• Max Current to be accelerated 460 µA
• Compute 0 and 25 Hz detuning power curves
• 75 MV/cryomodule (18.75 MV/m)
• Therefore matched power is 4.3 kW
(Scale increase 7.4 kW tube spec)
• Qext adjustable to 3.18×107(if not need more RF
power!)
51
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
0
2000
4000
6000
8000
10000
12000
14000
1 10 100
Power vs. Qext
Qext (×106)
Pg (kW)
25 Hz detuning
0 detuning3.2×107
52
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
RF Cavity with Beam and Microphonics
The detuning is now: 0 0
0 0
0
0tan 2 tan 2
where is the static detuning (controllable)
and is the random dynamic detuning (uncontrollable)
m
L L
m
f f fQ Q
f f
f
f
-10
-8
-6
-4
-2
0
2
4
6
8
10
90 95 100 105 110 115 120
Time (sec)
Fre
qu
en
cy
(H
z)
Probability Density
Medium CM Prototype, Cavity #2, CW @ 6MV/m
400000 samples
0
0.05
0.1
0.15
0.2
0.25
-8 -6 -4 -2 0 2 4 6 8
Peak Frequency Deviation (V)
Pro
ba
bilit
y D
en
sit
y
53
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Qext Optimization with Microphonics2 2
2 (1 )1 cos tan sin
4
c tot L tot Lg tot tot
L c c
V I R I RP
R V V
0
tan 2 L
fQ
f
where f is the total amount of cavity detuning in Hz, including static
detuning and microphonics.
Optimizing the generator power with respect to coupling gives:
where Itot is the magnitude of the resultant beam current vector in the
cavity and tot is the phase of the resultant beam vector with respect
to the cavity voltage.
2
2
0
0
( 1) 2 tan
where cos
opt tot
tot atot
c
fb Q b
f
I Rb
V
54
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Correct Static Detuning
To minimize generator power with respect to tuning:
The resulting power is
00
0
tan2
tot
ff b
Q
22
2
0
0
1(1 ) 2
4
c mg
a
V fP b Q
R f
22
2 2
0
0
2
11 2 1 2
4
12
c mg opt opt
a opt
copt
a
V fP b b Q
R f
Vb
R
55
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Optimal Qext and Power
Condition for optimum coupling:
and
In the absence of beam (b=0):
and
2
2
0
0
22
2
0
0
( 1) 2
1 ( 1) 22
mopt
opt c mg
a
fb Q
f
V fP b b Q
R f
2
0
0
22
0
0
1 2
1 1 22
mopt
opt c mg
a
fQ
f
V fP Q
R f
56
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Problem for the Reader
Assuming no microphonics, plot opt and Pgopt as function
of b (beam loading), b=-5 to 5, and explain the results.
How do the results change if microphonics is present?
57
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Example
ERL Injector and Linac:
fm=25 Hz, Q0=1x1010 , f0=1300 MHz, I0=100 mA,
Vc=20 MV/m, L=1.04 m, Ra/Q0=1036 ohms per cavity
ERL linac: Resultant beam current, Itot = 0 mA (energy
recovery)
and opt=385 QL=2.6x107 Pg = 4 kW per cavity.
ERL Injector: I0=100 mA and opt= 5x104 ! QL= 2x105
Pg = 2.08 MW per cavity!
Note: I0Va = 2.08 MW optimization is entirely
dominated by beam loading.
58
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
RF System Modeling
To include amplitude and phase feedback, nonlinear
effects from the klystron and be able to analyze transient
response of the system, response to large parameter
variations or beam current fluctuations
• We developed a model of the cavity and low level
controls using SIMULINK, a MATLAB-based program
for simulating dynamic systems.
Model describes the beam-cavity interaction, includes a
realistic representation of low level controls, klystron
characteristics, microphonic noise, Lorentz force detuning
and coupling and excitation of mechanical resonances
59
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
RF System Model
60
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
RF Modeling: Simulations vs.
Experimental Data
Measured and simulated cavity voltage and amplified gradient
error signal (GASK) in one of CEBAF’s cavities, when a 65 A,
100 sec beam pulse enters the cavity.
61
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Conclusions
We derived a differential equation that describes to a very
good approximation the rf cavity and its interaction with
beam.
We derived useful relations among cavity’s parameters and
used phasor diagrams to analyze steady-state situations.
We presented formula for the optimization of Qext under
beam loading and microphonics.
We showed an example of a Simulink model of the rf
control system which can be useful when nonlinearities
can not be ignored.
62
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Higher Order Modes (HOMs) Excitation
• HOMs are excited due to beam cavity interaction
• As bunch traverses a cavity, it deposits electromagnetic energy,
which is described in terms of wakefields
• Wakefields, in turn, can be presented as a sum of cavity
eigenmodes (fundamental and HOMs)
• Various components such as vacuum chamber, cavities,
bellows, dielectric coated pipes, and other obstacles that beam
passes through generates wakefields
• If a charge passes a cavity exactly on axis, it excites only
monopole modes
• Wakefields can act back on the beam and lead to instabilities
– This may limit the achievable current per bunch, the total
current, or even both
63
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Beam Cavity Interaction
64
The charge bunch leave a trail of wakefields that corresponds
to HOMs in the cavity.
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Beam Cavity Interaction
• If a charge passes a cavity exactly on axis, it excites only
monopole modes Longitudinal wakefield
– For a point charge this excitation depends only on the
amount of charge and the cavity shape.
– Subsequent bunches may be affected by these fields
and at high beam currents one must consider beam
instabilities and additional heating of accelerator
components.
– Causes energy loss and energy spread within the bunch
• If a charge is off axis, it excites dipole and quadrupole
modes Transverse wakefield
– Deflects particle trajectory
– Leads to beam instabilities
65
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Wakefields
66
For a relativistic point
charge (q) that traverses a
cavity parallel to the z-axis
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Longitudinal Wake Function and Loss Factor• For a bunch with charge Q:
• Energy loss (gain) by a single particle in the bunch
• Total energy loss by the bunch:
• Loss factor:
67
Gaussian charge distribution
o Can be represented as a sum of individual loss factors of cavity modes
o R/Q is in circuit definition
• Power loss due to HOMs: HOM l avgP k NqI
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Impedance
• Impedance is the Fourier transform of the wakefield potential
• Damp impedance with HOM couplers and absorbers to reduce
impedance below threshold
68
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
HOM Damping Methods
• Antenna couplers
• Waveguide dampers on
beam tubes
• Absorbers in cavity-
interconnecting beam
tubes (or cryomodule-
connecting beam tubes)
• Coupling through
Fundamental Power
Coupler (FPC)
69
Coaxial coupler
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
RF Focussing
In any RF cavity that accelerates longitudinally, because of
Maxwell Equations there must be additional transverse
electromagnetic fields. These fields will act to focus the beam
and must be accounted properly in the beam optics, especially in
the low energy regions of the accelerator. We will discuss this
problem in greater depth in injector lectures. Let A(x,y,z) be the
vector potential describing the longitudinal mode (Lorenz gauge)
tcA
1
2
22
2
22
cA
cA
70
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
...,
...,
2
10
2
10
rzzzr
rzAzAzrA zzz
For cylindrically symmetrical accelerating mode, functional form
can only depend on r and z
Maxwell’s Equations give recurrence formulas for succeeding
approximations
12
2
2
1
22
1,2
2
2
1,
22
2
2
nn
n
nz
nz
zn
cdz
dn
Acdz
AdAn
71
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Gauge condition satisfied when
nzn
c
i
dz
dA
in the particular case n = 0
00
c
i
dz
dAz
Electric field is
t
A
cE
1
72
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
So the magnetic field off axis may be expressed directly in terms
of the electric field on axis
00,0 zz A
c
i
dz
dzE
102
2
2
0
2
4,0 zzz
z AAcdz
AdzE
c
i
And the potential and vector potential must satisfy
zEc
rirAB zz ,0
2 2 1
73
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
And likewise for the radial electric field (see also )
dz
zdErzrE z
r
,0
2 2 1
Explicitly, for the time dependence cos(ωt + δ)
tzEtzrE zz cos,0,,
tdz
zdErtzrE z
r cos,0
2,,
tzEc
rtzrB z sin,0
2,,
0 E
74
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 202075
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 202076
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 202077
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Motion of a particle in this EM field
B
c
VEe
dt
md
V
''
'sin'2
''
'cos'
'
2
'
z
-z
dz
ztzGc
zxz
ztdz
zdGzx
zz
zz
xx
78
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
The normalized gradient is
2
0,
mc
zeEzG z
and the other quantities are calculated with the integral equations
z
z
zz dzztz
zGzz ''cos
'
'
z
dzztzGz ''cos'
z
zzz cz
dz
c
zzt
'
'lim 0
0
79
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
These equations may be integrated numerically using the
cylindrically symmetric CEBAF field model to form the Douglas
model of the cavity focussing. In the high energy limit the
expressions simplify.
z
a zz
x
z
a z
x
dzztzz
zGzxazax
dzzz
zzaxzx
''cos''
'
2
'
'''
''
2
80
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Transfer Matrix
E
EI
L
E
E
TG
G
21
4
21
For position-momentum transfer matrix
dzczzG
dzczzGI
/sinsin
/coscos
222
222
81
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Kick Generated by mis-alignment
E
EG
2
82
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Damping and Antidamping
By symmetry, if electron traverses the cavity exactly on axis,
there is no transverse deflection of the particle, but there is an
energy increase. By conservation of transverse momentum, there
must be a decrease of the phase space area. For linacs NEVER
use the word “adiabatic”
0
Vtransverse dt
md
x xz z
83
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Conservation law applied to angles
, 1
/ /
x y z
x x z x y y z y
z
x x
z
z
y y
z
zz z
zz z
84
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Phase space area transformation
z
x x
z
z
y y
z
dx d z dx dz z
dy d z dy dz z
Therefore, if the beam is accelerating, the phase space area after
the cavity is less than that before the cavity and if the beam is
decelerating the phase space area is greater than the area before
the cavity. The determinate of the transformation carrying the
phase space through the cavity has determinate equal to
Det
z
cavity
z
Mz z
85
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
By concatenation of the transfer matrices of all the accelerating
or decelerating cavities in the recirculated linac, and by the fact
that the determinate of the product of two matrices is the product
of the determinates, the phase space area at each location in the
linac is
000
000
y
z
zy
x
z
zx
ddyzz
zddy
ddxzz
zddx
Same type of argument shows that things like orbit fluctuations
are damped/amplified by acceleration/deceleration.
86
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Transfer Matrix Non-Unimodular
edeterminat daccumulateby normalize and
)before! (as part" unimodular" track theseparatelycan
detdetdet
!unimodular
det
21
2
2
1
1
21
MPMPM
M
M
M
M
MMP
MP
M
MMP
MMM
tot
tottot
tot
87
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
LCLS II
Subharmonic
Beam Loading
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Subharmonic Beam Loading
• Under condition of constant incident RF power, there is a
voltage fluctuation in the fundamental accelerating mode when
the beam load is sub harmonically related to the cavity
frequency
• Have some old results, from the days when we investigated
FELs in the CEBAF accelerator
• These results can be used to quantify the voltage fluctuations
expected from the subharmonic beam load in LCLS II
89
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
CEBAF FEL Results
Krafft and Laubach, CEBAF-TN-0153 (1989)
Bunch repetition
time τ1 chosen to
emphasize physics
90
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Model
• Single standing wave accelerating mode. Reflected power
absorbed by matched circulator.
• Beam current
• (Constant) Incident RF (β coupler coupling)
2 2 cosg cV ZP t
22
2
bc c c cc c
L c
d ZId V dV dVV
dt Q dt Q dt dt
b
l l
I t q t l I t l
91
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Analytic Method of Solution
• Green function
• Geometric series summation
• Excellent approximation
2ˆ ˆexp sin 1 1/ 42
c
c c c L
L
t tG t t t t Q
Q
/22cos cos
1c Lt n Q
c c L c
RP RV t t IQ e t
Q
/2
/ /22ˆ ˆcos cos cos 1
1 2
c L
c L c L
t n Q
Q Qcc c c c
RP Rq eV t t e t n e t n
Q D
92
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
2
1
RP
L
RIQ
Q
cV
cV
Phasor Diagram of Solution
93
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Single Subharmonic Beam
0
0.2
0.4
0.6
0.8
1
1.2
0 200 400 600 800 1000 1200
Voltage Deviation1
2
c
c
V
qR
Q
94
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Beam Cases
• Case 1
• Case 2
• Case 3
Beam Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µ A)
HXR 1 MHz 145 145
Straight 10 kHz 145 1.45
SXR 1 MHz 145 145
Beam Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µ A)
HXR 1 MHz 295 295
Straight 10 kHz 295 2.95
SXR 100 kHz 20 2
Beam Beam Pulse Rep. Rate Bunch Charge (pC) Average Current (µ A)
HXR 100 kHz 295 295
Straight 10 kHz 295 2.95
SXR 100 kHz 20 2
95
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Case 1
Straight Current
HXR Current
SXR Current
Total Voltage cV t
t2 µsec
100 µsec
96
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Case 1Volt
s
t/20 nsec
. . .
ΔE/E=10-4
97
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Case 2Volt
s
t/20 nsec
. . .ΔE/E=10-4
98
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Case 3
t/100 nsec
Volt
s
ΔE/E=10-4
99
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Summaries of Beam Energies
• Case 1
• Case 2
• Case 3
Beam Minimum (kV) Maximum (kV) Form
HXR 0.690 1.814 Linear 10
Straight 1.574 1.574 Constant
SXR 0.753 1.876 Linear 10
Beam Minimum (kV) Maximum (kV) Form
HXR 0.631 1.943 Linear 100
Straight 1.550 1.550 Constant
SXR 0.788 1.911 Linear 10
Beam Minimum (kV) Maximum (kV) Form
HXR 0.615 1.222 Linear 100
Straight 0.908 0.908 Constant
SXR 0.618 1.225 Linear 100
For off-crest cavities, multiply by cos φ
100
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Summary
• Fluctuations in voltage from constant intensity subharmonic
beams can be computed analytically
• Basic character is a series of steps at bunch arrival, the step
magnitude being (R/Q)πfcq
• Energy offsets were evaluated for some potential operating
scenarios. Spread sheet provided that can be used to
investigate differing current choices
101
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Case I’
Zero Crossing
Left Deflection
Right Deflection
Total Voltage
cV t
t20 µsec
100 µsec
102
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Case 2 (100 kHz contribution minor)
Zero Crossing
Left Deflection
Right Deflection
Total Voltage
cV t
t2 µsec
100 µsec
103
Accelerator Physics – USPAS, San Diego, CA, January 13-24, 2020
Case 3
Zero Crossing
Left Deflection
Right Deflection
Total Voltage cV t
t20 µsec
100 µsec
104