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Accelerator Physics
Particle Acceleration
S. A. Bogacz, G. A. Krafft, S. DeSilva, R. Gamage
Old Dominion University
Jefferson Lab
Lecture 6
USPAS Accelerator Physics June 2016
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
RF Focusing
Betatron Damping and Anti-damping
USPAS Accelerator Physics June 2016
Terminology
5 Cell Cavity
1 CEBAF Cavity
“Cells”
9 Cell Cavity
1 DESY Cavity
USPAS Accelerator Physics June 2016
Modern Jefferson Lab Cavities (1.497 GHz) are
optimized around a 7 cell design
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
USPAS Accelerator Physics June 2016
USPAS Accelerator Physics June 2016
Some Fundamental Cavity Parameters
• Energy Gain

d  mc 2
  eE x  t  , t  v


dt
• For standing wave RF fields and velocity of light particles
E  x , t   E  x  cos RF t        mc

2
  e  E  0, 0, z  cos  2 z / 
z
RF

=
eEz  2 / RF  e i  c.c.
Vc  eEz  2 / RF 
2
• Normalize by the cavity length L for gradient
E acc  MV/m  
Vc
L
USPAS Accelerator Physics June 2016
   dz
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)
Vc2
R

Q RF  stored energy 
• 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
USPAS Accelerator Physics June 2016
Unloaded Quality Factor
• As is usual in damped harmonic motion define a quality
factor by
Q
2  energy stored in oscillation 
energy dissipated in 1 cycle
• Unloaded Quality Factor Q0 of a cavity
Q0 
RF  stored energy 
heating power in walls
• Quantifies heat flow directly into cavity walls from AC
resistance of superconductor, and wall heating from
other sources.
USPAS Accelerator Physics June 2016
Loaded Quality Factor
• When add the input coupling port, must account for the energy
loss through the port on the oscillation
1
1
total power lost
1
1




Qtot QL RF  stored energy  Qext Q0
• Coupling Factor
Q0
Q0

1 for present day SRF cavities, QL 
Qext
1 
• 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)
USPAS Accelerator Physics June 2016
Q0 vs. Gradient for Several 1300 MHz
Cavities
Q0
1011
1010
AC70
AC72
AC73
AC78
AC76
AC71
AC81
Z83
Z87
Courtesey: Lutz Lilje
109
0
10
20
Eacc [MV/m]
USPAS Accelerator Physics June 2016
30
40
Eacc vs. time
45
40
35
BCP
EP
10 per. Mov. Avg. (BCP)
10 per. Mov. Avg. (EP)
Eacc[MV/m]
30
25
20
15
10
5
Courtesey: Lutz Lilje
0
Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06
USPAS Accelerator Physics June 2016
RF Cavity Equations





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
USPAS Accelerator Physics June 2016
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
USPAS Accelerator Physics June 2016
RF Cavity Fundamental Quantities
 Quality Factor Q0:
Q0 
0W
Pdiss

Energy stored in cavity
Energy dissipated in cavity walls per radian
 Shunt impedance Ra (accelerator definition);
Vc2
Ra 
Pdiss
 Note: Voltages and currents will be represented as complex
quantities, denoted by a tilde. For example:

Vc  t   Re Vc  t  eit
where Vc  Vc
varying phase.

Vc  t   Vc ei t 
is the magnitude of Vc
USPAS Accelerator Physics June 2016
and  is a slowly
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.
USPAS Accelerator Physics June 2016
Equivalent Circuit for an RF Cavity
 An rf cavity can be represented by a parallel LCR circuit:
V c (t )  V c eit
 Impedance Z of the equivalent circuit:
 Resonant frequency of the circuit:
 Stored energy W:
1
W  CVc2
2
USPAS Accelerator Physics June 2016
1
1

Z  
 iC 
 R iL

0  1/ LC
1
Equivalent Circuit for an RF Cavity
 Average power dissipated in resistor R:
Vc2
Pdiss 
2R
 From definition of shunt impedance
Vc2
Ra 
Pdiss

 Ra  2 R
Quality factor of resonator:
Q0 

0W
Pdiss

  0  
Note: Z  R 1  iQ0    
 0   

1
  0CR
For   0 ,

   0  
Z  R 1  2iQ0 


0



USPAS Accelerator Physics June 2016
1
Wiedemann
19.11
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
USPAS Accelerator Physics June 2016
Cavity with External Coupling
Consider the rf cavity after the rf is turned off.
Stored energy W satisfies the equation: dW / dt   Ptot
Total power being lost, Ptot, is: P  P  P  P
tot
diss
e
t
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
Q0 
 0W
Pdiss
QL 
 0W
Ptot
t
 0
dW
0W

 W  W0e QL
dt
QL
Energy in the cavity decays exponentially with time constant:
 L  QL / 0
USPAS Accelerator Physics June 2016
Decay rate equation
Ptot
Pdiss  Pe

 0W
 0W
suggests that we can assign a quality factor to each loss mechanism,
such that
1
1
1


QL
Q0 Qe
where, by definition,
Qe 
 0W
Pe
Typical values for CEBAF 7-cell cavities: Q0=1x1010, Qe QL=2x107.
USPAS Accelerator Physics June 2016
 Have defined “coupling parameter”:

and therefore
Pe
Q
 0
Pdiss Qe
1 (1   )

QL
Q0
Wiedemann
19.9
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.
USPAS Accelerator Physics June 2016
Cavity Coupled to an RF Source
 The system we want to model. A generator producing power Pg
transmits power to cavity through transmission line with
characteristic impedance Z0
Z0
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.
USPAS Accelerator Physics June 2016
Transmission Lines
I n2 L
Vn 1
C
I n 1
L Vn
C
…
In
L VN I N
C
Inductor Impedance and Current Conservation
Vn 1  Vn  i LI n 1
I n 1  I n  iCVn
USPAS Accelerator Physics June 2016
Z
Transmission Line Equations
• Standard Difference Equation with Solution
Vn  V0 e in , I n  I 0e in
V0  ei  1  i LI 0ei
I 0  ei  1  iCV0
2sin   /2    LC
V   L / C I  ei /2
V    L / C I ei /2
• Continuous Limit ( N → ∞)
   LC
k   LC 
V   x, t   V0 eit kx 
v  1/ LC 
L / C I 0 eit  kx  Z 0 I 0 eit  kx
V   x, t   V0 eit  kx   L / C I 0eit  kx  Z 0 I 0eit  kx
USPAS Accelerator Physics June 2016
Cavity Coupled to an RF Source
 Equivalent Circuit
Z0
ik  I   I 
V , I  
V , I  
1: k
RF Generator + Circulator
Coupler
Cavity
 Coupling is represented by an ideal (lossless) transformer of turns
ratio 1:k
USPAS Accelerator Physics June 2016
Cavity Coupled to an RF Source
 From transmission line equations, forward power from RF source
V
2
/ 2Z 0
 Reflected power to circulator
V
2
/ 2Z 0
 Transformer relations
Vc  kVk  k V  V 
ic  ik / k   I   I   / k  2 I  / k  Vc /  k 2 Z 0 
 Considering zero forward power case and definition of β

  V
2
 
/ 2Z0 / Vc
2

/ 2R  R /  k 2 Z0 
USPAS Accelerator Physics June 2016
Cavity Coupled to an RF Source
 Loaded cavity looks like
ig (t ) 
2I  t 
Wiedemann
Fig. 19.1
k
 Re  ig eit 


R
R
R

 2
 Zg 
Z g k Z0

 Effective and loaded resistance
1
1 1 1 
 

Reff R Z g
R
Wiedemann
19.1
Ra
RL  2 Reff 
1 
 Solving transmission line equations
1  Vc

V    kZ 0ic 
2 k

1  Vc

V    kZ 0ic 
2 k

USPAS Accelerator Physics June 2016
Powers Calculated
 Forward Power
  0  
V
Vc2 1   
1
Pg 
1  1  iQ0     
2
8Z 0 k

Ra 4
 0   
2
2
c
2
2

0  
2 
1  QL    

0   



 Reflected Power
  0  
V
Vc2 1   
1
Prefl 
1  1  iQ0     
2
8Z 0 k

Ra 4
 0   
2
2
c
2
2
    12
0  
2 

 QL    
2
 1   
0   



 Power delivered to cavity is
V 1   
Pg  Prefl 
Ra 4
2
c
2
    12  V 2
c

 Pdiss
1 

2
 1     Ra
as it must by energy conservation!
USPAS Accelerator Physics June 2016
Some Useful Expressions
 Total energy W, in terms of cavity parameters
Q0
P
0 diss
W
4

Pg
Pdiss 1    2

W  4

0  W
Q0
0
 Total impedance
1
 
 
1 Q 
 0
 
 0
1
2
2
L
Pg
2


 0
(1   ) 2  Q02 


 
 0
Q0
4
1
Pg
2
(1   ) 2 0

  0 
1   2QL
0 

ZTOT
 1
1

 
Z 
 Z g
ZTOT
R
 a
2
1


0  
(1


)

iQ



0 


0



USPAS Accelerator Physics June 2016
1
When Cavity is Detuned
 Define “Tuning angle” :
  
  0
tan   QL   0   2QL
0
 0  

4 Q0
1
W=
Pg
2
2
(1+ ) 0 1+tan 
 Note that:
Pdiss 
4
1
Pg
2
(1   ) 1  tan 2 
USPAS Accelerator Physics June 2016
for   0
Wiedemann
19.13
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:
Prefl  Pg  Pdiss


4
1
Prefl  Pg 1 

2
2
 (1   ) 1  tan  
.
Pdiss
1
0.9
0.8
On resonance:
W 
Dissipated and Reflected Power
4
Q0
Pg
(1   ) 2  0
4

Pg
2
(1   )
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
 1  
Prefl  
 Pg
 1  
0
1
USPAS Accelerator Physics June 2016
2
3
4
5

6
7
8
9
10
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:
iC  C
dVc
,
dt
iR 
Vc
,
RL / 2
Vc  L
diL
dt
ig the current induced by generator, ib beam current
 Differential equation that describes the dynamics of the system:
d 2Vc 0 dVc
0 RL d
2



V

ig  ib 
0 c
2
dt
QL dt
2QL dt
USPAS Accelerator Physics June 2016
Cavity with Beam
 Kirchoff’s law:
iL  iR  iC  ig  ib
 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
Vc  t   Re Vc eit 
ig  t   Re  ig eit 
ib  t   Re  ib eit 
where  is the generator angular frequency and
are complex quantities.
USPAS Accelerator Physics June 2016
Vc , ig , ib
Voltage for a Cavity with Beam
 Steady state solution
RL
(1  i tan  )Vc 
(ig  ib )
2
where  is the tuning angle.
 Generator current
2 Pg
2 2 Pg
| ig | 2 I 
2 
4 
k Z0
R

 For short bunches:
beam current.
| ib | 2 I 0
Pg
Ra
where I0 is the average
Wiedemann 19.19
USPAS Accelerator Physics June 2016
Voltage for a Cavity with Beam
 At steady-state:
or
RL / 2
RL / 2
ig 
ib
(1  i tan  )
(1  i tan  )
R
R
Vc  L ig cos ei  L ib cos ei
2
2
Vc 
or
Vc  Vgr cos ei  Vbr cos ei
or
Vc 
RL


V

i
g


 gr

2


R
L
V  
ib 
br



2

Vg

Vb
are the generator and beam-loading
voltages on resonance
and
Vg

Vb



are the generator and beam-loading voltages.
USPAS Accelerator Physics June 2016
Voltage for a Cavity with Beam
 Note that:
| Vgr |
2 
1 
Pg RL  2 Pg RL for large 
| Vbr | RL I 0
Wiedemann
19.16
Wiedemann
19.20
USPAS Accelerator Physics June 2016
Voltage for a Cavity with Beam
Im(Vg )
Vg  Vgr cos e
i
Vgr cos ei
Vb  Vbr cos ei
Re(Vg )
Vgr

As  increases, the magnitudes of both Vg and Vb
decrease while their phases rotate by .
USPAS Accelerator Physics June 2016
Example of a Phasor Diagram

Vb
Vbr
Vg
b
I acc
Vc
Ib
I dec
Wiedemann
Fig. 19.3
USPAS Accelerator Physics June 2016
On Crest/On Resonance Operation
 Typically linacs operate on resonance and on crest in order
to receive maximum acceleration.
 On crest and on resonance
Ib
Vbr

Vc
Vc  Vgr  Vbr
where Vc is the accelerating voltage.
USPAS Accelerator Physics June 2016
Vgr
More Useful Equations
 We derive expressions for W, Va, Pdiss, Prefl in terms of  and the
loading parameter K, defined by: K=I0Ra/(2  Pg )
 2 
Vc  Pg RL 
 1  
From:
| Vgr |
2 
1 
| Vbr | RL I 0
Vc  Vgr  Vbr

K  
1 
 


 
2
4  Q0 
K 
W
1 
 Pg
(1   ) 2 0 

Pg RL
2

Pdiss
4 
K 

1 
 Pg
(1   ) 2 
 
I 0Va  I 0 Ra Pdiss

2 
I 0Vc
K 


2 K 1 


Pg
1 


2
Prefl  Pg  Pdiss  I 0Va  Prefl
USPAS Accelerator Physics June 2016
(   1)  2 K  
 P

g
2
(   1)
Clarifications
 On Crest,
Vc 
2 
1 
Pg RL  RL I 0 

RL I 0 1  

Pg RL 1 

Pg RL 2 

2 
1 

Ra I 0
K 

2 Pg

 Off Crest with Detuning
Pg 
Z0 I 
2
2
2
=
Z 0 k ig
2
8
2Vc
ig 
1  i tan    2 I 0  cos b  i sin b 
RL
2
Pg 
Z 0 k Vc
2 RL2
2
2
2
 I R




I
R
1  0 L cos b    tan   0 L sin b  
Vc
Vc

 
 
USPAS Accelerator Physics June 2016
More Useful Equations
 For  large,
Pg
1
(Vc  I 0 RL ) 2
4 RL
Prefl
1
(Vc  I 0 RL ) 2
4 RL
 For Prefl=0 (condition for matching) 
VcM
RL  M
I0
and
Pg
M
0
M
c
I V
4
 Vc
I0 

 M
M 
V
I
0 
 c
USPAS Accelerator Physics June 2016
2
Example
 For Vc=14 MV, L=0.7 m, QL=2x107 , Q0=1x1010 :
I0 = 0
I0 = 100 A
I0 = 1 mA
3.65 kW
4.38 kW
14.033 kW
Pdiss
29 W
29 W
29 W
I0Vc
0W
1.4 kW
14 kW
Prefl
3.62 kW
2.951 kW
~ 4.4 W
Power
Pg
USPAS Accelerator Physics June 2016
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
I b  2 I 0ei b
Vc  Vcei c
and set  c  0
 The generator power can then be expressed as:
2
2
 
 
Vc2 (1   ) 
I 0 RL
 I 0 RL

Pg 
1

cos


tan


sin


b
b 

RL 4  
Vc
Vc


 


USPAS Accelerator Physics June 2016
Wiedemann
19.30
Off Crest/Off Resonance Operation
 Condition for optimum tuning:
tan  
I 0 RL
sin b
Vc
 Condition for optimum coupling:
 opt  1 
I 0 Ra
cos b
Vc
 Minimum generator power:
Pg ,min 
Vc2  opt
Ra
USPAS Accelerator Physics June 2016
Wiedemann
19.36
Bettor Phasor Diagram
 Off crest,  s synchrotron phase
Vc 1  i tan   
Vc
ei  ig  ib
cos 
s
Vc
Vc i
e
cos
ig
ib
USPAS Accelerator Physics June 2016
ig ,opt
Vc tan   ib sin s
C75 Power Estimates
G. A. Krafft
USPAS Accelerator Physics June 2016
12 GeV Project Specs
7-cell, 1500 MHz, 903 ohms
20
18
16
13 kW
14
P (kW)
12
10
8
Delayen
and
Krafft
TN-07-29
21.2 MV/m, 460 uA, 25 Hz, 0 deg
6
21.2 MV/m, 460 uA, 20 Hz, 0 deg
4
21.2 MV/m, 460 uA, 15 Hz, 0 deg
21.2 MV/m, 460 uA, 10 Hz, 0 deg
2
21.2 MV/m, 460 uA, 0 Hz, 0 deg
0
1
10
100
Qext (106)
460 µA*15 MV=6.8 kW
USPAS Accelerator Physics June 2016
7-cell, 1500 MHz, 903 ohms
20
18
16
P (kW)
14
12
10
8
25.0 MV/m, 460 uA, 25 Hz, 0 deg
6
24.0 MV/m, 460 uA, 25 Hz, 0 deg
4
23.0 MV/m, 460 uA, 25 Hz, 0 deg
22.0 MV/m, 460 uA, 25 Hz, 0 deg
2
21.0 MV/m, 460 uA, 25 Hz, 0 deg
0
1
10
100
6
Qext (10 )
USPAS Accelerator Physics June 2016
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!)
USPAS Accelerator Physics June 2016
Power vs. Qext
14000
12000
25 Hz detuning
10000
Pg (kW)
8000
6000
4000
3.2×107
0 detuning
2000
0
1
10
Qext (×106)
USPAS Accelerator Physics June 2016
100
RF Cavity with Beam and Microphonics
The detuning is now: tan   2QL
 f0   fm
tan  0  2QL
f0
 f0
f0
where  f0 is the static detuning (controllable)
Probability Density
Medium  CM Prototype, Cavity #2, CW @ 6MV/m
400000 samples
10
8
6
4
0.25
2
0
-2
-4
-6
-8
-10
90
95
100
105
Time (sec)
110
115
120
Probability Density
Frequency (Hz)
and  f m is the random dynamic detuning (uncontrollable)
0.2
0.15
0.1
0.05
0
-8
-6
-4
-2
0
2
Peak Frequency Deviation (V)
USPAS Accelerator Physics June 2016
4
6
8
Qext Optimization with Microphonics
2
2

 
 
V (1   )  Itot RL
Itot RL

Pg 
1

cos


tan


sin


tot 
tot  

RL 4  
Vc
Vc
 
 


2
c
tan   2QL
f
f0
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:


f
 opt  (b  1)2   2Q0
 b tan tot 
f0


I R
where b  tot a cos tot
Vc
2
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.
USPAS Accelerator Physics June 2016
Correct Static Detuning
 To minimize generator power with respect to tuning:
f0
 f0  
b tan tot
2Q0
2



Vc2 1 

f


2
m
Pg 
(1  b   )   2Q0
 
Ra 4 
f
0  



 The resulting power is
Vc2 1
Pg 
Ra 4  opt
2





f
2
2
m
1  b   2 1  b   opt   opt   2Q0
 
f
0  



Vc2

1  b   opt 
2 Ra
USPAS Accelerator Physics June 2016
Optimal Qext and Power
 Condition for optimum coupling:
2
 opt

 fm 
2
 (b  1)   2Q0

f
0 

Pgopt
2



Vc2 

f

b  1  (b  1) 2   2Q0 m  
2 Ra 
f0  



and
 In the absence of beam (b=0):
and
2
 opt

 fm 
 1   2Q0

f
0 

Pgopt
2



V 
f

1  1   2Q0 m  
2 Ra 
f0  



2
c
USPAS Accelerator Physics June 2016
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?
USPAS Accelerator Physics June 2016
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.
USPAS Accelerator Physics June 2016
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
USPAS Accelerator Physics June 2016
RF System Model
USPAS Accelerator Physics June 2016
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.
USPAS Accelerator Physics June 2016
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.
USPAS Accelerator Physics June 2016
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)

1 
 A  
c t
2 
2



2
2
 A 2 A
   2 
c
c
USPAS Accelerator Physics June 2016
For cylindrically symmetrical accelerating mode, functional form
can only depend on r and z
Az r , z   Az 0 z   Az1 z r 2  ...
 r , z   0 z   1 z r 2  ...
Maxwell’s Equations give recurrence formulas for succeeding
approximations
2
2n2 Azn 
d Az ,n1
2

2
2
Az ,n1
dz
c
2
2
d


2
2n n  n21   2 n1
dz
c
USPAS Accelerator Physics June 2016
Gauge condition satisfied when
dAzn
i
  n
dz
c
in the particular case n = 0
dAz 0
i
  0
dz
c
Electric field is



1 A
E   
c t
USPAS Accelerator Physics June 2016
And the potential and vector potential must satisfy
d0 i
E z 0, z   

Az 0
dz
c
d Az 0 
i

Ez 0, z  
 2 Az 0  4 Az1
2
c
dz
c
2
2
So the magnetic field off axis may be expressed directly in terms
of the electric field on axis
i r
 B  2rAz1 
E z 0, z 
2 c
USPAS Accelerator Physics June 2016

And likewise for the radial electric field (see also   E  0)
r dE z 0, z 
 Er  2r1 z   
2
dz
Explicitly, for the time dependence cos(ωt + δ)
Ez r, z, t   Ez 0, z cost   
r dE z 0, z 
Er r , z , t   
cost   
2
dz
B r , z , t   
r
2c
E z 0, z sin t   
USPAS Accelerator Physics June 2016
USPAS Accelerator Physics June 2016
USPAS Accelerator Physics June 2016
USPAS Accelerator Physics June 2016
Motion of a particle in this EM field




  V 
d mV
 e E   B 
dt
c


 z  x z       x   
xz ' dG z '



cost z '    
z 
dz '
2
dz '
 

 z z 'xz '
 z  z '

- 

G z 'sin t z '   


2c
USPAS Accelerator Physics June 2016
The normalized gradient is
eE z z ,0
Gz  
mc2
and the other quantities are calculated with the integral equations
z
 z         G z ' cost  z '   dz '

G  z '
 z  z z       z     
cost  z '   dz '
  z '
 z
z
z
z0
dz '
t  z   lim

z0      c
  z 'c
z
 z
USPAS Accelerator Physics June 2016
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 ' x z '
xz   xa   
dz '
 z ' z z '
a
z
 x   
x  z ' G  z '
z  a   
 xa  
cost z '   dz '
2
 z   
2  z ' z z '
a
z
USPAS Accelerator Physics June 2016
Transfer Matrix
For position-momentum transfer matrix
L 
 EG
1 

2E
 

T
EG 
I

  4 1  2 E 



I  cos 2    G 2  z  cos 2 z / c dz


 sin 2    G 2 z sin 2 z / c dz

USPAS Accelerator Physics June 2016
Kick Generated by mis-alignment

EG
 
2E
USPAS Accelerator Physics June 2016
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”



d mVtransverse
0
dt
  z   x  z       x   
USPAS Accelerator Physics June 2016
Conservation law applied to angles
 x ,  y   z  1
x  x / z
x  y   y / z
y
     z   
x  z  
 x   
  z  z  z 
     z   
y  z 
 y   
  z  z  z 
USPAS Accelerator Physics June 2016
Phase space area transformation
     z   
dx  d x   
dx  d x  z  
  z  z  z 
     z   
dy  d y   
dy  d y  z  
  z  z  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
     z   
Det  M cavity  
  z  z  z 
USPAS Accelerator Physics June 2016
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
 0 z 0
dx  d x z  
dx  d x 0
 z  z z 
 0 z 0
dy  d y z  
dy  d y 0
 z  z z 
Same type of argument shows that things like orbit fluctuations
are damped/amplified by acceleration/deceleration.
USPAS Accelerator Physics June 2016
Transfer Matrix Non-Unimodular
M tot  M 1  M 2
M
PM  
det M
PM 
unimodular !
M tot
M1
M2
PM tot  

 PM 1   PM 2 
det M tot det M 1 det M 2
 can separately track the " unimodular part" (as before! )
and normalize by accumulate d determinat e
USPAS Accelerator Physics June 2016
LCLS II
Subharmonic
Beam Loading
USPAS Accelerator Physics June 2016
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
USPAS Accelerator Physics June 2016
CEBAF FEL Results
Bunch repetition
time τ1 chosen to
emphasize physics
Krafft and Laubach, CEBAF-TN-0153 (1989)
USPAS Accelerator Physics June 2016
Model
• Single standing wave accelerating mode. Reflected power
absorbed by matched circulator.
d 2Vc c dVc
c  dV d  ZIb  
2

 c Vc 



2
dt
QL dt
Qc  dt
dt 
• Beam current
Ib  t  


l 
l 
 q  t  l     I   t  l 
• (Constant) Incident RF (β coupler coupling)
V  2  2ZPg cos ct    
USPAS Accelerator Physics June 2016
Analytic Method of Solution
• Green function
 c  t  t   
G  t  t    exp  
 sin ˆ c  t  t  
2QL 

• Geometric series summation
ˆ c  c 1  1/ 4QL2

2  RP
c Rq  ec t n  /2QL c / QL
c /2 QL
ˆ
ˆ


Vc  t  
cos ct     
e
cos

t

n


e
cos

t

n

1









c
c
1 
2Q 
D


• Excellent approximation
2  RP
R

Vc  t  
cos ct      IQL ec t  n   /2QL cos ct
1 
Q
USPAS Accelerator Physics June 2016

Phasor Diagram of Solution
R
IQL
Q
Vc  


2  RP
1 
USPAS Accelerator Physics June 2016
Vc   
Single Subharmonic Beam
1.2
Voltage Deviation
1
1
 Vc
 R c q 
Q 2 


0.8
0.6
0.4
0.2
0
0
200
400
600
USPAS Accelerator Physics June 2016
800
1000
1200
Beam Cases
• Case 1
Beam
Beam Pulse Rep. Rate
Bunch Charge (pC)
Average Current (µA)
HXR
1 MHz
145
145
Straight
10 kHz
145
1.45
SXR
1 MHz
145
145
• Case 2
Beam
Beam Pulse Rep. Rate
Bunch Charge (pC)
Average Current (µA)
1 MHz
10 kHz
100 kHz
295
295
20
295
2.95
2
Beam Pulse Rep. Rate
Bunch Charge (pC)
Average Current (µA)
HXR
100 kHz
295
295
Straight
10 kHz
295
2.95
SXR
100 kHz
20
2
HXR
Straight
SXR
• Case 3
Beam
USPAS Accelerator Physics June 2016
Case 1
Straight Current
HXR Current
SXR Current
Total Voltage
 Vc  t 
2 µsec
t
100 µsec
USPAS Accelerator Physics June 2016
Case 1
Volts
ΔE/E=10-4
...
t/20 nsec
USPAS Accelerator Physics June 2016
Volts
Case 2
ΔE/E=10-4
...
t/20 nsec
USPAS Accelerator Physics June 2016
Case 3
Volts
ΔE/E=10-4
t/100 nsec
USPAS Accelerator Physics June 2016
Summaries of Beam Energies
For off-crest cavities, multiply by cos φ
• Case 1
Beam
HXR
Straight
SXR
Minimum (kV)
Maximum (kV)
Form
0.615
0.908
0.618
1.222
0.908
1.225
Linear 100
Constant
Linear 100
Minimum (kV)
Maximum (kV)
Form
0.631
1.550
0.788
1.943
1.550
1.911
Linear 100
Constant
Linear 10
Minimum (kV)
Maximum (kV)
Form
0.690
1.574
0.753
1.814
1.574
1.876
Linear 10
Constant
Linear 10
• Case 2
Beam
HXR
Straight
SXR
• Case 3
Beam
HXR
Straight
SXR
USPAS Accelerator Physics June 2016
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
USPAS Accelerator Physics June 2016
Case I’
Zero Crossing
Left Deflection
Right Deflection
Total Voltage
 Vc  t 
20 µsec
t
100 µsec
USPAS Accelerator Physics June 2016
Case 2 (100 kHz contribution minor)
Zero Crossing
Left Deflection
Right Deflection
Total Voltage
 Vc  t 
2 µsec
t
100 µsec
USPAS Accelerator Physics June 2016
Case 3
Zero Crossing
Left Deflection
Right Deflection
Total Voltage
 Vc  t 
20 µsec
t
100 µsec
USPAS Accelerator Physics June 2016