Accelerator Physics · Graduate Accelerator Physics Fall 2015 Q ext Optimization with Microphonics...
Transcript of Accelerator Physics · Graduate Accelerator Physics Fall 2015 Q ext Optimization with Microphonics...
Graduate Accelerator Physics Fall 2015
Accelerator Physics Particle Acceleration
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 4
Graduate Accelerator Physics Fall 2015
Clarifications from Last Time
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
Graduate Accelerator Physics Fall 2015
Off Crest/Off Resonance Power
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
16.31
Graduate Accelerator Physics Fall 2015
Optimal Detuning and Coupling
Condition for optimum tuning with beam:
Condition for optimum coupling with beam:
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
16.36
Graduate Accelerator Physics Fall 2015
C75 Power Estimates
G. A. Krafft
Graduate Accelerator Physics Fall 2015
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 muA*15 MV=6.8 kW
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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!)
Graduate Accelerator Physics Fall 2015
0
2000
4000
6000
8000
10000
12000
14000
1 10 100
Power vs. Qext
Qext (×106)
Pg (kW)
25 Hz detuning
0 detuning 3.2×107
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
Qext Optimization with Microphonics 2 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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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?
Graduate Accelerator Physics Fall 2015
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.
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
RF System Model
Graduate Accelerator Physics Fall 2015
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.
Graduate Accelerator Physics Fall 2015
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.
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
...,
...,
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
Graduate Accelerator Physics Fall 2015
Graduate Accelerator Physics Fall 2015
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
Transfer Matrix
E
EI
L
E
E
TG
G
21
4
21
For position-momentum transfer matrix
dzczzG
dzczzGI
/sinsin
/coscos
222
222
Graduate Accelerator Physics Fall 2015
Kick Generated by mis-alignment
E
EG
2
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
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.
Graduate Accelerator Physics Fall 2015
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
Graduate Accelerator Physics Fall 2015
Solenoid Focussing
Can also have continuous focusing in both transverse directions by applying solenoid
magnets:
B z
z
Graduate Accelerator Physics Fall 2015
Busch’s Theorem
For cylindrical symmetry magnetic field described by a vector potential:
Conservation of Canonical Momentum gives Busch’s Theorem:
1ˆ, , is nearly constant
0, 0,,
2 2
z
z z
r
A A z r B rA z rr r
B r z r B r z rA z r B
2
22
for particle with 0 where 0, 0
2 2
z
czLarmor
P mr qrA const
B P
qr Bmr
Beam rotates at the Larmor frequency which implies coupling
Graduate Accelerator Physics Fall 2015
Radial Equation
2 2
2
2 2
2 2
2 2 22
2
thin lens focal length
1 weak compared to quadrupole for high
4
L z L
L
z
z
z
dmr mr qr B mr
dt
kc
e B dz
f m c
If go to full ¼ oscillation inside the magnetic field in the “thick” lens case, all particles
end up at r = 0! Non-zero emittance spreads out perfect focusing!
x
y
Graduate Accelerator Physics Fall 2015
Larmor’s Theorem
This result is a special case of a more general result. If go to frame that rotates with the
local value of Larmor’s frequency, then the transverse dynamics including the
magnetic field are simply those of a harmonic oscillator with frequency equal to the
Larmor frequency. Any force from the magnetic field linear in the field strength is
“transformed away” in the Larmor frame. And the motion in the two transverse
degrees of freedom are now decoupled. Pf: The equations of motion are
2
2
2 2
2
2 2
2
2
2-D Harmonic Oscillator
z
L L z L z
L
dmr mr qr B
dt
mr qA cons P
dmr mr mr mr qr B qr B
dt
mr P
dmr mr mr
dt
mr P