Upload
gilbert-bond
View
216
Download
0
Embed Size (px)
Citation preview
Room temperature RFPart 2.1: Strong beam-cavity coupling
(beam loading)
30/10/2010A.Grudiev
5th IASLC, Villars-sur-Ollon, CH
Outline1. Linear collider and rf-to-beam efficiency2. Beam loading effect: what is good and what is bad about it
a. Beam-cavity couplingi. Steady-state power flowii. Efficiency in steady-state and in pulsed regimeiii. Compensation of the transient effect on the acceleration
b. Standing wave structures (SWS)c. Travelling wave structure (TWS)
i. Steady-state power flowii. Efficiency in steady-state and in pulsed regimeiii. Compensation of the transient effect on the acceleration
3. Examples:a. CLIC main linac accelerating structureb. CTF3 drive beam accelerating structure
Linear collider and rf-to-beam efficiency
Modulator
RF power source
Linac
AC power from the grid: PAC
High voltage DC power: PDC
RF power: Prf
Beam power: Pb=IbEE
Ib
AC-to-DC conversion efficiency: ηAC-DC = PDC/PAC
DC-to-rf conversion efficiency: ηDC-rf = Prf/PDC
rf-to-beam efficiency: ηrf-to-beam = Pb/Prf
Two main parameters of a collider are center of mass collision energy E and luminosity L
For a fixed E: L ~ Ib ~ Pb ~ PACηAC-DCηDC-rfηrf-to-beam
All efficiencies are equally important. In this lecture, we will focus on the ηrf-to-beam
Beam loading in steady-state (power flow)
W
V
Pin
P0Cavity parameters w/o beam (reminder):
1;
;:condition Matching
;;;
00
0
00
2
0
extext
in
inext
Q
QQQ
PP
P
WQ
P
WQ
R
VP
W
V
Pin
P0
Cavity parameters with beam:
loading beam relativewhere
111;111
:condition Matching
;
0
0
0
0
V
RIY
YQ
Q
QQQ
PPP
constRI
QV
P
WQIVP
bb
bbbext
bin
bbbbb
PbIb
satisfied at any V
satisfied only if V = IbR
Beam loading and efficiency
In steady-state regime which correspond to CW (Continues Wave) operation V = const:
b
b
b
ext
in
bSWSbeamtorf Y
Y
Q
Q
P
P
1
W
V
Pin
P0
PbIb
The higher is the beam loading the higher is the rf-to-beam efficiency
In pulse regime, V ≠ const is a function of time:
V(t)
t
Pin
tf tp
V
p
fpbeamtorf
pin
bbpulsedbeamtorf t
tt
tP
tP
Ib
Transient beam loading effectEnergy conservation in a cavity without beam (reminder):
factor loaded 1
:where
;11
2)(ˆ
;0,1
0,01)(ˆ
:excitationfunction -step afor
ˆ2ˆ11
1ˆ2
in resultswhat
ˆ;
ˆ
ˆˆ;)ˆˆ(
;ˆ
where
0
2
0
2
0
2
0
22
0
Q
etV
t
t
Z
RtV
dt
VdQV
Z
RV
Z
V
Q
W
Q
W
R
VP
VVZ
VVP
Z
VP
dt
dWPPP
L
Q
t
in
in
rad
ext
refinrefrad
outin
in
outin
L
W
V
Pin
P0
Pout
IinVin
IrefVref
Vrad
Irad
·
Pin
Pout
Short-CircuitBoundaryCondition:
Matching condition: Vrad = Vin, only if β=1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.5
1
1.5
2
/2
V
=10=2=1=1/2
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.2
0.4
0.6
0.8
1
1.2
1.4
/2
V
tinj
=1000
tinj
=TinjM
tinj
=2000
tinj
=4000
tinj
=6000
no beam
TinjM
Transient beam loading effectEnergy conservation in a cavity with beam:
inj
injbin
bin
bb
outbin
tt
ttRtI
t
t
Z
RtV
dt
VdQV
RI
Z
RV
IVP
dt
dWPPPP
,1
,0)(ˆ;
0,1
0,01)(ˆ
:excitationfunction -step afor
ˆ2ˆ11ˆ1ˆ2
in resultswhat
ˆˆwhere
0
0
Depending on the tinj, there is transient beam loading which is variation of the voltage gained in the cavity along the upstream part of the beam. It is compensated only when:
W
V
Pin
P0
PbIb
Pout
fLM
injinj tQ
Tt
1
2ln
2
Solution for β=2 and different beam injection times tinj
From a cavity to a linacFor obvious reasons multi cell accelerating structures are used in linacs instead of single cell cavitiesWe will distinguish two types of accelerating structures:
Standing Wave Structures (SWS) Travelling Wave Structures (TWS)
In linacs, it useful to use some parameters normalized to a unit of length:
Voltage V [V] -> Gradient G [V/m] = V/Lc
Shunt impedance: R [Ω] -> R’ [Ω/m] = R/Lc
Stored energy: W [J] -> W’ [J/m] = W/Lc
Power loss: P [W] -> P’ [W/m] = P/Lc
where Lc – cell length
SWS
TWS
Beam loading in SWS
Here it is assumed that the coupling between cells is much stronger than the input coupling and the coupling to the beam then the beam loading behavior in SWS is identical to the beam loading behavior in a single cell cavity.
SWS parameters w/o beam (reminder):
1;
;:condition Matching
;;;
00
0
00
00
2
0
extext
L
in
inext
Q
QQQ
dlPPP
P
WQ
P
WQ
R
GP
s
SWS parameters with beam:
loading beam relativewhere
111;111
:condition Matching
;
0
0
0
0
G
RIY
YQ
Q
QQQ
PPP
constRI
QG
P
WQIGP
bb
bbbext
bin
bbbbb
satisfied at any G
satisfied only if G = IbR’
z
g
bl
dzzvzQ
g
g
g
inz
g
b
g
g
g
bb
dzzQzv
zR
zG
IzGzG
eR
zR
zQ
Q
zv
vGzG
Qv
RPGG
Qv
RI
Qvdz
Rd
Rdz
dQ
Qdz
dv
v
G
dz
dG
GIR
GPP
dz
dP
z
g
0 0
)()(2
1
0
00
000
00
0
0
2
0
)()(2
)(
)(1)()(
)0(
)(
)(
)0(
)(
)0()(
:[*] form closed ain obtained isSolution
conditionboundary input ;with
2
111
2
0 0
Beam loading in TWS in steady-state
velocitygroup;:where
TWS along flowpower ;
losses wall;;
0
2
00
2
0
W
Pv
QR
GvP
P
WQ
R
GP
g
g
In a cavity or SWS field amplitude is function of time only f(t). In TWS, it is function of both time and longitudinal coordinate f(z,t). Let’s consider steady-state f(z).
z
TWS parameters are function of z:
Energy conservation law in steady-state yields:(it is assumed hereafter that power flow is matched and there are no reflections)
[*] – A. Lunin, V. Yakovlev, unpublished
Efficiency in steady-state:
Pin Pout
sL
in
b
in
bCTWSbeamtorf dzzG
P
I
P
P
0
)(
constantn attenuatio TWS;2
where
1)(
0
0
g
bl
vQ
zeRIzeGzG
TWS example 1: constant impedance TWSIn constant impedance TWS, geometry of the cells is identical and group velocity, shunt impedance and Q-factor are constant along the structure.
This simplifies a lot the equations:
z
Efficiency in steady-state:
Pin Pout
loading beam relative TWS;
where
1212
00
200
G
RIY
sL
eLYsL
eY
bb
sbbCITWSbeamtorf
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
z/Ls
G/G
0
= 0.45
unloaded
loaded
Gradient unloaded and loaded for Yb0=1
•The higher is the beam loading the higher is the rf-to-beam efficiency•Beam loading reduces the loaded gradient compared to unloaded
Full beam loadingIn TWS, depending on the beam current, beam can absorb all available rf power so that Pout = Pin – P0 – Pb = 0 This is the case of full beam loading where rf-to-beam efficiency is closed to maximum.
z
Pin Pout=0
Pb
Ib Ib
0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
z/Ls
G/G
0
Yb0
= 0
Yb0
= 1
Yb0
= 2
Yb0
= 3
Yb0
= 4
Yb0
= 5
Yb0
= 6
unloaded
loaded
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Yb0
Structure is overloaded if the beam current is even higher
00
00
000
2
1
2
10
02
1
0
21;)0(:TWS aperedStronger t
)0(:where;1ln)(;2;)(:TWSgradient Const
loaded, ;111
)()( unloaded; ;1)(00
aLGzG
aza
RIGzGaGzG
aza
RIzGzGazGzG
s
bl
a
a
bla
TWS example 2: tapered TWSIn tapered TWS, geometry is different in each cell. If cell-to-cell difference is small, it is a weakly tapered structure and smooth approximation (no reflections) can still be applied:
Let’s consider a case of linear vg-tapering such that Q0=const, R’=const but vg = vg0(1+az):
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z/Ls
G/G
0
= 0.5
unloaded
loaded
vg/v
g0
Yb0=1, a=-2α
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z/Ls
G/G
0
= 0.53
unloadedloaded
vg/v
g0
Yb0=1, a=-3α
W’ = P/ vg ≈ const
;2:for 111ln12
12
1111
11
2111
2
02000
2
1
02
102
1
202
02
1
02
1
00
00
aaLaLYLY
aLaa
YaLa
Y
ssbsbCGTWSbeamtorf
as
aa
bas
a
bTWS
beamtorf
Tapering and rf-to-beam efficiencyFor the case of linear vg-tapering such that Q0=const, R’=const but vg = vg0(1+az)Efficiency in steady-state:
For low beam loading tapering helps to increase rf-to-beam efficiency
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
Yb0
SWS
TWS-CI: a = 0TWS-CG: a = -2
0
TWS: a = -30
The stronger is the tapering the higher is the efficiency at low beam loading The stronger is the tapering the low is the Yb0 for the maximum efficiency (full beam loading)SWS has higher efficiency than TWS at low beam loading BUT lower efficiency than fully beam loaded TWS
TWS efficiency in pulsed regime
timefilling;)(
:where0
sL
gf
p
fpbeamtorf
pin
bbpulsedbeamtorf
zv
dzt
t
tt
tP
tP
V(t)=∫G(z,t)dz
t
Pin
tf tp
V
In pulse regime, V ≠ const is a function of time. In simplest case of vg=const, R’/Q0=const, Q0=∞
G(z,t)
zLs
G0G(z,t1) G(z,t3)G(z,t2) G(z,tf)
z3z1 z2
tn = zn/vg; tf = Ls/vg In general, in TWS:
Transient beam loading in TWSLet’s continue with the simplest case of vg=const, R’/Q0=const, Q0=∞
tftp
G(z,t)
zLs
G0 G(z,t1) G(z,t3)G(z,t2) G(z,tf)
z3z1 z2
Ib = 0; 0 < t ≤ tf
0)( GzG zRIGzG bl 0)(
tinj tinj+tf
Ib
Ib > 0; tinj < t ≤ tinj+tfG(z,t)
zLs
G0G(z,t1)
G(z,t3)G(z,t2)
G(z,t4)
z3z1 z2
G1=G0-IbR’αLs
Transient beam loading result in different voltage gained by the bunches during tinj < t ≤ tinj+tf
t
Pin
V
Ib
Vl
tp+tf
Compensation of the transient beam loading in TWS
Let’s continue with the simplest case of vg=const, R’/Q0=const, Q0=∞
tf tp
tinj=tf
Two conditions:1. Ramping the input power
during the structure filling in such a way that the loaded gradient profile along the structure is formed.
2. The beam is injected right at the moment the structure is filled. (same as for SWS)
t
Pin
V
Ib
VL
tp+tf
G(z,t)
zLs
G0
G(z,t1) G(z,t3)G(z,t2) G(z,tf)
z3z1 z2
Structure filling process:Ib = 0; 0 < t ≤ tf
G1
G(z,t)
zLs
G0G(z,t1)
G(z,t3)G(z,t2)
G(z,tf)
z3z1 z2
Injection right after filling:Ib > 0; t > tinj = tf
G1
Pin-ramp
Ib
Transient beam loading in TWSLet’s consider general case of vg(z), R’(z), Q0(z)
),(),(),(
toarrive to time;)(
)(:
)(
)()()(),(
)()(),(
)();(:modeldependent 2.Time
)(
)()()()()()(
)()(
;:solution StateSteady 1.
0
0
0
00
0
0
0
0
tzGtzGtzG
zzv
dzzwhere
dzzg
zfztIzgtzG
zgztGtzG
tIItGG
dzzg
zfIzgzgGzGzGzG
zgGzG
constIconstG
bl
z
g
z
bb
bb
z
bbl
b
Model approximations:1. Smooth propagation of
energy along the structure (no reflections)
2. No effects related to the finite bandwidth of the signal and the structure (TWS bandwidth >> signal bandwidth)
3. Time of flight of the beam through the structure is much shorter than the filling time (Ls/c << tf)
Transient beam loading in TWS (cont.)
where:
voltagebeam;),()(
voltageunloaded;),()(
voltageloaded);()()(
0
0
dztzGtV
dztzGtV
tVtVtV
z
bb
z
bl
Compensation of the transient beam loading in TWS
In general case of vg=const, R’=const, Q0=const
;)(
)(ofsolution theis )(:where
)(
)()(
0;0);(
: timefilling duringgradient input Then
solution loaded is )(
andsolution unloaded is )(
:state-steadyin If
0
0
00
0
z
g
f
fl
fb
l
zv
dzzttz
ttzg
ttzGtG
ttItGG
zG
g(z)GzG
Compensation of the transient beam loading in constant impedance TWS
Let’s consider the case of constant impedance TWS: vg=const, R’=const, Q0=const
Uncompensated case
1)(
)()(:0for function -ramp when the
)()(andsolution statesteady ;1)(;)(
00
0
00
ttveRItGtG
v
Ltt
v
z
zv
dzztzeRIzeGzGzeGzG
fgbf
g
sf
z
ggbl
Compensated case
Compensation of the transient beam loading in constant gradient TWS
Constant gradient: a=-2α0
Let’s consider a case of linear vg-tapering such that Q0=const, R’=const but vg = vg0(1+az):
a=-3α0
CLIC main linac accelerating structure in steady-state
Parameters: input – outputf [ GHz] 12Ls [m] 0.25
vg/c [%] 1.7 – 0.8
Q0 6100 – 6300
R’ [MΩ/m] 90 – 110Ib [A] 1.3
tb [ns] 156
<G> [MV/m] 100
In reality, vg ≠ const, R’ ≠ const, Q0 ≠ const and general expressions must be applied but often a good estimate can be done by averaging R’ and Q0 and assuming linear variation of vg
In this case: <Q0>=6200; <R’>=100 MΩ/m; and vg/c = 1.7 - 0.9/0.25·z
Parameters calculated
Pin [MW] 67.6
ηCW [%] 48
tf [ns] 70
η [%] 33
0 0.05 0.1 0.15 0.2 0.250
2
4
6
8
10
12
14x 10
7
z[m]
G[V
/m];
vg[m
/s]
CW = 0.48
GG
l
vg
CLIC main linac accelerating structure during transient
Parameters: input – outputf [ GHz] 3Ls [m] 1.22
vg/c [%] 5.2 – 2.3
Q0 14000 – 11000
R’ [MΩ/m] 42 – 33Ib [A] 4
Pin [MW] 33
SICA - CTF3 3GHz accelerating structure in steady-state
Parameters calculatedηCW [%] 95
This structure is designed to operate in full beam loading regime
0 0.2 0.4 0.6 0.8 1 1.20
0.5
1
1.5
2x 10
7
z [m]
G [
V/m
], v
g[m
/s]
CW = 0.95GG
l
vg
Experimental demonstration of full beam loading