Short-Range Wakes in Elliptical Pipe Geometry
description
Transcript of Short-Range Wakes in Elliptical Pipe Geometry
Short-Range Wakes in Elliptical Pipe Geometry
Igor Zagorodnov
Beam Dynamics Group Meeting
05.09.05
1w mm width of the slot 5L m length of the slot
Pumping slot
25w m
Theoretical results (Bethe theory) only for .w
Pumping slot
Effect of the slot is small.
•Accuracy estimation of the numerical results.•Wake scaling for geometry parameters.
Used tools: ECHO (time-domain), CST Microwave Studio (modeling, meshing), Matlab (pre- and postprocessing).
0 lnhi
Z bZ
a 2
hiloss
z
Z ck
250 m
25 m
1( ln( / ))lossk O b a
37.2 /lossk V pC
372 /lossk V pC
in outW W W
Out In
2( ) /(2 )L b a
ba
[m]
dx
[dy,dz
[Loss,2D
[V/pC]
Error, 2D
[V/pC]
Loss, 3D
[V/pC]
Error, 3D
[V/pC]
250 1/40 1/40 36.3097 ref.
250 1/20 1/20 36.2932 0.016
250 1/5 1/5 36.2369 0.07 36.2265 0.1
25 1/20 1/20 370.77 ref.
25 1/10 1/10 370.60 0.17
25 1/5 1/5 369.96 0.8
25 1/5 2 369.78 1
outlossk
1( ln( / ))outlossk O b a
Accuracy of 3D calculations
-5 0 5-15
-10
-5
0
5x 10-3
out V mW
pC
/s
250 m
25 m
1( )outW O
[m]
dx
[dy,dz
[Loss,2D
[V/pC]
Error, 2D
[V/pC]
Loss, 3D
[V/pC]
Error, 3D
[V/pC]
250 1/40 1/40 -0.8606 ref.
250 1/10 1/10 -0.8678 0.001
250 1/5 1/5 -0.8802 0.012 -0.8478 0.013
25 1/20 1/20 0.8639 ref.
25 1/10 1/10 -0.8619 0.002
25 1/5 1/5 -0.8859 0.02
25 1/5 2 -0.9380 0.17
inlossk
1( )inlossk O a
in outloss lossk k
Accuracy of 3D calculations
-5 0 50
0.5
1
1.5
2in V
WpC
/s
250
(3 )
m
D
(1), 0inW O
25
(3 )
m
D
25
(2 )
m
D
1 lnoutloss
bk O
a
1inlossk O
a
Scaling laws
Out In
2( ) /(2 )L b a
ba
( )w
1 lnoutloss
bk O
a
inlossk O
a
, ?
Scaling laws
w
a
b
w10h mm
4a mm
Out In
w10h mm
4a mm
w
[mm]
[m]
Loss
[V/pC]
7 0.339 250 3.04+-0.1
4 0.167 250 0.91+-0.1
2 0.083 250 0.23+-0.1
7 0.339 25 31.8+-1
4 0.167 25
2 0.083 25 2.8+-1
outlossk
1( )outlossk O
1.8( ) ?!outlossk O
w
-5 0 5-1.5
-1
-0.5
0
0.51.8 1.8
outW V
pC rad
/s
0.3392
1.8( ), 0,outW O w
0.1672
0.0832
250 m
w10h mm
4a mm
w
[mm]
[m]
Loss
[V/pC]
7 0.339 250 -0.162+-0.013
4 0.167 250 -0.065+-0.013
2 0.083 250 -0.01+-0.013
7 0.339 25 -0.247+-0.17
4 0.167 25
2 0.083 25
inlossk
( ) ?!inlossk O
(1), 0inlossk O
in outloss lossk k
w
-5 0 5-0.05
0
0.05
0.1
0.15
0.2inW V
pC rad
/s
0.3392
( ), 0,inW O w
0.1672
0.0832
250 m
?error
w
1 1.8 lnoutloss
bk O
a
inlossk O
a
w
[mm]
a,
[mm]
[m]
Loss
[V/pC]
7 4 0.339 250 -0.16
1 7.5 0.021 25 -0.005
0.021 40.16 0.005
0.339 7.5inloss
Vk
pC
w
[mm]
b,
[mm]
a,
[mm]
[m]
Loss
[V/pC]
7 10 4 0.339 250 3
1 20 7.5 0.021 25 0.2
1.820
ln0.021 7.5
3 10 0.2100.339
ln4
outloss
Vk
pC
0.2loss
Vk
pC
Scaling laws and extrapolation
in outW W W 2out in eW W W
2 2 2 , 0in e e inW W W W if W
in outloss lossk k
( )ez ellipse ellipse slot zW w w w
(K. Zotter et al. , Impedances and Wakes …, 1998 andK.L.F.Bane, I.A.Zagorodnov, SLAC-PUB-11388, 2005)
w
Energy method
w
[mm]
[m]K_loss from time-domain
[V/pC]
K_loss from energy
[V/pC]
Error,
[V/pC]
7 250 2.88 3.10 0.22
7 25 31.5 31.0 0.5
2 250 0.22 0.26 0.04
2 25 2.75 2.63 0.12
Accuracy of the energy method
[mm] Mesh step in
Ellipse Ellipse with slot
0.25 2 2.89161337 2.89165659 4.32e-5
0.25 1 2.90553777 2.90560728 6.95e-5
0
2w
Z c
0.03 0.12 /lossk V pC
, 0.001 /kick xk V pC( , 0, 0)
( , 0, 0)s
yx
W s x yW s x y ds
a
xy
xa
20h mm
2 2 5000 25 3 16 [ ]h L e mm
0.2 /lossk V pC
Energy method
3mm
3.3mm3.8mm
84 248 257 /loss s sk x y V pC
, 390 46 436 /kick x s wk x x x V pC
, 348 47 301 /kick y s wk y y y V pC
Energy method
85 /lossk V pC
Round (a=3.8, b=3.3)
0.5mm
15mm
3mm
7.6yw mm
2 351 /( )Dkickk V pC m
3 359 /( )Dkickk V pC m
Direct time-domain with ECHO
x
y
9yw mm
7.8yw mm 85 /lossk V pC
91 /lossk V pC
Old pipe?
The energy loss for the new smaller pipe is a little smaller!
2%Error
x
y0.5mm
15mm
3mm
7.6 mm
250 250 /84loss s sk x y V pC
, 390 46 /436kick x s wk x x x V pC
, 348 47 /301kick y s wk y y y V pC
1200 1200 /42loss s sk x y V pC
, 280 206 /486kick x s wk x x x V pC
, 78 208 /130kick y s wk y y y V pC
Loss,
V/pC
Spread,
V/pC
Peak,
V/pC
var1 84 33 -118
var2 42 16 -58
x
yKick
, /486kick xk x V pC
, /130kick yk y V pC
xy
xa
, 0.001 /slotkick xk V pC
, ,0.0001 0.01 /slot abskick x kick xk k V pC
pro section (6.1 m)
Loss,
V/pC
Spread,
V/pC
Peak,
V/pC
absorber 1 42 16 -58
pumping slot 1 <0.2 <0.1 >-0.3
pump 1 9 4 -13
BPM 1
bellow 1 13 5 -18
flange gap 1 6 2.4 -8.5
Total geom. 70 25 -95
resistive (Cu) 6.1m 220 279 -542
resistive (Al) 6.1m 303 325 -660
Longitudinal wake for the case of the elliptical pipe (3.8mm)
-5 0 5
-300
-200
-100
0
100
200
300
-5 0 5-300
-200
-100
0
100
200
300
Shape dependence
-5 0 50
1000
2000
3000
4000
5000
Loss,
V/pC
Spread,
V/pC
Peak,
V/pC
Gauss, cu/al 36/50 46/53 -89/108
Rect, cu/al 79/93 90/72 -256/243
-1 -1: 5.8e+7[ m ], =2.46e-14[sec]cond relaxcu -1 -1: 3.66e+7[ m ], =7.1e-15[sec]cond relaxal
copper aluminum
I A
VW
pC
VW
pC
/ zs
/ zs / zs
25z m
4000 [ ]bN bunches
10 [ ]repf Hz
1[ ]q nC
2b rep lossP N f q k
0.52geom
WP
m
, 1.4res Cu
WP
m
, 2res Al
WP
m
Power Loss (Gaussian bunch)
Conclusion
• Loss and kick of the pumping slot are negligible • Wake of the absorber is reduced by factor 2• Resistive wake is dominant• Non-smooth bunch shape (rectangular) increases
the energy spread less than by factor 2