long range wake potential of bpm in undulator section
DESCRIPTION
Long Range Wake Potential of BPM in Undulator Section. Igor Zagorodnov and Martin Dohlus Beam Dynamics Group Meeting 29.01.08. BPM Geometry (Dirk Lipka). Longitudinal wake. Transverse wake. For structure with symmetry group of rotations C 4. FD - Frequency Domain TD - Time Domain - PowerPoint PPT PresentationTRANSCRIPT
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Long Range Wake Potential of BPM in Undulator Section
Igor Zagorodnov and Martin Dohlus
Beam Dynamics Group Meeting
29.01.08
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BPM Geometry (Dirk Lipka)
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|| 2 cosis
ci i
i
sw s k e
c
2
2 22
20.5
|| || 2( ) ( ) ( ) 2 cos ,
i i iss
c ci i i i
i
sW s w s s s ds k e e s
c c
2
221( )
2
s
s e
( )||( ) 2 cos ,
is
cii
i
sW s k e s
c
2 22 22 2 2
10.5 1 0.5
4( )i i
ic Q ci iik k e k e
(1) (2)*(1) (2),
4ii
V r V rk r r
U
2i
iiQ
Longitudinal wake
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( ) sin ,is
ci
ii
R sw s c e s
Q c
( , ) ( )w s r r w s
For structure with symmetry group of rotations C4
2 i
i
kR
Q
*
24i
i
V r V rk
U r
(1) (2) (1) (2), , (0,0) ( ) ( ) (2)QDw s r r w W s r W s r O ��
( )
( ) sinis
ci
ii
R sW s c e
Q c
2
22
( ) 0.5 ic
i i
R Re
Q Q
Transverse wake
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FD - Frequency DomainTD - Time Domain
MWS - CST Microwave StudioPS - CST Particle Studio
ECHO – time-domain wakefield codeProny – Prony-Pisarenko method
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0 10 20 30 400
0.2
0.4
0.6
0.8
65 70 75 80 85-3
-2
-1
0
1
2
3
f, GHz k, V/pC
4.3744 0.4696
11.0772 0.6294
15.7603 0.3466
21.7338 0.0577
2.4048 ~ 23GHz2cutc
fa
f, GHz k, V/pC
4.3713 0.4721
11.066 0.6505
15.7501 0.36574
21.677 0.0566
ECHO + Prony
CST PS + Prony
||V
pCW
cms
GHzf
V
pCk
Reference Cavity (PEC)
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f,
GHz
k,
V/(pC*m*m)
6.9594 14.1565e3
12.8695 34.5164e3
17.1830 40.8118e3
1.841 ~ 17.6GHz2cutc
fa
5 10 15 20 250
10
20
30
40
50
65 70 75 80 85-600
-400
-200
0
200
400
600
Reference Cavity (PEC)V
pC*mW
cms GHzf
V
pC*m*mmk
ECHO + Prony
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f, GHz
(MWS, FD)
k, V/pC
(MWS, FD)
k, V/pC
(PS, TD)
k, V/pC
(ECHO, TD)
Q
(MWS, FD)
Q
(PS, TD)
sec-1
4.37 0.46 0.47 0.47 636 602 2.2e7
11.06 0.60 0.65 0.63 995 890 3.5e7
15.75 0.33 0.36 0.35 1113 1075 4.4e7
21.72 0.16 0.07 0.06 1712 1126 4.0e7
f,
GHz
(MWS, FD)
k, V/(pC*m*m)(MWS, FD)
k, V/(pC*m*m)
(PS, TD)
k, V/(pC*m*m)(ECHO, TD)
R/Q,
/mm2
Q
(MWS, FD)
Q
(PS, TD)
sec-1
6.96 13e3 15e3 14e3 3.74 797 1989 2.7e7
12.86 31e3 32e3 34e3 4.82 1000 1129 4.0e7
17.22 39e3 48e3 41e3 4.53 1117 608 4.8e7
24.43 25e3 32e3 10e3 2.05 1586 2353 4.8e7
transverse (dipole)
Reference Cavity (Lossy)
Stainless Steel 304
conductivity1,40E+06longitudinal (monopole)
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longitudinal
transverse
f, GHz
(MWS, FD)
K, 1e12
(MWS, FD)
K, 1e12
(PS, TD)
Q
(MWS, FD)
sec-1
3.51 0.12 0.11 288 3.8e7
8.56 0.23 0.19 428 6.3e7
13.44 0.22 0.19 543 7.8e7
18.31 0.16 0.17 683 8.4e7
f, GHz
(MWS, FD)
k, V/(pC*m*m(MWS, FD)
k,
V/(pC*m*m
(PS, TD)
Q
(MWS, FD)
R/Q,
/mm2
sec-1
4.46 5.1e3 6.5e3 367 2.3 3.8e7
7.67 2.0e3 2.7e3 469 0.5 5.1e7
9.3 2.3e3 2.7e3 470 0.5 6.2e7
10.46 6.1e3 6.1e3 549 1.2 6.0e7
14.07 11.1e3 12.5e3 607 1.6 7.3e7
15.16 2.4e3 2.5e3 719 0.3 6.6e7
19.17 13.5e3 17e3 715 1.4 8.4e7
20.4 17.7e3 4.9e3 902 1.7 7.1e7
Resonator Cavity (Lossy)Stainless Steel 304
conductivity1,40E+06
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( )( )|| ||
1 1
( )
( ) ( ) 2 cos
2 2
1 1
ij s
cii
j j i
s
c cii
i ic c
j sW s W j s k e
c
e ek k
e e
min i
( ) ( )
1
c
iic
e RW s c
Qe
Asymptotic multi-bunch loss
Asymptotic multi-bunch kick
1c
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1|| 2.2 7 [sec ]e
2 7 [sec]ec
||
||
0.01
1
c
c
e
e
( )||
kV( ) 0.01 2 0.05
nCii
W s k
12.7 7 [sec ]e
( ) kV( ) 0.005 0.0066
nC×mmii
RW s c
Q
Multi-bunch effect
- bunch spacing
0.005
1
c
c
e
e
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( )||
kV( ) 0.01 2 0.05
nCii
W s k
( ) kV( ) 0.005 0.0066
nC×mmii
RW s c
Q
22
225μm||
kV16.6 2
nC
ic
ii
k k e
0.503 2.5
2 3Kick ( )
4
Z cg O
a
0.502.5
1Loss ( )
44
Z c gO
a
25μm kV0.045 ( )
nC×mm iii
Rk c D
Q
Long Range Wakes
Short Range Wakes
Long Range vs. Short Range
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( )||
kV( ) 0.05
nCW s ( ) kV
( ) 0.0066nC×mm
W s
Comparison with beam parameters in the undulator
( ) rad0.0004
mm
r eQW
r E
( )
|| 0.05keVE eQ W
0
50eV3 9
17.5GeV
Ee
E
1.4mm*mrad1 rad
34247*45mn
x
0 17.5GeVE
4 4
mmx
r e
r
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Geometry scaling
,sii
aQ F i
d
(see R.E.Collin „Foundations for microwave engineering“)
If we scale the cavity by factor as
, ,a d a d then
ii
i iQ Q
1.5i
i
|| ||1
i ik k
21
i i
R R
Q Q
2a
d
2s
i
31
i ik k
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1 2 3 4 50
5
10
15
Geometry scaling
If we scale only the cavity by factor as
, ,a d a d then
1.5
( )||
(0.01)( , ) 2 i
i
W s k
1.5
( )2
(0.005)( , )
ii
RW s c
Q
ii
( )||
( )||
( , )
( )
W s
W s
( )
( )
( , )
( )
W s
W s
2a
d
1.5i
i
1i ik k