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Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 1CHESS / LEPPCHESS / LEPP
Single HOM, two-pass analysis
y
z
y
x
E B
injectedbeam
2nd passdeflected
beam
TM110
1
10
100
1000
10000
100000
4.1 4.2 4.3 4.4 4.5 4.6 4.7
recirculation time [ns]
thre
sho
ld [
mA
]
theory
simulation
Motivation:
explain the plot produced by BBU code bi
R/Q = 100 OhmQ = 10000m12 = 10–6 m(c/eV) = 2 2 GHzt0 = (1.3 GHz)–1
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 2CHESS / LEPPCHESS / LEPP
Frequency of the instability• Dipole mode is excited by first current moment thru interaction with
longitudinal field of the mode
• Infinite number of bunches with finite number of passes (as opposed to finite number of bunches with infinite number of passes for BBU in storage rings)
• Potentially, any frequency can be present in FT of the current moment for infinite delta-function current train
• Instability occurs with frequency close to that of HOM, where impedance is maximal
xI
t
0t
0n0)1( n 0)1( nFT
)sin()/(2
1)( 2 tekQRtW Q
t
')'()'( '' dtetWiZ ti
2
2
'1)/(
2)'()'(
Q
ikQR
ixIV
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 3CHESS / LEPPCHESS / LEPP
Getting the master equation
t
dttxtIttWtV ')'()'()'()( )2()2(
t
rr dtttVmc
ettIttWtV ')'()'()'()( 12)1(
m
mtttItI )()( 000)1(
Sample solution: tieVtV 0)(
Summing geometric series 1)cos(2
)sin(1
02
0
t
tKe ti
0
0
20012 ,2
)/( tiQ
t
eekQRtIm
c
eK
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 4CHESS / LEPPCHESS / LEPP
Perturbative approach
Solve the master equation for instability frequency treating K as small parameter:
...2210 KK
The frequency up to the first order in K:
Kt
ee
Qi
t
n Q
t
tir
r
0
2
0 22
2
Requiring Im() = 0 yields famous
Q
t
tQkmQRe
cI r
rth 2
exp)sin()/(
2
12
)1(
Problem 1: Half solution is missing
Problem 2: Unphysical exponent
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 5CHESS / LEPPCHESS / LEPP
4.2 10 -94.3 10 -94.4 10 -94.5 10 -94.6 10 -94.7 10 -9
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
Second order perturbative term
The second order term is found to be:
2222
0
20
0
2
0 )1(4
)]1([
22
20
0
Keet
ettieK
t
ee
Qi
t
ntiti
tir
Q
t
Q
t
ti
r
rr
r
Im() = 0 yields quadratic equation for the threshold current
)]2(sin[)1()]2(sin[
)2sin()cot()2()(sin)1()(cos))(sin(sin(
)/(
4
00
0022
02
12
)2(
0
00
rtt
r
rrrQt
rQt
rQ
t
rth
tttt
tttttttte
mtQkQRe
cI
r
r
1st order2nd order
rt
thI
Observation: Clearly, the other half of the solution is not a 2nd order effect
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 6CHESS / LEPPCHESS / LEPP
Complex current approach
Solving master equation directly for current gives the following:
)cos(2)sin()/(
20
22)(2
00120
0
0
0
0
0
0
teeeeeettmkQRe
cI tiQ
t
tiQ
t
ttiQ
t
r
Re(I0)
Im(I0)
solution space
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 7CHESS / LEPPCHESS / LEPP
Max and min currents
-750 -500 -250 250 500 750
-750
-500
-250
250
500
750
)sin()/(
])cos(1[4
0120
0
tmtkQRe
tcIout
-0.04 -0.02 0.02 0.04
-0.04
-0.02
0.02
0.04
12)/(
2
mQkQRe
cI in
)1)cos((2...max ,)cos()cos(2... 0001
20
ttt
Q
t
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 8CHESS / LEPPCHESS / LEPP
Obtaining complete first order solution
In the following limit (HOM damping is small of timescale of t0)
(instability frequency shift is small compared to bunching frequency, or as seen later, equivalent
to
number of bunches in recirculating loop >> 1)
Solving Im(I0) = 0, yields the threshold and instability frequency
12
0 Q
t
1)( 00 tt
Qie
mkQRe
cI rtti
2)/(
4 )(
120
0
)cot(2
,)sin()/(
2
120
r
r
tQ
tQkmQRe
cI
Note: It’s sin(tr) not sin(tr)Unphysical exponent is gone.
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 9CHESS / LEPPCHESS / LEPP
4.2 10 -94.3 10 -94.4 10 -94.5 10 -94.6 10 -94.7 10 -9
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
Linearized solutions for instability frequency
Transcendental equation for instability frequency can be linearized for two important cases:
)cot(2 rtQ
rrt ,2look at solutions closest to
r
r
tQ
tΔωQ
/2 ,12
r
r
r t
t
tΔωQ
2 ,12
2
120 21
)/(
2
Q
QkmQRe
cI
rt0rt0rrr ttt
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 10CHESS / LEPPCHESS / LEPP
Comparison with tracking
10
100
1000
10000
100000
4.05 4.1 4.15 4.2 4.25 4.3 4.35 4.4 4.45 4.5 4.55 4.6 4.65 4.7
recirculation time [ns]
thre
sh
old
[m
A]
bi code
solution 1
solution 2
-1.5
-1
-0.5
0
0.5
1
1.5
4.05 4.15 4.25 4.35 4.45 4.55 4.65
recirculation tim e [ns]
Del
ta o
meg
a [G
Hz]
solution 1
solution 2
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 11CHESS / LEPPCHESS / LEPP
-0.1 -0.05 0.05 0.1
-0.1
-0.05
0.05
0.1
-40 -20 20 40
-40
-20
20
40
zoomed in
Solving Im(I0)=0 numerically
with increasing tr
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 12CHESS / LEPPCHESS / LEPP
Comparison of tracking with numeric solution of Im(I0)=0
10
100
1000
10000
100000
1000000
3.9E-09 4E-09 4.1E-09 4.2E-09 4.3E-09 4.4E-09 4.5E-09 4.6E-09 4.7E-09 4.8E-09
recirculation time [s]
thre
shol
d [m
A]
bi code
solution 1
solution 2
-1
-0.5
0
0.5
1
1.5
2
3.9E-09
4.0E-09
4.1E-09
4.2E-09
4.3E-09
4.4E-09
4.5E-09
4.6E-09
4.7E-09
4.8E-09
recirculation time [s]
Del
ta o
meg
a [G
Hz]
solution 1
solution 2
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 13CHESS / LEPPCHESS / LEPP
Large tr (r << /2Q)
0
20
40
60
80
100
120
140
160
180
200
4030.1 4030.2 4030.3 4030.4 4030.5 4030.6
recirculation time [ns]
thre
sho
ld [
mA
]
complex current method
1st order perturbation in Omega
tr does not matter as opposed to small accelerators case, threshold is approx. given by Iin
-1.5
-1
-0.5
0
0.5
1
1.5
4030.1 4030.2 4030.3 4030.4 4030.5 4030.6
recirculation time [ns]
De
lta
om
eg
a [
MH
z]
Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 14CHESS / LEPPCHESS / LEPP
A word on quad HOM BBU
ede xzWeqdsFx )( : dipole
teetyxeeqe yyxxyxzWeqdsFxee 00
2220
20 2)(2 : quad
coupling term
Wake functions are identical in the form, except for the loss factor difference
44,
22,
or
or
bkk
bkk
qloss
dloss
In the approximation that alignment error of cavity transverse position dominates and causes dipole-like BBU (b is beam pipe radius)
2
2
,,2b
II dthqth , i.e. ~ 2 orders of magnitude bigger