BBU Codes OverviewERL2005 BBU Codes Overview Masaru Sawamura and Ryoichi Hajima JAERI Outline •...
Transcript of BBU Codes OverviewERL2005 BBU Codes Overview Masaru Sawamura and Ryoichi Hajima JAERI Outline •...
ERL2005
BBU Codes OverviewMasaru Sawamura and Ryoichi Hajima
JAERI
Outline• Introduction• Beam transport Equation• How to solve (BBU-R, TDBBUU, bi, MATBBU etc.)
• Comparison of BBU codes
ERL2005
IntroductionERL current limited by • Beam Breakup
– Transversedeflect beam
– Longitudinalarrival time differenceHOM-induced energy spread
HOM
T12 or T34
HOM
T56
ERL2005
Beam Transport Equation
∑ ∑=
−−+
=−−− +−+−+−=
pn
r
MrpM
knkrn
ppnnp
ppnnp skMrpMnUIZGTMnUTMnU
1
1)(
1011,1,
0
)())(,1(),1(),( τω
n-th Cavity (n-1)-th Cavity
(p+1)-th pass
p-th pass
(p-1)-th pass
M
0
M-
M+
M
MM-
Transport
Kick due to Induceded HOMsby Previous Bunches
Induced HOM Decay&
Phase shift
•transport •kick by HOM
currentaverageIsitescavityofnumbern
passesnkes
lengthcavitylionrecirculatoneinbunchesofnumberM
impedancetransverseZ
G
matrixtransferT
p
nQ
k
nk
n
pqmn
n
::
:)sin()(
::
:
:
0
2
0
,
0100
τωτωτω
−
=
⎟⎠⎞⎜
⎝⎛=
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How to Solve
• Beam Tracking (Beam position vs. Time)– BBU-R (JAERI)– TDBBU (JLab)– bi (Cornell Univ.)– new code (JLab)
• Eigenvalue Solution (Current vs. Frequency)– MATBBU (JLab)
ERL2005
BBU-Rfeature• Transverse BBU• Beam position vs. time (time increment Trf/2)• Point-like bunch• X,Y HOM orientation• X- & Y-axis independently• Impulse of HOM kick• Using 2X2 Transfer Matrix• Two-pass recirculation
ERL2005
BBU-R Algorism
0 n m nloc-1
0
Trf/2
k Trf/2
nloc Trf/2
Cavity Cavity
Expand beam line into a consecutive time-step array
Set initial bunch
Move Trf/2 step
Bunch in cavity? Update bunch position to next cavity
Update HOM power Decay & Phase shift
Bunch in cavity? Bunch kicked by HOM
Last location? Create new bunch
Y
Y
Y
ERL2005
Threshold current of JAERI ERL-FEL• layout
-4
-2
0
2
4
0 500 1000 1500 2000
x (m
m)
Time ( sec)µ
3.42A
-50
-25
0
25
50
0 500 1000 1500 2000
x (m
m)
Time ( sec)
6.11A
µ
Beam Dump
Main 5-cell SCAElectron Gun
SHB
1-cell SCA
Injection Merger
2nd Arc1st ArcUndulator
Half Chicane
ERL2005
Example for future ERL design
β-function in the linac
Threshold current vs. HOM randomization
Threshold current vs. Cavity gradient
6GeV 1.3GHz 15MV/m
48 cryomodules 47 external QT
ERL2005
TDBBU
• Transverse BBU• (a) Move bunch
(b) calculate position of bunch entering the cavity(c) update all HOM excitation level
• Turn the current up/down to find instability• X- &Y-axis entirely independent• Suitable for large HOM numbers or
short characteristic time (2Q/ω)
ERL2005
Example of TDBBU results
[K.Beard et al., PAC2003 Proc. 332 (2003)]
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bi
• Transverse or longitudinal BBU• Allow any ERL topology• transient effect for arbitrary bunch pattern• Arbitrary HOM orientation
ERL2005
Example of bi results
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 50000 100000 150000
-3
-2
-1
0
1
2
3
0 50000 100000 150000
-10-8-6-4-202468
10
0 50000 100000 150000
bunch #
arriv
al ti
me
diff
eren
ce [p
s]
I = 22.0 mA
I = 23.0 mA
I = 22.1 mA
Longitudinal instability
ERL2005
New code at JLab
• Transverse BBU• Two-path machine• Full 2D particle tracking (4X4) or 1D (2X2)• Arbitrary HOM angle• Decoupled transverse motion and coupled motion
effect of rotated HOMeffects of rotated HOMs and of rotated optics
• Include FB for BBU suppression
ERL2005
Example of new code results
[C.Tennant and E.Pozdeyev, JLAB-TN-02-020 (2004)]
Beam displacement as a function of time for Ib>Ith
FFTs of the indicated”slices”of the beam displacement
Threshold current versus HOM orientation for a single cavity with one HOM and a decoupled recirculation matrix
ERL2005
MATBBU
• Solve eigenvalue of matrix representing system
• Transverse BBU• X,Y axis treated sequentially,
entirely independently
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Algorism of MATBBU
Ω is also unknown Scan Ω and find Im(I)=0 & I>0
∑∑∑∑∑=
−
== < =
−Ω Ω+Ω=pp n
p
i
llll
ppli
n
p pr
n
llll
rpMprlii DhZTIDhZeTID
1
1
112,
2 1
)(12, )()()()(
00 τ
0&)(),( >>= Ω MnVeMnU pMi
pτ
∑=
Ω−−−− −Ω+−=
pn
rrn
Mrpippnnnp
ppnnp nVheGTIZnVTnV
1
)(1,11, )1()()1()( 0 τ
∑ ∑=
−−+
=−−− +−+−+−=
pn
r
MrpM
knkrn
ppnnp
ppnnp rskMrpMnUIZGTMnUTMnU
1
1)(
1011,1,
0
)())(,1(),1(),( ω
∑=
=pn
kk
ikM nVenV0
)()( 0τ
assume a steady state solution
sum over all passes
)(iVofcomponentxDi −
DI
DM 1)( =ΩEigenvalue problem
Ω : coherent frequency
)cos()(2)(1)sin()()( 2 τω
τω
nnn
nnn HH
HhΩ−Ω+
Ω=Ω
ττω
Ω−−
=Ω iQn eeH
n
2)(
ERL2005
Example of MATBBU results
[K.Beard et al., PAC2003 Proc. 332 (2003)]
[J.J.Bisognano et al., 1987 PAC Proc. 1078 (1987)]
Stability plot of complex threshold current (amperes) for one site, two-pass configuration
Sweep coherent frequency and search Positive real current
ERL2005
Comparison of BBU codes
JLabJLabCornell Univ.JLabJAERIDeveloper
Fortran/CC++C++Fortran/CCProgrammingLanguage
Arbitrary2ArbitraryArbitrary2No. of Recirculation
X,YArbitraryArbitraryX,YX,YHOM direction
EigenvalueTrackingTrackingTrackingTrackingSolve
1D1D/2D2D1D1DDimension
TTT/LTTTransverse/Longitudinal
MATBBUNew code @JLabbiTDBBUBBU-RName
ERL2005
Other problems
• Thin cavityImpulse kickWhere does a kick work -entrance, middle or exit? Single- or multi- kick for low energy?
• Faster methods to determine whether stable or instablebeam displace or HOM voltage?