CAS Zakopane October 2006cas.web.cern.ch/.../zakopane-2006/bruning-lin-imp.pdfCAS Zakopane October...
Transcript of CAS Zakopane October 2006cas.web.cern.ch/.../zakopane-2006/bruning-lin-imp.pdfCAS Zakopane October...
CAS Zakopane October 2006
Oliver Bruning / CERN AP−ABP
Imperfections
Linear
integer resonances
local orbit bumps
BPMs & dipole correctors
orbit correction
quadrupole perturbations
beta−beat
half−integer resonances
one−turn map & tune error
Linear Imperfections
equation of motion in an accelerator
Hills equation
sine and cosine like solutions
closed orbit
sources for closed orbit perturbations
dipole perturbations
closed orbit response
dispersion orbit
+ K(s) x = G(s);
d sd x
K(s) =
K(s) = K(s + L);2
2+ K(s) x = 0;
d sd x2
2 G(s) =
dds= d
dtdsdt
v
Hill’s Equation:
p[GeV]B[T/m]
ρ 2
0.3 quadrupole
dipole1/
drift0
v p0
F(s)Lorentz
Variable Definition
x s
ρ
sy
Perturbations:
Variables in moving coordinate system:
ddsx = x
x = p
p
0
x
Y (0) =
cos ( K s)
− K sin ( K s)
y (s) = a Y (s) + b Y (s)
=
0 K y = 0 y + y =
x
x
2Y (s) =
2
2
2
=
1
1Y (0) = 1
1
=
0
cos ( K [s−s ])
− K sin ( K [s−s ])0
sin ( K s)
K cos ( K s)
0
1
y (s) = M(s − s ) y (s ) 0 0
1
sin ( K [s−s ])
K cos ( K [s−s ])0
0
2
Y (s) =
0
Y
Y
Y
Y
10
1
K = const
initial conditions:
and
Sinelike and Cosinelike Solutions
system of first order linear differential equations:
general solution:
transport map:
with:
s 0φ( ) + φ β( )s
β( )ss 0φ( ) + φ
s 0φ( ) + φ
Y (s) =1
s 0φ( ) + φ
= (s )0
(s )00S(s ) =
(s )0
(s )0=
0C (s ) =
β( ) s 0s φ( ) + φsin ( )
α( )
β( )s cos ( )Y (s) =2
2
s 0φ( ) + φ[cos ( ) + s sin( )] /
α( )−[sin ( )+ s cos( )]/
1C (s) = a Y (s) + b Y (s) 1 2S(s) = c Y (s) + d Y (s)
1
C
S
S0
C 0
1
β1φ (s) = ds; β1
2
Floquet theorem:
Sinelike and Cosinelike Solutions
α (s) =− (s)
´sinelike´ and ´cosinelike´ solutions:
β β(s) = (s + L);
and
to the case with constant K(s)!one can generate a transport matrix in analogy
with:
0
C (s) =0φ( ) + φ s 0φ( ) + φ
s 0φ( ) + φ
0(α −α)
1
1 0
0 I =
−α
β
−γJ =
α2; γ = [1 + α ] / β
M = I cos(2 Q) + J sin(2 Q) π π
β( )s
s sβ( )β( )
α( )s 0φ( ) + φφ( ) + φ
sin ( )
[cos ( ) + s sin( )]/
0
[sin ( )+ cos( )]/
[cos ( ) + sin( )]/
0 y (s) = M(s, s ) y (s ) 0 0
M = C (s)
C (s)
S (s)
S (s)
sβ( )0
sβ( )0
S (s) =β( )ss 0φ( ) + φ
s 0φ( ) + φ
β β0
s 0φ( ) + φ α( )s0
0−(1+αα ) s
transport map for s = s + L:
´sinelike´ and ´cosinelike´ solutions:
transport map from s to s:
Sinelike and Cosinelike Solutions
with:
B = g x + g x
B DB
FBDB
dipole component
x = x + x0y
x
y
x
0
FB
DB F
B = g y
B = g x
B = g y
Closed Orbit
particles oscillate around an ideal orbit:
additional dipole fields perturb the orbit:
energy error
α = ρl = 1− q B l
error in dipole field
offset in quadrupole field
Δ p + pq B l Δ p
p p
d sd x2
2+ K(s) x = G(s); G(s) =
g [T/m]p [GeV]
p[GeV]
g[T/m]Δ
B [T]
p [GeV]
x
k (s) = 0.3
k (s) = 0.3
k (s) = 0.3
v p0
00
x
y x
yF = −q v B
F = q v BN S
NS
y
x R
x
y
B = g y
B = g x
F(s)Lorentz
normalized fields:
quadrupole misalignment:
0
0
0
dipole:
quadrupole: 10
0
Quadrupole Magnet
the perturbation adds up
Q: βnumber of −oscillations per turn
Q = N + 0.5
the perturbation cancelsafter each turn
Kick
watch out for integer tunes!
Dipole Error and Orbit Stability
Q = N
Kick
amplitude growth and particle loss
Quadrupole Error:
amplitude increase
1. Turn: x > 0
Q = N + 0.5
F
kick
amplitude increase
beam offset in quadrupole
F
x < 02. Turn:
watch out for half integer tunes!
Orbit Stability
Quadrupole Error and
orbit kick proportional to
momentum distribtion
power supplies
calibration
civil engineeringseasonsmoon
slow drift
civilisation
injection energy (RF off)
RF frequency
Energy error of particles
ground motion
Sources for Orbit Errors
alignment +/- 0.1 mm
Quadrupole offset:
Error in dipole strength
Example Quadrupole Alignment inLEP
x−y coupling
aperture beam losses
filamentation beam size
aperture
energy error
field imperfections
beam separation
dispersion beam size at IP
rms < 0.5 mm x, y < 4 mmΔ
beam monitors and
injection errors:
orbit correctors
Δ
closed orbit errors:
Problems Generated by Orbit Errrors
Aim:
π
t
V
γ
a)
b)
assume: L > design orbit
the orbit determines the particle energy!
Synchrotron:
RFf = h f
rev
f = 12
qm
Brev
energy increase
Equilibrium:
E depends on orbit and magnetic field!
Moon
tide
<
Earth
orbit and beam energy modulation:
E 10 MeVΔ
0.02%
aim: Δ E 0.003%
requires correction!
tidal motion of the earth:
modf = 24 h; 12 h
Δ E
[M
eV]
November 11th, 1992
Δ E
[M
eV]
August 29th, 1993
Daytime
October 11th, 1993
-5
0
5
23:00 3:00 7:00 11:00 15:00 19:00 23:00 3:00
-5
0
5
11:0
0
13:0
0
15:0
0
17:0
0
19:0
0
21:0
0
23:0
0
18:0
0
20:0
0
22:0
0
24:0
0
2:00
4:00
6:00
8:00
Δ
energy modulation due to tidal motion of earth
E 10 MeV
orbit and energy perturbations
the position of the LEP tunnel and thus the
quadrupole positions
Δ E 20 MeV
energy modulation due to lake level changes
changes in the water level of lake Geneva change
TGV line between Geneva and Bellegarde
energy modulation due current perturbations in
the main dipole magnets
RAIL
TGV
Gen
eve
Mey
rin
Zim
eysa
LEP Polarization Team17.11.1995
LEP beam pipe
LEP NMR
E 5 MeV for LEP operation at 45 GeVΔ
with the voltage on the TGC train tracks
correlation of NMR dipole field measurements
ground motion due to human activity
quadrupole motion in HERA−p (DESY Hamburg)
RMS
peak to peak
d x0
0 1 0 K
ψ( )s
s = c(s) S(s) + d(s) C(s) ψ( )
y(s) = a S(s) + b C(s) +
G
0 y = G; G = y +
G(s) =+ K(s) x = G(s);2
2
d sk (s)
Closed Orbit Response
Δ
inhomogeneous equation:
we need to find only one solution!
variation of the constant:
φ( )φ( )s
φ( ) G(t) dt −1
s
0
s s u(s);
t u(s) =
φ( )s
ψ( ) = φ( )
s
s
G(t) dt −1
s
0
u (s) = G(s)
φ( )=s
y(s) = a S(s) + b C(s) +
S(s)
S (s) C (s)
C(s)
t
Closed Orbit Response
with
substitute into differential equation:
variation of the constant in matrix form:
φ( )
s πt β( )2 sin( Q)
s β( )π
s
y(s) =x(s)x (s)
s φ( )
t x(s) = G(t) cos[ − Q] dt
y(s) = a S(s) + b C(s) +
φ( )−φ( )
t −1
G(t) dt
s0
s0
s0+circ
x (s) = x (s + L)
Closed Orbit Response
periodic boundary conditions determinecoefficients anda b
periodic boundary conditions:
with
; x(s) = x(s + L);
t
G(t) cos[ − Q] dt s t
Δ1ρ( )t
π
1ρ( )
x(s) =
p p
φ( )−φ( )t s π cos[ − Q] dt t β( )
p[GeV]
B[T]
G(t) = 1
ρ( )t
Δ p p
β( )
ρ( )2 sin( Q)
s β( )π
D(s) =
k (s) = 0.3
t
k (s) =
φ( )−φ( )
−
Δ p p x(s) = D(s)
t 2 sin( Q)
s β( )π
Closed Orbit Response
0
Dispersion Orbit
with
Example:
normalized dipole strength:
0
particle momentum error
0
0
from the quadrupole alignment errors
magnets is dominated by the contributions
the orbit error in a storage ring with conventional
β -functions at the location of the dipole error
Orbit Correction
orbit perturbation is proportional to the local
alignment errors at QD causes mainly
horizontal orbit errors
alignment errors at QF cause mainly
vertical orbit errors
place orbit corrector and BPM next
to the main quadrupoles
vertical BPM and corrector next to QD
QF MB
horizontal BPM and corrector next to QF
QD
Orbit Correction
BPM
HX
BPM
HV
aim at a local correction of the dipole error due
to the quadrupole alignment errors
orbit in the opposite plane?
relative alignment of BPM and quadrupole?
Lake Geneva
LEP Orbit
Horizontal Orbit:
beam offset in quadrupoles
moon
energy error
Vertical Orbit:
beam separationorbit deflection depends onparticle energyvertical dispersion [D(s)]
small vertical beam size relieson good orbit
corrections in physics1994: 13000 vertical orbit
beam offset in quadrupoles:
yσ = ε β + δ 2Dy
2
y
π
2
M = I cos(2 Q) + J sin(2 Q) π
2; γ = [1 + α ] / βJ = α
1
1
1 0
0 I =
−α
β
n+1z = M z n
x x
z =
sin( 2 Q) ππM − I cos( 2 Q)
−γ
cos( 2 Q) = trace M π
can be generated by matrix multiplication:
provide the optic functions at s
remember:
0
Quadrupole Gradient Error
the coefficients of:
one turn map:
and can be expressed in terms of the C and S solutions
1
0
1
1 m =
0
M = m m M 00
−1
Δ
−k
0 ππ π
l−[k + k ]
l
lcos(2 Q) = cos(2 Q ) − k sin(2 Q )
11
1
β Δ1 2
1 0
1 m =
0
one turn matrix with quadrupole error:
Quadrupole Gradient Error
matrix for single quadrupole with error:
transfer matrix for single quadrupole:
trace M
4
π0cos(2 Q) = cos(2 Q ) −
2 0
πsin(2 Q )
β Δ
Δ
k ds
Δ1 π4
Δ Q = − β k ds
β Δk ds
Δ
Δ
Q =
= ξ
1 π
Δ
π
distributed perturbation:
momentum error k = − k
chromaticity: k = e g p 1
1
1
1
1
1
p p
p p
p p
Quadrupole Gradient Error
2 sin( Q)
s β( )
2πt β( ) φ( )−φ( )t s = Δ 2πk(t) cos[2[ ]− Q] dt
M = I cos(2 Q) + J sin(2 Q) π π
n+1z = M z n
2; γ = [1 + α ] / β−α
β
−γJ =
α
m
m 21
1211
m 22
m
12
M =
1
1 0
0 I =
β
s Δβ( )
2π
s0+circ
msin( Q)
calculate:
s0
− Beat
quadrupole error:
with
β
− beat oscillates with twice the betatron frequency
i θ i
θ =
β β( )s − φi
φ( )s sin[ ] x(s) =
θ i
− φi
φ( )s cos[ ] s β( )iβ /x (s) =
= G (t) dt p[GeV]
i0.3 B [T] l
deflection angle:
trajectory response:
[no periodic boundary conditions]
Local Orbit Bumps I
dipole
ii
closure of the perturbation within one turn
possibility to correct orbit errors locally
local orbit excursion
π
closure with one additional corrector magnet
closure with two additional corrector magnets
three corrector bump
Local Orbit Bumps II
closed orbit bump:
- bump
compensate the trajectory perturbation with
additional corrector kicks further down stream
1
β
β 1 θ = θ
22
3 θ θ 1 2 θ
orequires 90 lattice
closure depends on lattice phase advance
sensitive to lattice errors
sensitive to BPM errors
requires large number of correctors
requires horizontal BPMs at QF and QD
s
limits / problems:
(quasi local correction of error)
QF QD QF QD
- bump:π
QF
Local Orbit Bumps III
x
β 1β
θ
β
β 2 θ = −
Δφ3-1
Δφ 3-2
Δφ 1
θ θ θ 1 2 3
θ 1
3-1 θ = 3
1
2
Δφ3-1
Δφ 3-2
3
QF QD QF
sin( )sin( )
QF QD
can not control x
- cos( )sin( )tan( )
requires only horizontal BPMs at QF
works for any lattice phase advance
limits / problems:
Local Orbit Bumps IV
x
s
3 corrector bump: (quasi local correction of error)
sensitive to BPM errors
large number of correctors
Summary Linear Imperfections
they amplifies the response to quadrupole field errors
avoid storage ring tunes near half−integer resonances:
dispersion = eigenvector of extended ´1−turn´ map
closed orbit = fixed point of ´1−turn´ map
tune is given by the trace of the ´1−turn´ map
twiss functions are given by the matrix elements
avoid machine tunes near integer resonances:
they amplifies the response to dipole field errors
a closed orbit perturbation propagates with the
betatron phase around the storage ring
discontinuities in the derivative of the closed
orbit response at the location of the perturbation
the betatron phase advance around the storage ring
betafunction perturbations propagate with twice
integral expressions are mainly used for estimates
numerical programs mainly rely on maps