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Transcript of Fundamentals of Accelerator 2012 - USPASuspas.fnal.gov/materials/12MSU/Lecture Day 3.pdf · o o...
US Particle Accelerator School
Fundamentals of Accelerator2012Day 3
William A. BarlettaDirector, US Particle Accelerator School
Dept. of Physics, MITEconomics Faculty, University of Ljubljana
US Particle Accelerator School
Lumped circuit analogy of resonant cavity
V(t)
I(t)
CL
R
!
Z(") = j"C + ( j"L + R)#1[ ]#1
!
The resonant frequency is "o = 1LC!
Z(") =1
j"C + ( j"L + R)#1
=( j"L + R)
( j"L + R) j"C +1=
( j"L + R)
(1#" 2LC) + j"RC
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Q of the lumped circuit analogy
The width is
!
"#
#0
=R
L /.C
!
Z(") ~ 1#" 2
"o
2
$
% &
'
( )
2
+ ("RC)2
*
+ , ,
-
. / /
#1
Converting the denominator of Z to a real number we see that
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More basics from circuits - Q
!
Q = "o o Energy stored
Time average power loss
!
= 2" o Energy stored
Energy per cycle
!
E = 1
2L I
oIo
* and
!
P = i2(t) R =
1
2IoIo
*Rsurface
!
" Q =
LC
R =
#$
$o
%
& '
(
) *
+1
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Translate circuit model to a cavity model:Directly driven, re-entrant RF cavity
Outer region: Large, single turn Inductor
Central region: Large plate Capacitor
Beam (Load) current
I
B
EDisplacement current
Wall current
a
Rd
!
L =µo"a
2
2"(R + a)
!
C = "o
#R2
d
!
"o
= 1LC
= c2((R + a)d
#R2a2$
% & '
( )
12
Q – set by resistance in outer region
!
Q =LCR
Expanding outer region raises Q
Narrowing gap raises shunt impedance
Source: Humphries, Charged ParticleAccelerators
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Properties of the RF pillbox cavity
We want lowest mode: with only Ez & Bθ
Maxwell’s equations are:
and
Take derivatives
==>
!
1
r
"
"rrB#( ) =
1
c2
"
"tEz
!
"
"rEz =
"
"tB#
!
"
"t
1
r
"
"rrB#( )
$
% & '
( ) ="
"t
"B#"r
+B#
r
$
% & '
( ) =1
c2
" 2Ez"t 2
!
"
"r
"Ez
"r="
"r
"B#
"t
!
" 2Ez
"r 2+1
r
"Ez
"r=1
c2
" 2Ez
"t 2
d
Ez
b
Bθ
!
"walls
=#
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For a mode with frequency ω
Therefore,
(Bessel’s equation, 0 order)
Hence,
For conducting walls, Ez(R) = 0, therefore
!
Ez r, t( ) = Ez (r) ei"t
!
" " E z +" E z
r+#
c
$
% &
'
( )
2
Ez = 0
!
Ez (r) = Eo Jo"
cr
#
$ %
&
' (
!
2"f
cb = 2.405
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E-fields & equivalent circuit: Ton1o mode
Ez
Bθ
Rel
ativ
e in
tens
ity
r/R
T010
C L
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E-fields & equivalent circuitsfor To2o modes
T020
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E-fields & equivalent circuitsfor Tono modes
T030
T0n0 has n coupled, resonant
circuits; each L & C reduced by 1/n
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Simple consequences of pillbox model
L
Ez
R
Bθ
Increasing R lowers frequency ==> Stored Energy, E ~ ω-2
E ~ Ez2
Beam loading lowers Ez for thenext bunch
Lowering ω lowers the fractionalbeam loading
Raising ω lowers Q ~ ω -1/2
If time between beam pulses,Ts ~ Q/ω
almost all E is lost in the walls
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The beam tube makes field modes(& cell design) more complicated
Ez
Bθ
Peak E no longer on axis Epk ~ 2 - 3 x Eacc
FOM = Epk/Eacc
ωo more sensitive to cavitydimensions Mechanical tuning & detuning
Beam tubes add length & €’sw/o acceleration
Beam induced voltages ~ a-3
Instabilities
a
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Cavity figures of merit
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Figure of Merit: Accelerating voltage
The voltage varies during time that bunch takes to cross gap reduction of the peak voltage by Γ (transt time factor)
!
" =sin #
2( )#
2
where # =$d%c
2TrfFor maximum acceleration ==> Γ = 2/π
dV
t
Epk
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Figure of merit from circuits - Q
!
E =µo
2H
v
"2
dv = 1
2L I
oIo
*
!
P = Rsurf
2H
s
"2
ds = 1
2IoIo
*Rsurf
!
" Q =
LC
Rsurf
= #$
$o
%
& '
(
) *
+1
!
Q = "o o Energy stored
Time average power loss=
2# o Energy stored
Energy lost per cycle
!
!
Rsurf =1
Conductivity o Skin depth~ "1/ 2
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Measuring the energy stored in the cavityallows us to measure Q
We have computed the field in the fundamental mode
To measure Q we excite the cavity and measure the E fieldas a function of time
Energy lost per half cycle = UπQ
Note: energy can be stored in the higher order modes thatdeflect the beam
!
U = dz0
d
" dr2#r 0
b
" $Eo
2
2
%
& '
(
) * J1
2(2.405r /b)
= b2d $E
o
2/2( ) J1
2(2.405)
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Keeping energy out of higher order modes
d/b
ωb/c
0 1 2 3 4
TM020
TM010
TE011
TE111Source bandwidth
(green)
Dependence of mode frequency on cavity geometry
Choose cavity dimensions to stay far from crossovers
10
5
1
TE111 modeEnd view
Side view
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Figure of merit for accelerating cavity:Power to produce the accelerating field
Resistive input (shunt) impedance at ωο relates power dissipated in walls toaccelerating voltage
Linac literature commonly defines “shunt impedance” without the “2”
Typical values 25 - 50 MΩ
!
Rin
= V
2(t)
P =
Vo
2
2P = Q L
C
!
Rin = Vo
2
P ~
1
Rsurf
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Computing shunt impedance
!
P = Rsurf
2H
s
"2
ds
!
Rin
= Vo
2
P
!
Rsurf =µ"
2# dc
= $Zo
%skin
&rf where Zo =
µo
'o= 377(
The on-axis field E and surface H are generally computed with acomputer code such as SUPERFISH for a complicated cavity shape
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To make a linacUse a series of pillbox cavities
Power the cavities so that Ez(z,t) = Ez(z)eiωt
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Improve on the array of pillboxes?
Return to the picture of the re-entrant cavity
Nose cones concentrate Ez near beam for fixed stored energy
Optimize nose cone to maximize V2; I.e., maximize Rsh/Q
Make H-field region nearly spherical; raises Q & minimizesP for given stored energy
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In warm linacs “nose cones” optimize the voltageper cell with respect to resistive dissipation
Thus, linacs can be considered to bean array of distorted pillbox cavities…
!
Q =LC
R surface
Rf-power in
a
Usually cells are feed in groups not individually…. and
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Linacs cells are linked to minimize cost
==> coupled oscillators ==>multiple modes
Zero mode π mode
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Modes of a two-cell cavity
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9-cavity TESLA cell
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Example of 3 coupled cavities
!
x j = i j 2Lo and " = normal mode frequency
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Write the coupled circuit equationsin matrix form
Compute eigenvalues & eigenvectors to find the three normal modes
!
Lxq =1
"q
2xq where
!
L =
1/"o
2k /"
o
20
k /2"o
21/"
o
2k /2"
o
2
0 k /"o
21/"
o
2
#
$
% % %
&
'
( ( ( and xq =
x1
x2
x3
#
$
% % %
&
'
( ( (
!
Mode q = 0 : zero mode "0 =#
o
1+ k x
0=
1
1
1
$
%
& & &
'
(
) ) )
!
Mode q =1: "/2 mode #1 =$o x1 =
1
0
%1
&
'
( ( (
)
*
+ + +
!
Mode q = 2 : " mode #2 =$
o
1% k x2 =
1
%1
1
&
'
( ( (
)
*
+ + +
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For a structure with N coupled cavities
==> Set of N coupled oscillators N normal modes, N frequencies
From the equivalent circuit with magnetic coupling
where B = bandwidth (frequency difference betweenlowest & high frequency mode)
Typically accelerators run in the π-mode!
"m
="
o
1# Bcosm$
N
%
& '
(
) *
1/ 2+"
o1+ Bcos
m$
N
%
& '
(
) *
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Magnetically coupled pillbox cavities
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5-cell π-mode cell with magnetic coupling
The tuners change the frequencies by perturbing wall currents ==> changes the inductance==> changes the energy stored in the magnetic field
!
"#o
#o
="U
U
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Dispersion diagram for 5-cell structure
!
Emcell n( ) = A
nsin
m" (2n #1)
2N
$
% &
'
( )
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Schematic of energy flowin a standing wave structure
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What makes SC RF attractive?
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Comparison of SC and NC RF
Superconducting RF High gradient
==> 1 GHz, meticulous care
Mid-frequencies==> Large stored energy, Es
Large Es
==> very small ΔE/E
Large Q==> high efficiency
Normal Conductivity RF High gradient
==> high frequency (5 - 17 GHz)
High frequency==> low stored energy
Low Es
==> ~10x larger ΔE/E
Low Q==> reduced efficiency
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Recall the circuit analog
V(t)
I(t)
CL
Rsurf
As Rsurf ==> 0, the Q ==>∞.
In practice,
Qnc ~ 104 Qsc ~ 1011
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Figure of merit for accelerating cavity:power to produce the accelerating field
Resistive input (shunt) impedance at ωο relates power dissipated in walls toaccelerating voltage
Linac literature more commonly defines “shunt impedance” without the “2”
For SC-rf P is reduced by orders of magnitude
BUT, it is deposited @ 2K
!
Rin
= V
2(t)
P =
Vo
2
2P = Q L
C
!
Rin = Vo
2
P ~
1
Rsurf
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Surfing analogy of the traveling waveacceleration mechanism
To “catch” the wave the surfer must besynchronous with the phase velocity of the wave
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Typically we need a longitudinal E-field toaccelerate particles in vacuum
Example: the standing wave structure in a pillbox cavity
What about traveling waves? Waves guided by perfectly conducting walls can have Elong
But first, think back to phase stability To get continual acceleration the wave & the particle must stay in
phase Therefore, we can accelerate a charge with a wave with a
synchronous phase velocity, vph ≈ vparticle < c
EE E
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Can the accelerating structure be a simple(smooth) waveguide?
Assume the answer is “yes”
Then E = E(r,θ) ei(ωt-kz) with ω/k = vph < c
Transform to the frame co-moving at vph < c
Then, The structure is unchanged (by hypothesis) E is static (vph is zero in this frame)==> By Maxwell’s equations, H =0==> ∇E = 0 and E = -∇φ But φ is constant at the walls (metallic boundary conditions)==> E = 0
The assumption is false, smooth structures have vph > c
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To slow the wave, add irises
In a transmission line the irises
a) Increase capacitance, C
b) Leave inductance ~ constant
c) ==> lower impedance, Z
d) ==> lower vph
Similar for TM01 mode in the waveguide
!
"
k=
1
LC
Z =L
C
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Ultra-relativistic particles (v ≈ c) can“surf” an rf field traveling at c
RF- in = Po
z
RF-out= PL
Egap
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RF-cavities in metal and in plasmaThink back to the string of pillboxes
Plasma cavity100 µm1 m
RF cavity
Courtesy of W. Mori & L. da Silva
G ~ 30 MeV/m G ~ 30 GeV/m
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End of unit