Download RF Cavities - CERN Accelerator School

CERN Accelerator School
RF Cavities
Erk Jensen
CERN BE-RF
CERN Accelerator School, Varna 2010 - "Introduction to Accelerator Physics"
What is a cavity?
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
2
Lorentz force

A charged particle moving with velocity
p

v=
mγ
through an
electromagnetic field experiences a force

  
dp
= q ( E + v × B)
dt
W=
= mc 2 (γ − 1)
The total energy of this particle is
kinetic energy is
Wkin
(mc ) + ( pc )
2 2
2
= γ mc 2 , the
The role of acceleration is to increase the particle energy!
Change of W by differentiation:
  

 
2
2
W dW = c p ⋅ dp = q c p ⋅ (E + v × B ) d t = q c p ⋅ Ed t
 
dW = qv ⋅ Edt
2
Note: Only the electric field can change the particle energy!
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
3
Maxwell’s equations
The electromagnetic fields inside the “hollow place” obey these equations:
 1 ∂ 

∇× B − 2 E = 0 ∇⋅ B = 0
c ∂t
 ∂ 

∇× E + B = 0 ∇⋅ E = 0
∂t
With the curl of the 3rd, the time derivative of the 1st equation and the
vector identity



∇ × ∇ × E ≡ ∇∇ ⋅ E − ∆E
this set of equations can be brought in the form
 1 ∂2 
∆E − 2 2 E = 0
c ∂t
which is the Laplace equation in 4 dimensions.
With the boundaries of the “solid body” around it (the cavity walls), there
exist eigensolutions of the cavity at certain frequencies (eigenfrequencies).
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
4
Homogeneous plane wave
 
 
E ∝ u y cos(ωt − k ⋅ r )
 
 
B ∝ u x cos(ωt − k ⋅ r )
  ω
k ⋅ r = (cos(ϕ )z + sin (ϕ )x )
c

Wave vector k : 
the direction of k is the direction of
propagation, 
the length of k is the phase shift per
unit
 length.
k behaves like a vector.
x
k⊥ =
ωc
k=
c
ω
c
Ey
φ
z
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
ω
 ωc 
kz =
1−  
c
ω 
2
5
Wave length, phase velocity

The components of k are related to the wavelength in the direction of that
component as λz =
ω
2π
etc. , to the phase velocity as vϕ , z = = f λz
kz
kz
k⊥ =
ωc
k=
c
ω
c
Ey
x
z
k⊥ =
ωc
c
k=
ω
c
ω 
kz =
1−  c 
c
ω 
ω
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
2
6
Superposition of 2 homogeneous plane waves
Ey
x
z
+
=
Metallic walls may be inserted where E y = 0
without perturbing the fields.
Note the standing wave in x-direction!
This way one gets a hollow rectangular waveguide
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
7
Rectangular waveguide
Fundamental (TE10 or H10) mode
in a standard rectangular waveguide.
Example: “S-band” : 2.6 GHz ... 3.95 GHz,
Waveguide type WR284 (2.84” wide),
dimensions: 72.14 mm x 34.04 mm.
Operated at f = 3 GHz.





1 

*
power flow: Re  ∫∫ E × H ⋅ d A
2  cross

section
23-Sept-2010
electric field
magnetic field
CAS Varna/Bulgaria 2010- RF Cavities
power flow
power flow
8
Waveguide dispersion
What happens with different waveguide
dimensions (different width a)?
kz
k=
c
3
ω 
=
kz =
1−  c 
λg c
ω 
ω
2
ω
ωc
1
cutoff
23-Sept-2010
fc =
1:
a = 52 mm,
f/fc = 1.04
2:
a = 72.14 mm,
f/fc = 1.44
ω
2π
f = 3 GHz
2
3:
a = 144.3 mm,
f/fc = 2.88
c
2a
CAS Varna/Bulgaria 2010- RF Cavities
9
Phase velocity
f = 3 GHz
The phase velocity is the speed with
which the crest or a zero-crossing travels
in z-direction.
Note on the three animations on the
right that, at constant f, it is ∝ λg .
Note that at f = f c , vϕ , z = ∞ !
With f → ∞ , vϕ , z → c !
kz
k=
2:
a = 72.14 mm,
f/fc = 1.44
ω
c
3
1:
a = 52 mm,
f/fc = 1.04
ω
ω 
kz =
=
1−  c  =
λg c
vϕ , z
ω 
2
kz
ω
=
1
cutoff
23-Sept-2010
fc =
c
2a
2π
ω
2
1
vϕ , z
ω
ωc
3:
a = 144.3 mm,
f/fc = 2.88
CAS Varna/Bulgaria 2010- RF Cavities
10
Rectangular waveguide modes
TE10
23-Sept-2010
TE20
TE01
TE11
TM11
TE21
TM21
TE30
TE31
TM31
TE40
TE02
TE12
TM12
TE41
TM41
TE22
TM22
TE50
TE32
CAS Varna/Bulgaria 2010- RF Cavitiesplotted:
E-field
11
Radial waves
Also radial waves may be interpreted as
superpositions of plane waves.
The superposition of an outward and an
inward radial wave can result in the field of a
round hollow waveguide.
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
12
Round waveguide
TE11 – fundamental
fc
87.9
=
GHz a / mm
23-Sept-2010
TM01 – axial field
fc
114.8
=
GHz a / mm
CAS Varna/Bulgaria 2010- RF Cavities
f/fc = 1.44
TE01 – low loss
fc
182.9
=
GHz a / mm
13
Circular waveguide modes
TE11
TE21
TE31
23-Sept-2010
TE11
TM01
TE21
TE31
TE01
TM11
CAS Varna/Bulgaria 2010- RF Cavitiesplotted:
E-field
14
General waveguide equations:
Transverse wave equation (membrane equation):
TE (or H) modes
TM (or E) modes
boundary condition:
longitudinal wave equations
(transmission line equations):
propagation constant:
characteristic impedance:
ortho-normal eigenvectors:
transverse fields:
longitudinal field:
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
15
TE (H) modes:
TM (E) modes:
b
a
TE (H) modes:
Ø = 2a
TM (E) modes:
where
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
16
Waveguide perturbed by notches
“notches”
Signal flow chart
Reflections from notches lead to a superimposed standing wave pattern.
“Trapped mode”
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
17
Short-circuited waveguide
TM010 (no axial dependence)
23-Sept-2010
TM011
CAS Varna/Bulgaria 2010- RF Cavities
TM012
18
Single WG mode between two shorts
short
circuit
a
e
−1
− jk z 
short
circuit
Signal flow chart
e
−1
− jk z 
Eigenvalue equation for field amplitude a:
a = e − jk z 2  a
Non-vanishing solutions exist for
With
23-Sept-2010
kz =
ω 
1−  c 
c
ω 
ω
2k z  = 2π m :
2
, this becomes
 m
f 02 = f c2 +  c 
 2 
CAS Varna/Bulgaria 2010- RF Cavities
2
19
Simple pillbox
(only 1/2 shown)
TM010-mode
electric field (purely axial)
23-Sept-2010
magnetic field (purely azimuthal)
CAS Varna/Bulgaria 2010- RF Cavities
20
Pillbox cavity field (w/o beam tube)
χ 01 ρ 

J0 
h

1
 a  χ = 2.40483...
T (ρ ,ϕ ) =
01
π χ J  χ 01 

01 1 
 a 
Ø 2a
The only non-vanishing field components :
χ 01 ρ 

J0 

1 χ 01 1  a 
Ez =
χ 01 
jωε 0 a π

a J1 

 a 
χ 01 ρ 

J1 

1  a 
Bϕ = µ 0
π a J  χ 01 

1
 a 
23-Sept-2010
ω0
pillbox
Q pillbox
=
χ 01 c
η=
a
2aησχ 01
=
a
2 1 + 
 h χ h
sin 2 ( 01 )
R
4η
= 3 2
Q pillbox χ 01 π J1 ( χ 01 )
CAS Varna/Bulgaria 2010- RF Cavities
µ0
= 377 Ω
ε0
2 a
ha
21
Pillbox with beam pipe TM
010-mode
(only 1/4 shown)
One needs a hole for the beam pipe – circular waveguide below cutoff
electric field
23-Sept-2010
magnetic field
CAS Varna/Bulgaria 2010- RF Cavities
22
A more practical pillbox cavity
TM010-mode
Round of sharp edges
(field enhancement!)
electric field
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
(only 1/4 shown)
magnetic field
23
Stored energy
The energy stored in the electric field is
ε 2
∫∫∫ 2 E
1.0
WE
0.5
dV
1
2
3
4

E
5
6
5
6
cavity
0.5
1.0
WM
1.0
The energy stored in the magnetic field is
µ 
∫∫∫ 2 H
2
0.5
dV
1
2
3
cavity
0.5
4

H
1.0


Since E and H are 90° out of phase, the stored energy continuously swaps
from electric energy to magnetic energy. On average, electric and magnetic
energy must be equal.
The (imaginary part of the) Poynting vector describes this energy flux.
ε 2 µ  2

In steady state, the total stored energy W = ∫∫∫  E + H dV is
2
2

cavity
constant in time.
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
24
Stored energy & Poynting vector
electric field energy
23-Sept-2010
Poynting vector
CAS Varna/Bulgaria 2010- RF Cavities
magnetic field energy
25
Losses & Q factor
The losses
Ploss are proportional to the stored energy W.
The cavity quality factor Q is defined as the ratio Q =
ω0 W
Ploss
.
In a vacuum cavity, losses are dominated by the ohmic losses due to the finite
conductivity of the cavity walls.
If the losses are small, one can calculate them with a perturbation method:
•
The tangential magnetic field at the surface leads to a surface current.
•
This current will see a wall resistance RA =
•
ωµ
2σ
{ RA is related to the skin depth δ by δ σ RA = 1 . }
•
The cavity losses are given by Ploss =
∫∫ RA H t dA
2
wall
•
23-Sept-2010
If other loss mechanisms are present, losses must be added.
Consequently, the inverses of the Q ‘s must be added!
CAS Varna/Bulgaria 2010- RF Cavities
26
Acceleration voltage & R-upon-Q
I define
Vacc = ∫ E z e
ω
z
βc
j
dz . The exponential factor accounts for the
variation of the field while particles with velocity β c are traversing the gap
(see next page).
With this definition,
Vacc
is generally complex – this becomes important
with more than one gap. For the time being we are only interested in
Vacc .
Attention, different definitions are used!
The square of the acceleration voltage is proportional to the stored energy W .
The proportionality constant defines the quantity called R-upon-Q:
2
R Vacc
=
Q 2 ω0 W
Attention, also here different definitions are used!
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
27
Transit time factor
The transit time factor is the ratio of the acceleration voltage to the (non-physical)
voltage a particle with infinite velocity would see.
TT =
Vacc
∫ E z dz
=
∫ Ez e
ω
z
βc
j
dz
∫ E z dz
The transit time factor of an ideal pillbox cavity (no axial field dependence) of
height (gap length) h is:
Field rotates by 360°
during particle passage.
χ h
χ h
TT = sin  01   01 
 2a   2a 
h/λ
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
28
Shunt impedance
The square of the acceleration voltage is proportional to the power loss Ploss .
The proportionality constant defines the quantity “shunt impedance”
2
Vacc
R=
2 Ploss
Attention, also here different definitions are used!
Traditionally, the shunt impedance is the quantity to optimize in order to
minimize the power required for a given gap voltage.
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
29
Equivalent circuit
Simplification: single mode
IG
IB
Vacc
P

Generator
R
β:
coupling factor
C
β
L
R
Cavity
R: Shunt impedance
23-Sept-2010
L
C
Beam
L=R/(Qω0)
C=Q/(Rω0)
: R-upon-Q
CAS Varna/Bulgaria 2010- RF Cavities
30
Resonance
100
50
Q=100
20
|ZRQ
()|/(/)
ω
10
5
Q=10
2
Q=1
1
0.5
23-Sept-2010
Q=1
1
CAS Varna/Bulgaria 2010- RF Cavities
1.5
ω/ω0
2
31
Reentrant cavity
Nose cones increase transit time factor, round outer shape minimizes losses.
Example: KEK photon factory 500 MHz
- R probably as good as it gets -
nose cone
23-Sept-2010
R/Q:
Q:
R:
this cavity
optimized
pillbox
111 Ω
44270
4.9 MΩ
107.5 Ω
41630
4.47 MΩ
CAS Varna/Bulgaria 2010- RF Cavities
32
Loss factor
Impedance seen by the beam
V (induced)
kloss
ω0 R
IB
2
Vacc
1
=
=
=
2 Q 4 W 2C
Beam
Energy deposited by a single
2
charge q: kloss q
R/β
C
L
L=R/(Qω0)
R
C=Q/(Rω0)
Cavity
Voltage induced by a single
charge q:
1
0
-1
0
5
10
15
20
t f0
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
33
Summary: relations between Vacc, W, Ploss
gap voltage
R-upon-Q
2
R Vacc
=
Q 2 ω0 W
kloss
ω0 R
Shunt impedance
2
Vacc
R=
2 Ploss
2
Vacc
=
=
2 Q 4W
Energy stored inside the
cavity
Q=
ω0 W
Power lost in the cavity
walls
Ploss
Q factor
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
34
Beam loading – RF to beam efficiency
The beam current “loads” the generator, in the equivalent circuit
this appears as a resistance in parallel to the shunt impedance.
If the generator is matched to the unloaded cavity, beam loading
will cause the accelerating voltage to decrease.
1
The power absorbed by the beam is − Re{ Vacc I B* } ,
2
2
V
the power loss Ploss = acc .
2R
For high efficiency, beam loading should be high.
IB
1
The RF to beam efficiency is η =
.
=
Vacc
IG
1+
R IB
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
35
Characterizing cavities
•
Resonance frequency
•
Transit time factor
field varies while particle is traversing the gap
Circuit definition
•
Linac definition
Shunt impedance
gap voltage – power relation
•
Q factor
•
R/Q
independent of losses – only geometry!
•
loss factor
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
36
Higher order modes
external dampers
R1, Q1,ω1
R2, Q2,ω2
R3, Q3,ω3
...
...
n1
n2
n3
IB
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
37
Higher order modes (measured spectrum)
without dampers
with dampers
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
38
Pillbox: dipole mode
TM110-mode
electric field
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
(only 1/4 shown)
magnetic field
39
CERN/PS 80 MHz cavity (for LHC)
inductive (loop) coupling,
low self-inductance
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
40
Higher
order
modes
Example shown:
80 MHz cavity PS
for LHC.
Color-coded:
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
41
What do you gain with many gaps?
• The R/Q of a single gap cavity is limited to some 100 Ω.
Now consider to distribute the available power to n identical
cavities: each will receive P/n, thus produce an accelerating
voltage of
.
The total accelerating voltage thus increased, equivalent to a
total equivalent shunt impedance of
.
P/n
P/n
1
23-Sept-2010
P/n
2
P/n
3
n
CAS Varna/Bulgaria 2010- RF Cavities
42
Standing wave multicell cavity
• Instead of distributing the power from the amplifier, one might
as well couple the cavities, such that the power automatically
distributes, or have a cavity with many gaps (e.g. drift tube
linac).
• Coupled cavity accelerating structure (side coupled)
• The phase relation between gaps is important!
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
43
Brillouin diagram
Travelling wave
structure
π
ω L/c
2π/3
2π
speed of light line,
ω = β /c
π
synchronous
π/2
π/2
0
0
23-Sept-2010
π/2
βL
π
CAS Varna/Bulgaria 2010- RF Cavities
44
Examples of cavities
PEP II cavity
476 MHz, single cell,
1 MV gap with 150 kW,
strong HOM damping,
23-Sept-2010
LEP normal-conducting Cu RF cavities,
350 MHz. 5 cell standing wave + spherical cavity
for energy storage, 3 MV
CAS Varna/Bulgaria 2010- RF Cavities
45
CERN PS 200 MHz cavities
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
46
PS 19 MHz cavity (prototype, photo: 1966)
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
47
CERN PS 80 MHz Cavity (1997)
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
48
Ferrite cavity – CERN PSB, 0.6 ... 1.8 MHz
PS Booster, ‘98
0.6 – 1.8 MHz,
< 10 kV gap
NiZn ferrites
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
49
CERN PS 10 MHz cavity (1 of 10)
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
50
Drift-tube linac (JPARC JHF, 324 MHz)
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
51
CERN SPS 200 MHz TW cavity
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
52
Travelling wave cavities
CLIC “T18”, 12 GHz
CLIC “HDS”, 12 GHz
“Shintake” structure, 5.7 GHz
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
53
Side-coupled cavity (JHF, 972 MHz)
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
54
Single- and multi-cell SC cavities (1.3 GHz)
SC cavity lab KEK,
Japan
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
55
SC cavities in a cryostat (CERN LHC 400 MHz)
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
56
SC deflecting cavity (KEK-B, 508 MHz)
Asymmetric shape
to split the two
polarizations.
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
57
Summary
RF Cavities
• The EM fields inside a hollow cavity are superpositions of
homogeneous plane waves.
• When operating near an eigenfrequency, one can profit from
a resonance phenomenon (with high Q).
• R-upon-Q, Shunt impedance and Q factor were are useful
parameters, which can also be understood in an equivalent
circuit.
• The perturbation method allows to estimate losses and
sensitivity to tolerances.
• Many gaps can increase the effective impedance.
23-Sept-2010
CAS Varna/Bulgaria 2010- RF Cavities
58