f acceleratordivision introduction to rf for particle accelerators part 2: rf cavities dave mcginnis
TRANSCRIPT
f AcceleratorDivision
Introduction to RF for Particle AcceleratorsIntroduction to RF for Particle AcceleratorsPart 2: RF CavitiesPart 2: RF Cavities
Dave McGinnis
Introduction to RF - McGinnis 2
f AcceleratorDivision RF Cavity TopicsRF Cavity Topics
Modes Symmetry Boundaries Degeneracy
RLC model Coupling
Inductive Capacitive Measuring
Q Unloaded Q Loaded Q Q Measurements
Impedance Measurements Bead Pulls Stretched wire
Beam Loading De-tuning Fundamental Transient
Power Amplifiers Class of operation Tetrodes Klystrons
Introduction to RF - McGinnis 3
f AcceleratorDivision RF and Circular AcceleratorsRF and Circular Accelerators
For circular accelerators, the beam can only be accelerated by a time-varying (RF) electromagnetic field.
Faraday’s Law
SC
SdBt
ldE
The integral
C
ldEq
is the energy gained by a particle with charge q during one trip around the accelerator.
For a machine with a fixed closed path such as a synchrotron, if
0t
B
then 0ldEqC
Introduction to RF - McGinnis 4
f AcceleratorDivision RF CavitiesRF Cavities
Beam
RF Input
Power Amplifier
Coupler
Gap
New FNAL Booster Cavity
Multi-cell superconducting RF cavity
Transmission Line Cavity
Introduction to RF - McGinnis 5
f AcceleratorDivision Cavity Field PatternCavity Field Pattern
For the fundamental mode at one instant in time:
Electric Field
Magnetic Field
Out
In
Wall Current
Introduction to RF - McGinnis 6
f AcceleratorDivision Cavity ModesCavity Modes
We need to solve only ½ of the problem
For starters, ignore the gap capacitance.
The cavity looks like a shorted section of transmission line
x=L x=0
xjxjo
xjxj
eVeVIZ
eVeVV
where Zo is the characteristic impedance of the transmission line structure of the cavity
Introduction to RF - McGinnis 7
f AcceleratorDivision Cavity Boundary ConditionsCavity Boundary Conditions
Boundary Condition 1:
At x=0: V=0
xcosjVIZ
xsinVV
oo
o
Boundary Condition 2:
At x=L: I=0
4
1n2L
L4
c1n2f
21n2L
0Lcos
n
n
n
...3,2,1,0n Different values of n are called modes. The lowest value of n is usually called the fundamental mode
Introduction to RF - McGinnis 8
f AcceleratorDivision Cavity ModesCavity Modes
n=0
n=1
Introduction to RF - McGinnis 9
f AcceleratorDivision Even and Odd Mode De-CompositionEven and Odd Mode De-Composition
inV
inin V2
1V
2
1 inin V
2
1V
2
1
Introduction to RF - McGinnis 10
f AcceleratorDivision Even and Odd Mode De-CompositionEven and Odd Mode De-Composition
inV2
1inV
2
1
inV2
1inV
2
1
Even Mode
No Gap Field
Odd Mode
Gap Field!
Introduction to RF - McGinnis 11
f AcceleratorDivision Degenerate ModesDegenerate Modes
The even and odd decompositions have the same mode frequencies.
Modes that occur at the same frequency are called degenerate.
The even and odd modes can be split if we include the gap capacitance.
In the even mode, since the voltage is the same on both sides of the gap, no capacitive current can flow across the gap.
In the odd mode, there is a voltage difference across the gap, so capacitive current will flow across the gap.
inV2
1inV
2
1inV
2
1inV
2
1
Introduction to RF - McGinnis 12
f AcceleratorDivision Gap CapacitanceGap Capacitance
Boundary Condition 2:
At x=L: VCjI g where Cg is the gap capacitance
Boundary Condition 1:
At x=0: V=0 xcosjVIZ
xsinVV
oo
o
Lsin
LcosZC og
Introduction to RF - McGinnis 13
f AcceleratorDivision RF Cavity ModesRF Cavity Modes
Consider the first mode only (n=0) and a very small gap capacitance.
2
ZC
Lsin
Lcos2
L
o
o
ogo
The gap capacitance shifts the odd mode down in frequency and leaves the even mode frequency unchanged
Introduction to RF - McGinnis 14
f AcceleratorDivision Multi-Celled CavitiesMulti-Celled Cavities
Each cell has its own resonant frequency
For n cells there will be n degenerate modes
The cavity to cavity coupling splits these n degenerate modes.
The correct accelerating mode must be picked
Introduction to RF - McGinnis 15
f AcceleratorDivision Cavity QCavity Q
If the cavity walls are lossless, then the boundary conditions for a given mode can only be satisfied at a single frequency.
If the cavity walls have some loss, then the boundary conditions can be satisfied over a range of frequencies.
The cavity Q factor is a convenient way the power lost in a cavity.
The Q factor is defined as:
L
HEo
cycle/lost
stored
P
WW
W
WQ
Introduction to RF - McGinnis 16
f AcceleratorDivision Transmission Line Cavity QTransmission Line Cavity Q
We will use the fundamental mode of the transmission line cavity as an example of how to calculate the cavity Q.
Electric Energy:
o
2o
o
L
0
2l
vol
2E
Z
V
8
dxxVC4
12
dvolE4
1W
o
2o
o
L
0
2l
vol
2H
Z
V
8
dxxIL4
12
dvolH4
1W
Magnetic Energy:
Both Halves:
Introduction to RF - McGinnis 17
f AcceleratorDivision Transmission Line Cavity QTransmission Line Cavity Q
Assume a small resistive loss per unit length rL/m along the walls of the cavity.
Also assume that this loss does not perturb the field distribution of the cavity mode.
o
2o
l
L
0
2lloss
Z
VLr
2
1
dxxIr2
12P
The cavity Q for the fundamental mode of the transmission line cavity is:
Lr
Z
2Q
l
o
Time average
Less current flowing along walls
Less loss in walls
Introduction to RF - McGinnis 18
f AcceleratorDivision RLC Model for a Cavity ModeRLC Model for a Cavity Mode
Around each mode frequency, we can describe the cavity as a simple RLC circuit.
Igen
Vgap
Leq Req Ceq
Req is inversely proportional to the energy lost
Leq is proportional to the magnetic stored energy
Ceq is proportional to the electric stored energy
Introduction to RF - McGinnis 19
f AcceleratorDivision RLC Parameters for a Transmission Line CavityRLC Parameters for a Transmission Line Cavity
For the fundamental mode of the transmission line cavity:
Lr
Z4R
R
V
2
1P
l
2o
eq
eq
2gap
loss
o
oeq
eq2
o
2gap
H
Z8L
L
V
4
1W
ooeq
2gapeqE
Z
1
8C
VC4
1W
The transfer impedance of the cavity is:
eqeqeqc
gen
gapc
CjLj
1
R
1
Z
1
I
VZ
Introduction to RF - McGinnis 20
f AcceleratorDivision Cavity Transfer ImpedanceCavity Transfer Impedance
Since:
eqeqo
CL1
eq
eqeq
C
L
Q
R
eqeqo CRQ
2
oo2
oeqc
Qjj
j
Q
RjZ
Function of geometry only
Function of geometry and cavity material
Introduction to RF - McGinnis 21
f AcceleratorDivision Cavity Frequency ResponseCavity Frequency Response
2 1.5 1 0.5 0 0.5 1 1.5 21
0.5
0
0.5
1
MagnitudeRealImaginary
Frequency
Imp
ed
an
ce
fo
2 Q
fo
2 Q
2 1.5 1 0.5 0 0.5 1 1.5 290
45
0
45
90
Frequency
Ph
ase
(d
eg
ree
s)
fo
2 Q
fo
2 Q
2 1.5 1 0.5 0 0.5 1 1.5 290
45
0
45
90
Frequency
Ph
ase
(d
eg
ree
s)
fo
2 Q
fo
2 Q
Referenced to o
2
oo2
oeqc
Qjj
j
Q
RjZ
45Q2
Zarg
Q2ZRe
Q2ZIm
oo
oo
oo
Peak of the response is at
o
2o
2o
o Z2
1
Q2Z
Introduction to RF - McGinnis 22
f AcceleratorDivision
Mode Spectrum Example – Pill Box Mode Spectrum Example – Pill Box CavityCavity
The RLC model is only valid around a given mode
Each mode will a different value of R,L, and C
Introduction to RF - McGinnis 23
f AcceleratorDivision Cavity Coupling – Offset CouplingCavity Coupling – Offset Coupling
As the drive point is move closer to the end of the cavity (away from the gap), the amount of current needed to develop a given voltage must increase
Therefore the input impedance of the cavity as seen by the power amplifier decreases as the drive point is moved away from the gap
Introduction to RF - McGinnis 24
f AcceleratorDivision Cavity CouplingCavity Coupling
We can model moving the drive point as a transformer Moving the drive point away from the gap increases the
transformer turn ratio (n)
cZpacZ
n:1
c2pac Zn
1Z
Igen
Vgap
Leq Req Ceq
Introduction to RF - McGinnis 25
f AcceleratorDivision Inductive CouplingInductive Coupling
For inductive coupling, the PA does not have to be directly attached to the beam tube.
The magnetic flux thru the coupling loop couples to the magnetic flux of the cavity mode
The transformer ratio n = Total Flux / Coupler Flux
Out
In
Wall Current
Magnetic Field
Introduction to RF - McGinnis 26
f AcceleratorDivision Capacitive CouplingCapacitive Coupling
If the drive point does not physically touch the cavity gap, then the coupling can be described by breaking the equivalent cavity capacitance into two parts.
Vgap
Igen
LeqReq
C1
C2
c
eq2pac Z
CC
1Z
21eq C
1
C
1
C
1
As the probe is pulled away from the gap, C2 increases and the impedance of the cavity as seen by the power amp decreases
pacZ
Introduction to RF - McGinnis 27
f AcceleratorDivision Power Amplifier Internal ResistancePower Amplifier Internal Resistance
So far we have been ignoring the internal resistance of the power amplifier. This is a good approximation for tetrode power
amplifiers that are used at Fermilab in the Booster and Main Injector
This is a bad approximation for klystrons protected with isolators
Every power amplifier has some internal resistance
0LIgen
gengen I
VR
genRgenI
Introduction to RF - McGinnis 28
f AcceleratorDivision Total Cavity CircuitTotal Cavity Circuit
cZpacZ
n:1
Vgap
Leq Req CeqRgen
Igen/nLeq Req Ceq
n2Rgen
Vgap
Total circuit as seen by the cavity
Igen
Introduction to RF - McGinnis 29
f AcceleratorDivision Loaded QLoaded Q
The generator resistance is in parallel with the cavity resistance.
The total resistance is now lowered.
The power amplifier internal resistance makes the total Q of the circuit smaller (d’Q)
gen2
eqL Rn
1
R
1
R
1
extoL Q
1
Q
1
Q
1
eqgen2
oext
eqeqoo
eqLoL
CRnQ
CRQ
CRQ
Loaded Q
Unloaded Q
External Q
Introduction to RF - McGinnis 30
f AcceleratorDivision
Vgap
Leq Req Ceq
Rgen
Igen
Zo
Cavity CouplingCavity Coupling
The cavity is attached to the power amplifier by a transmission line. In the case of power amplifiers mounted directly on the
cavity such as the Fermilab Booster or Main Injector, the transmission line is infinitesimally short.
The internal impedance of the power amplifier is usually matched to the transmission line impedance connecting the power amplifier to the cavity. As in the case of a Klystron protected by an isolator As in the case of an infinitesimally short transmission line
ogen ZR
n:1
Introduction to RF - McGinnis 31
f AcceleratorDivision Cavity CouplingCavity Coupling
Look at the cavity impedance from the power amplifier point of view:
Vgap/n
Leq/n2
n2Ceq
Rgen
Igen
Zo
Req/n2
Assume that the power amplifier is matched (Rgen=Zo) and define a coupling parameter as the ratio of the real part of the cavity impedance as seen by the power amplifier to the characteristic impedance.
o
2eq
cpl Zn
R
r 1r
1r
1r
cpl
cpl
cpl
under-coupled
Critically-coupled
over-coupled
Introduction to RF - McGinnis 32
f AcceleratorDivision Which Coupling is Best?Which Coupling is Best?
Critically coupled would provide maximum power transfer to the cavity.
However, some power amplifiers (such as tetrodes) have very high internal resistance compared to the cavity resistance and the systems are often under-coupled. The limit on tetrode power amplifiers is dominated
by how much current they can source to the cavity
Some cavities a have extremely low losses, such as superconducting cavities, and the systems are sometimes over-coupled.
An intense beam flowing though the cavity can load the cavity which can effect the coupling.
Introduction to RF - McGinnis 33
f AcceleratorDivision Measuring Cavity CouplingMeasuring Cavity Coupling
2
oo2
oeqc
Qjj
j
Q
RjZ
The frequency response of the cavity at a given mode is:
which can be re-written as:
jeqc ecosRjZ
o
22oQtan
Introduction to RF - McGinnis 34
f AcceleratorDivision Cavity CouplingCavity Coupling
The reflection coefficient as seen by the power amplifier is:
o2
c
o2
c
ZnZ
ZnZ
1ecosr
1ecosrj
cpl
jcpl
This equation traces out a circle on the reflection (u,v) plane
Introduction to RF - McGinnis 35
f AcceleratorDivision Cavity CouplingCavity Coupling
1 0.5 0 0.5 1
1
0.5
0.5
1
Re
Im
1r
r
cpl
cpl
1r
1
cpl
1r
1r
cpl
cpl
1r
r
cpl
cpl
Radius =
0,1r
1
cplCenter =
0,1Left edge (=+/-/2) =
0,
1r
1r
cpl
cplRight edge (=0) =
Introduction to RF - McGinnis 36
f AcceleratorDivision Cavity CouplingCavity Coupling
The cavity coupling can be determined by: measuring the
reflection coefficient trajectory of the input coupler
Reading the normalized impedance of the extreme right point of the trajectory directly from the Smith Chart
Re
Im
1r
1r
cpl
cpl
1 0.5 0 0.5 1
1
0.5
0.5
1
Introduction to RF - McGinnis 37
f AcceleratorDivision Cavity CouplingCavity Coupling
Re
Im
Under coupled
Critically coupled
Over coupled
Introduction to RF - McGinnis 38
f AcceleratorDivision Measuring the Measuring the LoadedLoaded Q of the Cavity Q of the Cavity
The simplest way to measure a cavity response is to drive the coupler with RF and measure the output RF from a small detector mounted in the cavity.
Because the coupler “loads” the cavity, this measures the loaded Q of the cavity which depending on the coupling, can be much different than the unloaded Q Also note that changing the coupling in the cavity, can change the cavity
response significantly
Beam
RF Input
Power Amplifier
Coupler
Gap
RF Output
2 1.5 1 0.5 0 0.5 1 1.5 21
0.5
0
0.5
1
MagnitudeRealImaginary
Frequency
Imp
ed
an
ce
fo
2 Q
fo
2 Q
2 1.5 1 0.5 0 0.5 1 1.5 290
45
0
45
90
Frequency
Ph
ase
(d
eg
ree
s)
fo
2 Q
fo
2 Q
S21
Introduction to RF - McGinnis 39
f AcceleratorDivision Measuring the Measuring the UnloadedUnloaded Q of a Cavity Q of a Cavity
If the coupling is not too extreme, the loaded and unloaded Q of the cavity can be measured from reflection (S11) measurements of the coupler.
At:
cc
o
oo
ZReZIm4
Q
Unloaded Q!
For:
21vu
ZReZIm
22
cc
jvu
where:
Circles on the Smith Chart
Introduction to RF - McGinnis 40
f AcceleratorDivision Measuring the Measuring the UnloadedUnloaded Q of a Cavity Q of a Cavity
Measure frequency (-) when:
Measure resonant frequency (o)
Measure
frequency (+) when:
Compute
cc ZReZIm
cc ZReZIm
cc ZReZIm
cc ZReZIm
ooQ
Introduction to RF - McGinnis 41
f AcceleratorDivision Measuring the Measuring the LoadedLoaded Q of a Cavity Q of a Cavity
Measure the coupling parameter (rcpl)
Measure the unloaded Q (Qo)
eqo2
oext
eqocpl2
oo
ocpl2
eq
CZnQ
CZrnQ
ZrnR
Vgap
Leq Req Ceq
Rgen
Igen
Zo
n:1
1r
Q
1
Q
1
Q
1
cpl
oL
extoL
Introduction to RF - McGinnis 42
f AcceleratorDivision NEVER – EVER!!NEVER – EVER!!
2 1.5 1 0.5 0 0.5 1 1.5 240
30
20
10
0
r=0.5r=1r=2
Frequency
Re
flect
ion
Co
eff
icie
nt
(dB
)
Introduction to RF - McGinnis 43
f AcceleratorDivision Bead PullsBead Pulls
The Bead Pull is a technique for measuring the fields in the cavity and the equivalent impedance of the cavity as seen by the beam In contrast to
measuring the impedance of the cavity as seen by the power amplifier through the coupler
Introduction to RF - McGinnis 44
f AcceleratorDivision Bead Pull SetupBead Pull Setup
Set to unperturbed resonant frequency of the cavity
Phase Detector
Gap Detector
Bead
string
Introduction to RF - McGinnis 45
f AcceleratorDivision Bead PullsBead Pulls
In the capacitor of the RLC model for the cavity mode consider placing a small dielectric cube Assume that the small cube will not
distort the field patterns appreciably
The stored energy in the capacitor will change
dvE4
1dvE
4
1VC
4
1W 2
cor2
co2
gapeqE Total original Electric energy
Original Electric energy in the cube
new Electric energy in the cube
Where Ec is the electric field in the cube
dv is the volume of the cube
o is the permittivity of free space
r is the relative permittivity of the cube
dvE4
1W
2
volE
Introduction to RF - McGinnis 46
f AcceleratorDivision Bead PullsBead Pulls
The equivalent capacitance of the capacitor with the dielectric cube is:
2gapeq
2gapE VCC
4
1CV
4
1W
2
gap
cro V
Edv1C
The resonant frequency of the cavity will shift
CCL
1
eqeq
2o
Introduction to RF - McGinnis 47
f AcceleratorDivision Bead PullsBead Pulls
For << o and C << Ceq
T
E
o
2gapeq
2cro
eqo
W
W
VC21
dvE141
C
C
2
1
Introduction to RF - McGinnis 48
f AcceleratorDivision Bead PullsBead Pulls
Had we used a metallic bead (r>1) or a metal bead:
Also, the shape of the bead will distort the field in the vicinity of the bead so a geometrical form factor must be used.
For a small dielectric bead of radius a
For a small metal bead with radius a
T
HE
o W
WW
T
2b
r
ro
3
o W
E
2
1a
2
bo2
boT
3
oH
2E
W
aA metal bead can be used to measure the E field only if the bead is placed in a region where the magnetic field is zero!
Introduction to RF - McGinnis 49
f AcceleratorDivision Bead PullsBead Pulls
In general, the shift in frequency is proportional to a form factor F
T
2b
o W
EF
o3
r
ro
3
aF
2
1aF
Dielectric bead
Metal bead
Introduction to RF - McGinnis 50
f AcceleratorDivision Bead PullsBead Pulls
From the definition of cavity Q:
oeqo
2
gap
b
eq
2gap
oT
eq
2gap
L
L
To
z,y,x
QR
1
2
1
F
1
V
z,y,xE
QR
V
2
1W
R
V
2
1P
P
WQ
Introduction to RF - McGinnis 51
f AcceleratorDivision Bead PullsBead Pulls
Since:
2
gap o
gapgap
o
eq
gapgap
gapgap
dzz,y,x
2
1
F
1
Q
R
Vdzz,y,xE
Introduction to RF - McGinnis 52
f AcceleratorDivision Bead PullsBead Pulls
For small perturbations, shifts in the peak of the cavity response is hard to measure.
Shifts in the phase at the unperturbed resonant frequency are much easier to measure.
2 1.5 1 0.5 0 0.5 1 1.5 20
0.25
0.5
0.75
1
UnperturbedShifted
Frequency
Imp
ed
an
ce
2 1.5 1 0.5 0 0.5 1 1.5 290
45
0
45
90
UnperturbedShifted
Frequency
Ph
ase
(d
eg
ree
s)
Introduction to RF - McGinnis 53
f AcceleratorDivision Bead PullsBead Pulls
Since:
o
o
o
Q2
Qtan
2
gapgapgap
oeq dzz,y,xtan
2
1
2
1
F
1R
Introduction to RF - McGinnis 54
f AcceleratorDivision Bead PullsBead Pulls
Set to unperturbed resonant frequency of the cavity
Phase Detector
Gap Detector
Bead
string