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Introduction to RF for Particle AcceleratorsPart 2: RF CavitiesIntroduction to RF for Particle AcceleratorsPart 2: RF Cavities
Dave McGinnis
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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
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RF and Circular AcceleratorsRF and Circular Accelerators
For circular accelerators, the beam can only be accelerated by atime-varying (RF) electromagnetic field.
Faraday’s Law
The integral
is the energy gained by a particle with charge q during one triparound the accelerator.
For a machine with a fixed closed path such as a synchrotron, if
then
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RF CavitiesRF Cavities
Beam
RF Input
PowerAmplifier
Coupler
Gap
New FNAL Booster Cavity
Multi-cellsuperconductingRF cavity
Transmission Line Cavity
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Cavity Field PatternCavity Field Pattern
For the fundamental mode at one instant in time:
Electric Field
Magnetic Field
Out
In
Wall Current
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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
where Zo is the characteristic impedance of the transmissionline structure of the cavity
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Cavity Boundary ConditionsCavity Boundary Conditions
Boundary Condition 1:
At x=0: V=0
Boundary Condition 2:
At x=L: I=0
Different valuesof n are calledmodes. The lowestvalue of n isusually called thefundamental mode
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Cavity ModesCavity Modes
n=0
n=1
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Even and Odd Mode De-CompositionEven and Odd Mode De-Composition
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Even and Odd Mode De-CompositionEven and Odd Mode De-Composition
Even Mode
No Gap Field
Odd Mode
Gap Field!
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Degenerate ModesDegenerate Modes
The even and odd decompositions have the same modefrequencies.
Modes that occur at the same frequency are calleddegenerate.
The even and odd modes can be split if we include the gapcapacitance.
In the even mode, since the voltage is the same on bothsides of the gap, no capacitive current can flow across thegap.
In the odd mode, there is a voltage difference across thegap, so capacitive current will flow across the gap.
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Gap CapacitanceGap Capacitance
Boundary Condition 2:
At x=L:
where Cg is the gap capacitance
Boundary Condition 1:
At x=0: V=0
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RF Cavity ModesRF Cavity Modes
Consider the first mode only (n=0) and a very small gapcapacitance.
The gap capacitance shifts the odd mode down in frequency andleaves the even mode frequency unchanged
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Multi-Celled CavitiesMulti-Celled Cavities
Each cell has itsown resonantfrequency
For n cells therewill be ndegenerate modes
The cavity tocavity couplingsplits these ndegenerate modes.
The correctaccelerating modemust be picked
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Cavity QCavity Q
If the cavity walls are lossless, then the boundaryconditions for a given mode can only be satisfiedat a single frequency.
If the cavity walls have some loss, then theboundary conditions can be satisfied over a rangeof frequencies.
The cavity Q factor is a convenient way the powerlost in a cavity.
The Q factor is defined as:
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Transmission Line Cavity QTransmission Line Cavity Q
We will use the fundamental mode of the transmission linecavity as an example of how to calculate the cavity Q.
Electric Energy:
Magnetic Energy:
Both Halves:
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Transmission Line Cavity QTransmission Line Cavity Q
Assume a small resistive loss per unit length rL/m along thewalls of the cavity.
Also assume that this loss does not perturb the fielddistribution of the cavity mode.
The cavity Q for the fundamental mode of the transmission linecavity is:
Time average
Less current flowing along walls
Less loss in walls
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RLC Model for a Cavity ModeRLC Model for a Cavity Mode
Around each mode frequency, we can describe the cavity as asimple 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
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RLC Parameters for a Transmission Line CavityRLC Parameters for a Transmission Line Cavity
For the fundamental mode of the transmission line cavity:
The transfer impedance of the cavity is:
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Cavity Transfer ImpedanceCavity Transfer Impedance
Since:
Function of geometry only
Function of geometry and cavity material
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Cavity Frequency ResponseCavity Frequency Response
Referenced to o
Peak of theresponse is ato
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Mode Spectrum Example – Pill Box CavityMode Spectrum Example – Pill Box Cavity
The RLC model is only validaround a given mode
Each mode will a differentvalue of R,L, and C
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Cavity Coupling – Offset CouplingCavity Coupling – Offset Coupling
As the drive point is movecloser to the end of thecavity (away from the gap),the amount of currentneeded to develop a givenvoltage must increase
Therefore the inputimpedance of the cavity asseen by the power amplifierdecreases as the drive pointis moved away from the gap
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Cavity CouplingCavity Coupling
We can model moving the drive point as a transformer
Moving the drive point away from the gap increases thetransformer turn ratio (n)
Igen
Vgap
Leq
Req
Ceq
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Inductive CouplingInductive Coupling
For inductive coupling, the PA does not have to be directlyattached to the beam tube.
The magnetic flux thru the coupling loop couples to themagnetic flux of the cavity mode
The transformer ratio n = Total Flux / Coupler Flux
Out
In
Wall Current
Magnetic Field
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Capacitive CouplingCapacitive Coupling
If the drive point doesnot physically touch thecavity gap, then thecoupling can be describedby breaking the equivalentcavity capacitance intotwo parts.
Vgap
Igen
Leq
Req
C1
C2
As the probe is pulled awayfrom the gap, C2 increases andthe impedance of the cavity asseen by the power ampdecreases
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Power Amplifier Internal ResistancePower Amplifier Internal Resistance
So far we have been ignoring the internalresistance of the power amplifier.
This is a good approximation for tetrode poweramplifiers that are used at Fermilab in the Booster andMain Injector
This is a bad approximation for klystrons protected withisolators
Every power amplifier has some internal resistance
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Total Cavity CircuitTotal Cavity Circuit
Vgap
Leq
Req
Ceq
Rgen
Igen/n
Leq
Req
Ceq
n2Rgen
Vgap
Total circuit as seen by the cavity
Igen
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Loaded QLoaded Q
The generator resistance is in parallel with thecavity resistance.
The total resistance is now lowered.
The power amplifier internal resistance makes thetotal Q of the circuit smaller (d’Q)
Loaded Q
Unloaded Q
External Q
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Vgap
Leq
Req
Ceq
Rgen
Igen
Zo
Cavity CouplingCavity Coupling
The cavity is attached to the power amplifier by atransmission line.
In the case of power amplifiers mounted directly on the cavitysuch as the Fermilab Booster or Main Injector, thetransmission line is infinitesimally short.
The internal impedance of the power amplifier is usuallymatched to the transmission line impedance connecting thepower 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
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Cavity CouplingCavity Coupling
Look at the cavity impedance from the power amplifier point ofview:
Vgap/n
Leq/n2
n2Ceq
Rgen
Igen
Zo
Req/n2
Assume that the power amplifier is matched (Rgen=Zo) anddefine a coupling parameter as the ratio of the real part of thecavity impedance as seen by the power amplifier to thecharacteristic impedance.
under-coupled
Critically-coupled
over-coupled
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Which Coupling is Best?Which Coupling is Best?
Critically coupled would provide maximum powertransfer to the cavity.
However, some power amplifiers (such astetrodes) have very high internal resistancecompared to the cavity resistance and the systemsare often under-coupled.
The limit on tetrode power amplifiers is dominated byhow much current they can source to the cavity
Some cavities a have extremely low losses, such assuperconducting cavities, and the systems aresometimes over-coupled.
An intense beam flowing though the cavity canload the cavity which can effect the coupling.
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Measuring Cavity CouplingMeasuring Cavity Coupling
The frequency response of the cavity at a given mode is:
which can be re-written as:
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Cavity CouplingCavity Coupling
The reflection coefficient as seen by the power amplifier is:
This equation traces out a circle on the reflection (u,v) plane
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Cavity CouplingCavity Coupling
Radius =
Center =
Left edge(=+/-/2) =
Right edge(=0) =
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Cavity CouplingCavity Coupling
The cavitycoupling can bedetermined by:
measuring thereflectioncoefficienttrajectory ofthe inputcoupler
Reading thenormalizedimpedance ofthe extremeright point ofthe trajectorydirectly fromthe Smith Chart
1
0.5
0
0.5
1
1
0.5
0.5
1
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Cavity CouplingCavity Coupling
Under coupled
Critically coupled
Over coupled
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Measuring the Loaded Q of the CavityMeasuring the Loaded Q of the Cavity
The simplest way to measure a cavity response is to drive the coupler withRF 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 thecavity
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 cavityresponse significantly
Beam
RF Input
PowerAmplifier
Coupler
Gap
RF Output
S21
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Measuring the Unloaded Q of a CavityMeasuring the Unloaded Q of a Cavity
If the coupling is not too extreme, the loaded andunloaded Q of the cavity can be measured fromreflection (S11) measurements of the coupler.
At:
Unloaded Q!
For:
where:
Circles on theSmith Chart
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Measuring the Unloaded Q of a CavityMeasuring the Unloaded Q of a Cavity
Measurefrequency (-)when:
Measure resonantfrequency (o)
Measurefrequency (+)when:
Compute
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Measuring the Loaded Q of a CavityMeasuring the Loaded Q of a Cavity
Measure the coupling parameter (rcpl)
Measure the unloaded Q (Qo)
Vgap
Leq
Req
Ceq
Rgen
Igen
Zo
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NEVER – EVER!!NEVER – EVER!!
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Bead PullsBead Pulls
The Bead Pull is atechnique formeasuring the fields inthe cavity and theequivalent impedanceof the cavity as seenby the beam
In contrast tomeasuring theimpedance of the cavityas seen by the poweramplifier through thecoupler
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Bead Pull SetupBead Pull Setup
Set to unperturbed resonantfrequency of the cavity
Phase Detector
Gap Detector
Bead
string
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Bead PullsBead Pulls
In the capacitor of the RLC modelfor the cavity mode consider placinga small dielectric cube
Assume that the small cube will notdistort the field patterns appreciably
The stored energy in the capacitorwill change
Total originalElectric energy
OriginalElectric energyin the cube
new Electricenergy in thecube
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
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Bead PullsBead Pulls
The equivalent capacitance of the capacitor with the dielectriccube is:
The resonant frequency of the cavity will shift
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Bead PullsBead Pulls
For << o and C << Ceq
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Bead PullsBead Pulls
Had we used a metallic bead (r>1) or a metalbead:
Also, the shape of the bead will distort the fieldin the vicinity of the bead so a geometrical formfactor must be used.
For a small dielectric bead of radius a
For a small metal bead with radius a
A metal bead can be usedto measure the E field onlyif the bead is placed in aregion where the magneticfield is zero!
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Bead PullsBead Pulls
In general, the shift in frequency is proportionalto a form factor F
Dielectric bead
Metal bead
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Bead PullsBead Pulls
From the definition of cavity Q:
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Bead PullsBead Pulls
Since:
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Bead PullsBead Pulls
For small perturbations, shifts in the peak of thecavity response is hard to measure.
Shifts in the phase at the unperturbed resonantfrequency are much easier to measure.
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Bead PullsBead Pulls
Since:
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Bead PullsBead Pulls
Set to unperturbed resonantfrequency of the cavity
Phase Detector
Gap Detector
Bead
string