introduction to rf for particle accelerators part 2: rf

fAcceleratorDivision

Introduction to RF for Particle AcceleratorsIntroduction to RF for Particle AcceleratorsPart 2: RF CavitiesPart 2: RF Cavities

Dave McGinnis

Introduction to RF – Part 2 – Cavities - McGinnis 2

fAcceleratorDivision RF Cavity TopicsRF Cavity Topics

ModesSymmetryBoundariesDegeneracy

RLC modelCoupling

InductiveCapacitiveMeasuring

QUnloaded QLoaded QQ Measurements

Impedance MeasurementsPower Amplifiers

Class of operationTetrodesKlystrons

Beam LoadingDe-tuningFundamentalTransient


fAcceleratorDivision 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

∫∫∫ •∂∂

−=•SC

SdBt

ldErrrr

The integral

∫ •C

ldEqrr

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

0tB=

∂∂r

then 0ldEqC

=•∫rr


fAcceleratorDivision RF CavitiesRF Cavities

Beam

RF Input

Power Amplifier

Coupler

Gap

New FNAL Booster Cavity

Multi-cell superconducting RF cavity

Transmission Line Cavity


fAcceleratorDivision Cavity Field PatternCavity Field PatternFor the fundamental mode at one instant in time:

Electric Field

Magnetic Field

Out

In

Wall Current


fAcceleratorDivision Cavity ModesCavity ModesWe 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

eVeVVβ+−β−+

β+−β−+

−=

+=

where Zo is the characteristic impedance of the transmission line structure of the cavity


fAcceleratorDivision Cavity Boundary ConditionsCavity Boundary Conditions

Boundary Condition 1:

At x=0: V=0( )

( )xcosjVIZxsinVV

oo

oβ−=

β=


At x=L: I=0 ( )

( )

( )

( )4

1n2L

L4c1n2f

21n2L

0Lcos

n

n

n

λ+=

+=

π+=β

=β

...3,2,1,0n =Different values of n are called modes. The lowest value of n is usually called the fundamental mode


fAcceleratorDivision Cavity ModesCavity Modes

n=0

n=1


fAcceleratorDivision Even and Odd Mode DeEven and Odd Mode De--CompositionComposition

inV

inin V21V

21

+ inin V21V

21

−


fAcceleratorDivision Even and Odd Mode DeEven and Odd Mode De--CompositionComposition

inV21

inV21

inV21

inV21

−

Even Mode

No Gap Field

Odd Mode

Gap Field!


fAcceleratorDivision 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.

inV21

inV21

inV21

inV21

−


fAcceleratorDivision Gap CapacitanceGap Capacitance


At x=L: VCjI gω= where Cg is the gap capacitance


At x=0: V=0 ( )( )xcosjVIZxsinVV

oo

oβ−=

β=

( )( )Lsin

LcosZC og ββ

=ω


fAcceleratorDivision RF Cavity ModesRF Cavity ModesConsider the first mode only (n=0) and a very small gap capacitance.

( )( )

δπ

=ωω∆

ω−=δ

δ−≈ββ

δ+π

=β

2

ZCLsinLcos

2L

o

o

ogo

The gap capacitance shifts the odd mode down in frequency and leaves the even mode frequency unchanged


fAcceleratorDivision MultiMulti--Celled CavitiesCelled Cavities

Each cell has its own resonant frequencyFor n cells there will be ndegenerate modesThe cavity to cavity coupling splits these ndegenerate modes.The correct accelerating mode must be picked


fAcceleratorDivision 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

PWW

WWQ

+ω=

=


fAcceleratorDivision Transmission Line Cavity QTransmission Line Cavity QWe 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

ZV

8

dxxVC412

dvolE41W

ωπ

=

=

ε=

∫

∫∫∫r

( )

o

2o

o

L

0

2l

vol

2H

ZV

8

dxxIL412

dvolH41W

ωπ

=

=

µ=

∫

∫∫∫r

Magnetic Energy:

Both Halves:


fAcceleratorDivision Transmission Line Cavity QTransmission Line Cavity QAssume a small resistive loss per unit length rLΩ/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

ZVLr

21

dxxIr212P

=

= ∫

The cavity Q for the fundamental mode of the transmission line cavity is:

LrZ

2Q

l

oπ=

Time average

Less current flowing along walls

Less loss in walls


fAcceleratorDivision 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


fAcceleratorDivision RLC Parameters for a Transmission Line CavityRLC Parameters for a Transmission Line Cavity

For the fundamental mode of the transmission line cavity:

LrZ4R

RV

21P

l

2o

eq

eq

2gap

loss

=

=

o

oeq

eq2

o

2gap

H

Z8L

L

V41W

ωπ=

ω=

ooeq

2gapeqE

Z1

8C

VC41W

ωπ

=

=

The transfer impedance of the cavity is:

eqeqeqc

gen

gapc

CjLj1

R1

Z1

IV

Z

ω+ω

+=

=


fAcceleratorDivision Cavity Transfer ImpedanceCavity Transfer Impedance

Since:

eqeqo

CL1=

ω eq

eqeqCL

QR

=

eqeqo CRQ ω=

( )( ) 2

oo2

oeqc

Qjj

jQ

RjZ

ω+ω

ω+ω

ωω=ω

Function of geometry only

Function of geometry and cavity material


fAcceleratorDivision 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

Impe

danc

e

fo−

2 Q⋅

fo2 Q⋅

2 1.5 1 0.5 0 0.5 1 1.5 290

45

0

45

90

Frequency

Pha

se (d

egre

es)

fo−

2 Q⋅

fo2 Q⋅

Referenced to ωo

( )( ) 2

oo2

oeqc

Qjj

jQ

RjZ

ω+ω

ω+ω

ωω=ω

°=⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω±ω

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω±ω=

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω±ω

45Q2

Zarg

Q2ZRe

Q2ZIm

oo

oo

oo

m

m

Peak of the response is at

ωo

( )2o

2o

o Z21

Q2Z ω=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ω±ω


fAcceleratorDivision Mode Spectrum Example Mode Spectrum Example –– Pill Box CavityPill Box Cavity

The RLC model is only valid around a given modeEach mode will a different value of R,L, and C


fAcceleratorDivision Cavity Coupling Cavity Coupling –– Offset CouplingOffset 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 increaseTherefore the input impedance of the cavity as seen by the power amplifier decreases as the drive point is moved away from the gap


fAcceleratorDivision Cavity CouplingCavity Coupling

We can model moving the drive point as a transformerMoving the drive point away from the gap increases the transformer turn ratio (n)

cZpacZ

n:1

c2pac Zn1Z =

Igen

Vgap

Leq Req Ceq


fAcceleratorDivision 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 modeThe transformer ratio n = Total Flux / Coupler Flux

Out

In

Wall Current

Magnetic Field


fAcceleratorDivision 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 C1

C1

C1

+=

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


fAcceleratorDivision 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 InjectorThis is a bad approximation for klystrons protected with isolators

Every power amplifier has some internal resistance

0LIgen

gengen I

VR

=∆

∆=

genRgenI


fAcceleratorDivision 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


fAcceleratorDivision 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)

gen2eqL Rn

1R

1R1

+=

extoL Q1

Q1

Q1

+=

eqgen2

oext

eqeqoo

eqLoL

CRnQ

CRQ

CRQ

ω=

ω=

ω= Loaded Q

Unloaded Q

External Q


fAcceleratorDivision

VgapLeq 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 isolatorAs in the case of an infinitesimally short transmission line

ogen ZR =

n:1



Look at the cavity impedance from the power amplifier point of view:

Vgap/nLeq/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 thecavity 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


fAcceleratorDivision 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.


fAcceleratorDivision Measuring Cavity CouplingMeasuring Cavity Coupling

( )( ) 2

oo2

oeqc

Qjj

jQ

RjZ

ω+ω

ω+ω

ωω=ω

The frequency response of the cavity at a given mode is:

which can be re-written as:

( ) ( ) φφ=ω jeqc ecosRjZ

( )ωωω−ω

=φo

22oQtan



The reflection coefficient as seen by the power amplifier is:

o2

c

o2

cZnZZnZ

+

−=Γ

( )( ) 1ecosr

1ecosrj

cpl

jcpl

+φ

−φ=Γ

φ

φ

This equation traces out a circle on the reflection (u,v) plane



1 0.5 0 0.5 1

1

0.5

0.5

1

ΓRe

ΓIm

1rr

cpl

cpl+

1r1

cpl +

1r1r

cpl

cpl+

−

1rr

cpl

cpl+

Radius =

⎟⎟⎠

⎞⎜⎜⎝

⎛

+− 0,

1r1

cplCenter =

( )0,1−Left edge (φ=+/-π/2) =

⎟⎟⎠

⎞⎜⎜⎝

⎛

+

−0,

1r1r

cpl

cplRight edge (φ=0) =



The cavity coupling can be determined by:

measuring the reflection coefficient trajectory of the input couplerReading the normalized impedance of the extreme right point of the trajectory directly from the Smith Chart

ΓRe

ΓIm

1r1r

cpl

cpl+

−

1 0.5 0 0.5 1

1

0.5

0.5

1



ΓRe

ΓIm

Under coupled

Critically coupled

Over coupled


fAcceleratorDivision Measuring the Measuring the LoadedLoaded Q of the CavityQ 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 QAlso 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 2

MagnitudeRealImaginary

Frequency

fo−

2 Q⋅

fo2 Q⋅S21


fAcceleratorDivision Measuring the Measuring the UnloadedUnloaded Q of a CavityQ 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

±=

π±=φ

ωω=ω m

Unloaded Q!

For:

( ) 21vu

ZReZIm22

cc

=±+

±=jvu +=Γ

where:

Circles on the Smith Chart


fAcceleratorDivision Measuring the Measuring the UnloadedUnloaded Q of a CavityQ of a Cavity

Measure frequency (ω-) when:

Measure resonant frequency (ωo)

Measure frequency (ω+) when:

Compute

cc ZReZIm =

cc ZReZIm −=

cc ZReZIm =

cc ZReZIm =

−+ ω−ωω

= ooQ


fAcceleratorDivision Measuring the Measuring the LoadedLoaded Q of a CavityQ of a Cavity

Measure the coupling parameter (rcpl)Measure the unloaded Q (Qo)

eqo2

oext

eqocpl2

oo

ocpl2

eq

CZnQ

CZrnQ

ZrnR

ω=

ω=

=

VgapLeq Req Ceq

Rgen

Igen

Zo

n:1

1rQQ

Q1

Q1

Q1

cpl

oL

extoL

+=

+=


fAcceleratorDivision NEVER NEVER –– EVER!!EVER!!

2 1.5 1 0.5 0 0.5 1 1.5 240

30

20

10

0

r=0.5r=1r=2

Frequency

Ref

lect

ion

Coe

ffici

ent (

dB)


fAcceleratorDivision 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


fAcceleratorDivision Bead Pull SetupBead Pull Setup

Set to unperturbed resonant frequency of the cavity Phase Detector

Gap Detector

Bead

string



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

dvE41dvE

41VC

41W 2

cor2

co2

gapeqE εε+ε−=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

εo is the permittivity of free space

εr is the relative permittivity of the cube

dvE41W

2

volE ∫∫∫ε=

r



The equivalent capacitance of the capacitor with the dielectric cube is:

( ) 2gapeq

2gapE VCC

41CV

41W ∆+==

( )2

gap

cro V

Edv1C ⎟⎟⎠

⎞⎜⎜⎝

⎛−εε=∆

The resonant frequency of the cavity will shift

( ) ( )CCL1

eqeq

2o ∆+

=ω∆+ω



For ∆ω << ωo and ∆C << Ceq

( )

T

E

o

2gapeq

2cro

eqo

WW

VC21

dvE141

CC

21

∆=

ωω∆

−εε=

∆=

ωω∆



Had we used a metallic bead (µr>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 WWW ∆−∆

=ωω∆

T

2b

r

ro

3

o WE

21a ⎟⎟⎠

⎞⎜⎜⎝

⎛+ε−ε

επ−=ωω∆

⎥⎦⎤

⎢⎣⎡ µ

+επ

−=ωω∆ 2

bo2

boT

3

oH

2E

Wa

A 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!



In general, the shift in frequency is proportional to a form factor F

T

2b

o WEF−=

ωω∆

o3

r

ro

3

aF

21aF

επ=

⎟⎟⎠

⎞⎜⎜⎝

⎛+ε−ε

επ= Dielectric bead

Metal bead



From the definition of cavity Q:

( ) ( )oeqo

2

gap

b

eq

2gap

oT

eq

2gap

L

L

To

z,y,x

QR

12

1F1

Vz,y,xE

QRV

21W

RV

21P

PWQ

ωω∆

ω=⎟

⎟⎠

⎞⎜⎜⎝

⎛

ω=

=

ω=



Since:

( )

( ) 2

gap o

gapgap

o

eq

gapgap

gapgap

dzz,y,x

21

F1

QR

Vdzz,y,xE

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

ω

ω∆

ω=

=

∫

∫



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

Impe

danc

e

2 1.5 1 0.5 0 0.5 1 1.5 290

45

0

45

90

UnperturbedShifted

Frequency

Pha

se (d

egre

es)



Since:

( )

o

o

o

Q2

Qtan

ωω∆

≈

⎟⎟⎠

⎞⎜⎜⎝

⎛ωω

−ωω

=φ

( )( )2

gapgapgap

oeq dzz,y,xtan

21

21

F1R

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡φ

ω= ∫



Set to unperturbed resonant frequency of the cavity Phase Detector

Gap Detector

Bead

string


fAcceleratorDivision Power Amplifiers Power Amplifiers -- TriodeTriode

The triode is in itself a miniature electron acceleratorThe filament boils electrons off the cathodeThe electrons are accelerated by the DC power supply to the anodeThe voltage on the grid controls how many electrons make it to the anodeThe number of electrons flowing into the anode determines the current into the load.The triode can be thought of a voltage controlled current sourceThe maximum frequency is inversely proportional to the transit time of electrons from the cathode to the anode.

Tetrodes are typically used at frequencies below 300 MHz


fAcceleratorDivision TetrodesTetrodes

Courtesy of Tim Berenc


fAcceleratorDivision KlystronsKlystrons

The filament boils electrons off the cathodeThe velocity (or energy) of the electrons is modulated by the input RF in the first cavityThe electrons drift to the cathodeBecause of the velocity modulation, some electrons are slowed down, some are sped up.If the output cavity is placed at the right place, the electrons will bunch up at the output cavity which will create a high intensity RF field in the output cavityKlystrons need a minimum of two cavities but can have more for larger gain.A Klystron size is determined by the size of the bunching cavities.

Klystrons are used at high frequencies (>500 MHz))


fAcceleratorDivision KlystronsKlystrons


fAcceleratorDivision Traveling Wave TubeTraveling Wave Tube

Cutaway view of a TWT. (1) Electron gun; (2) RF input; (3) Magnets; (4) Attenuator; (5) Helix coil; (6) RF output; (7) Vacuum tube; (8) Collector.


fAcceleratorDivision Traveling Wave TubesTraveling Wave Tubes

Traveling wave tubes (TWTs) can have bandwidths as large as an octave (fmax = 2 x fmin)TWTs have a helix which wraps around an electron beam

The helix is a slow wave electromagnetic structure.The phase velocity of the slow wave matches the velocity of the electron beam

At the input, the RF modulates the electron beam.The beam in turn strengthens the RFSince the velocities are matched, this process happens all along the TWT resulting in a large amplification at the output (40dB = 10000 x)


fAcceleratorDivision Power Amplifier BiasPower Amplifier Bias

The power amplifier converts DC energy into RF energy.With no RF input into the amplifier, the Power amplifier sits at its DC bias.The DC bias point is calculated from the intersection of the tube characteristics with the outside load line

Anode Volts

Ano

de C

urre

nt

Ibias

Vbias

oAAoA

A IR

VVI +

−=


fAcceleratorDivision Class A BiasClass A Bias

In Class A, the tube current is on all the time – even when there is no input.The tube must dissipate

The most efficient the power amplifier can be is 50%

acAacAdcAdcAdis V

21VIP −=

Anode Volts

Ano

de C

urre

nt

Ibias

Vbias

Time

Ano

de C

urre

nt

Anode Volts

Tim

e


fAcceleratorDivision Class B BiasClass B Bias

In Class B, the tube current is on ½ of the timeThe tube dissipates no power when the there is no driveThe output signal harmonics which must be filteredThe efficiency is much higher (>70%)

Anode Volts

Ano

de C

urre

nt

Ibias

Vbias

Time

Ano

de C

urre

nt

Anode Volts

Tim

e

introduction to rf for particle accelerators part 2: rf

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