F AcceleratorDivision Introduction to RF for Particle Accelerators Part 2: RF Cavities Dave McGinnis

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


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


f AcceleratorDivision RF CavitiesRF Cavities

Beam

RF Input

Power Amplifier

Coupler

Gap

New FNAL Booster Cavity

Multi-cell superconducting RF cavity

Transmission Line Cavity


f AcceleratorDivision Cavity Field PatternCavity Field Pattern

For the fundamental mode at one instant in time:

Electric Field

Magnetic Field

Out

In

Wall Current


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


f AcceleratorDivision Cavity Boundary ConditionsCavity Boundary Conditions

Boundary Condition 1:

At x=0: V=0

xcosjVIZ

xsinVV

oo

o


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


f AcceleratorDivision Cavity ModesCavity Modes

n=0

n=1


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


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!


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


f AcceleratorDivision Gap CapacitanceGap Capacitance


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


At x=0: V=0 xcosjVIZ

xsinVV

oo

o

Lsin

LcosZC og


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


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


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


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:


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


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


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


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


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



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


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


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


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


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


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


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


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



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



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


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.


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



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



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) =



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



Re

Im

Under coupled

Critically coupled

Over coupled


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


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


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


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

QQ

Q

1

Q

1

Q

1

cpl

oL

extoL


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

)


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


f AcceleratorDivision 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

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



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



For << o and C << Ceq

T

E

o

2gapeq

2cro

eqo

W

W

VC21

dvE141

C

C

2

1



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!



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



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



Since:

2

gap o

gapgap

o

eq

gapgap

gapgap

dzz,y,x

2

1

F

1

Q

R

Vdzz,y,xE



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)



Since:

o

o

o

Q2

Qtan

2

gapgapgap

oeq dzz,y,xtan

2

1

2

1

F

1R



Set to unperturbed resonant frequency of the cavity

Phase Detector

Gap Detector

Bead

string

Documents

F AcceleratorDivision Introduction to RF for Particle Accelerators Part 2: RF Cavities Dave McGinnis