rf cavity design - cern...cas daresbury '07 — rf cavity design 4 lorentz force a charged...

CAS Daresbury '07 — RF Cavity Design 1

RF Cavity Design

Erk Jensen

CERN AB/RF

CERN Accelerator SchoolDaresbury 2007


Overview• DC versus RF

– Basic equations: Lorentz & Maxwell, RF breakdown• Some theory: from waveguide to pillbox

– rectangular waveguide, waveguide dispersion, standing waves … waveguide resonators, round waveguides, Pillbox cavity

• Accelerating gap– Induction cell, ferrite cavity, drift tube linac, transit time factor

• Characterizing a cavity– resonance frequency, shunt impedance, – beam loading, loss factor, RF to beam efficiency,– transverse effects, Panofsky-Wenzel, higher order modes, PS 80 MHz cavity

(magnetic coupling)• More examples of cavities

– PEP II, LEP cavities, PS 40 MHz cavity (electric coupling), • RF Power sources• Many gaps

– Why?– Example: side coupled linac, LIBO

• Travelling wave structures– Brillouin diagram, iris loaded structure, waveguide coupling

• Superconducting Accelerators• RFQ’s


DC accelerator

RF accelerator

DC versus RF

potential ΔΦ−=⋅=Δ ∫ qsEqW rr

d


Lorentz forceA charged particle moving with velocity through an electro-magnetic field experiences a force

( )BvEqtp rrrr

×+=dd

vr

The energy of the particle is ( ) ( ) 2222 mcpcmcW γ=+=

Note: no work is done by the magnetic field.

Change of W due to the this force (work done) ; differentiate: ( ) tEpcqtBvEpcqppcWW dddd 222 rrrrrrrr ⋅=×+⋅=⋅=

tEvqW ddrr ⋅=

( )12 −= γmcW kin

γmpvr

r =


Maxwell’s equations (in vacuum)

ρμ

μ

20

02

0

01

cEBt

E

BJEtc

B

=⋅∇=∂∂

+×∇

=⋅∇=∂∂

−×∇

rvr

rrrr

DC ( ): which is solved by0=×∇ Er

Φ−∇=Er

Limit: If you want to gain 1 MeV, you need a potential of 1 MV!

Circular machine: DC acceleration impossible since 0d =⋅∫ sE rr

0≡∂∂t

why not DC?

Bt

Evr

∂∂

−=×∇

With time-varying fields:

∫∫∫ ⋅∂∂

−=⋅ AtBsE

rr

rrdd

1)

2)

00

0012

=⋅∇=∂∂

+×∇

=⋅∇=∂∂

−×∇

EBt

E

BEtc

Brvr

rrr

(source-free)


Maxwell’s equation in vacuum (contd.)

012

2

2 =∂∂

+×∇×∇ Etc

Err

EEErrr

Δ−⋅∇∇=×∇×∇

012

2

2 =∂∂

−Δ Etc

Err

vector identity:

curl of 3rd and of 1st equation:t∂

∂

with 4th equation :

i.e. Laplace in 4 dimensions

00

0012

=⋅∇=∂∂

+×∇

=⋅∇=∂∂

−×∇

EBt

E

BEtc

Brvr

rrr


Another reason for RF: breakdown limit in vacuum,Cu surface,room temperature

f in GHz, Ec in MV/m

cEc eEf

25.4

67.24−

=


Some theory: from waveguide to pillbox

ck ω

=

2

1 ⎟⎠⎞

⎜⎝⎛−=

ωωω c

z ck

z

x

Polarization: Ey

ck cω

=⊥

frequency domain: ωj≡∂∂t


2 superimposed plane waves

Er


Waveguides

Fundamental (TE10 or H10) modein a standard rectangular waveguide.E.g. forward wave

electric field

magnetic field

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⋅×∫∫sectioncross

* dRe21 AHE

rrrpower flow:

z

z

-y

power flowx

x


Waveguide dispersion

e.g.: TE10-wave in rectangular waveguide:

a

Z

ac

2

j

j

cutoff

0

22

=

=

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

λγωμ

πωγ

ωcω

{ }g

zkλπγ 2Im ==

TMfor j

TE,for j

j

00

22

ωεγ

γωμ

ωγ

==

−⎟⎠⎞

⎜⎝⎛= ⊥

ZZ

kc

general cylindrical waveguide:

In a hollow waveguide: phase velocity > c, group velocity < c

free space, ω/c

“slow” wave

“fast” wave


Waveguide dispersion (continued)

{ }γIm=zk

ω

cω

free space, ω/c

TE10

TE10

TE01

TE20


General waveguide equations:

02

=⎟⎠⎞

⎜⎝⎛+Δ T

cT cωTransverse wave equation (membrane equation):

TM (or E) modesTE (or H) modes

boundary condition:

longitudinal wave equations (transmission line equations):

0

propagation constant:

characteristic impedance:

ortho-normal eigenvectors:

transverse fields:

longitudinal field:

=∇⋅ Tnr 0=T( ) ( )

( ) ( ) 0d

d

0d

d

0

0

=+

=+

zUZz

zI

zIZzzU

γ

γ

γωμj

0 =Zωεγ

j0 =Z

Tue z ∇×= rr Te −∇=r

( )( ) euzIH

ezUE

zrrr

rr

×==

( )ωμ

ωj

2 zUTc

H cz ⎟

⎠⎞

⎜⎝⎛= ( )

ωεω

j

2 zITc

E cz ⎟

⎠⎞

⎜⎝⎛=

2

1j ⎟⎠⎞

⎜⎝⎛−=

ωωωγ c

c


Rectangular waveguide : transverse eigenfunctions

TM (E) modes:

TE (H) modes:( ) ( ) ⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

+= y

bnx

am

nambabT nmH

mnππεε

πcoscos1

22)(

( ) ( )⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

+= y

bnx

am

nambabT E

mnππ

πsinsin2

22)(

⎩⎨⎧

≠=

=0201

iforifor

iε

( )( )( )⎭

⎬⎫

⎩⎨⎧⎟

⎠⎞

⎜⎝⎛

=− ϕ

ϕχχ

ρχ

πε

mmaT

mnmmn

mnmmE

mn cossin

J

J

1

)(

( ) ( )( )( )⎭

⎬⎫

⎩⎨⎧

′

⎟⎠⎞

⎜⎝⎛ ′

−′=

ϕϕ

χ

ρχ

χπε

mma

mT

mnm

mnm

mn

mHmn sin

cosJ

J

22)(

Round waveguide : transverse eigenfunctions

TE (H) modes:

TM (E) modes:

where

Ø = 2a

ab

22

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

bn

am

cc ππω

acmnc χω

=


Standing wave – resonator

Same as above, but twocounter-running waves of identical amplitude.

electric field

magnetic field(90º out of phase)

0dRe21

sectioncross

* =⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧⋅×∫∫ AHE

rrrno net power flow:


TE11: fundamental mode

mm/85.87

GHz afc =

mm/74.114

GHz afc =

TE01: lowest losses!

mm/74.334

GHz afc =

Round waveguide

TM01: axial electric field

Er

Br

parameters used in calculation: f = 1.43, 1.09, 1.13 fc , a: radius


Pillbox cavity

electric field magnetic field

(only 1/8 shown)TM010-mode


Pillbox cavity field (w/o beam tube)

( )⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

a

aT01

101

010

J

J1,

χχ

ρχ

πϕρ ...40483.201 =χ

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

aa

aB

aa

aa

Ez

011

011

0

011

010

01

0

J

J1

J

J1

j1

χ

ρχ

πμ

χ

ρχ

πχ

ωε

ϕ

ac

pillbox01

0χω =

( ) ahah

QR

pillbox

)2

(sin

J4

012

0121

301

χ

χπχη

=

⎟⎠⎞

⎜⎝⎛ +

=

ha

aQ pillbox

12

2 01ησχ

The only non-vanishing field components :

h

Ø 2a

Ω== 3770

0

εμη

for later:


dielectric guide – transversely damped wave

There is another solution to :2220

2

zkkkc

+==⎟⎠⎞

⎜⎝⎛

⊥ω

20

2 kkjk z −±=⊥

0εε >

εμω 0=kreal :1⊥k

zk

zkk <= 000 εμω

amplitude of Ez

symmetry plane


Accelerating gap


Accelerating gap

gap voltage

• We want a voltage across the gap!

• The limit can be extended with a material which acts as “open circuit”!

• Materials typically used:– ferrites (depending on f-range)– magnetic alloys (MA) like Metglas®, Finemet®,

Vitrovac®…

• resonantly driven with RF (ferrite loaded cavities) – or with pulses (induction cell)

∫∫∫ ⋅−=⋅ AtBsE

rr

rrd

ddd

• It cannot be DC, since we want the beam tube on ground potential.

• Use

• The “shield” imposes a– upper limit of the voltage pulse duration or –

equivalently –– a lower limit to the usable frequency.


Linear induction accelerator

∫∫∫ ⋅∂∂

−=⋅ AtBsE

rr

rrdd

compare: transformer, secondary = beam

Acc. voltage during B

ramp.


Ferrite cavity

PS Booster, ‘980.6 – 1.8 MHz,< 10 kV gapNiZn ferrites


Gap of PS cavity (prototype)


Drift Tube Linac (DTL) – how it works

For slow particles –protons @ few MeV e.g. – the drift tube lengthscan easily be adapted.

electric field


Drift tube linac – practical implementations


Transit time factor

If the gap is small, the voltage is small.∫ zEzd

∫

∫zE

zeE

z

zc

z

d

dj ω

If the gap large, the RF field varies notably while the particle passes.

Define the accelerating voltage ∫= zeEVz

czgap d

jω

Example pillbox:transit time factor vs. h

h/λ

Transit time factor

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

ah

ah

22sin 0101 χχ


Characterizing a cavity


Cavity resonator – equivalent circuit

RR/β

Cavity

Generator

IG

P

Vgap

C=Q/(Rω0)

Beam

IB

L=R/(Qω0)LC

β: coupling factor

R: Shunt impedance CL : R-upon-Q

Simplification: single mode


Resonance


Reentrant cavity

Example: KEK photon factory 500 MHz - R as good as it gets -

this cavity optimizedpillbox

R/Q: 111 Ω 107.5 ΩQ: 44270 41630R: 4.9 MΩ 4.47 MΩ

nose cone

Nose cones increase transit time factor, round outer shape minimizes losses.


Loss factor

0 5 10 15 20

t f0

-1

0

1

qkV

loss

gap

2

Voltage induced by a single charge q: t

QLe 20ω

−

RR/β

Cavity

Beam

C=Q/(Rω0)

V (induced)IB

L=R/(Qω0)LCEnergy deposited by a single charge q: 2qkloss

CWV

QRk gap

loss 21

42

2

0 ===ω

Impedance seen by the beam


Summary: relations Vgap, W, Ploss

Energy stored inside the cavity Power lost in the cavity

walls

gap voltage

loss

gapshunt P

VR

2

2

=

lossPWQ 0ω

=

WV

QR gap

0

2

2ω=

WV

QRk gap

loss 42

2

0 ==ω


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.

• The power absorbed by the beam is ,

the power loss .

• For high efficiency, beam loading shall be high. • The RF to beam efficiency is .

{ }*Re21

Bgap IV−

RV

P gap

2

2

=

G

B

gap II

IRV =

+=

1

1

B

η


Characterizing cavities• Resonance frequency

• Transit time factorfield varies while particle is traversing the gap

• Shunt impedancegap voltage – power relation

• Q factor

• R/Qindependent of losses – only geometry!

• loss factor

lossshuntgap PRV 22

=

lossPQW =0ω

CL

WV

QR gap ==

0

2

2ω

CL ⋅=

10ω

WV

QRk gap

loss 42

2

0 ==ω

Linac definition

lossshuntgap PRV =2

WV

QR gap

0

2

ω=

WV

QRk gap

loss 44

2

0 ==ω

Circuit definition∫

∫zE

zeE

z

zc

z

d

dj ω


Example Pillbox:

ac

pillbox01

0χω =

( ) ahah

QR

pillbox

)2

(sin

J4

012

0121

301

χ

χπχη

=

⎟⎠⎞

⎜⎝⎛ +

=

ha

aQ pillbox

12

2 01ησχ Ω== 3770

0

εμη

4048.201 =χ

S/m108.5 7Cu ⋅=σ


Higher order modes

IB

R3, Q3,ω3R1, Q1,ω1 R2, Q2,ω2

......

external dampers

n1 n3n2


Higher order modes (measured spectrum)

without dampers

with dampers


Pillbox: Dipole mode

electric field (@ 0º) magnetic field (@ 90º)

(only 1/8 shown)(TM110)


Panofsky-Wenzel theorem

||j FFc ⊥⊥ ∇=rω

For particles moving virtually at v=c, the integrated transverse force (kick) can be determined from the transverse variation of the integrated longitudinal force!

W.K.H. Panofsky, W.A. Wenzel: “Some Considerations Concerning the Transverse Deflection of Charged Particles in Radio-Frequency Fields”, RSI 27, 1957]

Pure TE modes: No net transverse force !

Transverse modes are characterized by• the transverse impedance in ω-domain• the transverse loss factor (kick factor) in t-domain !


CERN/PS 80 MHz cavity (for LHC)

inductive (loop) coupling, low self-inductance


Example shown:80 MHz cavity PS for LHC.Color-coded:

Higher order modes

Er


More examples of cavities


PS 19 MHz cavity (prototype, photo: 1966)


Examples of cavities

PEP II cavity476 MHz, single cell,

1 MV gap with 150 kW, strong HOM damping,

LEP normal-conducting Cu RF cavities,350 MHz. 5 cell standing wave + spherical cavity for energy storage, 3 MV


example for capacitive coupling

cavity

coupling C

CERN/PS 40 MHz cavity (for LHC)


RF power sources


RF Power sources

Thales TH1801, Multi-Beam Klystron (MBK), 1.3 GHz, 117 kV. Achieved: 48 dB gain, 10 MW peak, 150 kW average, η = 65 %

> 200 MHz: Klystrons

Tetrode IOT UHF Diacrode

< 1000 MHz: grid tubes

pictures from http://www.thales-electrondevices.com

dB: 8.410=rinput poweeroutput pow


RF power sources

Typical ranges (commercially available)

0.1

1

10

100

1000

10000

10 100 1000 10000f [MHz]

CW

/Ave

rage

pow

er [k

W]

Transistors

solid state (x32)

grid tubes

klystrons

IOT CCTWTs


Example of a tetrode amplifier (80 MHz, CERN/PS)

22 kV DC anode voltage feed-through with λ/4 stub

18 Ω coaxial output (towards cavity)

coaxial input matching circuit

tetrode cooling water feed-throughs

400 kW, with fast RF feedback


Many gaps


What do you gain with many gaps?

( )PnRnPRnVacc 22 ==

• The R/Q of a single gap cavity is limited to some 100 Ω.Now consider to distribute the available power to nidentical 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 .

nPR2

nR

1 2 3 n

P/nP/n P/nP/n


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!


A 3 GHz Side Coupled Structure to accelerate protons out of cyclotrons from 62 MeV to 200 MeV

Medical application:treatment of tumours.

Prototype of Module 1built at CERN (2000)

Collaboration CERN/INFN/Tera Foundation

An example of Side Coupled Structure :LIBO (= Linac Booster)


LIBO prototype

This Picture made it to the title page of CERN Courier vol. 41 No. 1 (Jan./Feb. 2001)


Travelling wave structures


Brillouin diagram Travelling wave

structure

synchronous

ω L/c

speed of light line, ω = β /c

π

π/2

ππ/2 β L0

2π

0

π/2

π

2π/3


Iris loaded waveguide

1 cm

30 GHz structure (CLIC)

11.4 GHz structure (NLC)


Disc loaded structure with strong HOM damping“choke mode cavity”

Dimensions in mm


Waveguide coupling ¼ geometry shownInput coupler

Output coupler

Travelling wave structure(CTF3 drive beam, 3 GHz)

shown: Re {Poynting vector} (power density)


3 GHz Accelerating structure (CTF3)


Superconducting Linacs


LEP Superconducting cavities

10.2 MV/ per cavity

LEP was not a linac, but still …


LHC SC RF, 4 cavity module, 400 MHz


Small β superconducting cavities (example RIA, Argonne)

345 MHz β = 0.4 spoke cavity

pictures from Shepard et al.: “Superconducting accelerating structures for a multi-beam driver linac for RIA”, Linac 2000, Monterey

115 MHz split-ring cavity, 172.5 MHz β = 0.19 “lollipop” cavity

β = 0.021 fork cavity

57.5 MHz cavities:

β = 0.03 fork cavity

β= 0.06 QWR(quarter wave resonator)


More superconducting cavities for linacs with β < 1 (proton driver – heavy ion)

Need to standardise construction of cavities:only few different types of cavities are made for some β’smore cavities are grouped in cryostats

Example:CERN design, SC linac 120 - 2200 MeV


ILC high gradient SC Linac at 1.3 GHzILC 9-cell Niobium cavity

More on the ILC at

Technology has made big progress, > 40 MV/m accelerating gradient have been obtained.The plot below illustrates the effect of “buffered chemical polishing” and “electro-polishing”.

http://www.linearcollider.org/cms/

http://www.linearcollider.org/cms/


RFQ’s


Old preinjector 750 kV DC , CERN Linac 2 before 1990

All this was replaced by the RFQ …


RFQ of CERN Linac 2


The Radio Frequency Quadrupole (RFQ)

Minimum Energy of a DTL: 500 keV (low duty) - 5 MeV (high duty)At low energy / high current we need strong focalisationMagnetic focusing (proportional to β) is inefficient at low energy. Solution (Kapchinski, 70’s, first realised at LANL):

Electric quadrupole focusing + bunching + acceleration


RFQ electrode modulation

The electrode modulation creates a longitudinal field component that creates the“bunches” and accelerates the beam.


A look inside CERN AD’s “RFQD”

rf cavity design - cern...cas daresbury '07 — rf cavity design 4 lorentz force a charged...

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