high power rf - cern

RF Power Transportation, S. Choroba, DESY, CERN School on RF for Accelerators, 8-17 June 2010, Ebeltoft, Danmark

Stefan Choroba, DESY, Hamburg, Germany

RF Power Transportation

Part I


OverviewPart I• Introduction• Theory of Electromagnetic Waves in Waveguides• TE10-Mode

Part II• Reminder• Waveguide Elements • Waveguide Distributions• Limitations, Problems and Countermeasures


Introduction


RF Power Transportation

RF Generator(s)

Load(s)

•Purpose: Transmission of RF power of several kW up to several MW at frequencies from the MHz to GHz range. The RF power generated by an RF generator or a number of RF generators must be combined, transported and distributed to a load or cavity or a number of loads or cavities.

•Requirements: low loss, high efficieny, low reflections, high reliability, adjustment of phase and amplitude, ….


Transmission Lines for RF Transportation

• Two-wire lines (Lecher Leitung)– often used for indoor

antenna (e.g. radio or TV)– problem: radiation to the

environment, can not be used for high power transportation

• Strip-lines – often used for microwave

integrated circuits – problem: radiation to the

environment and limited power capability, can not be used for high power transportation

conductor

conductor

conductor

conductor

dielectric


Transmission Lines for RF Transportation (2)

• Coaxial transmission lines – often used for power RF

transmission and connection of RF components

– problem: high loss above a certain frequency due to heating of inner conductor and dielectric material and limited power capability at higher frequencies due to small dimensions

• Waveguides (rectangular,cylindrical or elliptical) – often used for high power

RF transmission (mostly rectangular)

– problem: waveguide plumbing, rigidity

conductor

conductor

conductor

dielectric, gas or vacuum

dielectric, gas or vacuum


Theory of Electromagnetic Waves in Waveguides


Strategy for Calculation of Fields in Lines for Power Transportation

• start with Maxwell equation• derive wave equation• Ansatz: separation into transversal and longitudinal field

components• wave equation for transversal and longitudinal

components • rewrite Maxwell equation in transversal and longitudinal

components• solve eigenvalue problem for three cases

TEM (Ez=Hz =0), TE (Ez=0, Hz≠0), TM (Hz=0, Ez≠0)• derive properties of the the solutions


Maxwell Equations

00

gets one ty)permittivi ( and ty)permeabili ( current), external (no 0charges), external (no 0with

field electric nt,displaceme electric intensity, magnetic field, magnetic

0 law Gauss

law Faradays

law Amperes

=⋅∇=⋅∇

∂∂−=×∇

∂∂=×∇

====

=⋅∇=⋅∇

∂∂−=×∇

+∂∂=×∇

HE

HE

EH

EDHBj

EDHB

BD

BE

jDH

t

t

t

t

µ

ε

εεµµρ

ρ


Wave Equation

( )

( )

HHH

EEE

E

EEE

EH

HE

HE

for equation wave0

Hfor way same in the

for equation ve wa0

0 use

and of use

curlapply and start with

equations. Maxwell thefrom derived becan equation waveThe

222

222

222

2

=∂

∂−∇

=∂

∂−∇⇒

=⋅∇∂

∂−=∇−⋅∇∇⇒

∇−⋅∇∇=∇××∇∂∂=×∇

∂∂×∇−=×∇×∇⇒

∂∂−=×∇

t

t

t

t

t

t

µε

µε

µε

ε

µ

µ


Ansatz for Wave Equation

s.coordinate rsalon transveonly dependbut they components al transversand allongitudin with vectorsstill are ),(),,(

).r,(or y x,e.g. only, scoordinate rsalon transve operates :

0)( and :with

0:in results

)(exp),(),,,()(exp),(),,,(

222

22

22

time.and coordinate allongitudinon dependingright the tomoving wavea andonly scoordinate rsalon transve dependingfunction ain fields of Separation

yxyx

Notek

ik

ztiyxtzyxztiyxtzyx

t

t

HE

EE

EE

HHEE

Φ∇

=++∇

==

=+∇

−=−=

γ

βγµεω

µεω

βωβω

xy

z


Derivation of Maxwell Equation for transversal and longitudinal Components

( ) ( ) ( )

tztztz

zttz

ztt

zz

zzt

ztz

ztt

ztzzztzt

ztzzzttztt

ztztzt

iEii

i

EE

iiii

iii

HeEeEEe

HE

EeeEeEeeE

eeE

HHEeEEeE

HHEEeEHE

ωµββ

ωµ

ωµωµββ

ωµβωµ

−=∇×−×−=×∇+×−

−=×∇

=×⊥×∇⊥×

×∇

∇×−=×∇=×∇

−−=×−×∇+×−×∇

+−=+×−∇=×∇−=×∇

components al transversand allongitudin of Separation0 ||

seecan one Now

With


Derivation Maxwell Equation for transversal and longitudinal Components (2)

ztt

ztt

tztztz

ztt

Ei

Hi

iHii

i

β

β

ωεβωε

ωε

=⋅∇=⋅∇

=⋅∇=⋅∇

−=∇×+×=×∇

=×∇

EE

HH

EeHeEH

EH

0With

0With

manner same in the With


Equations of transversal and longitudinal Components as Function of transversal

Coordinates

ztt

ztt

tztztz

ztt

tztztz

ztt

t

t

EiHi

iHii

iEii

kk

ββ

ωεβωε

ωµβωµ

γ

γ

=⋅∇=⋅∇

−=∇×+×=×∇

−=∇×−×−−=×∇

=++∇

=++∇

EH

EeHeEH

HeEeHE

HHEE

0)(

0)(222

222

Wave equation

Maxwell equation for

transversal and longitudinal components


TEM-, TE-, TM- Waves

wave)-E(or TM 0 and 0 3.wave)-H(or TE 0 and 0 2.

TEM 0 1.

≠=≠=

==

zz

zz

zz

EHHE

HE

On the next slides we will try to find solutions for TE-waves. The treatment for TM- modes is similar. For TEM modes the treatment is even easier, but TEM-modes do not exist in hollow transmission lines, because transversal E components require longitudinal H components and transversal H components require longitudinal E components. These are 0 in TEM fields. TEM-modes exist in coaxial lines since on the inner conductor we might have j≠0.


Derivation of TE-Wave Equations

0

0

equations Maxwellin 0With

0

0components allongitudin and ersalfor transvor

0

and 0)(equation eWith wav

22

22

22

222222

=⋅∇=⋅∇

−=∇×+×=×∇

−=×−−=×∇=

=+∇

=+∇

=+∇

=+=++∇

tt

ztt

tztztz

tt

ttz

ztt

z

zczt

tctt

ct

ct

HiiHi

iii

E

HkHk

kkkk

EH

EeHeH

HEeHE

HH

HHHH

βωεβ

ωµβωµ

γγ


Derivation of TE-Wave Equations(2)( )

( )

( )

( ) ( ) ( )( )

too. from thereforeand from and from calculated becan

Now

0

withand0

0 and With

0 gives 0on curl ofn Applicatio

2

2

2

222

zttzt

tzFtztzt

tztzzttzztzz

ztc

t

tczt

ztt

tcttt

tcttttttt

ttttt

HH

kZ

Hki

kHiHi

kk

HEH

HeHeHeE

HeEeeEEeeEee

H

HH

HHHH

HH

×−=×−=×−=

×=−=⋅−⋅=××

∇−=

=+∇

=⋅∇=+⋅∇∇⇒

=+∇∇−⋅∇∇=×∇×∇

=×∇×∇=×∇

ββωµε

εµ

βωµ

ωµβββ

ββ

β


Derivation of TE-Wave Equations(3)Impedance of a TE-Wave

right the tomoving wave-TEfor

wave)-(H wave-TE a of impedance

space freein waveneticelectromagan of impedance

x

y

y

xTE

FTE

F

HE

HEZ

kZZ

Z

−==

=

=

β

εµ


TE-Wave Equation in rectangular Waveguides

zc

TEy

zc

TEx

zc

y

zc

x

zczz

tzTEt

ztc

t

zczt

HxkiZE

HykiZE

HykiH

HxkiH

HkHyHx

Z

Hki

HkH

∂∂=

∂∂−=

∂∂−=

∂∂−=

=+∂

∂+∂

∂

×−=

∇−=

=+∇

2

2

2

2

22

22

2

2

22

problem Eigenvalue 0

0

β

β

β

β

β

HeE

H TE wave equation

TE wave equationwritten in components


Solution of TE- Wave Equation

222

22

2

22

2

22

22

2

22

22

2

1

1

011 and with

s.coordinatey and x into Separation :Ansatz

0

cyx

y

x

c

z

zczz

kkk

kgdyd

g

kfdxd

f

kgdyd

gfdx

df

g(y)gf(x)fgfH

HkHyHx

=+

−=

−=

=++

==⋅=

=+∂

∂+∂

∂


Solution of TE-Wave Equation(2)

byyHy

axxHx

byyHaxxH

ykBykBgxkAxkAf

z

z

y

x

yy

xx

===∂∂

===∂∂⇒

=======

+=+=

⇒

⊥

and 0for 0

and 0for 0

and 0for 0 and 0for 0

surfaceon 0H:H ofcomponent Normal :conditionsBoundary

sincossincos

21

21

b

x

y

z

a



0mnbut 0,1,2,..m 0,1,2..n with coscos),(

,....2,1,0

from

0 from 0manner same in the ,....2,1,0

from

0 from 0

and 0for 0cossin

2

2

21

≠====

=

==

==

=

==

==⇒

===+−

bym

axnHyxH

m

byb

mk

yB

n

axa

nk

xA

axxxkkAxkkA

nmz

y

x

xxxx

ππ

π

π



( )

decay lexponentiaimaginary is for

npropagatioreal is for

expi

with

number waveoffcut

222

222

222

222

2/122

⇒<

⇒>

−=

−=

−=

+=

+=

+

=

nmcnm

nmcnm

cnmnm

nmcnm

nmcnm

c

cnm

kkkk

kkkk

zti

kkkk

bm

ank

β

β

β

β

βωβγ

γ

γ

ππ


TEnm -Fields

)sin(sincos),,,(

)sin(cossin),,,(

)cos(coscos),,,(

)sin(cossin),,,(

)sin(sincos),,,(

0),,,(

2

2

,2

,2

ztb

yma

xnbk

mtzyx

ztb

yma

xnak

ntzyx

ztb

yma

xntzyx

ztb

yma

xnZak

ntzyx

ztb

yma

xnZbk

mtzyx

tzyx

nmnmcnm

nmy

nmnmcnm

nmx

nmnmz

nmnmnmTEcnm

nmy

nmnmnmTEcnm

nmx

z

HH

HH

HH

HE

HEE

βωπππβ

βωπππβ

βωππ

βωπππβ

βωπππβ

−

−=

−

−=

−

=

−

−=

−

−=

=


Cut Off Frequency and Wavelength

h wavelengtoffcut 2

frequency offcut 2

22 since

is 0for

2/122

2/122

+

=

+

=

==

==

bm

an

bm

ancf

fc

k

kk

cnm

cnm

cnmcnm

cnm

cnmnm

λ

πππ

πλπ

β

Waves with frequency lower than the cut off frequency (f<fcnm) or wavelength longer than the cut off wavelength (λ>λcnm) can not propagte in nm-mode.


Guide Wavelength

λλ

λλλ

λ

λλ

πβπλ

β

>

−

=−

=−

==

−=

gnm

gnm

cnmcnm

cnmnmgnm

cnmnm

kk

kk

. waveguidein the repeatspattern mode h theafter whic distance thegivesIt

nm. mode ofh wavelengtguide called is

111

122nm mode ofconstant n propagatio

2

2

22

22

222


TM-Waves

waveTM theof impedance

h wavelengtoffcut 2

frequency offcut 2

number waveoffcut

:obtains also One .components fieldother theand

1,2,...m and ... 1,2,n with sinsin

calculates one WavesTEfor asmanner same in the0 and 0

0TMnm

2/122

2/122

2/122

Zk

Z

bm

an

bm

ancf

bm

ank

byn

axnEE

EH

nm

cnm

cnm

cnm

nmz

zz

β

λ

πππ

ππ

ππ

=

+

=

+

=

+

=

===

−≠=


TMnm -Fields

)sin(sincos1),,,(

)sin(cossin1),,,(

0),,,(

)sin(cossin),,,(

)sin(sincos),,,(

)cos(sinsin),,,(

2

2

2

2

ztb

yma

xnZak

ntzyx

ztb

yma

xnZbk

ntzyx

tzyx

ztb

yma

xnbk

ntzyx

ztb

yma

xnak

ntzyx

ztb

yma

xntzyx

nmnmTMnmcnm

nmy

nmnmTMnmcnm

nmx

z

nmnmcnm

nmy

nmnmcnm

nmx

nmnmz

EH

EHH

EE

EE

EE

βωπππβ

βωπππβ

βωπππβ

βωπππβ

βωππ

−

=

−

−=

=

−

=

−

=

−

=


Rectangular Waveguide Mode Pattern

TE10 TE20

TE01 TE11


Rectangular Waveguide Mode Pattern(2)

TM11TE21


TE- and TM- Mode Pattern

Mode pattern images can be found for instance in

•N. Marcuvitz, Waveguide Handbook, MIT Radiation Laboratory Series, Vol. 10, McGraw Hill 1951•H. J. Reich, P. F. Ordung, H. L.Krauss, J. G. Skalnik, Microwave Theory and Techniques, D. van Nostrand 1953

and probably in other books, too.


TE10 (H10)-Mode


Waveguide Size and Modes

( )

propagatecan modeTE only the 2for 52h wavelengtoffcut with TM is eguideheight wav half a

in propagatecan which avegenth),(longest wfrequency lowest with mode-TM....... ,82 ,52

h wavelengtcutoff modesTM

propagate.can TEonly 2for ....... ,82 ,2

, ,52 , ,2

hs wavelengtcutoff modesTE

42

system. waveguideretangular ain 2 use common to isIt

10

1111

2111

101001

2102

20110110

2/122

−<<⇒=

==

−

=<<=⇒==

====

−+

=⇒

=

aaa

aa

aaaa

aaaa

mna

ba

c

cc

cc

cc

cccc

cnm

λλ

λλ

λλλλλ

λλλλ

λ


TE10 (H10) -Field

0),,,(

)sin(sin),,,(

)cos(cos),,,(

)sin(sin),,,(

0),,,(

0),,,(

=

−

=

−

=

−

=

=

=

tzyx

ztax

ktzyx

ztaxtzyx

ztax

kZtzyx

tzyx

tzyx

HHH

HH

HEEE

y

cnmx

nmz

cnmTEy

x

z

βωπβ

βωπ

βωπβ

•The mode with lowest frequency propagating in the waveguide is the TE10 (H10) mode. For a<λ<2a only this mode can propagate.

E-FieldH-Field


Some TE10 Properties

2

2

1

377 impedance

2.32but 1.231.3GHzfat WR650:example

h wavelengtspace free h wavelengtguide

1

h wavelengtguide

2h wavelengtoffcut 2

frequency offcut

−

Ω=

===

>⇒

−

=

=

=

λλ

λλ

λλλλ

λλ

λ

c

g

g

c

g

c

c

Z

cmcm

aacf

TE10 Impedance

0

2000

4000

6000

8000

10000

12000

0 10 20 30 40

wavelength/cm

impe

danc

e/oh

m

cut off


Some TE10 Properties (2)

cvcc

dd

v

c

a

cc

v

a

phg

gg

c

gph

g

cg

<==

=

>

−

=

−

==

⇒

−

=

−=

− 221

222

222

velocity group

but

212

2

velocity phase

dispersion :on depends

212constant n propagatio

βωω

β

πλπ

λλ

λπ

λπ

βω

λβ

πλπ

λλ

λπ

β


Some Standard Waveguide SizesWaveguide

typea

(in)b

(in)fc10

(GHz)frequency range

(GHz)

WR 2300 23.000 11.500 .256 .32–.49

WR 2100 21.000 10.500 .281 .35 –.53

WR 1800 18.000 9.000 .328 .41 –.62

WR975 9.750 4.875 .605 .75 – 1.12

WR770 7.700 3.850 .767 .96 – 1.45

WR650 6.500 3.250 .908 1.12 – 1.70

WR430 4.300 2.150 1.375 1.70 – 2.60

WR284 2.84 1.34 2.08 2.60 – 3.95

WR187 1.872 .872 3.16 3.95 – 5.85

WR137 1.372 .622 4.29 5.85 – 8.20

WR90 .900 .450 6.56 8.2 – 12.4

WR62 .622 .311 9.49 12.4 - 18

a

b


Some Waveguides of different Size

WR1800e.g. for 500MHz

P-Band

WR650e.g. for 1.3GHz

L-Band

WR284e.g. for 3GHz

S-Band

18in,46cm


Power in TE10-Mode

( )

[ ] [ ] [ ]( )cmVEcmbcmaP

mmnn

Zk

abHP

dtdxdyT

P

g

mm

nn

TEnmcnm

nm

mn

nmnm

T

z

a b

nmnmnm

/1063.6

:caluclatecan oneTEfor

0for 2 and 0for 1with 0for 2 and 0for 1with

2

1TEin sportedPower tran

S :Vector Poynting

2410

10

00

00

2

00

2

0 0 0nm

⋅=

≠===≠===

=

⇒

⋅×=

×=

−

∫ ∫ ∫

λλ

εεεε

βεε

eHE

HE


Theoretical Power Limit in TE10The maximum power which can be transmitted theoretically in a waveguide of certain size a, b and frequency f is determined by the electrical breakdown limit Emax.In air it is Emax=30kV/cm. From this one can find the maximum handling power in air filled waveguides.

1

10

100

1000

100 1000 10000Frequency/MHz

Pow

er/M

W

WR284WR650WR770WR1800WR2100WR2300


Attenuation in TE10• The walls of the waveguides are not perfect conductors.

They have finite conductivity σ, resulting in a skin depth of

• Due to current in the walls of the waveguides loss appears and the waves are attenuated.

• The attenuation constant for the TE10 is:

[ ][ ] [ ] 2/12

2

1

21

221

12026.0/

−

+

=

a

aab

cmcmbkmdB

λ

λ

λα

k1= 1.00 Ag, 1.03 Cu, 1.17 Au, 1.37 Al, 2.2 Brass

( ) 2/12/ −= ωµσδ s


Attenuation in Al-Waveguides in TE10

0,0001

0,001

0,01

0,1

1

100 1000 10000

Frequency/MHz

Atte

nuat

ion/

dB

/m WR284WR650WR770WR1800WR2100WR2300


Reflection and Impedance

ratio wavestanding voltage11

waveincoming and reflected theof amlitude and

t,coefficien reflection

12

12

ρρ

ρ

ρ

−+

=−

+=

+−

=

=

EEEEVSWR

ZZZZ

EEEE

rf

rf

fr

f

r

Z1 Z2

Ef1

Er1

Ef2


Travelling and Standing Wave

TE10 travelling wave TE10 standing wave due to full reflection ρ=1.

The maximum electrical field strength in the standing wave is double the strength of the travelling wave. The same field strength can only be found in a travelling wave of 4-times power.

high power rf - cern

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