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

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

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

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

=⋅∇=⋅∇

∂∂−=×∇

∂∂=×∇

=⋅∇=⋅∇

∂∂−=×∇

+∂∂=×∇

εεµµρ

Wave Equation

for equation wave0

Hfor way same in the

for equation ve wa0

and of use

curlapply and start with

equations. Maxwell thefrom derived becan equation waveThe

∂−∇

∂−∇⇒

=⋅∇∂

∂−=∇−⋅∇∇⇒

∇−⋅∇∇=∇××∇∂∂=×∇

∂∂×∇−=×∇×∇⇒

∂∂−=×∇

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),(),,,(

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

ztiyxtzyxztiyxtzyx

=++∇

−=−=

βγµεω

µεω

βωβω

Derivation of Maxwell Equation for transversal and longitudinal Components

( ) ( ) ( )

tztztz

ztzzztzt

ztzzzttztt

ztztzt

HeEeEEe

EeeEeEeeE

HHEeEEeE

HHEEeEHE

ωµββ

ωµωµββ

ωµβωµ

−=∇×−×−=×∇+×−

−=×∇

=×⊥×∇⊥×

∇×−=×∇=×∇

−−=×−×∇+×−×∇

+−=+×−∇=×∇−=×∇

components al transversand allongitudin of Separation0 ||

seecan one Now

Derivation Maxwell Equation for transversal and longitudinal Components (2)

tztztz

ωεβωε

=⋅∇=⋅∇

−=∇×+×=×∇

=×∇

EeHeEH

manner same in the With

Equations of transversal and longitudinal Components as Function of transversal

Coordinates

tztztz

ωεβωε

ωµβωµ

=⋅∇=⋅∇

−=∇×+×=×∇

−=∇×−×−−=×∇

=++∇

EeHeEH

HeEeHE

0)(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.

≠=≠=

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

equations Maxwellin 0With

0components allongitudin and ersalfor transvor

and 0)(equation eWith wav

222222

=⋅∇=⋅∇

−=∇×+×=×∇

−=×−−=×∇=

=+=++∇

tztztz

βωεβ

ωµβωµ

Derivation of TE-Wave Equations(2)( )

( ) ( ) ( )( )

too. from thereforeand from and from calculated becan

withand0

0 and With

0 gives 0on curl ofn Applicatio

tzFtztzt

tztzzttzztzz

tcttttttt

HeHeHeE

HeEeeEEeeEee

×−=×−=×−=

×=−=⋅−⋅=××

∇−=

=⋅∇=+⋅∇∇⇒

=+∇∇−⋅∇∇=×∇×∇

=×∇×∇=×∇

ββωµε

βωµ

ωµβββ

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

TE-Wave Equation in rectangular Waveguides

HxkiZE

HykiZE

HkHyHx

∂∂=

∂∂−=

∂+∂

×−=

∇−=

problem Eigenvalue 0

H TE wave equation

TE wave equationwritten in components

Solution of TE- Wave Equation

011 and with

s.coordinatey and x into Separation :Ansatz

g(y)gf(x)fgfH

HkHyHx

==⋅=

∂+∂

Solution of TE-Wave Equation(2)

byyHaxxH

ykBykBgxkAxkAf

===∂∂

===∂∂⇒

=======

and 0for 0

and 0for 0 and 0for 0

surfaceon 0H:H ofcomponent Normal :conditionsBoundary

sincossincos

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

,....2,1,0

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

0 from 0

and 0for 0cossin

≠====

===+−

axnHyxH

axxxkkAxkkA

decay lexponentiaimaginary is for

npropagatioreal is for

number waveoffcut

βωβγ

TEnm -Fields

)sin(sincos),,,(

)sin(cossin),,,(

)cos(coscos),,,(

)sin(cossin),,,(

)sin(sincos),,,(

0),,,(

xntzyx

nmnmcnm

nmnmnmTEcnm

βωπππβ

βωππ

βωπππβ

Cut Off Frequency and Wavelength

h wavelengtoffcut 2

frequency offcut 2

22 since

is 0for

cnmcnm

πππ

πλπ

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

λλλ

πβπλ

cnmcnm

cnmnmgnm

. waveguidein the repeatspattern mode h theafter whic distance thegivesIt

nm. mode ofh wavelengtguide called is

122nm mode ofconstant n propagatio

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

πππ

−≠=

TMnm -Fields

)sin(sincos1),,,(

)sin(cossin1),,,(

0),,,(

)sin(cossin),,,(

)sin(sincos),,,(

)cos(sinsin),,,(

xntzyx

nmnmTMnmcnm

nmnmcnm

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

system. waveguideretangular ain 2 use common to isIt

101001

20110110

−<<⇒=

=<<=⇒==

λλλλλ

λλλλ

TE10 (H10) -Field

0),,,(

)sin(sin),,,(

)cos(cos),,,(

)sin(sin),,,(

0),,,(

ztaxtzyx

kZtzyx

cnmTEy

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

377 impedance

2.32but 1.231.3GHzfat WR650:example

h wavelengtspace free h wavelengtguide

h wavelengtguide

2h wavelengtoffcut 2

frequency offcut

λλλλ

TE10 Impedance

0 10 20 30 40

wavelength/cm

cut off

Some TE10 Properties (2)

− 221

velocity group

velocity phase

dispersion :on depends

212constant n propagatio

βωω

πλπ

Some Standard Waveguide SizesWaveguide

(in)fc10

(GHz)frequency range

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

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

dtdxdyT

TEnmcnm

nmnmnm

/1063.6

:caluclatecan oneTEfor

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

1TEin sportedPower tran

S :Vector Poynting

0 0 0nm

≠===≠===

⋅×=

∫ ∫ ∫

εεεε

βεε

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.

100 1000 10000Frequency/MHz

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

12026.0/

cmcmbkmdB

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

100 1000 10000

Frequency/MHz

/m WR284WR650WR770WR1800WR2100WR2300

Reflection and Impedance

ratio wavestanding voltage11

waveincoming and reflected theof amlitude and

t,coefficien reflection

EEEEVSWR

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.

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