Shuichi Noguchi, KEK6-th ILC School, November 20111 Shuichi Noguchi, KEK6-th ILC School, November 20111 RF Basics; Contents Maxwell’s Equation Plane

Shuichi Noguchi, KEK 6-th ILC School, November 2011 1Shuichi Noguchi, KEK 6-th ILC School, November 2011 1

RF Basics; Contents Maxwell’s Equation Plane Wave Boundary Condition Wave Guide Cavity & RF Parameters Normal Mode Analysis Perturbation Theory Equivalent Circuit Coupled Cavity

Part-1


Literatures

J. C. Slator “Microwave Electronics”

Rev. Mod. Phys. 18,(1946)


Maxwell’s Equation ( MKS )

EJHBED

DdivJt

DHrot

Bdivt

BErot

,,

,

,0,0

Not a Beam Current

Faraday

Ampere


Pointing Vector & Power Flow

SdHEVdHEdiv

VdErotHHrotEVdHEt

VdJE

nSV

VV V

xx

2

1 22

From Maxwell’s Equation

Energy Loss + Change of Electric and Magnetic Energy= Power Flow at Boundary S

HEP

HES

xRe2

1;FlowPower

x ;Vector Pointing


Maxwell’s Equation - Wave Equation

EEdivEErotrotJt

Et

Et

Jt

Hrott

Hrott

Erotrot

222

2

2

2

,

;11

;

2

2

2

2

2

2

2

2

2

2

22

Azrr

rrr

Azyx

A

Cartesian Coordinate

Cylindrical Coordinate

= 0


Wave Equation

01

,01

2

2

22

2

2

22

t

H

cH

t

E

cE

0,0 2222 HkHEkE

222,,

ckeHE tj

0,


Wave Equation Helmholtz Equation

022 EkE

Mode Magnetic / Electric Transverse;TEM;0,0

Mode Magnetic Transverse;TM;0,0

Mode Electric Transverse;TE;0,0

zz

zz

zz

HE

HE

HE

Particular Solution for our Application

No TEM Modes in one closed Conductor


Maxwell’s Equation in Cartesian Coordinates

0,

,

,

,

z

H

y

H

x

H

z

E

y

E

x

E

EjJy

H

x

HHj

y

E

x

E

EjJx

H

z

HHj

x

E

z

E

EjJz

H

y

HHj

z

E

y

E

zyxzyx

zzxy

zxy

yyzx

yzx

xxyz

xyz


Maxwell’s Equation in Cylindrical Coordinates

011

,11

11,

11

,

1,

1

z

HH

rHr

rrz

EE

rEr

rr

EjJH

rHr

rrHj

E

rEr

rr

EjJr

H

z

HHj

r

E

z

E

EjJz

HH

rHj

z

EE

r

zr

zr

zzr

zr

zrzr

rrz

rz


Plane Wave in Uniform Medium

0,0

,

,

,

z

H

y

H

x

H

z

E

y

E

x

E

EjJy

H

x

HHj

y

E

x

E

EjJx

H

z

HHj

x

E

z

E

EjJz

H

y

HHj

z

E

y

E

zyxzyx

zzxy

zxy

yyzx

yzx

xxyz

xyz

0;exp;exp tjzHHtjzEE


Plane Wave in Uniform Medium

0;exp;exp tjzHHtjzEE

0,0

,

,

zz

xy

xy

yx

yx

HE

Ejz

HHj

z

E

Ejz

HHj

z

E

Frequency Time Dependence exp( jt )

No Boundary

TEM Mode


Plane Wave

,112

,112

,

expexp,

2

1

22

2

2

1

22

2

212

2

jjj

zEzEEEjjzd

Edxx

x

Propagation Constant

Attenuation Constant( Real Part )

Phase Constant( Imaginary Part )


Impedance ; E / H

jj

jjZ

zEzEZzd

Ed

jH

i

i

xy

,expexp11

21

Intrinsic Impedance

i

i

Zj

jZj

,,Dielectric;

21,

21Conductor,;


Boundary Condition221121112221 ,,, nnSttSnntt HHJHHEEEE

t

ldE

Medium 11, 1, Z1

Medium 22, 2, Z2

Medium 1 Medium 2

sJs

Et1 Et2 Ht1 Ht2

En1 En2 Hn1 Hn2

SJldH

E = H = 0 in Perfect Conductor ; Et =Hn = 0

Faraday Ampere

0


Reflection & Transmission

zjkii

zjki e

Z

EHeE 11

1

,

zjktt

zjkt e

Z

EHeE 22

2

,

Medium 11, 1, Z1

Medium 22, 2, Z2

z

x

zjkrr

zjkr e

Z

EHeE 11

1

,

Dielectric Boundary


iitiir

tritri

zjkty

zjkrzjkiy

zjktx

zjkr

zjkix

ETEZZ

ZEEE

ZZ

ZZE

Z

E

Z

EEEEE

eZ

EHe

Z

Ee

Z

EH

eEEeEeEE

12

2

12

12

21

22

111

21

2,

,

,

,

211

211


Metallic Boundary

111111

1

,

Z 21,21 2222222

22

jjjZ

zjziy

zjzix

iyix

itir

eZ

EHeE

Z

ZE

zkZ

EHzkEjE

EZ

ZEEEZZ

2222

12

1

22

11

111

1

221

2,2

cos2

,sin2

2,,


Metallic Boundary

111111

1

,

Z 21,21 2222222

22

jjjZ

z

x

DepthSkin ;21

222

Dielectric Metallic

E

H


Power Loss & Surface Impedance

LossPower ;2

1

Resistance Surface;1

2

Impedance Surface;

22

1Re

2

1Re

2

1

2

22

2

2

222

2

tS

S

SSS

tttt

HRP

R

XjRZ

HZHHEP


Wave Guide

Coaxial LineParallel ConductorStrip LineCircular Wave GuideRectangular Wave Guide Ridged Wave Guide


Traveling Wave Mode

0,

,

,

,

z

H

y

H

x

H

z

E

y

E

x

E

EjJy

H

x

HHj

y

E

x

E

EjJx

H

z

HHj

x

E

z

E

EjJz

H

y

HHj

z

E

y

E

zyxzyx

zzxy

zxy

yyzx

yzx

xxyz

xyz

0,;exp,;exp, zjtjyxHHzjtjyxEE


Traveling Wave Mode 0,;exp,;exp, zjtjyxHHzjtjyxEE

0,0

,

,

,

zyx

zyx

zxy

zxy

yz

xyz

x

xz

yxz

y

Hjy

H

x

HEj

y

E

x

E

Ejy

H

x

HHj

y

E

x

E

Ejx

HHjHj

x

EEj

Ejy

HHjHj

y

EEj


TE-Modes ; Ez = 0

0,0

0,

,

,

zyxyx

xyz

xy

yz

xyx

xz

yxy

Hjy

H

x

H

y

E

x

E

y

H

x

HHj

y

E

x

E

Ejx

HHjHjEj

Ejy

HHjHjEj


2222222

2

2

2

2222

2222

,0

,

,

kkHky

H

x

H

x

H

k

jE

y

H

k

jE

y

H

k

jH

x

H

k

jH

czczz

zy

zx

zy

zx

number wavecutoff / critical;ck


TE-mn Modes in Rectangular WG

22

21

222

2222

22

12

121

2

22

22

22

12

121

,

011

,

kk

Hyd

HdH

xd

Hd

kyd

Hd

Hxd

Hd

HyHxHH

c

zz

zz

cz

z

z

zzzz

x

z

y

a

b

1,,

expcoscos

21

nmb

n

a

m

zjyb

nx

a

mH z

From Boundary Condition


;12

;22

;22

;

2

22

2222

cg

cc

cc

bnamk

k

kkbnamk

Wave Length in Medium

Critical Wave Length

Guide Wave Length

If k < kc ( c ) wave can not propagate.


TE-mn Modes

zjz

zj

cy

zj

cx

z

zj

cy

zj

cx

eyb

nx

a

mH

eyb

nx

a

m

b

n

k

jH

eyb

nx

a

m

a

m

k

jH

E

eyb

nx

a

m

a

m

k

jE

eyb

nx

a

m

b

n

k

jE

sincos

,sincos

,cossin

,0

,cossin

,sincos

2

2

2

2


TM-Modes ; Hz = 0

0,0

,0

,

,

y

H

x

HEj

y

E

x

E

Ejy

H

x

H

y

E

x

E

EjHjHjx

EEj

EjHjHjy

EEj

yxz

yx

zxyxy

yxyz

x

xyxz

y


TM-mn Modes

0

,sincos

,cossin

,cossin

,cossin

,sincos

2

2

2

2

z

zj

cy

zj

cx

zjz

zj

cy

zj

cx

H

eyb

nx

a

m

a

m

k

jH

eyb

nx

a

m

b

n

k

jH

eyb

nx

a

mE

eyb

nx

a

m

b

n

k

jE

eyb

nx

a

m

a

m

k

jE


Power Loss

dSHRPS tSloss

2

2

1


TEM-Modes ; Ez, Hz = 0

0,0

0,0

,

,

y

H

x

H

y

E

x

E

y

H

x

H

y

E

x

E

EjHjHjEj

EjHjHjEj

yxyx

xyxy

yxyx

xyxy

0,,

Cutoff No , WavePlane;

2

22

yE

xE yx


Maxwell’s Equation in Cylindrical Coordinates

011

,11

11,

11

,

1,

1

z

HH

rHr

rrz

EE

rEr

rr

EjJH

rHr

rrHj

E

rEr

rr

EjJr

H

z

HHj

r

E

z

E

EjJz

HH

rHj

z

EE

r

zr

zr

zzr

zr

zrzr

rrz

rz


Traveling Wave Modes 0,;exp,;exp, zjtjrHHzjtjrEE

011

,11

11,

11

,

1,

1

z

HH

rHr

rrz

EE

rEr

rr

EjJH

rHr

rrHj

E

rEr

rr

EjJr

H

z

HHj

r

E

z

E

EjJz

HH

rHj

z

EE

r

zr

zr

zzr

zr

zrzr

rrz

rz


011

,011

11,

11

,

1,

1

zrzr

zr

zr

zr

zr

rz

rz

HjH

rHr

rrEj

E

rEr

rr

EjH

rHr

rrHj

E

rEr

rr

Ejr

HHjHj

r

EEj

EjHjH

rHjEj

E

r

0,;exp,;exp, zjtjrHHzjtjrEE

Traveling Wave Modes


TM-Modes ; Hz = 0

011

,011

11,0

11

,

,1

H

rHr

rrEj

E

rEr

rr

EjH

rHr

rr

E

rEr

rr

EjHjHjr

EEj

EjHjHjEjE

r

rzr

zrr

rz

r

rrz


22222

2

2

2222

,011

1,

,

kkEkr

Er

rr

E

r

E

rk

jE

r

E

k

jE

EHEH

czczz

zzr

rr


Function Bessel;

,cos

,011

,

;

1

2

122

21

21

2

22

22

2

21

rkJE

mE

Erk

m

rkd

Ed

rkrkd

Ed

Emd

Ed

ErEE

cmz

z

zcc

z

cc

z

zz

zzz


Boundary Conditionr = a

z

22

,

,0

,0

a

y

yak

akJ

arE

mn

mnc

cm

z

m n 1 2 3 4

0 2.4048 5.5201 8.6537 11.7915

1 3.8317 7.0156 10.1735 13.3237

2 5.1356 8.4172 11.6198 14.7960

3 6.3802 9.7610 13.0152 16.2235

4 7.5883 11.0647 14.3725 17.6160

ymn


TM-man Modes

0

,cos

,1

sin

,cos

,1

sin

,cos

'

2

2

'

z

zjmnm

mn

zjmnm

mn

r

zjmnmz

zjmnm

mn

zjmnm

mnr

H

era

yJm

ayjH

era

yJ

rm

ay

mjH

era

yJmE

era

yJ

rm

ay

mjE

era

yJm

ayjE


TE-Modes

011

,011

011

,11

,

1,

zrr

rz

r

zrr

rz

r

HjH

rHr

rr

E

rEr

rr

H

rHr

rrHj

E

rEr

rr

Ejr

HHjHjEj

EjHjH

rHjEj


TEM-Modes ; Ez = Hz = 0

011

,011

011

,011

,

,

H

rHr

rr

E

rEr

rr

H

rHr

rr

E

rEr

rr

EjHjHjEj

EjHjHjEj

rr

rr

rr

rr

off-Cut No,0,22 ck


Coaxial Waveguide

Impedance sticCharacteri;In2

A2,InA

A1,

A

0,0

,0,0

0

2

0

a

bZ

I

VZ

eZ

daJIea

bdrEV

erZ

Her

E

Hrr

Err

HE

i

zj

iS

zjb

a r

zj

i

zjr

r

r

a b


Power

baZ

PR

a

b

baZ

PR

baZRbdHadHRP

a

b

ZrdrdHEP

fS

i

fS

iSbrarSloss

i

b

a rf

11

2In

11

11A

2

1

InA

Re2

1

0

2

22

0

2

0

22

22

0


Resonator / Cavity


Can be solved Analyticallyor by Computer Codes

Boundary Condition Short-Circuited Plane S Open-Circuited Plane S’

S'on 0,0x

Son0,0x

EnHn

HnEn

・

・

S

S’ S’n

Media ;

wall

Cavity

; Perfect Conductor


Analytic Solution, Example

L

a

L

lLzz

at ;0sin


TM-01l Modes

0

,cos

,1

sin

,cos

,1

sin

,cos

'

2

2

'

z

zjmnm

mn

zjmnm

mn

r

zjmnmz

zjmnm

mn

zjmnm

mnr

H

era

yJm

ayjH

era

yJ

rm

ay

mjH

era

yJmE

era

yJ

rm

ay

mjE

era

yJm

ayjE

0

,cos

,0

,cos

,0

,sin

'

01

010

01'0

01

z

mnm

r

z

r

H

zL

lr

a

yJ

ayjH

H

zL

lr

a

yJE

E

zL

lr

a

yJ

ayjE


Cavity RF Parameters

22

2

0

2

2

2

00

222

0

,

,

22,

2

Cavityacc

Cavityacc

S

S

LW

E

Q

RL

P

ER

dSH

dVHG

R

G

P

WQ

dVEdVHWdSHR

P

dzztrzEL

EL

zCavity

acc cos0,1

0

Geometric Factor


Transit Time Factor ( TTF )

La

aQ

E

E

E

EL

c

L

L

c

dzrzE

dzztrzE

acc

Sp

Sp

acc

L

z

L

z

1

Cavity ngAccelerati;2

,2

TTF,2

,2

sin2

0,

cos0,TTF

0

0

TM010 Mode in Cylindrical Cavity

Shuichi Noguchi, KEK 6-th ILC School, November 2011 50

Calculate Skin Depth & Surface Resistance

using following Values.

m

mm

/mho10x6

/H10x4;/F10x36

1

7

79

Documents

Shuichi Noguchi, KEK6-th ILC School, November 20111 Shuichi Noguchi, KEK6-th ILC School, November 20111 RF Basics; Contents Maxwell’s Equation Plane