TRANSMISSION MEDIA

MAXWELL’S EQUATIONS AND

TRANSMISSION MEDIA CHARACTERISTICS

ENEE 482 Spring 2002DR. KAWTHAR ZAKI

ENEE482 2

MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS

Two conductorwire Coaxial line Shielded

Strip line

Dielectric

ENEE482 3

Rectangular guide

Circular guide

Ridge guide

Common Hollow-pipe waveguides

ENEE482 4

STRIP LINE CONFIGURATIONS

W

SINGLE STRIP LINE COUPLED LINES

COUPLED STRIPSTOP & BOTTOM

COUPLED ROUND BARS

ENEE482 5

MICROSTRIP LINE CONFIGURATIONS

TWO COUPLED MICROSTRIPS SINGLE MICROSTRIP

TWO SUSPENDED SUBSTRATE LINES

SUSPENDED SUBSTRATE LINE

ENEE482 6

TRANSMISSION MEDIA

• TRANSVERSE ELECTROMAGNETIC (TEM):– COAXIAL LINES– MICROSTRIP LINES (Quasi TEM)– STRIP LINES AND SUSPENDED SUBSTRATE

• METALLIC WAVEGUIDES:– RECTANGULAR WAVEGUIDES–CIRCULAR WAVEGUIDES

• DIELECTRIC LOADED WAVEGUIDES

ANALYSIS OF WAVE PROPAGATION ON THESETRANSMISSION MEDIA THROUGH MAXWELL’SEQUATIONS

ENEE482 7

Electromagnetic Theory Maxwell’s Equations

2

2

2

2

V/mdensity current s)(fictitiou Magnetic

A/mdensity current ElectricJ

A/mIntensity Field Magnetic

)Sec./m-Vor (Telsa TDensity Flux Magnetic

C/mDensity Flux Electric

V/mIntensity Filed Electric

:Equation Continuity

0 ;

; M-

M

H

B

D

Et

J

BD

Jt

DH

t

BE

ENEE482 8

Auxiliary Relations:

tyPermeabili Relative

H/m 104 ; 5.

Constant Dielectric Relative

F/m 10854.8 ; 4.

Current Convection ; 3.

Current Conduction ;ty Conductivi

Law) s(Ohm' 2.

Velocity ; Charge

Newton .1

r

12o

12

HHB

EED

JvJ

J

EJ

vq

BvEqF

or

r

oor

ENEE482 9

Maxwell’s Equations in Large Scale Form

SdDt

SdJldH

SdMSdBt

ldE

SdB

dvSdD

SSl

SSl

S

SV

0

ENEE482 10

Maxwell’s Equations for the Time - Harmonic Case

DjJHMBjE

BD

EEtEE

eEEejEEE

jEEa

jEEajEEazyxE

ezyxEtzyxE

xrxixixr

jtjxixr

tjxixrx

zizrz

yiyryxixrx

tj

tj

,

0,

)/(tan , )cos(

]Re[])Re[(

)(

)()(),,(

]),,(Re[),,,(

: then,variationse Assume

122

22

ENEE482 11

Boundary Conditions at a General Material Interface

s

s

s

sttnn

s

snn

stt

JHHn

MEEn

BBn

DDn

JHHBB

DD

MEE

)(ˆ

)(ˆ

0)(ˆ

)(ˆ

;

Density Charge Surface

21

21

21

21

2121

21

21

D1n

D2n

h

s

h

E1t

E2t

ENEE482 12

)(ˆ)(ˆ

)(ˆ)(ˆ

)(ˆ)(ˆ

)(ˆ)(ˆ

0 ;

0

0

21

21

21

21

2121

21

21

HnHn

EnEn

BnBn

DnDn

HHBB

DD

EE

ttnn

nn

tt

Fields at a Dielectric Interface

ENEE482 13

HnHJ

B

Dn

ts

n

ˆ

0Bn 0

ˆρD

0En oE

:ConductorPerfect aat ConditionsBoundary

sn

t

+ + +n

s

Js

Ht

ENEE482 14

The magnetic wall boundary condition

0)(ˆ

)(ˆ

0)(ˆ

0)(ˆ

Hn

MEn

Bn

Dn

s

ENEE482 15

2/ ; 0

; 0

:medium free Sourcea For

)(

)(

22

2222

2

vkHkH

kEkE

EjJj

HjEEE

Wave Equation

ENEE482 16

Plane Waves

zayaxar

kakakakAeE

kkk

zyxE

zyxiEkz

E

y

E

x

E

Ekz

E

y

E

x

EEkE

zyx

zzyyxxzjkyjkxjk

x

zy

x

iiii

zys

Let ,

k

variablesof separation Using),,,(for Solve

,,, 0

0

20

222x

202

2

2

2

2

2

202

2

2

2

2

220

2

ENEE482 17

space. free of admittance intrinsic theis

377 space free of impedance interensic theis

1

11

1

waveplane called issolution The

.kn propagatio ofdirection thelar toperpendicu is vector The

00 Since

,Similarly ,

0

0

00

0

0

0

0

00

00

00

0

0

00

Y

EnEnYEnEnk

eEkeEj

eEj

H

HjE

E

EkEeEE

CeEBeEAeE

rkjrkjrkj

rkj

rkjz

rkjy

rkjx

ENEE482 18

x

n

z

y

E

H is perpendicular to E and to n. (TEM waves)

U 2

1 U,

2

1U

:are TEM wavea of fields magnetic

and electric thein densitiesenergy average timeThe2

1)(Re

2

1Re

2

1

,2

real is if ),cos()Re(

e*

000m

*

000e

*

000*

0*

00

000

HHEE

EEYnEnEYnHEP

k

EtrkEeEE tjrkj

H

ENEE482 19

Plane Wave in a Good Conductor

s

j

1)1(

2j)(1

j

2

1

2j)(1

j jj

s

ENEE482 20

Boundary Conditions at the Surface of a Good Conductor

The field amplitude decays exponentially from its surface According to e-u/

s where u is the normal distance into theConductor, s is the skin depth

Hn ,1

: Impedance surface The

EJ , 2

msmts

m

s

ZJZEj

Z

ENEE482 21

Reflection From A Dielectric Interface

Parallel Polarization

z

x

Ei

Et

Er

3

1

2

n1

n3

n20

0

0000

202

101

Y,

,

, 120

110

k

EnYHeEE

EnYHeEE

rrrnjk

r

iirnjk

i

ENEE482 22

sin , cos

sin , cos

sin , cos

sin sin

cossin

cossin

cossin

, ,

,

331333

221222

111111

3121

333

222

111

3032010

00

3313

EEEE

EEEE

EEEE

n

aan

aan

aan

nnkknnknk

nkknYY

EnYHeEE

zx

zx

zx

zx

zx

zx

xxxx

ttrnjk

t

ENEE482 23

Energy and Power

Under steady-state sinusoidal time-varying Conditions, the time-average energy stored in theElectric field is

V

e

VV

e

dVEEW

dVEEdVDEW

*

**

4

thenreal, andconstant is If

4

1

4

1Re

ENEE482 24

S

V

V

m

dSHEP

dVHH

dVBHW

*

*

*

Re2

1

:by given is S surface

closeda across smittedpower tran average timeThe

constant and real is if 4

4

1Re

:is field magnetic thein storedenergy average Time

ENEE482 25

Poynting Theorem

dVMHJEdVDEHB

j

dVMHJEdVDEHBj

dSHEVdHE

EJJ

JEEDjHMBj

EHHEHE

V

s

V

s

VV

SV

s

)(2

1

442

)(2

1)(

2

2

1

2

1

)(

)()(

****

****

**

***

***

ENEE482 26

dSHEP

dVEEHHj

dVEEHHdVEE

dSHEdVMHJE

j

S

V

VV

S

S

V

Ss

*0

**

***

**

2

1

)(2

)(22

1

2

1)(

2

1-

ty conductivi and j- ,

:by zedcharacteri is medium theIf

ENEE482 27volume.

in the storedenergy reactive the times2 and )( volumein the

heat lost topower the,P surface he through tsmittedpower tran the

of sum the toequal is )(P sources by the deliveredpower The

)(2

)(2

442

2

1Im

)(2

1P

0

s

0

***

*s

P

WWjPPP

WW

dVEEHH

dSHE

dVMHJE

ems

em

VS

ss

V

s

losspower average Time

2

1)(

2***

dVEEdVEEHHPVV

ENEE482 28

Circuit Analogy

C

L RI

V

networka of impedance theof n DefinitioGeneral

2

1)(2P

)(2P

)4

1

4

1(2

2

1

)(2

1

2

1

2

1

*

2

***

***

II

WWjZ

WWjC

IILIIjRII

C

jLjRIIZIIVI

em

em

ENEE482 29

Potential Theory

22

22

22

22

2

0D

equation. HelmholtzousInhomogene

condition) (Lorentz or

Let ,

,

1

, 0

,Let

k

JAkA

jA

jAk

JjAkAAA

JjAJEjAH

AjE

AjEAjE

AjBjEAB

ENEE482 30

Solution For Vector Potential

(x,y,z)(x’,y’, z’) R

rr’

J

VdR

erJrA

rrzzyyxxR

VdR

erJrAzyxA

jkR

V

jkR

)(4

)(

)()()(

current alinfinitism anfor )(4

)(),,(

222

ENEE482 31

Waves on An Ideal Transmission LineRg

z

Ldz

Cdz

I(z,t)

V(z,t)V(z,t)+v/z dz

I(z,t)+I/z dz

Lumped element circuit model for a transmission line

ENEE482 32

Impedance sticCharacteri:

C

L , ,

)()(),(

)()(),(

1

0),(),(

0),(),(

2

2

2

2

2

2

2

2

c

ccc

Z

ZZ

VI

Z

VI

v

ztfI

v

ztfItzI

v

ztfV

v

ztfVtzV

LCv

t

tzILC

z

tzI

t

tzVLC

z

tzV

ENEE482 33

Steady State Sinusoidal Waves

LCC

L

YZ

VYIVYIeIeIzI

eVeVzV

vzV

vdz

zVd

zCVjz

zI

zLIjz

zV

tVtV

cc

cczjzj

zjzj

gg

, 1

, ,)(

)(

, 0)()(

)()(

)()(

cos)(

2

2

2

2

ENEE482 34

Transmission Line Parameters

C2

C1

S

22P ,2/

2P

isconductor the

ofty conductivi finite toduelength unit per lossPower

44W

44W

:line ofsection m 1for energy magnetic stored

average- timeThe .I becurrent theand V

:be conductors ebetween th voltageLet the

2

0*d

2

0*

c

2

0*

e

2

0*

m

00

21

VGdSEEIRdHH

R

VC

dSEE

IL

dSHH

ee

SCC

s

S

S

zjzj

ENEE482 35

Terminated Transmission Line

ZL

Zc

Z

To generator

1/

1/ ,

1

1

tcoefficien on Reflecti

)(1

L

cL

cL

c

L

L

L

L

cL

LL

L

ZZ

ZZ

Z

ZV

V

VVZZ

VIIII

VVVV

ENEE482 36

tan

tan

1

1

)2

(sin41

,

)1(2

1

)1)(1(Re2

1)Re(

2

1

2/122

22

*2*

Lc

cL

c

inin

jL

zjjzj

zjL

zj

Lc

LLcLL

jZZ

jZZ

Z

ZZ

S

lVV

eeVeeV

eVeVV

VY

VYIVP

ENEE482 37

Transmission Lines & Waveguides

Wave Propagation in the Positive z-Direction is Represented By:e-jz

,

,

)(

)()()(

,,

,,,,,,

,,

,,,,,,

zttztt

zttztt

zjztzzztzt

zjzt

zjztzt

zjz

zjt

zt

zjz

zjt

zt

ejehjh

ejhhje

ehhjeajeeaje

ehhjeeeajE

eyxheyxh

zyxHzyxHzyxH

eyxeeyxe

zyxEzyxEzyxE

ENEE482 38

Modes Classification:

1. Transverse Electromagnetic (TEM) Waves

0 zz HE

2. Transverse Electric (TE), or H Modes

0but , 0 zz HE

3. Transverse Magnetic (TM), or E Modes

0But , 0 zz EH

4. Hybrid Modes

0 , 0 zz EH

ENEE482 39

TEM WAVES

zjz

zjtt

zjt

zjtt

t

t

t

tt

ttt

eeaYehH

eyxeeE

yx

yxyxe

h

e

eh

ˆ

),(

0,

entialScalar Pot 0,,

eha , 0

hea , 0

0 , 0

0

2

t0tzt

t0tz

t

ENEE482 40

wavesTEMfor

0])([ , 0)(

, but , 0

:equation Helmholtzsatisfy must field The

direction z-or in then propagatio for wave

H

E

Impedance Wave , 1

0

20

2t

20

2

222t

20

2

0y

x

00

0

k

kEkE

ajEkE

H

E

ZY

tttt

tztt

x

y

ENEE482 41

TE WAVES

0

, 0

,

0

let , 0)(

0),()(

0

,

22

222222

222

22

ttztt

tzztztt

ttzztt

zczt

czt

zzt

ehjh

ejhajhah

heahje

hkh

kkhk

hkyxh

HkH

ENEE482 42

hx

y

h

tztzt

ztc

t

Zh

e

kZ

haZk

hae

hk

jh

y

x

0

000

2

h

e

Impedance Wave

; ˆˆ

ENEE482 43

Admittance Wave

ˆ

0

let , 0)(

0),()(

0

0

2

22

222222

222

22

Yk

Y

eaYh

ek

je

eke

kkeke

ekyxe

EkE

e

zet

ztc

t

zczt

czzt

zzt

TM WAVES

ENEE482 44

TEM TRANSMISSION LINES

Parallel -plate Two-wire Coaxial

ab

ENEE482 45

COAXIAL LINES

a b

0

jkz-00

jkz-r

0

0

021

2

2

2

e a )/ln(

and e a )/ln(

)/ln(

)/ln(

0at 0,at ln

0for 01

)(1

Y

abr

VYH

abr

VE

ba

brV

rarVCrCrr

rrr

ENEE482 46

)/ln(ˆRe

2

1

e)/ln(

2e

)/ln(I

eˆ)/ln(

ˆˆ

200

2

0

*

jkz-2

0

00

jkz-00

jkz-00

ab

VYrdrdaHEP

ab

VYad

aba

VY

aaba

VYHaHnJ

z

b

a

zrs

• THE CHARACTERISTIC IMPEDANCE OF A COAXIAL IS Z0

Ohms ln2

1

00

0

a

b

YI

VZ c

ENEE482 47

Zc OF COAXIAL LINE AS A FUNCTION OF b/a

r Zo=X

b/a

1

10

1000

20

40

60

80

100

120

140

160

180

200

220

240

260

ENEE482 48

Transmission line with small losses

EJ

YY

kkEaYHeE

kk

kkjjkjjjk

j

jYYjkk

rr

rzjkz

t

r

r

r

r

rr

r

rr

rrr

rrrr

ty conductivi the toequivalent is

, ˆ ,

, 2

2)1(

losses small For )(

)( and )(

0

0

00

00

2/10

0

2/10

2/10

ENEE482 490

0c

*

**

0

0

*

0

**

, )/ln(2

Re2

1

2Re

2

1 ,

1

:lossconductor the todue losspower The

222 ,

2

2- ,

22

1

:is lengthunit per losspower The

2121

YYab

ab

ab

YR

SdHEP

dHHR

dJJZPj

Z

kYY

dSEEY

P

PPz

PePP

dSEEdSJJP

m

s

SS

sm

s

SS

sms

m

r

r

r

d

S

z

SS

ENEE482 50

Qc OF COAXIAL LINE AS A FUNCTION OF Zo

Q-C

op

per

of

Coaxia

l Lin

e

2000

2200

2400

2600

2800

3000

3200

34000 10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

r Zc

GHz

c

fb

Q

ENEE482 51

Parallel Plate Waveguide

w

y

x

d

jkzww

s

jkzd

yjkz

jkzjkz

t

t

ed

wVdxzHydxzJI

edyEVed

VxEzyxH

ed

VyeyxeyxE

d

Vyyxe

d

yVyx

yx

0

00

00

0

0

00

0

2

ˆ)ˆ(ˆ

V ,ˆˆ),(

ˆ),(),(

ˆ),( , ),(

Vd)(x,

0,(x,0) By Ay)(x,

dy0

w,x0 0),(

TEM Modes

ENEE482 52

TM modes

zj-

zj-

22

22

2

e cos),,(

e sin),,(

sin),(

)(

,....3,2,1,0 , , 0B

d 0,yat 0),(

cossin ),(

0),(

yd

nA

k

jzyxH

yd

nAzyxE

yd

nAyxe

d

nK

nndk

yxe

ykBykAyxe

yxeky

nc

x

nz

nz

c

z

ccz

zc

ENEE482 53

0nfor 2

)Re(

0nfor 4

)Re(2

1ˆ

2

1

2 ,

Z

:is modes TM theof impedance waveThe

22

0E ,e cos),,(

2

2

2

2

*

0 00 0

*0

g

TM

xzj-

nc

nc

x

w

x

d

y y

w

x

d

y

p

x

y

cc

ync

y

Ak

d

Ak

d

dydxHEdydxzHEP

v

kH

E

d

kf

Hyd

nA

k

jzyxE

ENEE482 54

0nfor Np/m 22

22

2

lossconductor toduen Attenuatio

2

2

222

0

0c

d

kR

d

R

Ak

wRdxJ

RP

P

P

ssc

nc

sw

x ss

ENEE482 55

TE Modes

zj-

zj-

22

22

2

e sin),,(

e cos),,(

cos),(

)(

,....3,2,1 , , 0A

d 0,yat 0),(

cossin ),(

0),(

yd

nB

k

jzyxE

yd

nBzyxH

yd

nByxh

d

nk

nndk

yxe

ykBykAyxh

yxhky

nc

x

nz

nz

c

x

ccz

zc

ENEE482 56Np/m 2

0nFor )Re(4

2

1ˆ

2

1

2 ,

Z

:is modes TM theof impedance waveThe

22

0E ,e sin),,(

2

c

2

2

*

0 00 0

*0

g

TE

yzj-

dk

Rk

Bk

dw

dydxHEdydxzHEP

v

k

H

E

d

kf

Hyd

nB

k

jzyxH

sc

nc

y

w

x

d

y x

w

x

d

y

p

y

x

cc

xnc

y

ENEE482 57

COUPLED LINES EVEN & ODDMODES OF EXCITATIONS

AXIS OF EVEN SYMMETRY AXIS OF ODD SYMMETRY

P.M.C. P.E.C.

EVEN MODE ELECTRICFIELD DISTRUBUTION

ODD MODE ELECTRIC FIELD DISTRIBUTION

eZ0 oZ0=EVEN MODE CHAR. IMPEDANCE

=ODD MODE CHAR. IMPEDANCE

Equal currents are flowing in the two lines

Equal &opposite currents areflowing in the two lines

ENEE482 58

WAVEGUIDES

• HOLLOW CONDUCTORS RECTANGULAR OR CIRCULAR.

• PROPAGATE ELECTROMAGNETIC ENERGY ABOVE A CERTAIN FREQUENCY (CUT OFF)

• INFINITE NUMBER OF MODES CAN PROPAGATE, EITHER TE OR TM MODES

• WHEN OPERATING IN A SINGLE MODE, WAVEGUIDE CAN BE DESCRIBED AS A TRANSMISSION LINE WITH C/C IMPEDANCE Zc & PROPAGATION CONSTANT

ENEE482 59

WAVEGUIDE PROPERTIES• FOR A W/G FILLED WITH DIELECTRIC r

MODES TM FOR 377

MODES TE FOR 377

:IS ZIMPEDANCE WAVE

H WAVELENGTOFF CUT IS

TH WAVELENGGUIDE IS

SPACE FREE IN H WAVELENGTIS

DIELECTRIC IN TH WAVELENGIS

E WHER111

1

1

w

1

22221

g

r

g

r

W

C

g

cg

r

Z

ENEE482 60

• PROPAGATION PHASE CONSTANT:

LENGTH ITRADIANS/UN 2

g

• FOR RECTANGULAR GUIDE a X b, CUTOFFWAVELENGTH OF TE10 MODES ARE:

rc

cC fa

8.11 , 2

cf CUT OFF FREQUENCY IN GHz (c INCHES):

• FOR CIRCULAR WAVEGUIDE OF DIAMETER D CUTOFF WAVE LENGTH OF TE11 MODE IS:cD• DOMINANT MODES ARE TE10 AND TE11 MODE FOR RECTANGULAR & CIRCULAR WAVEGUIDES

ENEE482 61

RECTANGULAR WAVEGUIDE MODE FIELDS

y

xz

a

b

CONFIGURATION

ENEE482 62

TE modes

zjmnz

yyxxz

cyx

yx

c

z

c

zc

eb

yn

a

xmAzyxH

ykDykCxkBxkAyxh

kkk

kdy

Yd

Yk

dx

Xd

X

kdy

Yd

Ydx

Xd

X

yYxXyxh

kk

yxhkyx

coscos),,(

)sincos)(sincos(),(

1 ,

1

011

)()(),(

0),(

222

22

22

2

2

22

2

2

2

222

22

2

2

2

ENEE482 63

TEmn MODES

22

2/12222222

2220

2

2

)()(2

1

2

)(

2 ;

)()(k ;

e )sin( )cos(

e )cos( )sin(

,

0 , e )cos( )cos(

b

n

a

mkf

anbm

abkk

b

n

a

mZkZ

b

yn

a

m

bk

njH

b

yn

a

xm

ak

mjH

HZEHZE

Eb

yn

a

xmH

ccmn

cc

ch

zj

cy

zj

cx

xhyyhx

zzj

z

ENEE482 64

The dominant mode is TE10

10

10

10

2 2

2310*

10 20 0

2 32 2

10 2

2 3 23

cos

sin

sin

0

/ , k ( / )

1ˆRe Re( )

2 4

( )2 2 2

(2 )

j zz

j zy

j zx

x z y

c

a b

x y

ss s

C

sc

xH A e

aj a x

E A ea

j a xH A e

aE E H

k a a

a A bP E H zdydx

R a aP J d R A b

Rb a k

a b k

Np/m

ENEE482 65

2/12222222

2220

2

2

ee

)(

2 ;

)()(k ;

e )cos( )sin(

e )sin( )cos(

Z/ , /Z

0

e )sin( )sin(

anbm

abkk

b

n

a

m

k

ZZ

b

yn

a

m

bk

njE

b

yn

a

xm

ak

mjE

EHEH

Hb

yn

a

xmE

cc

ce

zj

cy

zj

cx

xyyx

z

zjz

TMmn MODES

ENEE482 66

TE Modes of a Partially Loaded Waveguide

x

y

m0

22

2

22

2

TE have no y - variation and the structure is uniform in the y-direction

0 for 0 x tx

0 for t x ax

, are the cutoff wavenumbers for dielectric and air regions

d z

a z

d a

k h

k h

k k

2 2 2 2r 0 0

cos sin for 0 x t

cos ( ) sin ( ) for t x a

d a

d dz

a a

k k k k

A k x B k xh

C k a x D k a x

ENEE482 67

toyieldscan hat equation t sticcharacteri theis This

0)(tantan

)(coscos

)(sinsink

A-

tat x continuous are ),(E , 0DB

a xand 0at x 0E that conditionsBoundary esatisfy th To

axfor t )(cos)(sin[

tx0for ]cossin[

d

y

y

0

0

takktkk

takCtkA

takk

Ctk

H

xakDxakCk

j

xkBxkAk

j

e

adda

ad

aa

d

x

aaa

ddd

y

ENEE482 68

CIRCULAR WAVEGUIDE MODES

x

y

r

a

z

ENEE482 69

TE Modes

0)(

0 , 1

0111

)()(),(

0),(11

),(),,(

0

2222

22

22

22

2

2

22

2

22

2

22

2

22

2

22

Rkkd

dR

d

Rd

kd

dk

d

d

kd

d

d

dR

Rd

Rd

R

Rh

hk

ehzH

HkH

c

c

z

zc

zjzz

zz

ENEE482 70

zjcn

c

cn

cn

cncn

cncn

c

ekJnBnAk

jz

E

kJnBnA

kY

kYkJ

kDYkJ

Rnkd

dR

d

Rd

nknBnA

)( ) cossin(),,(E

aat 0),( conditionboundary The

)( ) cossin(),(h

0D 0at infinite is )(

kinds. second andfirst of function Bessel theare )(),(

)()(C)R(

:is solution The equation. ial DifferentsBessel'

0)(

, cossin)(

z

2222

22

22

ENEE482 71

zjcn

c

zjcn

c

zjcn

c

zjcn

c

nmccnm

nmc

nmcnm

nnmnmncn

ekJnBnAk

njH

ekJnBnAk

jH

ekJnBnAk

jE

ekJnBnAk

njE

a

pkf

a

pkkk

a

pk

JppJakJ

)()sincos(

)()cossin(

)()cossin(

)()sincos(

22

)( ,

ofroot mth 0)( , 0)(

2

2

2222nm

ENEE482 72

0

)(cos

)(sin

)(sin

)(cos

)(sin

TE is ModeDominant

12

1

1

12

1

11

z

zjc

c

zjc

c

zjc

c

zjc

c

zjcz

TE

E

ekJAk

jH

ekJAk

jH

ekJAk

jE

ekJAk

jE

ekJAH

k

H

E

H

EZ

ENEE482 73

TEnm MODES

nmcc

nmh

nnm

hh

zj

c

nmn

zj

c

nmnnm

z

zjnmnz

pakk

apkZZ

xp

HZHZE

n

ne

rk

apJjnH

n

ne

ak

apJpjH

E

n

ne

a

pJH

/2 ;

/k ; /

)(J of zerosth m' theis

E ;

)cos(

)sin(

)/(

)sin(

)cos(

)/(

0

)sin(

)cos(

222

c0

2

2

ENEE482 74

nmcc

nme

nnm

ee

zj

c

nmn

zj

c

nmnnm

z

zjnmnz

pakk

apkZZ

xp

ZEZEH

n

ne

rk

apJjnE

n

ne

ak

apJpjE

H

n

ne

a

pJE

/2 ;

/k ; /

)(J of zerosth m' theis

/H ; /

)cos(

)sin(

)/(

)sin(

)cos(

)/(

0

)sin(

)cos(

222

c0

2

2

TMnmMODES

ENEE482 75

TM01

TE11

TM11

TE01TE21 TE31

TM21

0 1 fc/fcTE11

Cutoff frequencies of the first few TEAnd TM modes in circular waveguide

ENEE482 76

ATTENUATION IN WAVEGUIDES

• ATTENUATION OF THE DOMINANT MODES (TEm0) IN A COPPER RECTANGULAR WAVEGUIDE DIM. a X b, AND (TE11) CIRCULAR WAVEGUIDE, DIA. D ARE:

lengthdB/unit

1

42.0108.3

lengthdB/unit

1

21

109.1

2

2

4

)(c

2

2

4

(

11

)0

f

f

f

f

D

fx

f

f

f

f

a

b

b

fx

c

c

rTE

c

c

rTEc m

WHERE f IS THE FREQUENCY IN GHz

ENEE482 77

ATTENUATION IN COPPER WAVEGUIDESDUE TO CONDUCTOR LOSS

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

1 1.2 1.4 1.6 1.8 2 2.2

(f/fc)

Alfa

*a/S

qrt

(ep

sr*f

(GH

z))

dB

(G

Hz)

(-1

/2)

Alfa TE0m;b/a=.45

Alfa TEm0;b/a=.5

Alfa Circ. TE11

E

a

b

RectangularGuide

E

a

Circular Guide

ENEE482 78

Higher Order Modes in Coaxial Line

TE Modes:

c

cncncncn

cncn

cncn

c

cncnz

k

akYbkJbkYakJ

bkYDbkJC

akYDakJC

k

kDYkCJnBnAh

for solve toequation sticcharacteri theis This

)()()()(

0)()(

0)()(

ba,at 0)( EconditionsBoundary

))()()(cossin(),(

ENEE482 79

Grounded Dielectric Slab

d

x

zDielectric

Ground plane TM Modes

20

22220

2

zj-z

2202

2

2202

2

z-j

,Let

e),(),,(E

xdfor 0),(x

dx0for 0),(x

variatione Assume

khkk

yxezyx

yxek

yxek

rc

z

z

zr

ENEE482 80

., equations two theSolving

)1( tan

cos , sin

0, 0

0

dat x continuous

dat x continuous

at

0at 0

:are conditionsBoundary

xdfor ),(

dx0for cossin),(

20

22

r

hk

khkhdkk

eh

Ddk

k

ADedkA

C B

HEH

(x,y,z)H

(x,y,z)E

x(x,y,z)E

x(x,y,z)E

DeCeyxe

xkBxkAyxe

c

rcrcc

hdc

c

hdc

zyx

y

z

z

z

hxhxz

ccz

ENEE482 81

Stripline

w

x

y

b

z

Approximate Electrostatic Solution:

0-a/2 a/2

y

b0,y&

2/at 0),(

0),(2

axyx

yxt

b/2

ENEE482 82

1

1

1

1

byb/2for )(

coshcos

b/2y0for coshcos

b/2yat continuous bemust Potential

byb/2for )(

sinhcos

b/2y0for sinhcos),(

nn

nn

y

y

nn

nn

nn

a

ybn

a

xn

a

nA

a

yn

a

xn

a

nA

E

yE

BA

a

ybn

a

xnB

a

yn

a

xnA

yx

ENEE482 83

wdxQ

a

bnAdyyxEV

abnn

awnaA

a

bn

a

n

a

nA

byxDbyxDx

w

wx

w

w s

oddn

n

b

y

rn

oddn

nr

yy

2/

2/

1

2/

0

02

10

s

s

2sinh),0(

)2/cosh()(

)2/sin(2

2coshcos)(2

)2/,()2/,()(

2/xfor 0

2/xfor 1)(Let

ENEE482 84

impedance sticcharacteri theis

1

)2/cosh()(2

sinh)2/sin(2

0

0

1 02

Z

cCCvC

LZ

abnnabn

awna

w

V

QC

r

p

oddn r

ENEE482 85

Microstrip

d

w

x

y

solution ticElectrosta eApproximatAn

1

constant. dielectric effective theis

, 0

re

e

e

e

p kc

v

0,yat 0),(

, 2/at 0),(

0),(2

yx

axyx

yxt

-a/2 a/2

ENEE482 86

1

d)/a-(yn-

1

1

1

ydfor e(

sinhcos

dy0for coshcos

sindyat continuous bemust Potential

ydfor cos

dy0for sinhcos),(

nn

nn

y

y

a

dn

nn

n

a

yn

n

nn

a

dn

a

xn

a

nA

a

yn

a

xn

a

nA

E

yE

eBa

xnA

ea

xnB

a

yn

a

xnA

yx

ENEE482 87

wdxQ

a

dnAdyyxEV

adnadnn

awnaA

a

dn

a

dn

a

n

a

nA

dyxDdyxDx

w

wx

w

w s

oddn

n

d

y

rn

r

oddn

n

yy

2/

2/

10

02

10

s

s

sinh),0(

)/cosh()/[sinh()(

)2/sin(4

coshsinhcos)(2

),(),()(

2/xfor 0

2/xfor 1)(Let

ENEE482 88

impedance sticcharacteri theis

1

1)( dielectricair an

withline microstrip theoflength unit per eCapacitanc

constant dielectric a

with line microstrip theoflength unit per eCapacitanc

)/cosh()/[sinh()(

sinh)2/sin(2

1

0

0

0

r

0

r

1 02

Z

cCCvZ

C

C

C

C

adnadnwna

dnawnaV

QC

e

p

e

oddn rr

ENEE482 89

The Transverse Resonance Technique

yyy allfor 0)(Z)(Z

zero bemust sideeither tolookingseen impedances

input The line, on theanypoint at line,resonant aFor

inrin

TM Modes for the parallel plate waveguide

w

d

x

y

0

d

y

)(Zrin y

)(Zin y

ENEE482 90

0,1,2,..nfor

0cos)(cos

sin

0]tan)([tan

resonance ersefor transvCondition

tan)(

)(tan)(

/Z0

d

nkk

ykydk

ykjZ

ykydkjZ

ykjZyZ

ydkjZyZ

k

kkZ

yc

yy

yTM

yyTM

yTMin

yTMrin

yTM

ENEE482 91

b

a er1

MODES IN DIELTECTRIC LOADED WAVEGUIDE

er2

CATEGORIES OF FIELD SOLUTIONS:• TE0m MODES• TM0m MODES• HYBRID HEnm MODES

ENEE482 92

BOUNDARY CONDITIONS

FIELDS SATISFY THE WAVE EQUATION,SUBJECT TO THE BOUNDARY CONDITIONSEz , E , Hz , H ARE CONTINUOUS AT r=bEz , E VANISH AT r=a

)(

/)(

/

sin

)(

/)(

/

1cos

sin)(j

cos)( ar0for

1

1122

1

1

1122

1

1

1

1

1

rJ

rrJ

nk

nnAHj

E

rJ

rrJ

kn

nnAHjE

nrAJH

nrAJE

n

nr

n

nr

nz

nz

ENEE482 93

)(

/)(

/

sin

)(

/)(

/

1cos

sin)(j

cos)( brafor

2

1222

22

2

1222

22

2

2

rP

rrR

nk

nnAHj

E

rR

rrP

kn

nnAHjE

nrPAH

nrRAE

n

nr

n

nr

nz

nnz

WHERE A IS AN ARBITRARY CONSTANT

ENEE482 94

Characteristic equation

0220

2 nnnn WVakUG

Where z=a is the radial wave number in

)(

)(

)(

)(

11)(

k ; k ;

)( ;

2

2

1

1

222

221

1

0020

202

22

201

21

222

22

221

21

a

aP

a

aJV

aaanJU

kkk

kk

nnn

nn

rr

ENEE482 95

)()()()(

)()()()()()(

)()()()(

)()()()()()(

)()(

2222

222212

2222

222212

2

22

1

11

bKaIbIaK

bKrIbIrKaJrR

bKaIbIaK

bKrIbIrKaJrP

a

aR

a

aJW

nnnn

nnnnnn

nnnn

nnnnnn

nr

nrn

ENEE482 96

For n = 0, the Characteristic Equation Degenerates in twoSeparate Independent Equations for TE and TM Modes:

0)(

)(

)(

)(

2

2

1

1

a

aP

a

aJV nn

n

For TE ModesAnd:

0)()(

2

22

1

11

a

aR

a

aJW n

rn

rn

For TM Modes

ENEE482 97

COMPLEX MODES

• COMPLEX PROPAGATION CONSTANT :j• ONLY HE MODE CAN SUPPORT COMPLEX WAVES• PROPAGATION CONSTANT OF COMPLEX MODESARE CONJUGATE : j

• COMPLEX MODES DON’T CARRY REAL POWER• COMPLEX MODES CONSTITUTE PART OF THE COMPLETE SET OF ELECTROMAGNETIC FIELDSPACE• COMPLEX MODES HAVE TO BE INCLUDED IN THEFIELD EXPANSIONS FOR CONVERGENCE TO CORRECT SOLUTIONS IN MODE MATCHING TECHNIQUES.

ENEE482 98

OPTICAL FIBER

2a

Step-index fiber

IN CIRCULAR CYLINDRICAL COORDINATES:

)( ; )(

cos

sin)( H;

cos

sin)(H

sin

cos)( E;

sin

cos)(E

ar a r

arfor 0i

a,rfor 1i ; 0)(11

2/12202

2/1221

2z21z1

2z21z1

222

2

22

2

1

kkkk

n

nrkBK

n

nrkBJ

n

nrkAK

n

nrkAJ

EkE

rr

E

rr

E

cc

cncn

cncn

zizzz

ENEE482 99

For the symmetric case n=0, the solution break into Separate TE and TM sets. The continuity condition for Ez1= Ez2

and = Hat r=a gives for the TM set:

akK

akK

ak

ak

akJ

akJ

c

c

c

c

c

c

20

21

21

10

10

11 )(

)(

)(

The continuity condition for Hz1= Hz2

and = Eat r=a gives for the TE set:

akK

akK

k

k

akJ

akJ

c

c

c

c

c

c

20

21

2

1

10

11 )(

)(

)(

If n is different from 0, the fields do not separate into TM and TE types, but all the fields become coupled throughcontinuity conditions.

ENEE482 100

Parallel Plate Transmission Line

a

bc

Partially loaded parallel Plate waveguide

y

x

20

22

22

220222

k)1(

region dielectric for the

regionair for Let

region dielectric thein -k

regionair thein -k , 0 :modesTM

with variationno , variation Assume

r

c

czczt

zj

p

pk

keke

xe

ENEE482 101

regionair for

region dielectricFor )(

1at continuous is

at continuous is )( , ,0at 0)(

for 0

0for 0

2

2

22

22

2

22

2

y

e

p

jy

ej

ye

y

e

py

eayH

ayyebyye

byaepdy

ed

ayedy

ed

z

z

y

ay

z

ay

zrx

zz

zz

zz

ENEE482 102

pp

pcpa

pcCp

aC

pcCaC

byaybpCye

ayyCye

y

e

p

Yjky

eYkj

yh

r

r

r

z

z

z

zr

x

, k)1( usly withsimultaneo

solved bemust equation talTransceden tantan

cos1

cos

sin sin

)(sin)(

0 sin)(

regionair for

region dielectricFor )(

20

22

21

21

2

1

20

200

ENEE482 103

20

20

20

0000

0

00

)1(

tantan

: then of valueingcorrespond thebe toLet

ifoccur can and between of valueThe

kp

cppa

jppkk

r

r

lly.exponentia decays field theand is variationThe

imaginary is if tingnonpropaga be willmodes theofMost

k 22220

ze

lkp

ENEE482 104

Low Frequency Solution

When the frequency is low,

constant dielectric effective theis

k

is for solution The k)1(

or

k)1(

small very are and ,number small very is k

002

020

202

0

202

020

20

20

0020

e

er

r

r

r

rr

r

kkca

bp

ca

ap

a

cpp

cpa

p

ENEE482 105

W

bLWJI

LIWbJdxdyHW

W

CLC

zz

zz

b w

w xm

m

e

2 2

4

1

24

isenergy magnetic stored average timeThe

meter.per

ecapacitanc and inductance static theare , L

0

220

0

20

a

bc

y

x-W W

ENEE482 106

c

acCjYC

Ykjh

c

bjCC

je

yCe

ay

cpajCcpjaCC

r

rrrx

ry

z

)1(

)(

)1(

0

/sinh/sin

1010

00

110

01

0010012

LCac

bLC

ac

WC

a

WC

c

WC

CC

CCC

r

r

r

r

rda

da

da

2

2,

2 , ecapacitanc The

00

0

00

ENEE482 107

LCac

bLC

ac

WC

a

WC

c

WC

CC

CCC

r

r

r

r

rda

da

da

2

2,

2 , ecapacitanc The

00

0

00

c

bjCC

je

yCe

ay

cpajCcpjaCC

ry

z

)1(

0

/sinh/sin

:sexpression Field

110

01

0010012

ENEE482 108

xzz

rr

b

y

r

rrx

rry

z

r

rrrx

WHWJI

cac

bjCdyeV

c

acCjYC

Ykjh

c

bjCC

cp

aje

c

ybaCe

c

acCjYC

Ykjh

22

)()1(

)1(

)(

)1(

)(

:regionair theIn

)1(

)(

1

0

1010

00

1120

0

01

1010

00

ENEE482 109

mod Ean is npropagatio of mode thefrequency hight At

mode)TEM -(quasi modeTEM a becomed npropagatio

of modedominant thelimit,frequency low theIn

)(

20

C

Lbca

W

Z

I

VZ

r

rc

:is impedance sticcharacteri The

ENEE482 110

High Frequency Solution:

mode. wave

surface called is mode of typeThis sheet. dielectric by the guided is

field The large. is )( as long as b on dependnot does and

surface dielectric-air thefromaway lly exponentia decays field The

afor sinsin

)(sinh

)(sinhsin)(sinh)(

0for sin)(

oft independen is solution The )1(

)1(tanh

1tanh

large. are , and frequency highAt

00

)(01)(

)(

01

0

00102

01

20

20

20

20

20000

0

000

0

0

0

abpcp

byaeCe

eaC

abp

ybpaCybpjCye

ayyCye

bkp

kpa

cp

pk

aypabp

ybp

z

z

r

rrr

ENEE482 111

Microstrip Transmission Line

H

w

yyyzzxxr

s

s

aEaEaED

JDjH

AjE

AjBjEAB

HyzxJzyxJ

Hyzxzyx

ˆ)ˆˆ(

:dielectric canisotropiFor

,

)(),(),,(

)(),(),,(

00

x

y

ENEE482 112

zzrz

xxrx

yyyry

rr

r

yyyy

zxzzxxr

JAkyA

JAkyA

J

Jy

aAaj

AjjA

jA

yaaAj

za

xaaAaAjD

JDjAAA

020

2

020

2

00

0002

00

0

0

002

)(

)(

component a y havenot does

]ˆˆ)(

[-

condition) (Lorentz let

)ˆˆ(

]ˆˆ)ˆˆ([

ENEE482 113

y

AjHyHAj

kyyyx

D

HyHy

j

yrjAkyA

yryyy

ry

rry

ryr

yyy

)()()()1(

)()1)()(

)()(

0

20

22

2

2

2

r

00

0020

2

ENEE482 114

)()1(

)()1(

,

,

)(),(limlim

0

00

00

00

0

002

2

0

HAjyy

Hjy

A

Hy

AH

y

A

Jy

AJ

y

A

J

dyHyzxJy

Ady

y

A

yys

H

y

H

r

H

H

y

xz

zx

sz

H

H

zsx

H

H

x

sx

sx

H

H

H

H

xH

H

x

Boundary conditions:

ENEE482 115

yr

yr

yryr

r

y

ryyy

zr

xr

y

Ajk

yzx

yjAk

Ak

Ak

substrate isotropic anFor

1 regionair theIn

)(

)()(

0)(

0)(

:have weinterface thefromaway region substrate theIn

202

2

2

2

2

2

0020

2

20

2

20

2