TRANSMISSION MEDIA MAXWELL’S EQUATIONS AND TRANSMISSION MEDIA CHARACTERISTICS ENEE 482 Spring 2002...

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

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

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

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

ENEE482 9

Maxwell’s Equations in Large Scale Form

SdJldH

SdMSdBt

ENEE482 10

Maxwell’s Equations for the Time - Harmonic Case

DjJHMBjE

eEEejEEE

jEEajEEazyxE

ezyxEtzyxE

xrxixixr

jtjxixr

tjxixrx

yiyryxixrx

)/(tan , )cos(

]Re[])Re[(

)()(),,(

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

: then,variationse Assume

ENEE482 11

Boundary Conditions at a General Material Interface

Density Charge Surface

ENEE482 12

)(ˆ)(ˆ

Fields at a Dielectric Interface

ENEE482 13

0En oE

:ConductorPerfect aat ConditionsBoundary

+ + +n

ENEE482 14

The magnetic wall boundary condition

ENEE482 15

2/ ; 0

:medium free Sourcea For

Wave Equation

ENEE482 16

Plane Waves

zayaxar

kakakakAeE

zyxiEkz

zzyyxxzjkyjkxjk

variablesof separation Using),,,(for Solve

ENEE482 17

space. free of admittance intrinsic theis

377 space free of impedance interensic theis

waveplane called issolution The

.kn propagatio ofdirection thelar toperpendicu is vector The

00 Since

,Similarly ,

EnEnYEnEnk

eEkeEj

EkEeEE

CeEBeEAeE

rkjrkjrkj

ENEE482 18

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

:are TEM wavea of fields magnetic

and electric thein densitiesenergy average timeThe2

real is if ),cos()Re(

EEYnEnEYnHEP

EtrkEeEE tjrkj

ENEE482 19

Plane Wave in a Good Conductor

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

: Impedance surface The

EJ , 2

ENEE482 21

Reflection From A Dielectric Interface

Parallel Polarization

EnYHeEE

rrrnjk

iirnjk

ENEE482 22

sin , cos

sin sin

cossin

331333

221222

111111

3032010

nnkknnknk

nkknYY

EnYHeEE

ttrnjk

ENEE482 23

Energy and Power

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

dVEEdVDEW

thenreal, andconstant is If

ENEE482 24

:by given is S surface

closeda across smittedpower tran average timeThe

constant and real is if 4

:is field magnetic thein storedenergy average Time

ENEE482 25

Poynting Theorem

dVMHJEdVDEHB

dVMHJEdVDEHBj

dSHEVdHE

JEEDjHMBj

EHHEHE

ENEE482 26

dVEEHHj

dVEEHHdVEE

dSHEdVMHJE

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

WWjPPP

dVEEHH

dVMHJE

losspower average Time

dVEEdVEEHHPVV

ENEE482 28

Circuit Analogy

networka of impedance theof n DefinitioGeneral

IILIIjRII

jLjRIIZIIVI

ENEE482 29

Potential Theory

equation. HelmholtzousInhomogene

condition) (Lorentz or

JjAkAAA

JjAJEjAH

AjEAjE

AjBjEAB

ENEE482 30

Solution For Vector Potential

(x,y,z)(x’,y’, z’) R

rrzzyyxxR

erJrAzyxA

)()()(

current alinfinitism anfor )(4

)(),,(

ENEE482 31

Waves on An Ideal Transmission LineRg

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:

)()(),(

0),(),(

ztfItzI

ztfVtzV

ENEE482 33

Steady State Sinusoidal Waves

VYIVYIeIeIzI

eVeVzV

cczjzj

, 0)()(

ENEE482 34

Transmission Line Parameters

22P ,2/

isconductor the

ofty conductivi finite toduelength unit per lossPower

:line ofsection m 1for energy magnetic stored

average- timeThe .I becurrent theand V

:be conductors ebetween th voltageLet the

VGdSEEIRdHH

ENEE482 35

Terminated Transmission Line

To generator

tcoefficien on Reflecti

ENEE482 36

(sin41

)1)(1(Re2

eeVeeV

ENEE482 37

Transmission Lines & Waveguides

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

)()()(

,,,,,,

zttztt

zjztzzztzt

zjztzt

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

eeaYehH

eyxeeE

entialScalar Pot 0,,

eha , 0

hea , 0

ENEE482 40

wavesTEMfor

0])([ , 0)(

, but , 0

:equation Helmholtzsatisfy must field The

direction z-or in then propagatio for wave

Impedance Wave , 1

ENEE482 41

TE WAVES

let , 0)(

0),()(

222222

tzztztt

ttzztt

ejhajhah

heahje

ENEE482 42

Impedance Wave

; ˆˆ

ENEE482 43

Admittance Wave

let , 0)(

0),()(

222222

TM WAVES

ENEE482 44

TEM TRANSMISSION LINES

Parallel -plate Two-wire Coaxial

ENEE482 45

COAXIAL LINES

jkz-00

e a )/ln(

and e a )/ln(

0at 0,at ln

0for 01

rarVCrCrr

ENEE482 46

)/ln(ˆRe

e)/ln(

)/ln(I

eˆ)/ln(

jkz-00

VYrdrdaHEP

VYHaHnJ

• THE CHARACTERISTIC IMPEDANCE OF A COAXIAL IS Z0

Ohms ln2

ENEE482 47

Zc OF COAXIAL LINE AS A FUNCTION OF b/a

r Zo=X

ENEE482 48

Transmission line with small losses

kkEaYHeE

kkjjkjjjk

jYYjkk

ty conductivi the toequivalent is

, ˆ ,

losses small For )(

)( and )(

ENEE482 490

, )/ln(2

:lossconductor the todue losspower The

:is lengthunit per losspower The

dJJZPj

dSEEdSJJP

ENEE482 50

Qc OF COAXIAL LINE AS A FUNCTION OF Zo

Coaxia

34000 10

ENEE482 51

Parallel Plate Waveguide

jkzjkz

wVdxzHydxzJI

edyEVed

VxEzyxH

VyeyxeyxE

ˆ)ˆ(ˆ

V ,ˆˆ),(

ˆ),(),(

ˆ),( , ),(

Vd)(x,

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

w,x0 0),(

TEM Modes

ENEE482 52

TM modes

e cos),,(

e sin),,(

sin),(

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

d 0,yat 0),(

cossin ),(

nAzyxE

ykBykAyxe

ENEE482 53

0nfor 2

0nfor 4

:is modes TM theof impedance waveThe

0E ,e cos),,(

0 00 0

dydxHEdydxzHEP

ENEE482 54

0nfor Np/m 22

lossconductor toduen Attenuatio

ENEE482 55

TE Modes

e sin),,(

e cos),,(

cos),(

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

d 0,yat 0),(

cossin ),(

nBzyxH

ykBykAyxh

ENEE482 56Np/m 2

0nFor )Re(4

:is modes TM theof impedance waveThe

0E ,e sin),,(

0 00 0

dydxHEdydxzHEP

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

ENEE482 60

• PROPAGATION PHASE CONSTANT:

LENGTH ITRADIANS/UN 2

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

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

CONFIGURATION

ENEE482 62

TE modes

xmAzyxH

ykDykCxkBxkAyxh

yYxXyxh

yxhkyx

coscos),,(

)sincos)(sincos(),(

)()(),(

ENEE482 63

TEmn MODES

2/12222222

)()(k ;

e )sin( )cos(

e )cos( )sin(

0 , e )cos( )cos(

HZEHZE

xhyyhx

ENEE482 64

The dominant mode is TE10

10 20 0

2 32 2

2 3 23

/ , k ( / )

1ˆRe Re( )

( )2 2 2

xH A e

aj a x

E A ea

j a xH A e

aE E H

a A bP E H zdydx

R a aP J d R A b

Rb a k

ENEE482 65

2/12222222

)()(k ;

e )cos( )sin(

e )sin( )cos(

Z/ , /Z

e )sin( )sin(

TMmn MODES

ENEE482 66

TE Modes of a Partially Loaded Waveguide

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

2 2 2 2r 0 0

cos sin for 0 x t

cos ( ) sin ( ) for t x 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

tat x continuous are ),(E , 0DB

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

axfor t )(cos)(sin[

tx0for ]cossin[

takktkk

takCtkA

xakDxakCk

xkBxkAk

ENEE482 68

CIRCULAR WAVEGUIDE MODES

ENEE482 69

TE Modes

)()(),(

0),(11

),(),,(

ENEE482 70

ekJnBnAk

kJnBnA

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'

, cossin)(

ENEE482 71

nmccnm

nnmnmncn

ekJnBnAk

JppJakJ

)()sincos(

)()cossin(

)()sincos(

ofroot mth 0)( , 0)(

2222nm

ENEE482 72

TE is ModeDominant

ENEE482 73

TEnm MODES

zjnmnz

apJjnH

apJpjH

/k ; /

)(J of zerosth m' theis

ENEE482 74

zjnmnz

apJjnE

apJpjE

/k ; /

)(J of zerosth m' theis

/H ; /

TMnmMODES

ENEE482 75

TE01TE21 TE31

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

42.0108.3

lengthdB/unit

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 TE0m;b/a=.45

Alfa TEm0;b/a=.5

Alfa Circ. TE11

RectangularGuide

Circular Guide

ENEE482 78

Higher Order Modes in Coaxial Line

TE Modes:

cncncncn

akYbkJbkYakJ

bkYDbkJC

akYDakJC

kDYkCJnBnAh

for solve toequation sticcharacteri theis This

)()()()(

ba,at 0)( EconditionsBoundary

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

ENEE482 79

Grounded Dielectric Slab

zDielectric

Ground plane TM Modes

e),(),,(E

xdfor 0),(x

dx0for 0),(x

variatione Assume

yxezyx

ENEE482 80

., equations two theSolving

)1( tan

cos , sin

dat x continuous

:are conditionsBoundary

xdfor ),(

dx0for cossin),(

khkhdkk

ADedkA

(x,y,z)H

(x,y,z)E

x(x,y,z)E

DeCeyxe

xkBxkAyxe

ENEE482 81

Stripline

Approximate Electrostatic Solution:

0-a/2 a/2

2/at 0),(

ENEE482 82

byb/2for )(

coshcos

b/2y0for coshcos

b/2yat continuous bemust Potential

byb/2for )(

sinhcos

b/2y0for sinhcos),(

ENEE482 83

bnAdyyxEV

byxDbyxDx

2sinh),0(

)2/cosh()(

)2/sin(2

2coshcos)(2

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

2/xfor 0

2/xfor 1)(Let

ENEE482 84

impedance sticcharacteri theis

)2/cosh()(2

sinh)2/sin(2

abnnabn

oddn r

ENEE482 85

Microstrip

solution ticElectrosta eApproximatAn

constant. dielectric effective theis

0,yat 0),(

, 2/at 0),(

-a/2 a/2

ENEE482 86

d)/a-(yn-

ydfor e(

sinhcos

dy0for coshcos

sindyat continuous bemust Potential

ydfor cos

dy0for sinhcos),(

ENEE482 87

dnAdyyxEV

adnadnn

dyxDdyxDx

sinh),0(

)/cosh()/[sinh()(

)2/sin(4

coshsinhcos)(2

),(),()(

2/xfor 0

2/xfor 1)(Let

ENEE482 88

impedance sticcharacteri theis

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

adnadnwna

dnawnaV

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

TM Modes for the parallel plate waveguide

)(Zrin y

)(Zin y

ENEE482 90

0,1,2,..nfor

0cos)(cos

0]tan)([tan

resonance ersefor transvCondition

)(tan)(

ykydkjZ

ykjZyZ

ydkjZyZ

yTMrin

ENEE482 91

MODES IN DIELTECTRIC LOADED WAVEGUIDE

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)(j

cos)( ar0for

nnAHjE

ENEE482 93

sin)(j

cos)( brafor

nnAHjE

WHERE A IS AN ARBITRARY CONSTANT

ENEE482 94

Characteristic equation

2 nnnn WVakUG

Where z=a is the radial wave number in

k ; k ;

aaanJU

ENEE482 95

)()()()(

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

)()()()(

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

222212

bKaIbIaK

bKrIbIrKaJrR

bKaIbIaK

bKrIbIrKaJrP

nnnnnn

ENEE482 96

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

aJV nn

For TE ModesAnd:

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

Step-index fiber

IN CIRCULAR CYLINDRICAL COORDINATES:

)( ; )(

sin)( H;

sin)(H

cos)( E;

cos)(E

ar a r

arfor 0i

a,rfor 1i ; 0)(11

2/12202

2/1221

2z21z1

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:

The continuity condition for Hz1= Hz2

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

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

Partially loaded parallel Plate waveguide

220222

region dielectric for the

regionair for Let

region dielectric thein -k

regionair thein -k , 0 :modesTM

with variationno , variation Assume

ENEE482 101

regionair for

region dielectricFor )(

1at continuous is

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

0for 0

ayyebyye

byaepdy

ENEE482 102

byaybpCye

ayyCye

, k)1( usly withsimultaneo

solved bemust equation talTransceden tantan

sin sin

)(sin)(

0 sin)(

regionair for

region dielectricFor )(

ENEE482 103

tantan

: then of valueingcorrespond thebe toLet

ifoccur can and between of valueThe

lly.exponentia decays field theand is variationThe

imaginary is if tingnonpropaga be willmodes theofMost

k 22220

ENEE482 104

Low Frequency Solution

When the frequency is low,

constant dielectric effective theis

is for solution The k)1(

small very are and ,number small very is k

ENEE482 105

LIWbJdxdyHW

isenergy magnetic stored average timeThe

meter.per

ecapacitanc and inductance static theare , L

ENEE482 106

acCjYC

cpajCcpjaCC

/sinh/sin

0010012

2 , ecapacitanc The

ENEE482 107

2 , ecapacitanc The

cpajCcpjaCC

/sinh/sin

:sexpression Field

0010012

ENEE482 108

bjCdyeV

acCjYC

:regionair theIn

ENEE482 109

mod Ean is npropagatio of mode thefrequency hight At

mode)TEM -(quasi modeTEM a becomed npropagatio

of modedominant thelimit,frequency low theIn

: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

large. are , and frequency highAt

)(01)(

byaeCe

ybpaCybpjCye

ayyCye

aypabp

ENEE482 111

Microstrip Transmission Line

yyyzzxxr

aEaEaED

AjBjEAB

HyzxJzyxJ

Hyzxzyx

ˆ)ˆˆ(

:dielectric canisotropiFor

)(),(),,(

ENEE482 112

zxzzxxr

xaaAaAjD

JDjAAA

component a y havenot does

]ˆˆ)(

condition) (Lorentz let

)ˆˆ(

]ˆˆ)ˆˆ([

ENEE482 113

AjHyHAj

yrjAkyA

)()()()1(

)()1)()(

ENEE482 114

)(),(limlim

dyHyzxJy

Boundary conditions:

ENEE482 115

substrate isotropic anFor

1 regionair theIn

:have weinterface thefromaway region substrate theIn

TRANSMISSION MEDIA MAXWELL’S EQUATIONS AND TRANSMISSION MEDIA CHARACTERISTICS ENEE 482 Spring 2002...

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