electromagnetic waves maxwell`s equations, transmission...

Electromagnetic WavesMaxwell`s Equations,

Transmission Lines andWaveguides

By: Dr. Supreet Singh

POYNTING VECTOR

•This is a measure of power per area. Unitsare watts per meter2.

•Direction is the direction in which the waveis moving.

S E B 1

0S E B

1

0

POYNTING VECTOR

•However, since E and B are perpendicular,

S EB

E

Bc

Sc

Ec

B

1

1

0

0

2

0

2

and since

S EB

E

Bc

Sc

Ec

B

1

1

0

0

2

0

2

and since

MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS

E A E s

B A B s

dq

dd

dt

d d id

dt

B

E

0

0 00

E A E s

B A B s

dq

dd

dt

d d id

dt

B

E

0

0 00

AMPERE’S LAWOriginal:

As modified by Maxwell:

B s

B s

d i

d id

dtE

0

0 0

Original:

As modified by Maxwell:

B s

B s

d i

d id

dtE

0

0 0

Reasons for the Extra Term

SYMMETRY

CONTINUITY

SYMMETRY

A time varying magnetic field produces anelectric field.

A time varying electric field produces amagnetic field.

SYMMETRYMaxwell' s Equations in Free Space

q i

d dd

dt

d dd

dt

B

E

0 0

0

0 0 0

E A E s

B A B s

Maxwell' s Equations in Free Space

q i

d dd

dt

d dd

dt

B

E

0 0

0

0 0 0

E A E s

B A B s

CONTINUITY

CONTINUITYWithout Maxwell' s modification:

At

At

At

P d i

P d

P d i

1 0

2

3 0

0

:

:

:

B s

B s

B s

Without Maxwell' s modification:

At

At

At

P d i

P d

P d i

1 0

2

3 0

0

:

:

:

B s

B s

B s

CONTINUITYWith Maxwell's modification:

At

At

At

"Displacement current"

P d i

P dd

dti

P d i

d

dti

Ed

Ed

1 0

2 0 0 0

3 0

0

:

:

:

B s

B s

B s

With Maxwell's modification:

At

At

At

"Displacement current"

P d i

P dd

dti

P d i

d

dti

Ed

Ed

1 0

2 0 0 0

3 0

0

:

:

:

B s

B s

B s

CONTINUITYi i

q CV

CA

dV Ed

qA

dEd AE

idq

dt

d

dti

d

E

Ed

0

0 0 0

0

i i

q CV

CA

dV Ed

qA

dEd AE

idq

dt

d

dti

d

E

Ed

0

0 0 0

0

AMPERE’S LAW

AMPERE’S LAWFor path 1:

For path 2:

For path 3:

since

B s

B s

B s

d i

d i i

d i i

i id

d

0

0 0

0 0

( )

( )

For path 1:

For path 2:

For path 3:

since

B s

B s

B s

d i

d i i

d i i

i id

d

0

0 0

0 0

( )

( )

ELECTROMAGNETICWAVES


q i

d dd

dt

d dd

dt

B

E

0 0

0

0 0 0

E A E s

B A B s


q i

d dd

dt

d dd

dt

B

E

0 0

0

0 0 0

E A E s

B A B s

ELECTROMAGNETICWAVES

E

x

B

t

B

x

E

t

E

B

E

Bc

c

m

m

0 0

0 0

8130 10. m/s

E

x

B

t

B

x

E

t

E

B

E

Bc

c

m

m

0 0

0 0

8130 10. m/s

MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS

E A E s

B A B s

dq

dd

dt

d d id

dt

i i

B

E

d

0

0 0

0

0

( )

E A E s

B A B s

dq

dd

dt

d d id

dt

i i

B

E

d

0

0 0

0

0

( )

TRANSMISSION LINES ANDWAVEGUIDES

ReferencesTEXT BOOK

1. John D Ryder, “Networks lines and fields”, Prentice Hall of India,New Delhi, 2005

REFERENCES1.William H Hayt and Jr John A Buck, “Engineering Electromagnetics”Tata Mc Graw-Hill Publishing Company Ltd, New Delhi, 2008

2.David K Cheng, “Field and Wave Electromagnetics”, PearsonEducation Inc, Delhi, 2004

3.John D Kraus and Daniel A Fleisch, “Electromagnetics withApplications”, Mc Graw Hill Book Co,2005

4.GSN Raju, “Electromagnetic Field Theory and Transmission Lines”,Pearson Education, 2005

5.Bhag Singh Guru and HR Hiziroglu, “Electromagnetic FieldTheory Fundamentals”, Vikas Publishing House, New Delhi, 2001.

6. N. Narayana Rao, “ Elements of Engineering Electromagnetics” 6

Transmission Line

Has two conductors running parallel Can propagate a signal at any frequency (in theory) Becomes lossy at high frequency Can handle low or moderate amounts of power Does not have signal distortion, unless there is lossMay or may not be immune to interference Does not have Ez or Hz components of the fields (TEMz)

Properties

Coaxial cable (coax)Twin lead

(shown connected to a 4:1impedance-transforming balun)

3

Transmission Line (cont.)

CAT 5 cable(twisted pair)

The two wires of the transmission line are twisted to reduce interference andradiation from discontinuities.

4


Microstrip

h

w

er

er

w

Stripline

h

Transmission lines commonly met on printed-circuit boards

Coplanar strips

her

w w

Coplanar waveguide (CPW)

her

w

5


Transmission lines are commonly met on printed-circuit boards.

A microwave integrated circuit

Microstrip line

6

Fiber-Optic GuideProperties

Uses a dielectric rod Can propagate a signal at any frequency (in theory) Can be made very low loss Has minimal signal distortion Very immune to interference Not suitable for high power Has both Ez and Hz components of the fields

7

Waveguides

Has a single hollow metal pipe Can propagate a signal only at high frequency:>c The width must be at least one-half of a wavelength Has signal distortion, even in the lossless case Immune to interference Can handle large amounts of power Has low loss (compared with a transmission line) Has either Ez or Hz component of the fields (TMz or TEz)

Properties

http://en.wikipedia.org/wiki/Waveguide_(electromagnetism) 8

Lumped circuits: resistors, capacitors, inductors

neglect time delays (phase)

account for propagation andtime delays (phase change)

Transmission-Line Theory

Distributed circuit elements: transmission lines

We need transmission-line theory whenever the length ofa line is significant compared with a wavelength.

9

Transmission Line

2 conductors

4 per-unit-length parameters:

C = capacitance/length [F/m]

L = inductance/length [H/m]

R = resistance/length [/m]

G = conductance/length [ /m or S/m]

Dz

10

11


z

,i z t

+ + + + + + +- - - - - - - - - - ,v z tx x xB

11

RDz LDz

GDz CDz

z

v(z+z,t)

+

-

v(z,t)

+

-

i(z,t) i(z+z,t)

12

( , )( , ) ( , ) ( , )

( , )( , ) ( , ) ( , )

i z tv z t v z z t i z t R z L z

tv z z t

i z t i z z t v z z t G z C zt


12

RDz LDz

GDz CDz

z

v(z+z,t)

+

-

v(z,t)

+

-

i(z,t) i(z+z,t)

13

Hence

( , ) ( , ) ( , )( , )

( , ) ( , ) ( , )( , )

v z z t v z t i z tRi z t L

z ti z z t i z t v z z t

Gv z z t Cz t

Now let ∆z 0:

v iRi L

z ti v

Gv Cz t

“Telegrapher’sEquations”

TEM Transmission Line (cont.)

13

14

To combine these, take the derivative of the first one withrespect to z:

2

2

2

2

v i iR L

z z z t

i iR L

z t z

vR Gv C

t

v vL G C

t t

Switch theorder of thederivatives.


14

15

2 2

2 2( ) 0

v v vRG v RC LG LC

z t t

The same equation also holds for i.

Hence, we have:

2 2

2 2

v v v vR Gv C L G C

z t t t


15

16

Let

Convention:

Solution:

2 ZY

( ) z zV z Ae Be

1/2

( )( )R j L G j C

principal square root

2

2

2( )

d VV

dzThen


is called the "propagation constant."

/2jz z e

j

0, 0

attenuationcontant

phaseconstant

16

17


0 0( ) z z j zV z V e V e e

Forward travelling wave (a wave traveling in the positive z direction):

0

0

0

( , ) Re

Re

cos

z j z j t

j z j z j t

z

v z t V e e e

V e e e e

V e t z

g0t

z0

zV e

2

g

2g

The wave “repeats” when:

Hence:

17

18

Phase Velocity (cont.)

0

constant

t z

dz

dtdz

dt

Set

Hence pv

1/2

Im ( )( )p

vR j L G j C

In expanded form:

18

19

Characteristic Impedance Z0

0

( )

( )

V zZ

I z

0

0

( )

( )

z

z

V z V e

I z I e

so 00

0

VZ

I

+V+(z)-

I+ (z)

z

A wave is traveling in the positive z direction.

(Z0 is a number, not a function of z.)

19

20

Use Telegrapher’s Equation:

v iRi L

z t

sodV

RI j LIdz

ZI

Hence0 0

z zV e ZI e

Characteristic Impedance Z0 (cont.)

20

21

From this we have:

Using

We have

1/2

00

0

V Z ZZ

I Y

Y G j C

1/2

0

R j LZ

G j C

Characteristic Impedance Z0 (cont.)

Z R j L

Note: The principal branch of the square root is chosen, so that Re (Z0) > 0.21

22

00

0 0

j z j j z

z z

z j zV e e

V z V e V

V e e e

e

e

0

0 cos

c

, R

os

e j t

z

z

V e t

v z t V z

z

V z

e

e t

Note:

wave in +zdirection wave in -z

direction

General Case (Waves in Both Directions)

22

23

Backward-Traveling Wave

0

( )

( )

V zZ

I z

0

( )

( )

V zZ

I z

so

+V -(z)-

I - (z)

z

A wave is traveling in the negative z direction.

Note: The reference directions for voltage and current are the same asfor the forward wave.

23

24

General Case

0 0

0 00

( )

1( )

z z

z z

V z V e V e

I z V e V eZ

A general superposition of forward andbackward traveling waves:

Most general case:

Note: The referencedirections for voltageand current are thesame for forward andbackward waves.

24

+V (z)-

I (z)

z

Lossless Case

0, 0R G

1/ 2

( )( )j R j L G j C

j LC

so 0

LC

1/2

0

R j LZ

G j C

0

LZ

C

1pv

LC

pv

(indep. of freq.)(real and indep. of freq.)25

26

Lossless Case (cont.)1

pvLC

In the medium between the two conductors is homogeneous (uniform)and is characterized by (,), then we have that

LC

The speed of light in a dielectric medium is1

dc

Hence, we have that p dv c

The phase velocity does not depend on the frequency, and it is always thespeed of light (in the material).

(proof given later)

26

What if we know

@V V z and

0 0V V V e

z zV z V e V e

0V V e

0 0V V V e

Terminated Transmission Line (cont.)

0 0z zV z V e V e

Hence

Can we use z = - l asa reference plane?

Terminating impedance (load)

27

( ) ( )z zV z V e V e


0 0z zV z V e V e

Compare:

Note: This is simply a change of reference plane, from z = 0 to z = -l.


28

0 0z zV z V e V e

What is V(-l )?

0 0V V e V e

0 0

0 0

V VI e e

Z Z

propagatingforwards

propagatingbackwards


l distance away from load

The current at z = - l is then


29

20

0

1 L

VI e e

Z

200

00 0 1

VV eV V e ee

VV

Total volt. at distance lfrom the load

Ampl. of volt. wave prop.towards load, at the loadposition (z = 0).

Similarly,

Ampl. of volt. wave prop.away from load, at theload position (z = 0).

0

21 LV e e

L Load reflectioncoefficient


0,Z

l Reflection coefficient at z = -l

30

20

2

2

0

0

2

0

1

1

1

1

L

L

L

L

V V e e

VI e e

Z

V eZ Z

I e

Input impedance seen “looking” towards loadat z = -l .


0,Z

Z

31

Simplifying, we have

00

0

tanh

tanhL

L

Z ZZ Z

Z Z


202

0 0 00 0 2

2 0 00

0

0 00

0 0

00

0

1

1

cosh sinh

cosh sinh

L

L L L

L LL

L

L L

L L

L

L

Z Ze

Z Z Z Z Z Z eZ Z Z

Z Z Z Z eZ Ze

Z Z

Z Z e Z Z eZ

Z Z e Z Z e

Z ZZ

Z Z

Hence, we have

32

20

20

0

2

0 2

1

1

1

1

j jL

j jL

jL

jL

V V e e

VI e e

Z

eZ Z

e

Impedance is periodicwith periodg/2

2

/ 2g

g

Terminated Lossless Transmission Line

j j

Note: tanh tanh tanj j

tan repeats when

00

0

tan

tanL

L

Z jZZ Z

Z jZ

33

For the remainder of our transmission line discussion we will assume that thetransmission line is lossless.

20

20

0

2

0 2

00

0

1

1

1

1

tan

tan

j jL

j jL

jL

jL

L

L

V V e e

VI e e

Z

V eZ Z

I e

Z jZZ

Z jZ

0

0

2

LL

L

g

p

Z Z

Z Z

v

Terminated Lossless Transmission Line

0 ,Z

Z

34

Matched load: (ZL=Z0)

0

0

0LL

L

Z Z

Z Z

For any l

No reflection from the load

A

Matched Load

0 ,Z

Z

0Z Z

0

0

0

j

j

V V e

VI e

Z

35

Short circuit load: (ZL = 0)

0

0

0

01

0

tan

L

Z

Z

Z jZ

Always imaginary!Note:

B

2g

scZ jX

S.C. can become an O.C.with a g/4 trans. line

g/

Short-Circuit Load

0 ,Z

0 tanscX Z

36

Using Transmission Lines to Synthesize Loads

A microwave filter constructed from microstrip.

This is very useful is microwave engineering.

37

00

0

tan

tanL

inL

Z jZ dZ Z d Z

Z jZ d

inTH

in TH

ZV d V

Z Z

0Z

Example

Find the voltage at any point on the line.

38

Note: 021 j

LjV V e e

0

0

LL

L

Z Z

Z Z

20 1j d j d in

THin TH

LV dZ

Ze V

ZV e

2

2

1

1

jj din L

TH j dm TH L

Z eV V e

Z Z e

At l = d :

Hence

Example (cont.)

0 2

1

1j din

TH j din TH L

ZV V e

Z Z e

39

Some algebra: 2

0 2

1

1

j dL

in j dL

eZ Z d Z

e

2

20 20

2 220

0 2

20

20 0

2

0

20 0

0

111

1 111

1

1

1

j dL

j dj dLL

j d j dj dL TH LL

THj dL

j dL

j dTH L TH

j dL

j dTH THL

TH

in

in TH

eZ

Z ee

Z e Z eeZ Z

e

Z e

Z Z e Z Z

eZ

Z

Z

Z Z

Z Z Ze

Z Z

Z

2

0

20 0

0

1

1

j dL

j dTH THL

TH

e

Z Z Z Ze

Z Z

Example (cont.)

40

2

02

0

1

1

jj d L

TH j dTH S L

Z eV V e

Z Z e

20

20

1

1

j din L

j din TH TH S L

Z Z e

Z Z Z Z e

where 0

0

THS

TH

Z Z

Z Z

Example (cont.)

Therefore, we have the following alternative form for the result:

Hence, we have

41

2

02

0

1

1

jj d L

TH j dTH S L

Z eV V e

Z Z e

Example (cont.)

0Z

Voltage wave that would exist if there were no reflections fromthe load (a semi-infinite transmission line or a matched load).

42

2 2

2 2 2 20

0

1 j d j dL L S

j d j d j d j dTH L S L L S L S

TH

e eZ

V d V e e e eZ Z

Example (cont.)

0Z

Wave-bounce method (illustrated for l = d):

43

Example (cont.)

22 2

22 2 20

0

1

1

j d j dL S L S

j d j d j dTH L L S L S

TH

e e

ZV d V e e e

Z Z

Geometric series:

2

0

11 , 1

1n

n

z z z zz

2 2

2 2 2 20

0

1 j d j dL L S

j d j d j d j dTH L S L L S L S

TH

e eZ

V d V e e e eZ Z

2j dL Sz e

44

Example (cont.)

or

2

0

202

1

1

1

1

j dL s

THj dTH

L j dL s

eZV d V

Z Ze

e

2

02

0

1

1

j dL

TH j dTH L s

Z eV d V

Z Z e

This agrees with the previous result (setting l = d).

Note: This is a very tedious method – not recommended.

Hence

45

00

0

tan

tanL T

in TT L

Z jZZ Z

Z jZ

2

4 4 2g g

g

00

Tin T

L

jZZ Z

jZ

0

20

0

0in in

T

L

Z Z

ZZ

Z

Quarter-Wave Transformer

20T

inL

ZZ

Z

so

1/2

0 0T LZ Z Z

Hence

This requires ZL to be real.

ZLZ0 Z0T

Zin

46

20 1 Lj j

LV V e e

20

20

1

1 L

j jL

jj jL

V V e e

V e e e

max 0

min 0

1

1

L

L

V V

V V

max

min

V

VVoltage Standing Wave Ratio VSWR

Voltage Standing Wave Ratio

0 ,Z

1

1L

L

VSWR

z

1+ L

1

1- L

0

( )V z

V

/ 2z 0z

47

General Transmission Line Formulas

tanG

C

0 0 r rLC

0losslessL

ZC characteristic impedance of line (neglecting loss)(1)

(2)

(3)

Equations (1) and (2) can be used to find L and C if we know the materialproperties and the characteristic impedance of the lossless line.

Equation (3) can be used to find G if we know the material loss tangent.

a bR R R

tanG

C

(4)

Equation (4) can be used to find R (discussed later).

,iC i a b contour of conductor,

2

2

1( )

i

i s sz

C

R R J l dlI

48

General Transmission Line Formulas (cont.)

tanG C

0losslessL Z

0/ losslessC Z

R R

Al four per-unit-length parameters can be found from0 ,losslessZ R

49

Common Transmission Lines

0 0

1ln [ ]

2lossless r

r

bZ

a

Coax

Twin-lead

100 cosh [ ]

2lossless r

r

hZ

a

2

1 2

12

s

h

aR R

a h

a

1 1

2 2sa sbR R Ra b

a

b

,r r

h

,r r

a a

50

At high frequency, discontinuity effects can become important.

Limitations of Transmission-Line Theory

Bend

incident

reflected

transmitted

The simple TL model does not account for the bend.

ZTH

ZLZ0

+-

51

At high frequency, radiation effects can become important.

When will radiation occur?

We want energy to travel from the generator to the load, without radiating.

Limitations of Transmission-Line Theory (cont.)

ZTH

ZLZ0

+-

52

r a

bz

The coaxial cable is a perfectlyshielded system – there is neverany radiation at any frequency, orunder any circumstances.

The fields are confined to the regionbetween the two conductors.


53

The twin lead is an open type of transmissionline – the fields extend out to infinity.

The extended fields may causeinterference with nearby objects.(This may be improved by using“twisted pair.”)

+ -


Having fields that extend to infinity is not the same thing as having radiation, however.

54

The infinite twin lead will not radiate by itself, regardless of how far apartthe lines are.

h

incident

reflected

The incident and reflected waves represent an exact solution to Maxwell’sequations on the infinite line, at any frequency.

*1 ˆRe E H 02t

S

P dS

S


No attenuation on an infinite lossless line

55

A discontinuity on the twin lead will cause radiation to occur.

Note: Radiation effectsincrease as thefrequency increases.


h

Incident wavepipe

Obstacle

Reflected wave

Bend h

Incident wave

bend

Reflected wave 56

To reduce radiation effects of the twin lead at discontinuities:

h

1) Reduce the separation distance h (keep h <<).2) Twist the lines (twisted pair).


CAT 5 cable(twisted pair)

57

58

Rectangularguide

Circularguide

Ridge guide

Common Hollow -pipe waveguides

59

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

60

Auxiliary Relations:

tyPermeabiliRelative

H/m104;5.

ConstantDielectricRelative

F/m10854.8;4.

CurrentConvection;3.

CurrentConduction;tyConductivi

Law)s(Ohm'2.

Velocity;Charge

Newton.1

r

12o

12

HHB

EED

JvJ

J

EJ

vq

BvEqF

or

r

oor

61

Maxwell’s Equations in Large Scale Form

SdDt

SdJldH

SdMSdBt

ldE

SdB

dvSdD

SSl

SSl

S

SV

0

ENEE482 62

•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,variationseAssume

122

22

63

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

;

DensityChargeSurface

21

21

21

21

2121

21

21

D1n

D2n

h

Ds

h

E1t

E2t

m1,e1

m2,e2

64

)(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

65

HnHJ

B

Dn

ts

n

ˆ0Bn0

ˆρD

0EnoE

:ConductorPerfectaatConditionsBoundary

sn

t

+ + +n

rs

Js

Ht

66

0)(ˆ)(ˆ

0)(ˆ

0)(ˆ

Hn

MEn

Bn

Dn

s

67

2/;0

;0

:mediumfreeSourceaFor

)(

)(

22

2222

2

vkHkH

kEkE

EjJj

HjEEE

Wave Equation

68

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

variablesofseparation Using),,,(forSolve

,,,0

0

20

222x

202

2

2

2

2

2

202

2

2

2

2

220

2

69

space.freeofadmittanceintrinsic theis

377spacefreeofimpedanceinterensic theis

1

111

waveplanecalledissolutionThe

.knpropagatioofdirection thelar toperpendicuis vectorThe

00Since

,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

70

s

j

1)1(

2j)(1

j

21

2j)(1

jjj

s

71

The field amplitude decays exponentially from its surfaceAccording to e -u/d

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

Hn,1

:ImpedancesurfaceThe

EJ,2

msmts

m

s

ZJZEj

Z

72

Parallel Polarization

z

x

Ei

Et

Er

q3

q1

q2

n1

n3

n2e0

e

0

0000

202

101

Y,

,

,120

110

k

EnYHeEE

EnYHeEE

rrrnjk

r

iirnjk

i

73

sin,cos

sin,cos

sin,cos

sinsin

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

tt

rnjk

t

74

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

V

e

VV

e

dVEEW

dVEEdVDEW

*

**

4

thenreal,andconstantisIf

4

1

4

1Re

75

S

V

V

m

dSHEP

dVHH

dVBHW

*

*

*

Re2

1

:bygivenisSsurface

closedaacrosssmittedpower tranaverage timeThe

constantandrealisif4

4

1Re

:isfieldmagnetic theinstoredenergyaverageTime

76

dVMHJEdVDEHB

j

dVMHJEdVDEHBj

dSHEVdHE

EJJ

JEEDjHMBj

EHHEHE

V

s

V

s

VV

SV

s

)(2

1

442

)(2

1)(

2

2

1

2

1

)(

)()(

****

****

**

***

***

77

dSHEP

dVEEHHj

dVEEHHdVEE

dSHEdVMHJE

j

S

V

VV

S

S

V

Ss

*0

**

***

**

2

1

)(2

)(22

1

2

1)(

2

1-

tyconductiviandj-,

:byzedcharacteriismedium theIf

78

volume.

in thestoredenergyreactive the times2and)( volumein the

heatlost topower the,Psurfacehe through tsmittedpower tranthe

ofsum the toequalis)(Psourcesby thedeliveredpowerThe

)(2

)(2

442

2

1Im

)(2

1P

0

s

0

***

*s

P

WWjPPP

WW

dVEEHH

dSHE

dVMHJE

ems

em

VS

ss

V

s

losspoweraverageTime

2

1)(

2***

dVEEdVEEHHPVV

79

C

L RI

V

networkaofimpedance theofn DefinitioGeneral2

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

80

Transmission Lines & Waveguides

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

,

,

)(

)()()(

,,

,,,,,,

,,

,,,,,,

zttztt

zttztt

zjztzzztzt

zjzt

zjztzt

zjz

zjt

zt

zjz

zjt

zt

ejehjh

ejhhje

ehhjeajeeaje

ehhjeeeajE

eyxheyxh

zyxHzyxHzyxH

eyxeeyxe

zyxEzyxEzyxE

81

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

82

TEM WAVES

zjz

zjtt

zjt

zjtt

t

t

t

tt

ttt

eeaYehH

eyxeeE

yx

yxyxe

h

e

eh

ˆ),(

0,

entialScalar Pot0,,

eha,0

hea,0

0,0

0

2

t0tzt

t0tz

t

83

wavesTEMfor

0])([,0)(

,but,0

:equationHelmholtzsatisfymustfieldThe

directionz-orin thenpropagatiofor 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

84

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

electromagnetic waves maxwell`s equations, transmission...

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