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

POYNTING VECTOR

•However, since E and B are perpendicular,

and since

MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS

E A E s

B A B s

d d id

E A E s

B A B s

d d id

AMPERE’S LAWOriginal:

As modified by Maxwell:

Original:

As modified by Maxwell:

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

E A E s

B A B s

Maxwell' s Equations in Free Space

E A E s

B A B s

CONTINUITY

CONTINUITYWithout Maxwell' s modification:

Without Maxwell' s modification:

CONTINUITYWith Maxwell's modification:

"Displacement current"

2 0 0 0

With Maxwell's modification:

"Displacement current"

2 0 0 0

CONTINUITYi i

dEd AE

AMPERE’S LAW

AMPERE’S LAWFor path 1:

For path 2:

For path 3:

For path 1:

For path 2:

For path 3:

ELECTROMAGNETICWAVES

E A E s

B A B s

E A E s

B A B s

ELECTROMAGNETICWAVES

8130 10. m/s

MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS

E A E s

B A B s

d d id

E A E s

B A B s

d d id

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)

Transmission Line (cont.)

CAT 5 cable(twisted pair)

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

Microstrip

Stripline

Transmission lines commonly met on printed-circuit boards

Coplanar strips

Coplanar waveguide (CPW)

Transmission lines are commonly met on printed-circuit boards.

A microwave integrated circuit

Microstrip line

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

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.

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]

,i z t

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

RDz LDz

GDz CDz

v(z+z,t)

v(z,t)

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

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

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

RDz LDz

GDz CDz

v(z+z,t)

v(z,t)

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

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

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.)

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

v i iR L

z z z t

i iR L

vR Gv C

v vL G C

Switch theorder of thederivatives.

2 2( ) 0

v v vRG v RC LG LC

The same equation also holds for i.

Hence, we have:

v v v vR Gv C L G C

z t t t

Convention:

Solution:

( ) z zV z Ae Be

( )( )R j L G j C

principal square root

dzThen

is called the "propagation constant."

/2jz z e

attenuationcontant

phaseconstant

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

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

( , ) Re

z j z j t

j z j z j t

v z t V e e e

V e e e e

V e t z

The wave “repeats” when:

Hence:

Phase Velocity (cont.)

constant

Hence pv

Im ( )( )p

vR j L G j C

In expanded form:

Characteristic Impedance Z0

V z V e

I z I e

+V+(z)-

I+ (z)

A wave is traveling in the positive z direction.

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

Use Telegrapher’s Equation:

v iRi L

RI j LIdz

Hence0 0

z zV e ZI e

Characteristic Impedance Z0 (cont.)

From this we have:

We have

V Z ZZ

Y G j C

R j LZ

Characteristic Impedance Z0 (cont.)

Z R j L

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

j z j j z

z j zV e e

V z V e V

V e e e

v z t V z

wave in +zdirection wave in -z

direction

General Case (Waves in Both Directions)

Backward-Traveling Wave

+V -(z)-

I - (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.

General Case

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.

+V (z)-

Lossless Case

0, 0R G

( )( )j R j L G j C

R j LZ

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

Lossless Case (cont.)1

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

The speed of light in a dielectric medium is1

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)

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

Can we use z = - l asa reference plane?

Terminating impedance (load)

( ) ( )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.

0 0z zV z V e V e

What is V(-l )?

0 0V V e V e

V VI e e

propagatingforwards

propagatingbackwards

l distance away from load

The current at z = - l is then

VI e e

00 0 1

VV eV V e ee

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).

21 LV e e

L Load reflectioncoefficient

l Reflection coefficient at z = -l

V V e e

VI e e

V eZ Z

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

Simplifying, we have

Z ZZ Z

0 0 00 0 2

2 0 00

cosh sinh

Z Z Z Z Z Z eZ Z Z

Z Z Z Z eZ Ze

Z Z e Z Z eZ

Z Z e Z Z e

Hence, we have

V V e e

VI e e

Impedance is periodicwith periodg/2

Terminated Lossless Transmission Line

Note: tanh tanh tanj j

tan repeats when

Z jZZ Z

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

V V e e

VI e e

V eZ Z

Terminated Lossless Transmission Line

Matched load: (ZL=Z0)

For any l

No reflection from the load

Matched Load

Short circuit load: (ZL = 0)

Always imaginary!Note:

scZ jX

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

Short-Circuit Load

0 tanscX Z

Using Transmission Lines to Synthesize Loads

A microwave filter constructed from microstrip.

This is very useful is microwave engineering.

Z jZ dZ Z d Z

Z jZ d

ZV d V

Example

Find the voltage at any point on the line.

Note: 021 j

LjV V e e

20 1j d j d in

THin TH

jj din L

TH j dm TH L

Z eV V e

At l = d :

Example (cont.)

1j din

TH j din TH L

ZV V e

Some algebra: 2

in j dL

eZ Z d Z

j dj dLL

j d j dj dL TH LL

THj dL

j dTH L TH

j dTH THL

Z e Z eeZ Z

Z Z e Z Z

Z Z Ze

j dTH THL

Z Z Z Ze

Example (cont.)

jj d L

TH j dTH S L

Z eV V e

j din L

j din TH TH S L

Z Z Z Z e

where 0

Example (cont.)

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

Hence, we have

jj d L

TH j dTH S L

Z eV V e

Example (cont.)

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

2 2 2 20

1 j d j dL L S

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

V d V e e e eZ Z

Example (cont.)

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

Example (cont.)

22 2 20

j d j dL S L S

j d j d j dTH L L S L S

ZV d V e e e

Geometric series:

11 , 1

z z z zz

2 2 2 20

1 j d j dL L S

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

V d V e e e eZ Z

2j dL Sz e

Example (cont.)

j dL s

THj dTH

L j dL s

eZV d V

TH j dTH L s

Z eV d V

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

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

tanL T

in TT L

Z jZZ Z

4 4 2g g

0in in

Quarter-Wave Transformer

0 0T LZ Z Z

This requires ZL to be real.

ZLZ0 Z0T

20 1 Lj j

LV V e e

V V e e

V e e e

VVoltage Standing Wave Ratio VSWR

Voltage Standing Wave Ratio

( )V z

/ 2z 0z

General Transmission Line Formulas

0 0 r rLC

0losslessL

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

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

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

,iC i a b contour of conductor,

i s sz

R R J l dlI

General Transmission Line Formulas (cont.)

tanG C

0losslessL Z

0/ losslessC Z

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

Common Transmission Lines

1ln [ ]

2lossless r

Twin-lead

100 cosh [ ]

2lossless r

2 2sa sbR R Ra b

At high frequency, discontinuity effects can become important.

Limitations of Transmission-Line Theory

incident

reflected

transmitted

The simple TL model does not account for the bend.

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.)

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.

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.

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

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

No attenuation on an infinite lossless line

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

Note: Radiation effectsincrease as thefrequency increases.

Incident wavepipe

Obstacle

Reflected wave

Bend h

Incident wave

Reflected wave 56

To reduce radiation effects of the twin lead at discontinuities:

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

CAT 5 cable(twisted pair)

Rectangularguide

Circularguide

Ridge guide

Common Hollow -pipe waveguides

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

Auxiliary Relations:

tyPermeabiliRelative

H/m104;5.

ConstantDielectricRelative

F/m10854.8;4.

CurrentConvection;3.

CurrentConduction;tyConductivi

Law)s(Ohm'2.

Velocity;Charge

Newton.1

Maxwell’s Equations in Large Scale Form

SdJldH

SdMSdBt

ENEE482 62

•Maxwell’s Equations for the Time - Harmonic Case

DjJHMBjE

eEEejEEE

jEEajEEazyxE

ezyxEtzyxE

xrxixixr

jtjxixr

tjxixrx

yiyryxixrx

)/(tan,)cos(

]Re[])Re[(

)()(),,(

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

: then,variationseAssume

Boundary Conditions at a General Material Interface

)(ˆ)(ˆ

0)(ˆ)(ˆ

DensityChargeSurface

)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ

Fields at a Dielectric Interface

ˆ0Bn0

:ConductorPerfectaatConditionsBoundary

+ + +n

0)(ˆ)(ˆ

:mediumfreeSourceaFor

Wave Equation

zayaxar

kakakakAeE

zyxiEkz

zzyyxxzjkyjkxjk

variablesofseparation Using),,,(forSolve

space.freeofadmittanceintrinsic theis

377spacefreeofimpedanceinterensic theis

waveplanecalledissolutionThe

.knpropagatioofdirection thelar toperpendicuis vectorThe

00Since

,Similarly,

EnEnYEnEnk

eEkeEj

EkEeEE

CeEBeEAeE

rkjrkjrkj

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

:ImpedancesurfaceThe

Parallel Polarization

EnYHeEE

rrrnjk

iirnjk

sin,cos

sinsin

cossin

331333

221222

111111

3032010

nnkknnknk

nkknYY

EnYHeEE

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

dVEEdVDEW

thenreal,andconstantisIf

:bygivenisSsurface

closedaacrosssmittedpower tranaverage timeThe

constantandrealisif4

:isfieldmagnetic theinstoredenergyaverageTime

dVMHJEdVDEHB

dVMHJEdVDEHBj

dSHEVdHE

JEEDjHMBj

EHHEHE

dVEEHHj

dVEEHHdVEE

dSHEdVMHJE

tyconductiviandj-,

:byzedcharacteriismedium theIf

volume.

in thestoredenergyreactive the times2and)( volumein the

heatlost topower the,Psurfacehe through tsmittedpower tranthe

ofsum the toequalis)(Psourcesby thedeliveredpowerThe

WWjPPP

dVEEHH

dVMHJE

losspoweraverageTime

dVEEdVEEHHPVV

networkaofimpedance theofn DefinitioGeneral2

IILIIjRII

jLjRIIZIIVI

Transmission Lines & Waveguides

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

)()()(

,,,,,,

zttztt

zjztzzztzt

zjztzt

ejehjh

ejhhje

ehhjeajeeaje

ehhjeeeajE

eyxheyxh

zyxHzyxHzyxH

eyxeeyxe

zyxEzyxEzyxE

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

TEM WAVES

eeaYehH

eyxeeE

entialScalar Pot0,,

wavesTEMfor

0])([,0)(

,but,0

:equationHelmholtzsatisfymustfieldThe

directionz-orin thenpropagatiofor wave

Impedance Wave,1

TE WAVES

let,0)(

0),()(

222222

tzztztt

ttzztt

ejhajhah

heahje

electromagnetic waves maxwell`s equations, transmission...

Documents

electromagnetic waveselectromagnetic waves according to...

chapter 6 maxwell's equations for electromagnetic waves ......

chapter 34 maxwell’s equations; electromagnetic...

class 12th physics chapter 8 electromagnetic waves...

lecture 15.1 : electromagnetic waves · 1 lecture 15.1 :...

electromagnetic waves. james clerk maxwell scottish...

l 30 electricity and magnetism [7] electromagnetic waves...

lecture 18 rlc circuits transformer maxwell electromagnetic...

chapter 30: maxwell ’ s equations and electromagnetic...

ch. 22 electromagnetic waves - umass...

chapter 32: electromagnetic waves · chapter 32:...

james clerk maxwell (1831 – 1879). electromagnetic (light)...

maxwell equations and electromagnetic waves€¦ ·...

chapter 25 electromagnetic waves. units of chapter 25 the...

w13d2: maxwell ’ s equations and electromagnetic waves

electromagnetic waves what is light? · electromagnetic...

electromagnetic waves and lasers · electromagnetic waves...

18.1 electromagnetic waves how are ____________________...

maxwell equations: electromagnetic waves

electromagnetic waves & the electromagnetic spectrum