EC6503 TRANSMISSION LINES AND WAVEGUIDES BY H.UMMA HABIBA, Professor SVCE,Chennai.

transcript

EC6503 TRANSMISSION LINES AND WAVEGUIDES

H.UMMA HABIBA, Professor

SVCE,Chennai.

Overview of syllabusOBJECTIVES

To introduce the various types of transmission lines and to discuss the losses associated.

To give thorough understanding about impedance transformation and matching.

To use the Smith chart in problem solving.

To impart knowledge on filter theories and waveguide theories

Overview of syllabus

OUTCOMESUpon completion of the course, students will be able to Discuss the propagation of signals through transmission lines.

Analyze signal propagation at Radio frequencies.

Explain radio propagation in guided systems.

Utilize cavity resonators

Overview of syllabus TEXT 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”, Pearson Education Inc, Delhi, 2004

3.John D Kraus and Daniel A Fleisch, “Electromagnetics with Applications”, 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 Field Theory Fundamentals”, Vikas Publishing House, New Delhi, 2001.

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

UNIT I -TRANSMISSION LINE THEORY

General theory of Transmission lines - the transmission line - general solution - The infinite line - Wavelength, velocity of propagation - Waveform distortion - the distortion-less line - Loading and different methods of loading - Line not terminated in Z0 - Reflection coefficient - calculation of current, voltage, power delivered and efficiency of transmission - Input and transfer impedance - Open and short circuited lines - reflection factor and reflection loss.

UNIT II -HIGH FREQUENCY TRANSMISSION LINES

Transmission line equations at radio frequencies - Line of Zero dissipation - Voltage and current on the dissipation-less line, Standing Waves, Nodes, Standing Wave Ratio - Input impedance of the dissipation-less line - Open and short circuited lines - Power and impedance measurement on lines - Reflection losses - Measurement of VSWR and wavelength.

UNIT III-IMPEDANCE MATCHING IN HIGH FREQUENCY LINES

Impedance matching: Quarter wave transformer - Impedance matching by stubs - Single stub and double stub matching - Smith chart - Solutions of problems using Smith chart - Single and double stub matching using Smith chart.

UNIT IV-PASSIVE FILTERS

Characteristic impedance of symmetrical networks - filter fundamentals, Design of filters: Constant K - Low Pass, High Pass, Band Pass, Band Elimination, m- derived sections - low pass, high pass composite filters.

UNIT V -WAVE GUIDES AND CAVITY RESONATORS

General Wave behaviours along uniform Guiding structures, Transverse Electromagnetic waves, Transverse Magnetic waves, Transverse Electric waves, TM and TE waves between parallel plates, TM and TE waves in Rectangular wave guides, Bessel‟s differential equation and Bessel function, TM and TE waves in Circular wave guides, Rectangular and circular cavity Resonators.

UNIT I

TRANSMISSION LINE THEORY UNIFORM PLANE

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 loss May 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 4:1 impedance-transforming

balun)11

Transmission Line (cont.)

CAT 5 cable(twisted pair)

The two wires of the transmission line are twisted to reduce interference and radiation 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

Fiber-Optic Guide (cont.)Two types of fiber-optic guides:

1) Single-mode fiber

2) Multi-mode fiber

Carries a single mode, as with the mode on a transmission line or waveguide. Requires the fiber diameter to be small relative to a wavelength.

Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).

Fiber-Optic Guide (cont.)

http://en.wikipedia.org/wiki/Optical_fiber

Higher index core region

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) 18

Lumped circuits: resistors, capacitors,inductors

neglect time delays (phase)

account for propagation and time delays (phase change)

Transmission-Line Theory

Distributed circuit elements: transmission lines

We need transmission-line theory whenever the length of a 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

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

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

with respect to z:

v i iR L

z z z t

i iR L

vR Gv C

v vL G C

Switch the order of the derivatives.

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

2( ) ( ) 0

d VRG V RC LG j V LC V

2 2( ) 0

v v vRG v RC LG LC

Time-Harmonic Waves:

Note that

= series impedance/length

d VRG V j RC LG V LC V

2( ) ( )( )RG j RC LG LC R j L G j C

Z R j L

Y G j C

= parallel admittance/length

Then we can write:2

d VZY V

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

Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest.

0( , ) cos( )zv z t V e t z

vp (phase velocity)

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. 34

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 +z direction

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 as for the forward wave.

General Case

V z V e V e

I z V e V eZ

A general superposition of forward and backward traveling waves:

Most general case:

Note: The reference directions for voltage and current are the same for forward and backward waves. 37

+ V (z)-

V z V e V e

V VI z e e

j R j L G j C

R j LZ

V(z)+- z

[m/s]pv

guided wavelength g

phase velocity vp

Summary of Basic TL formulas

Lossless Case

0, 0R G

( )( )j R j L G j C

R j LZ

(indep. of freq.)(real and indep. of freq.)39

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 the speed of light (in the material).

(proof given later)

0 0z zV z V e V e

Where do we assign z = 0?

The usual choice is at the load.

V(z)+-

Terminating impedance (load)

Ampl. of voltage wave propagating in negative z direction at z = 0.

Ampl. of voltage wave propagating in positive z direction at z = 0.

Terminated Transmission Line

Note: The length l measures distance from the load: z41

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 as a reference plane?

V(z)+-

( ) ( )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.

V(z)+-

0 0z zV z V e V e

What is V(-l )?

0 0V V e V e

V VI e e

propagating forwards

propagating backwards

l distance away from load

The current at z = - l is then

V(z)+-

VI e e

00 0 1

VV eV V e ee

Total volt. at distance l from the load

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

Similarly,

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

21 LV e e

L Load reflection coefficient

Terminated Transmission Line (cont.)I(-l )

V(-l )+

l Reflection coefficient at z = - l

V V e e

VI e e

V eZ Z

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

I(-l )

V(-l )+

At the load (l = 0):

Z ZZ Z

Recall

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 periodic with period g/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 the transmission line is lossless.

V V e e

VI e e

V eZ Z

Terminated Lossless Transmission Line

I(-l )

V(-l )+

Matched load: (ZL=Z0)

For any l

No reflection from the load

Matched LoadI(-l )

V(-l )+

Short circuit load: (ZL = 0)

Always imaginary!Note:

scZ jX

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

0 1/4 1/2 3/4g/

inductive

capacitive

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

V(-l)+

+ZinV(-d)

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.)

V(-l)+

Voltage wave that would exist if there were no reflections from the 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.

V(-l)+

At a distance l from the load:

0 2 2 * 2*0

1Re 1 1

20 2 4

VP e e

If Z0 real (low-loss transmission line)

Time- Average Power Flow

V V e e

VI e e

*2 * 2

pure imaginary

Low-loss line

20 2 4

20 02 2* *0 0

VP d e e

V Ve e

power in forward wave power in backward wave

Lossless line ( = 0)

Time- Average Power Flow

V(-l)+

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

I(-l )

V(-l )+

( )V z

/ 2z 0z

Coaxial Cable

Here we present a “case study” of one particular transmission line, the coaxial cable.

Find C, L, G, R

We will assume no variation in the z direction, and take a length of one meter in the z direction in order top calculate the per-unit-length parameters.

For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics.

Coaxial Cable (cont.)

ˆ ˆ2 2 r

Find C (capacitance / length)

Coaxial cable

h = 1 [m]

From Gauss’s law:

V V E dr

Coaxial cable

h = 1 [m]

We then have

0 F/m2

Find L (inductance / length)

From Ampere’s law:

Coaxial cable

h = 1 [m]

center conductorMagnetic flux:

Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency.

Coaxial cable

h = 1 [m]

0 H/mln [ ]2

Observation:

0 F/m2

0 0 r rLC

This result actually holds for any transmission line.

0 H/mln [ ]2

For a lossless cable:

0 F/m2

1ln [ ]

376.7303 [ ]

ˆ ˆ2 2 r

Find G (conductance / length)

Coaxial cable

h = 1 [m]

From Gauss’s law:

V V E dr

We then have leakIG

leak a

2[S/m]

Observation:

This result actually holds for any transmission line.

2[S/m]

To be more general:

Note: It is the loss tangent that is usually (approximately) constant for a material, over a wide range of frequencies.

As just derived,

The loss tangent actually arises from both conductivity loss and polarization loss (molecular friction loss), ingeneral.

This is the loss tangent that would arise from conductivity effects.

General expression for loss tangent:

Effective permittivity that accounts for conductivity

Loss due to molecular friction Loss due to conductivity

Find R (resistance / length)

Coaxial cable

h = 1 [m]

a bR R R

2a saR Ra

2b sbR Rb

Rs = surface resistance of metal

General Transmission Line Formulas

0 0 r rLC

0losslessL

characteristic impedance of line (neglecting loss)(1)

Equations (1) and (2) can be used to find L and C if we know the material properties 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 from 0 ,losslessZ R

Common Transmission Lines

1ln [ ]

2lossless r

Twin-lead

100 cosh [ ]

2lossless r

R Ra h

2 2sa sbR R Ra b

Common Transmission Lines (cont.)

Microstrip

eff effr reff effr r

fZ f Z

0 / 1.393 0.667 ln / 1.444effr

Zw h w h

( / 1)w h

t hw w

Common Transmission Lines (cont.)

Microstrip ( / 1)w h

(0)(0)

effr reff eff

1 1 11 /0

2 2 4.6 /1 12 /

eff r r rr

w hh w

4 1 0.5 1 0.868ln 1r

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 perfectly shielded system – there is never any radiation at any frequency, or under any circumstances.

The fields are confined to the region between the two conductors.

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

The extended fields may cause interference 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 apart the lines are.

incident

reflected

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

*1ˆRe E H 0

No attenuation on an infinite lossless line

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

Note: Radiation effects increase as the frequency increases.

Incident wavepipe

Obstacle

Reflected wave

Bend h

Incident wave

Reflected wave 89

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)

Two conductorwire Coaxial line Shielded

Strip line

Dielectric

Two conductorwire Coaxial line Shielded

Strip line

Dielectric

Rectangular guide

Circular guide

Ridge guide

Common Hollow-pipe waveguides

STRIP LINE CONFIGURATIONS

SINGLE STRIP LINE COUPLED LINES

COUPLED STRIPSTOP & BOTTOM

COUPLED ROUND BARS

MICROSTRIP LINE CONFIGURATIONS

TWO COUPLED MICROSTRIPS SINGLE MICROSTRIP

TWO SUSPENDED SUBSTRATE LINES

SUSPENDED SUBSTRATE LINE

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:

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

Maxwell’s Equations in Large Scale Form

SdJldH

SdMSdBt

ENEE482 99

• Maxwell’s Equations for the Time - Harmonic Case

DjJHMBjE

eEEejEEE

jEEajEEazyxE

ezyxEtzyxE

xrxixixr

jtjxixr

tjxixrx

yiyryxixrx

)/(tan , )cos(

]Re[])Re[(

)()(),,(

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

: then,variationse Assume

Boundary Conditions at a General Material Interface

Density Charge Surface

)(ˆ)(ˆ

Fields at a Dielectric Interface

0En oE

:ConductorPerfect aat ConditionsBoundary

+ + +n

2/ ; 0

:medium free Sourcea For

Wave Equation

zayaxar

kakakakAeE

zyxiEkz

zzyyxxzjkyjkxjk

variablesof separation Using),,,(for Solve

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

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

Parallel Polarization

EnYHeEE

rrrnjk

iirnjk

sin , cos

sin sin

cossin

331333

221222

111111

3032010

nnkknnknk

nkknYY

EnYHeEE

ttrnjk

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

dVEEdVDEW

thenreal, andconstant is If

:by given is S surface

closeda across smittedpower tran average timeThe

constant and real is if 4

:is field magnetic thein storedenergy average Time

dVMHJEdVDEHB

dVMHJEdVDEHBj

dSHEVdHE

JEEDjHMBj

EHHEHE

dVEEHHj

dVEEHHdVEE

dSHEdVMHJE

ty conductivi and j- ,

:by zedcharacteri is medium theIf

volume.

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

networka of impedance theof n DefinitioGeneral

IILIIjRII

jLjRIIZIIVI

equation. HelmholtzousInhomogene

condition) (Lorentz or

JjAkAAA

JjAJEjAH

AjEAjE

AjBjEAB

Solution For Vector Potential

(x,y,z)(x’,y’, z’) R

rrzzyyxxR

erJrAzyxA

)()()(

current alinfinitism anfor )(4

)(),,(

I(z,t)

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

I(z,t)+I/z dz

Lumped element circuit model for a transmission line

Impedance sticCharacteri:

)()(),(

0),(),(

ztfItzI

ztfVtzV

VYIVYIeIeIzI

eVeVzV

cczjzj

, 0)()(

To generator

tcoefficien on Reflecti

(sin41

)1)(1(Re2

eeVeeV

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

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 Pot 0,,

eha , 0

hea , 0

wavesTEMfor

0])([ , 0)(

, but , 0

:equation Helmholtzsatisfy must field The

direction z-or in then propagatio for wave

Impedance Wave , 1

TE WAVES

let , 0)(

0),()(

222222

tzztztt

ttzztt

ejhajhah

heahje

Impedance Wave

; ˆˆ

Admittance Wave

let , 0)(

0),()(

222222

TM WAVES

TEM TRANSMISSION LINES

Parallel -plate Two-wire Coaxial

COAXIAL LINES

jkz-00

e a )/ln(

and e a )/ln(

0at 0,at ln

0for 01

rarVCrCrr

)/ln(ˆRe

e)/ln(

)/ln(I

eˆ)/ln(

jkz-00

VYrdrdaHEP

VYHaHnJ

• THE CHARACTERISTIC IMPEDANCE OF A COAXIAL IS Z0

Ohms ln2

Zc OF COAXIAL LINE AS A FUNCTION OF b/a

r Zo=X

kkEaYHeE

kkjjkjjjk

jYYjkk

ty conductivi the toequivalent is

, ˆ ,

losses small For )(

)( and )(

, )/ln(2

:lossconductor the todue losspower The

:is lengthunit per losspower The

dJJZPj

dSEEdSJJP

Qc OF COAXIAL LINE AS A FUNCTION OF Zo

Coaxia

34000 10

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

e cos),,(

e sin),,(

sin),(

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

d 0,yat 0),(

cossin ),(

nAzyxE

ykBykAyxe

0nfor 2

0nfor 4

:is modes TM theof impedance waveThe

0E ,e cos),,(

0 00 0

dydxHEdydxzHEP

0nfor Np/m 22

lossconductor toduen Attenuatio

e sin),,(

e cos),,(

cos),(

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

d 0,yat 0),(

cossin ),(

nBzyxH

ykBykAyxh

Np/m 2

0nFor )Re(4

:is modes TM theof impedance waveThe

0E ,e sin),,(

0 00 0

dydxHEdydxzHEP

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

EC6503 TRANSMISSION LINES AND WAVEGUIDES BY H.UMMA HABIBA, Professor SVCE,Chennai.

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