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

EC6503 TRANSMISSION LINES AND WAVEGUIDES

BY

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

WAVES

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.

12


Microstrip

h

w

r

r

w

Stripline

h

Transmission lines commonly met on printed-circuit boards

Coplanar strips

hr

w w

Coplanar waveguide (CPW)

hr

w

13


Transmission lines are commonly met on printed-circuit boards.

A microwave integrated circuit

Microstrip line

14

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

15

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).

16

Fiber-Optic Guide (cont.)

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

Higher index core region

17

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.

19

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]

z

20

21


z

,i z t

+ + + + + + +- - - - - - - - - -

,v z tx x xB

21

Rz Lz

Gz Cz

z

v(z+z,t)

+

-

v(z,t)

+

-

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

22

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

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

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


22

Rz Lz

Gz Cz

z

v(z+z,t)

+

-

v(z,t)

+

-

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

23

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.)

23

24

To combine these, take the derivative of the first one

with respect 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 the order of the derivatives.


24

25

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


25

26

2

2

2( ) ( ) 0

d VRG V RC LG j V LC V

dz

2 2

2 2( ) 0

v v vRG v RC LG LC

z t t


Time-Harmonic Waves:

26

27

Note that

= series impedance/length

2

2

2( )

d VRG V j RC LG V LC V

dz

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

2( )

d VZY V

dz


27

28

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

28

29


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:

29

30

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

z

vp (phase velocity)

30

31

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:

31

32

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.)

32

33

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.)

33

34

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. 34

35

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 +z direction

wave in -z direction

General Case (Waves in Both Directions)

35

36

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 as for the forward wave.

36

37

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 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)-

I (z)

z

38

1

2

12

0

0 0

0 0

0 0

z z

z z

V z V e V e

V VI z e e

Z

j R j L G j C

R j LZ

G j

Z

C

I(z)

V(z)+- z

2mg

[m/s]pv

guided wavelength g

phase velocity vp

Summary of Basic TL formulas

38

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.)39

40

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 the speed of light (in the material).

(proof given later)

40

41

0 0z zV z V e V e

Where do we assign z = 0?

The usual choice is at the load.

I(z)

V(z)+-

zZL

z = 0

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

Hence

Can we use z = - l as a reference plane?

I(z)

V(z)+-

zZL

z = 0


42

( ) ( )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.

I(z)

V(z)+-

zZL

z = 0


43

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

propagating forwards

propagating backwards


l distance away from load

The current at z = - l is then

I(z)

V(z)+-

zZL

z = 0


44

20

0

1 L

VI e e

Z

200

00 0 1

VV eV V e ee

VV

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

ZL-

0,Z

l Reflection coefficient at z = - l

45

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 load at z = -l .


I(-l )

V(-l )+

l

ZL-

0,Z

Z

46

At the load (l = 0):

0

10

1L

LL

Z Z Z

Thus,

20

00

20

0

1

1

L

L

L

L

Z Ze

Z ZZ Z

Z Ze

Z Z


0

0

LL

L

Z Z

Z Z

2

0 2

1

1L

L

eZ Z

e

Recall

47

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

48

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 periodic with period g/2

2

/ 2

g

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

49

For the remainder of our transmission line discussion we will assume that the transmission 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

I(-l )

V(-l )+

l

ZL-

0 ,Z

Z

50

Matched load: (ZL=Z0)

0

0

0LL

L

Z Z

Z Z

For any l

No reflection from the load

A

Matched LoadI(-l )

V(-l )+

l

ZL-

0 ,Z

Z

0Z Z

0

0

0

j

j

V V e

VI e

Z

51

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

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

XSC

inductive

capacitive

Short-Circuit Load

l

0 ,Z

0 tanscX Z

52

Using Transmission Lines to Synthesize Loads

A microwave filter constructed from microstrip.

This is very useful is microwave engineering.

53

00

0

tan

tanL

inL

Z jZ dZ Z d Z

Z jZ d

inTH

in TH

ZV d V

Z Z

I(-l)

V(-l)+

l

ZL

-0Z

ZTH

VTH

d

Zin

+

-

ZTH

VTH

+ZinV(-d)

+

-

Example

Find the voltage at any point on the line.

54

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

55

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 11

1

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.)

56

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

57

2

02

0

1

1

jj d L

TH j dTH S L

Z eV V e

Z Z e

Example (cont.)

I(-l)

V(-l)+

l

ZL

-0Z

ZTH

VTH

d

Zin

+

-

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

58

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.)

ZL0Z

ZTH

VTH

d

+

-

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

59

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

60

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

61

I(-l)

V(-l)+

l

ZL-

0 ,Z

At a distance l from the load:

*

*

2

0 2 2 * 2*0

1Re 1 1

1R

2

e2

L L

Ve e

Z

V I

e

P

2

20 2 4

0

11

2 L

VP e e

Z

If Z0 real (low-loss transmission line)

Time- Average Power Flow

20

20

0

1

1

L

L

V V e e

VI e e

Z

j

*2 * 2

*2 2

L L

L L

e e

e e

pure imaginary

Note:

62

Low-loss line

2

20 2 4

0

2 2

20 02 2* *0 0

11

2

1 1

2 2

L

L

VP d e e

Z

V Ve e

Z Z

power in forward wave power in backward wave

2

20

0

11

2 L

VP d

Z

Lossless line ( = 0)

Time- Average Power Flow

I(-l)

V(-l)+

l

ZL-

0 ,Z

63

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

64

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

I(-l )

V(-l )+

l

ZL-

0 ,Z

1

1L

L

VSWR

z

1+ L

1

1- L

0

( )V z

V

/ 2z 0z

65

Coaxial Cable

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

a

b ,r

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.

66

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.)

-l0

l0

a

b

r0 0

0

ˆ ˆ2 2 r

E

Find C (capacitance / length)

Coaxial cable

h = 1 [m]

r

From Gauss’s law:

0

0

ln2

B

AB

A

b

ra

V V E dr

bE d

a

67

-l0

l0

a

b

r

Coaxial cable

h = 1 [m]

r

0

0

0

1

ln2 r

QC

V ba

Hence

We then have

0 F/m2

[ ]ln

rCba


68

ˆ2

IH

Find L (inductance / length)

From Ampere’s law:

Coaxial cable

h = 1 [m]

r

I

0ˆ

2 r

IB

(1)b

a

B d S

h

I

I z

center conductorMagnetic flux:


69

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]

r

I

0

0

0

1

2

ln2

b

r

a

b

r

a

r

H d

Id

I b

a

0

1ln

2r

bL

I a

0 H/mln [ ]2

r bL

a

Hence


70

0 H/mln [ ]2

r bL

a

Observation:

0 F/m2

[ ]ln

rCba

0 0 r rLC

This result actually holds for any transmission line.


71

0 H/mln [ ]2

r bL

a

For a lossless cable:

0 F/m2

[ ]ln

rCba

0

LZ

C

0 0

1ln [ ]

2r

r

bZ

a

00

0

376.7303 [ ]


72

-l0

l0

a

b

0 0

0

ˆ ˆ2 2 r

E

Find G (conductance / length)

Coaxial cable

h = 1 [m]

From Gauss’s law:

0

0

ln2

B

AB

A

b

ra

V V E dr

bE d

a


73

-l0

l0

a

b

J E

We then have leakIG

V

0

0

(1) 2

2

22

leak a

a

r

I J a

a E

aa

0

0

0

0

22

ln2

r

r

aa

Gba

2[S/m]

lnG

ba

or


74

Observation:

F/m2

[ ]ln

Cba

G C

This result actually holds for any transmission line.

2[S/m]

lnG

ba

0 r


75

G C

To be more general:

tanG

C

tanG

C

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.

76

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

General expression for loss tangent:

c

c c

j

j j

j

tan c

c

Effective permittivity that accounts for conductivity

Loss due to molecular friction Loss due to conductivity


77

Find R (resistance / length)

Coaxial cable

h = 1 [m]


,b rb

a

b

,a ra

a bR R R

1

2a saR Ra

1

2b sbR Rb

1sa

a a

R

1

sbb b

R

0

2a

ra a

0

2b

rb b

Rs = surface resistance of metal

78

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 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

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

79

General Transmission Line Formulas (cont.)

tanG C

0losslessL Z

0/ losslessC Z

R R

Al four per-unit-length parameters can be found from 0 ,losslessZ R

80

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

ha

R Ra h

a

1 1

2 2sa sbR R Ra b

a

b

,r r

h

,r r

a a

81

Common Transmission Lines (cont.)

Microstrip

0 0

1 00

0 1

eff effr reff effr r

fZ f Z

f

0

1200

0 / 1.393 0.667 ln / 1.444effr

Zw h w h

( / 1)w h

21 ln

t hw w

t

h

w

r

t

82

Common Transmission Lines (cont.)

Microstrip ( / 1)w h

h

w

r

t

2

1.5

(0)(0)

1 4

effr reff eff

r rfF

1 1 11 /0

2 2 4.6 /1 12 /

eff r r rr

t h

w hh w

2

0

4 1 0.5 1 0.868ln 1r

h wF

h

83

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

+-

84

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

+-

85

ra

bz

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.


86

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.

87

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

h

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

2t

S

P dS

S

+ -


No attenuation on an infinite lossless line

88

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

Note: Radiation effects increase as the frequency increases.


h

Incident wavepipe

Obstacle

Reflected wave

Bend h

Incident wave

bend

Reflected wave 89

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)

90

91

Two conductorwire Coaxial line Shielded

Strip line

Dielectric

92

Two conductorwire Coaxial line Shielded

Strip line

Dielectric

93

Rectangular guide

Circular guide

Ridge guide

Common Hollow-pipe waveguides

94

STRIP LINE CONFIGURATIONS

W

SINGLE STRIP LINE COUPLED LINES

COUPLED STRIPSTOP & BOTTOM

COUPLED ROUND BARS

95

MICROSTRIP LINE CONFIGURATIONS

TWO COUPLED MICROSTRIPS SINGLE MICROSTRIP

TWO SUSPENDED SUBSTRATE LINES

SUSPENDED SUBSTRATE LINE

96

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

97

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

98

Maxwell’s Equations in Large Scale Form

SdDt

SdJldH

SdMSdBt

ldE

SdB

dvSdD

SSl

SSl

S

SV

0

ENEE482 99

• 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

100

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

101

)(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

102

HnHJ

B

Dn

ts

n

ˆ

0Bn 0

ˆρD

0En oE

:ConductorPerfect aat ConditionsBoundary

sn

t

+ + +n

s

Js

Ht

103

0)(ˆ

)(ˆ

0)(ˆ

0)(ˆ

Hn

MEn

Bn

Dn

s

104

2/ ; 0

; 0

:medium free Sourcea For

)(

)(

22

2222

2

vkHkH

kEkE

EjJj

HjEEE

Wave Equation

105

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

106

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

107

s

j

1)1(

2j)(1

j

2

1

2j)(1

j jj

s

108

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

109

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

110

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

111

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

112

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

113

dVMHJEdVDEHB

j

dVMHJEdVDEHBj

dSHEVdHE

EJJ

JEEDjHMBj

EHHEHE

V

s

V

s

VV

SV

s

)(2

1

442

)(2

1)(

2

2

1

2

1

)(

)()(

****

****

**

***

***

114

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

115

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

)(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

116

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

117

22

22

22

22

2

0D

equation. HelmholtzousInhomogene

condition) (Lorentz or

Let ,

,

1

, 0

,Let

k

JAkA

jA

jAk

JjAkAAA

JjAJEjAH

AjE

AjEAjE

AjBjEAB

118

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

119

Rg

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

120

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

121

LCC

L

YZ

VYIVYIeIeIzI

eVeVzV

vzV

vdz

zVd

zCVjz

zI

zLIjz

zV

tVtV

cc

cczjzj

zjzj

gg

, 1

, ,)(

)(

, 0)()(

)()(

)()(

cos)(

2

2

2

2

122

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

123

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

124

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

125

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

126

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

127

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

128

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

129

hx

y

h

tztzt

ztc

t

Zh

e

kZ

haZk

hae

hk

jh

y

x

0

000

2

h

e

Impedance Wave

; ˆˆ

130

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

131

TEM TRANSMISSION LINES

Parallel -plate Two-wire Coaxial

ab

COAXIAL LINES

132

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

133

)/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

Zc OF COAXIAL LINE AS A FUNCTION OF b/a

134

r Zo=X

b/a

1

10

1000

20

40

60

80

100

120

140

160

180

200

220

240

260

135

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

136

00c

*

**

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

Qc OF COAXIAL LINE AS A FUNCTION OF Zo

137

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

10

0

11

0

12

0

13

0

14

0

15

0

16

0

17

0

18

0

r Zc

GHz

c

fb

Q

138

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

139

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

140

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

141

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

142

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

143

Np/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

144

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

Documents

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