2.Transmission Line Theory

7/24/2019 2.Transmission Line Theory

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TRANSMISION LINE THEOR


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Outline2

1. The Lumped-Element Model for a Transmission L

2. Field Analysis of Transmission Lines

3. The Terminated Lossless Transmission Line

4. The Smith Chart

5. The QuarterWave Transformer

6. Generator and Load Mismatches

7. Lossy Transmission Line

8. Transients on Transmission Lines


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Transmission Line3

Transmission Line: two-port network with sending end an

receiving end

Sending end: Generator circuit

Receiving end: Load circuit

Vg

Rg

RL

A

A

B

B

Sending-end

portReceiving-end

port

Generator Circuit

Load Circuit

Transmission Line


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General consideration4

Transmission Line are commonly wires in Low-frequencyelectrical circuits

We often analyze circuits without considering dispersive effect

of wire.

Question: When do we need to consider transmission line effe

Ans. It depends on length of line land frequencyf

of generator circuit


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General consideration5

Vg Rg

RL

A

A

B

B

Generator Circuit

Load Circ

Transmission Line

xl0

Phas


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Outline7


Wave Propagation on a Transmission Line

The Lossless Line


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Lumped-Element Model8

R: series resistance per unitlength, for both conductors

[/m]

L: series inductance per unit

length, for both conductors

[H/m] G: shunt conductance per unit

length [S/m]

C: shunt capacitance per unit

length [F/m]


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10

Transmission Line Equation (Cont

For the sinusoidal steady-state condition, with cosine-based phasors, the t

equations simplify to


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11

Wave Propagation on a Transmissio

The two equations can be solve simultaneously to give wave equations for

where is the complex propagation constant which is the function of freque


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12

Wave Propagation on a Tx. Line (C

Traveling wave solution can be found as:

where the e z

term represents wave propagation in the +z direction, and therepresents wave propagation in the z direction. The current on the line:


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Characteristic Impedance13

The characteristic impedance relate to the voltage and current on the li


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Voltage Waveform14

f is the phase angle of the complex voltage V

The wavelength and the phase velocity are


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In many practical cases the loss of the line is very small so can be n

resulting in a simplification of the results. SettingR = G = 0 the propagation constant is given

The characteristic impedance reduce to

The Lossless Line16


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Voltage and Current on a Lossless T17

The wave length and the phase velocity are


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Outline18


2. Field Analysis of Transmission Lines

3. The Terminated Lossless Transmission Line

4. The Smith Chart


6. Generator and Load Mismatches




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Field Analysis of Transmission Lines19

The self-inductance per unit length:

The time-average stored electric energy per unit length:

The capacitance per unit length:

The time-average stored magnetic energy per unit length:


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Field Analysis of Transmission Lines (C20

The time-average power dissipated per unit length in a lossy dielectri

The power loss per unit length due to the finite conductivity of the co

The series resistance R per unit length:

The shunt conductance per unit length:


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Voltage Reflection Coefficient23

The time-average power flow along the line at the positionz


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Return Loss24

When the load is mismatched, not all of the available power from the

delivered to the load. The loss cause by the reflection at load is call the return loss and is de

Matched load (=0): RL=

Total reflection (=1): RL= 0


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Standing Wave Ratio25

The magnitude of the voltage on the line:

Standing wave ratio is the ratio of Vmax to Vmin as follow


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Reflection Coefficient and Input Impeda26

The reflection coefficient atz = - l

The imput impedance atz = - l


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Transmission Line Impedance E27


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Outline28

1. The Lumped-Element Model for a Transmission Line

2. Field Analysis of Transmission Lines3. The Terminated Lossless Transmission Line:

Terminated lossless transmission line

4. The Smith Chart

5. The QuarterWave Transformer6. Generator and Load Mismatches



Special Cases of Lossless Terminated


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Special Cases of Lossless Terminated Short-circuited Transmission Line

29

Short-circuited Tx. Line:

The voltage and current on the line:

The input impedance:


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Short-circuited Transmission Line30

Open-circuited Transmission Line


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Open-circuited Tx. Line:

The voltage and current on the line:

Open-circuited Transmission Line31

The input impedance:

(


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Open-circuited Transmission Line (Co32

H lf l h T Li


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Half-wavelength Tx. Line33

If the length of the line is half of the wave length

Q f


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Quarter-wavetransformer34

If the length of the line is a quarter wavelength, then the in

impedance is given by

a quarter wavelength transformer

O i H k


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Onsite Homework35

P 2.3, 2.8, 2.10,

O tli


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Outline36


2. Field Analysis of Transmission Lines3. The Terminated Lossless Transmission Line

4. The Smith Chart


6. Generator and Load Mismatches7. Lossy Transmission Line



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N li d L d I d


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Normalized Load Impedance38

0ZZz LL

The nomalized load impedance:

G

G 1

1Lz

LLL jxrz

The nomalized load impedance is a complex quantity compose

normalized load resistance and normalized load reactance

ir

irL

j

jz

GG

GG

1

1

GG

G

GG

GG

22

22

22

1

2

1

1

ir

i

L

ir

ir

L

x

r


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Normali ed Load Impedance (cont


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Normalized Load Impedance (cont41

0LrWhich circle yield

The Smith chart

simply a depiction onr- i plane, of the

families of circles

family rL and xL fa

plotted for sele

values ofrL andxL

Normalized Load Impedance


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42

Consider norma

impedance (1+j2)

Point A: the interse

of the rL = 1 con

resistance circle

the xL = 2 con

reactance circle segm

The norma

impedance of point

Normalized Load Impedance

E Fi d th fl ti ffi i t


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Find point P represents normalized load impedance

zL= (2-j).

Find the correcsponding reflection coefficient.

Ex: Find the reflection coefficient43

Point P: zL= (2-j)OR

OP G

r .26

Input Impedance


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Input Impedance44

G

G

zj

zj

in

e

eZzZ

2

2

0

1

1)(

G

G

zj

zj

inin

e

e

Z

Zzz

2

2

0 1

1)(

L

L

in zz G

G

1

1

)( G

G

1

1

Lz

The same form!!!

On Smith chart, transforming to l means maintainingconstant and decreasing the phase r

Input Impedance on Smith Chart


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Input Impedance on Smith Chart

Consider a 50 Ohm lossless transmission line terminated in a load

impedance (100j50) Ohm. Find Zin at a distance l = 0.1 from l

45

Z

Zz LL 2

0

Point A repre

zL = 2

Point B repre

zin =0.6j0.

SWR Voltage Maxima and Minim


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SWR, Voltage Maxima, and Minim46

Point A represents

zL = 2+j

S = 2.6 (at Pmax

lmax =(0.25-0.213)

=0.037

lmin =(0.037+0.25)

=0.287

A

0.287

SWR Circle

0 Pmax

0.213

0.037

Pmin

Impedance to Admittance Transform


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Impedance to Admittance Transform47

The admittance Y is the reciprocal ofZ

2222211

RXj

XRR

XRjXR

jXRZY

Conductance GSusceptan

The normalized admittancey:

jbgY

Bj

Y

G

Y

Yy

000

Impedance to Admittance Transform


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Impedance to Admittance Transform48

The admittanceyL:

Rotation /4 on the Smith chart transformszL intoyL

GG

111

L

Lz

y

Lj

j

in ye

elz

G

G

G

G

1

1

1

1)4/(


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Single-stub Matching (Cont.)


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g g ( )50

The normalized impedance is

jj

Z

Zz LL

5.0

50

5025

0

(point A in Smith chart)


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g g ( )52

In the admittance domain , rL circles becomeyL circles,

xL circles become bL circles

The normalized admittance is

8.04.05025

500 jjZ

Zy

L

L

(point B in Smith chart,

0.25 rotate from A)

It is easier to work with admittances

than with impedance

Single-stub Matching


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g g53



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Single stub Matching54

Solution for Point C:

At C,yd = 1+j1.58

Distance between B and C

d1 = (0.178-0.115)=0.063

Need:ys = - j1.58 (point F)

l1 = (0.34-0.25)=0.09

( distance from E to F)

Step 1: Select dso that

yLbecomes :

yd = 1+b

There are 2 solutions:

Point C and Point D



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Single stub Matching55

Solution for Point D:

At D,yd = 1-j1.58Distance between B and D

d2 = (0.178-0.115)

=0.207

Need:

ys = j1.58 (point G)l2 = (0.25+0.16)=0.41

(distance from E to G)



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56

Two Solutions



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d1 = 0.063

l1 = 0.09

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d2 = 0.207

l2 = 0.41

Outline


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58



4. The Smith Chart




Quarter-wave transformer


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Q59

Ex: Quarter-wave transformer


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Q60

A 50 Ohm lossless Tx line is to be matched to a resistive loa

impedance 100 Ohm via a quarter-wave section thereb

eliminating reflections along the feedline. Find the characterist

impedance of the quarter-wave transformer

Ex: Quarter-wave transformer (cont


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Q (61

Solution: To eliminate reflections at terminal AA, the in

impedance Zin looking into the quarter-wave line should be equa

Z01, from following Eq.,

L

inZZZ

2

02

7.701005002 LinZZZ

Outline


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62



4. The Smith Chart




Input impedance


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Voltage reflection coefficient


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is determined by the ratio:

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Is real or complex?! complex

For passive load | | 1

Passive load Re[ZL] 0

Also, lossless Z0= Real 0

Re[ZL /Z0] 0

11/

1/

0

0

0

0

G

ZZ

ZZ

ZZ

ZZ

L

L

L

L

Standing point of transmission line


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65

VSWR


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VSWR: Voltage standing wave ratio

66

);1[ S

S=1 ||=0 No reflected power

S= ||=1 All power reflect

1


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67

Let

Then we will have

Matched Case


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68

Load match to the line

Generator matched to loaded line

Conjugate matching

Outline


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69



4. The Smith Chart


6. Generator and Load Mismatches7. Lossy Transmission Line (self study)


Transients on transmission line


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Treatment of wave propagation on Tx line: Focus on the analysis of s

frequencies, time-harmonic signals under steady-state conditions

Tool: impedance matching & the use of Smith chart

Useful for a wide range of applications

Inappropriate for dealing with digital or wideband signals

Need to exam the transient behavior as a function of time

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Basic approach


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0,0

0,1)(

t

ttU

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elsewhere,0

0,)(

0 tVtV

)()(

)()()(

00

21

tUVtUV

tVtVtV

V(t) is the sum of 2 unit step functions :

Analyze the response of Tx line

to a unit step function ( DC turn on voltage) V0 U(t)

From the linearity of the line, the pulse transient response is then

obtained by superposition

Consider the simple case of a single

rectangular pulse of amplitude V0 duration :

Transient response


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Consider a step voltage V0 U(t) applied to a Tx line ofZ0terminated in a real loadZL

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g

g

VZR

ZV

0

0

1

0

1ZR

VI

g

g

Switch close at t = 0+

Initial voltage V1+ & currentI1

+

Transient response (cont.)


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What happens as this voltage step echoes off the ends of the

line?

Consider the line withRg = 4Z0 &ZL = 2Z0

73

3

1

0

0

G

ZZ

ZZ

L

LL

5

3

0

0

G

ZR

ZR

g

g

g

The length of the line is l

One way transit time : T=l/up

Transient response (cont.)


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Steady state voltage V


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G

GGGGG

GGGGGGGGG

2

1

22

1

23222

1

332211

11

11

1

xxV

V

V

VVVVVVV

L

gLgLL

gLgLgLgLL

Lx GG

The series inside the square bracket is the binominal series of

funtion

1for11

1 2

xxxxThen we will get

gL

LVVGG

G

1

1

1

Lg

Lg

ZR

ZVV

Bounce diagram


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A simple graphical way to keep track of the bounce history

Axes of bounce diagram represent

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z

t

Bounce diagram (Cont.)


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Voltage vs time atz=l/4 for a circuit

withg =3/5 &L=1/3

Bounce diagram


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Onsite Homework & Homework79


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Onsite P 2.8, 2.12

Home work: P 2.3, 2.10, 2.16, 2.20, 2.21,2.31

Documents

2.Transmission Line Theory