Chapter-1-Transmission-Line-Theory

5 Characteristic Impedance Zo At a given location along the line, find:  Current, voltage and power  Reflection coefficient, impedance, VSWR  Design real TLs, such as micro-strip l

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Huynh Phu Minh Cuong hpmcuong@hcmut.edu.vn

Department of Telecommunications Faculty of Electrical and Electronics Engineering

Ho Chi Minh city University of Technology

Chapter 1

Transmission Line Theory

MICROWAVE ENGINEERING

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Chapter 1: Transmission Line Theory

Contents

1 Introduction

2 Lumped-Element Circuit Model for Transmission Lines

3 Transmission Line Equations and Solutions

4 Characteristic Impedance of Transmission Line

5 Propagation constant and velocity

6 Lossless and Lossy Transmission Lines

7 Reflection Coefficient

8 Transmission Line Impedance and Admittance

9 Power Transmission on Transmission Lines

10 Standing Wave and Standing Wave Ratio

11 Practical Transmission Lines

Problems

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1 Introduction

 Transmission line theory bridges the gap between field analysis and basic circuit theory and therefore is of significant importance in the analysis of microwave circuits and devices

 The key difference between circuit theory and transmission line theory is electrical size

 At low frequencies, an electrical circuit is completely characterized by the electrical parameters like resistance, inductance etc and the physical size of the electrical components plays no role

in the circuit analysis

 As the frequency increases however, the size of the components becomes important The voltage and currents exist in the form of waves Even a change in the length of a simple connecting wire may alter the behavior of the circuit

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1 Introduction

 The circuit approach then has to be re-investigated with inclusion

of the space into the analysis This approach is then called the transmission line approach

 Although the primary objective of a transmission line is to carry electromagnetic energy efficiently from one location to other, they find wide applications in high frequency circuit design

 Also at high frequencies, the transit time of the signals can not be ignored In the era of high speed computers, where data rates are approaching to few Gb/sec, the phenomena related to the electromagnetic waves, like the bit distortion, signal reflection, impedance matching play a vital role in high speed communication networks

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5

Characteristic Impedance Zo

At a given location along the line, find:

 Current, voltage and power

 Reflection coefficient, impedance, VSWR

 Design real TLs, such as micro-strip lines, CPW lines

General problems of the chapter

1 Introduction

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 A transmission line is basically a two-conductor line for

guiding the signal power from one point to another

 A transmission line is a distributed parameter network, where

voltages and currents can vary in magnitude and phase over its length

 While ordinary circuit analysis deals with lumped elements,

where voltage and current do not vary appreciably over the

physical dimension of the elements

 TLs are analyzed using transmission-line theory or Distributed-circuit theory not traditional lumped-circuit theory

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x x + ∆x

Load Source

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R, L, G, and C are per-unit-length quantities defined as follows:

 R = series resistance per unit length, for both conductors, in Ω/m

 L = series inductance per unit length, for both conductors, in H/m

 G = shunt conductance per unit length, in S/m

 C = shunt capacitance per unit length, in F/m

 Series inductance L represents the total self-inductance of the two conductors,

 Shunt capacitance C is due to the close proximity of the two conductors

 Series resistance R represents the resistance due to the finite conductivity of the individual conductors

 Shunt conductance G is due to dielectric loss in the material between the conductors

 R and G, therefore, represent loss

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Applying Kirchoff for voltage:

Applying Kirchoff for currrent:

( , ) ( , ) ( , ) ( , ) i x t

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

I x x I x

G j C V x x x

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I x

G j C V x x

I x

R j L G j C I x x

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Where γ ω ( ) = ( R + j L G ω )( + j C ω )

2

2 2

I x

R j L G j C I x x

2

2 2

I x

I x x

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2 2

are constant with a given source, load and TL

V and V

How to calculate V+ and V- ?

Incident voltage Reflected voltage

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Source Reflected wave Incident wave Load

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x

X = L

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2 2

Where is Characteristic Impedance

Chapter 1: Transmission Lines

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Chapter 1: Transmission Lines

0

R j L R j L Z

What is the physical meaning of Z o ?

 What are V + and I + at t = 0 with an infinity long TL ?

 Can an infinity long TL be replaced by a Z 0 ?

 What is the input impedance of a TL terminated by a Z 0 ?

 What is the input impedance of a infinitesimal TL, ∆x length,

terminated by a Z 0 ?

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 In practice, Z0 is always real

 In communication systems: Z0 = 50 Ω

 Telecommunication: Z0 = 75 Ω - Why 50Ω or 75Ω ?

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5 Propagation Constant and Velocity

: Attenuation constant - unit: [Np/m] or [dB/m]

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 Lossless Transmission Line:

0, 0

( ) 0 ( )

6 Lossless and Low-loss Transmission Lines

 In practice, transmission lines have losses due to finite conductivity and/or lossy dielectric, but these losses are usually small

 In most practical microwave:

 Loss may be neglected  Lossless T.L

 Loss may be assumed to be very small  Low-loss T.L

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 When the loss is small, some approximations can be made to simplify the expressions for the general transmission line parameters

of γ = α + jβ and Z0

 Low-loss Transmission Line:

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 Low-loss Transmission Line:

For a low-loss line both conductor and dielectric loss will be small, and we can assume that R << ωL and G >>ωC Then, RG << ω2LC

Using the Taylor series expansion for

So:

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γ

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7 Reflection Coefficient – At Load -

− +

Γ = 2 l

l

V

e V

1

1 ( )

1 ( ) 1

−

− +

+

− +

0 0

−

Γ =

+

L L

L

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At load − 2

+

Γ = l l

V e V

7 Reflection Coefficient – At any location x - Γx

2

. −

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 TL is terminated by Z 0

0 0

There is no reflected wave

7 Reflection Coefficient – Some special cases

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0 0

 TL is shorted

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Incident and reflected currents are out of phase  I l ( ) = 0

 TL is opened

0 0

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0 0

 TL is terminayed by reactance

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7 Reflection Coefficient – Presentation on a complex plane

Source Ref wave Inc wave Load

/ 22

λ

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x

V e V

γ

2. −

0 0

( ) L

L

Z Z l

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2( )( )

0 0

1 ( ) ( ) ( ) ( ) ( )

0

0 0

0

0 0

0

1 ( )( )

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+

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ZL = ⇒ 0 Z x ( ) = j R tg ( β d ) = j X d ( ) , pure reactance

Shorted Open

⇒ Shorted-circuit transmission lines can be used to realize inductors or

capacitors at specific frequencies  Distributed components

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0 0 0

0 0

⇒ Opened-circuit transmission lines can be used to realize inductors or

capacitors at specific frequencies  Distributed components

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ZL = ∞ ⇒ Z x ( ) = − j R cotg( β d ) = j X d ( ) , pure reactance

Nối tắt Hở Mạch

Shorted Open

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transformation

2 0

Z

0 0

0 2 0

Z ( ) ( )

0 0

Z

β β

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+ +

( ) ( )

 Reflection coefficient and line impedance

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( ) ( )

Y x

γ γ

+

0 0

γ γ

+

0 0

+

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⇒ Voltage and Current Calculation

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{ } { }

_ 0

2 2 2 . 2 . 2 2

2 2 2 2 0

2 2 2 0

ef

1

1 2

+ Γ

− +

α α

α

Source

inc

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2

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 Standing wave

x x

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t = 0

t = T/8 t = 3T/8 t = T/2

x

x Total wave

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Ex 3.13 p86

0

ax min

1SWR

1

l m

l

V V

V

+ Γ

− Γ

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Ex 3.13 p86

( ) ( )

V x

I x

x

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Ex 3.13 p86

( ) ( )

V x

I x

x

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 Prior to 1965 nearly all communication microwave equipment

utilized microwave tubes connected with coaxial lines or

waveguides

 In recent years - with the advance of microwave solid-state

electronics and the introduction of microwave integrated circuits (MIC) & monolithic RF/microwave integrated circuits

(RFIC/MMIC) - microstrip lines (), strip line, and

coplanar-waveguide (CPW) lines have been used extensively, on which state devices can be placed

solid-Chapter 1: Transmission Line Theory

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Wave Guide

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Problems

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