H63MCM Microwave Communications

Unit 2 – Transmission Line Theory

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K.T. Selvan

Department of Electrical & Electronic Engineering

The University of Nottingham Malaysia Campus

Unit Objectives

To discuss

� Transmission line fundamentals

� Lossless lines

Special cases

2

� Special cases

� Low-loss lines

� Distortion

Transmission Lines

� When to bother?

3

� Consider transmission line effects for / 0.01l λ ≥

� Reflection, power loss, dispersion, distortion

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Lumped-Element Model

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Figure 2.1 (p. 50)

Voltage and current definitions and equivalent circuit for an incremental

length of transmission line. (a) Voltage and current definitions. (b)

Lumped-element equivalent circuit.

Wave Propagation on a Transmission Line

� The traveling wave solutions are

( ) z zo oz V e V e

γ γ+ − −= +V

( ) z zo oz I e I e

γ γ+ − −= +I

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� The complex propagation constant γ is

( )( )j R j L G j Cγ α β ω ω= + = + +

� The characteristic impedance Zo is:

� The wavelength on the line is:

oR j L R j L

ZG j C

ω ω

γ ω

+ += =

+

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� The phase velocity is:

2πλ

β=

pv fω

λβ

= =

The Lossless Line

� Condition:

� One gets

0, LCα β ω= =

0R G= =

8

� The other parameters are:

oL

ZC

=2 2

LC

π πλ

β ω= =

1pv

LC

ω

β= =

The voltage reflection coefficient at the load

Terminated Lossless Line

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� The voltage reflection coefficient at the load

is:/ 1

/ 1

o L o L o

L o L oo

V Z Z Z Z

Z Z Z ZV

−

+

− −Γ = = =

+ +

ljel

β2)0()( −Γ=Γ

� Power flow:

� Return loss:

2

2

oiav

o

VP

Z

+

=2

2r iav avP P= − Γ

2

21

2

o

avo

VP

Z

+

= − Γ

10

� Standing wave ratio (SWR) is defined as

max

min

1

1

VS

V

+ Γ= =

− Γ

RL 20log dB= − Γ

� The input impedance of a length of

transmission line with an arbitrary load

impedance is:

in

cos sin

cos sin

L oo

o L

Z l jZ lZ Z

Z l jZ l

β β

β β

+= +

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Special Cases of Lossless Terminated Lines

� Half-wave line

In this case,

/ 2l mλ=

2( 0,1,2,...)

ml m m

π λβ π= = =

12

Then:

Implication?

2( 0,1,2,...)

2

ml m m

π λβ π

λ= = =

( / 2)in LZ l m Zλ= =

� Quarter-wave transformer

For this case, then,

2(2 1) (2 1) ( 0,1,2,...)

4 2l m m m

π λ πβ

λ= + = + =

2

13

Can be used for matching two impedances Zo1 and Zo3, when the transformer has an

impedance

2

in ( / 4) o

L

ZZ l

Zλ= =

2 1 3o o oZ Z Z=

� Short-circuited line

0LZ =

1Γ = −

s = ∞

in tanoZ jZ lβ= (Purely reactive input impedance)

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Inductive for

Application in microwave and high-speed ICs

tan 0 : taneq ol j L jZ lβ ω β> =

eq

1tan 0 : tanol jZ l

j Cβ β

ω< =Capacitive for

� Open-circuited line

LZ = ∞

1Γ =

15

s = ∞

in cotoZ jZ lβ= −

The Low-Loss Line

� We can assume R << ωL and G << ωC

� To deduce attenuation and phase constants,

let us start with propagation constant:

( )( )j R j L G j Cγ α β ω ω= + = + +

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� Rearranging,

( )( ) 1 1R G

j L j Cj L j C

γ ω ωω ω

= + +

� Since for a low-loss line RG << ω2LC

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R G RGj LC j

L C LCγ ω

ω ω ω

= − + −

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1R G

j LC jL C

γ ωω ω

= − +

12

j R Gj LC

L Cω

ω ω

≈ − +

� Therefore:

1 1

2 2o

o

C L RR G GZ

L C Zα

≈ + = +

LCβ ω≈

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� By the same order of approximation:

� Thus Zo and γ for low-loss lines can be closely approximated to that of lossless lines

oL

ZC

≈

The Dispersionless Line

� β in general not a linear function of frequency (when loss is present)

� This means various frequency components

travel with different phase velocities

This leads to dispersion.

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� This leads to dispersion.

� In turn, dispersion leads to the concept of

group velocity

� Consider a lossy line satisfying the relation:

� Under this condition:

R G

L C=

CR j LC j

Lγ ω α β= + = +

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� Thus, though a constant attenuation is present, β is a linear function of frequency. Hence no dispersion!

� Realizing this condition requires L to be increased

by loading series loading coils along the line

R j LC jL

γ ω α β= + = +

Summary

� Fundamental equations for characterizing

transmission lines

� Lossless lines

� Reflection at discontinuities

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� Special cases of transmission lines

� Low-loss lines

� Dispersionless lines