TRANSMISSION LINES By

Dr. Sikder Sunbeam Islam

EEE, IIUC.

INTRODUCTION

Transmission lines (TL) are interconnections that convey

electromagnetic energy from one point to another. Example

include two-wire, coaxial, strip, microstrip, waveguides, optical

fiber lines etc.

Transmission lines basically consists of two or more parallel

conductors used to connect a source to load. Source may be

generator, transmitter, oscillator and load may be factory,

antenna, oscilloscope, etc.

(c)

Fig.1. Example TL: (a)coaxial line, (b)two-wire line, (c) waveguide. 2

TRANSMISSION LINE: CIRCUIT THEORY

Transmission line (TL) can be described by circuit parameters

that are distributed throughout its length.

Consider a differential length ∆z of a TL that is described by the

following four parameters:

R= Series Resistance per unit length (both conductor),ohm/m

L= Series Inductance per unit length (both conductor),H/m

G=Shunt Conductance per unit length (both conductor),S/m

C=Shunt Capacitance per unit length (both conductor),H/m

Fig.2: Distributed parameters of two conductor TL

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TRANSMISSION LINE: GENERAL EQUATION

Fig.3 shows a equivalent electric circuit of such line segment.

The quantities v(z, t) and v( z+∆z, t) denotes the instantaneous

voltages at z and z+∆z, respectively. Similarly, i(z,t) and i(z+∆z, t)

denotes the instantaneous currents at z and z+∆z, respectively.

Applying Kirchhoff’s voltage law, we find

Fig.3:

Equivalent

circuit of

differential

length ∆z

of two wire

TL.

------------------(1) Or, 4

TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)

In the limit as ∆z→0, from equ.(1),

Now, applying Kirchhoff’s Current law to node N, we find

On dividing by ∆z and ∆z→0, from equ.(3),

Equation (2) and (4) are General Transmission Line Equations.

------------------(2)

------------------(3)

------------------(4)

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TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)

For simplifying the TL equations Assuming Harmonic Time

dependence,

Where, V(z) and I(z) are the phasors form of v(z, t) and i(z, t)

respectively. Therefore, equ. (2) and (4) becomes,

Equations (6.1) and (6.2) are Time harmonic TL-Equations. Equ.(6.1)

and (6.2) can be combined to solve V(z) and I(z) . Now taking second

derivative to equ. (6.1),

------------------(5.1)

------------------(5.2)

------------------(6.1)

------------------(6.2)

6 −𝑑2𝑉(𝑧)

𝑑𝑧2 = (R+j𝜔𝐿) 𝑑𝐼(𝑧)

𝑑𝑧

𝑑2𝑉(𝑧)

𝑑𝑧2 = (R+j𝜔𝐿) (G+j𝜔𝐶)V(z)

So,

Here, 𝛾 is propagation constant and α and β are attenuation

(Np/m) and phase (rad/m) constant respectively.

Solution of Equ. (7) and (8), are,

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------------------(7)

and similarly, ------------------(8)

TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)

Where,

------------------(9)

------------------(10.1)

------------------(10.2)

Here, +ve and –ve sign indicates the wave travelling in the +z and –z

direction respectively.

Thus we obtain instantaneous voltage expression,

The ratio of the voltage and the current at any (point of) z for an

infinitely long line is independent of z and is called the

characteristic impedance (Zo) of the line.

Lossless Line ( R=G=0): A TL is said to be lossless if

conductors of the lines are perfect (σ≈∞) and the dielectric

medium separating them is lossless (σ ≈ 0) .

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TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)

------------------(10.3)

------------------(11)

------------------(12.1)

------------------(12.2)

------------------(12.3)

Distortion less Line (R/L=G/C): Distortion less line is one in

which attenuation constant is frequency independent where

phase constant is linearly dependent on frequency. So in this

case,

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TRANSMISSION LINE: GENERAL EQUATION (CONTINUE.)

𝛽

------------------(12.4)

------------------(12.5)

TRANSMISSION LINE PARAMETERS

Comparison of Parameters of Two-wire and coaxial

cable:

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PROBLEM.1:

So,

Problem.2: It is found that attenuation on a 50 Ω distortion less

transmission line is 0.01 (dB/m). The line has a capacitance of 0.1

(nF/m).

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1Np=8.686 dB

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Voltage Standing Wave Ratio(VSWR): It is ratio of maximum to minimum

value of voltage is called VSWR. ------------------(13)

Voltage Reflection Coefficient (ГL): Voltage Reflection Coefficient at any

point on the line is the ratio of magnitude of reflected wave to that of incident

wave. ------------------(14)

Where, Zo= Characteristics Impedance; ZL= Load Impedance

INPUT IMPEDANCE OF TL

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------------------(15)

------------------(16)

Zin = Input impedance,

l= line length or

distance from the load.

ZL= Load Impedance

Zo=Characteristics Impedance

MICROSTRIP LINE

Microstrip line (ML) is a form of parallel plate TL consists of dielectric substrate sitting on a grounded conducting plane with a thin narrow metal strip on top of the substrate (Seen in Fig.5).

Few Equations for ML:

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Zo= Characteristics Impedance

(infinite long TL)

------(17)

Fig.5

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IMPEDANCE MATCHING: SMITH CHART

Smith Chart is a valuable graphical tool. It is a graphical

indications of the impedance of TL as one moves along the line.

Invented by Phillip H. Smith in 1939

Used to solve a variety of transmission line and waveguide problems.

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Fig.6:

SMITH CHART: BASIC USES

For evaluating the rectangular components, or the magnitude

and phase of an input impedance or admittance, voltage,

current, and related transmission functions at all points

along a transmission line, including:

Complex voltage and current reflections coefficients

Complex voltage and current transmission coefficients

Power reflection and transmission coefficients

Reflection Loss

Return Loss

Standing Wave Loss Factor

Maximum and minimum of voltage and current, and SWR

Shape, position, and phase distribution along voltage and

current standing waves. 18

Smith Chart constructed within a circle of unit radius (Г≤1). The chart is based on the equ.(14) , Where, Г𝑟and Г𝑖 are real and imaginary part of reflection coefficient (Г).

For a TL, the all impedance in the chart is normalized by a characterized impedance Zo. So, normalized load impedance ZL will be 𝑧𝐿,

The r and x are normalized resistance and reactance respectively. Different values of r yields circles having centers on Г𝑟-axis and centers of all x-circles lie on Г𝑟=1. Those, x >0 (inductive reactance)lie aboveГ𝑟-axis and x <0 (capacitivereactance)lie b𝐞𝐥𝐨𝐰 Г𝑟-axis. [See Fig.7]

Thex=0 circle becomes the Г𝑟-axis line.

All r-circles pass through (Г𝑟=1 , Г𝑖=0) point. Centers of all r-circles lie on Г𝑟-axis . The r=0 circle is the largest , centered at origin (of unity radius ).

At Psc on the chart r=0,x=0,(ZL=0+j0) represents short circuit on the TL. At Pocon the chart r=∞,x=∞,(ZL=∞+j∞)represents opencircuit on the TL.

The complete round-trip (3600) around the Smith Chart represents a

distance of 𝝀

𝟐 on the TL. The 𝛌 distance on the TL correspond to

𝟕𝟐𝟎𝟎movement on the chart.

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SMITH CHART: FEATURES

------(14) Or,

The clockwise movement on the chart regarded as moving towards

generator (away from the load).Similarly, The counter-clockwise

movement on the chart regarded as moving towards Load (away from

the source) [See Fig.8].

There are 3-scales around the periphery of the smith chart, seen in

Fig.8. The Outer most scale-determines the distance on the line

from Generator end in terms of 𝛌. The next one- determines the

distance on the line from Load end. The innermost scale is used for

determining 𝜽𝒓.

The𝑽𝒎𝒂𝒙occur where 𝒁𝒊𝒏,𝒎𝒂𝒙 is located on the chart; that is on the

positive Г𝑟-axis or on OPoc . The𝑽𝒎𝒊𝒏occur where 𝒁𝒊𝒏,𝒎𝒊𝒏 is located on the

chart; that is on the negative Г𝑟-axis or on OPsc . 𝑽𝒎𝒂𝒙 and 𝑽𝒎𝒊𝒏 are

𝟏𝟖𝟎𝟎apart .

Smith chart can be used both as impedance and admittance (Y=1/Z)

chart. In admittance chart (normalized impedance y= Y/Yo =G+jB), the

g and b correspond to r and x-circles respectively.

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SMITH CHART: FEATURES

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SMITH CHART: FEATURES

Fig.7:

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SMITH CHART: FEATURES

PROBLEM:4

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REFERENCE

Engineering Electromagnetics; William Hayt &

John Buck, 7th & 8th editions; 2012

Electromagnetics with Applications, Kraus and

Fleisch, 5th edition, 2010

Elements of Electromagnetics ; Matthew N.O.

Sadiku

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