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COURSE MATERIAL

Electromagnetic Waves and

SUBJECT
Transmission lines (19A04401)

UNIT V

COURSE B.TECH

DEPARTMENT ECE

SEMESTER 2-2

K. UPENDRA RAJU
PREPARED BY
D. SRILATHA
(Faculty Name/s)
OMKAR NAIDU. V
K.R. SURENDRA

Version V-1

PREPARED / REVISED DATE 01-04-2021

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TABLE OF CONTENTS – UNIT 1
S. NO CONTENTS PAGE NO.
1 COURSE OBJECTIVES 3
2 PREREQUISITES 3
3 SYLLABUS 3
4 COURSE OUTCOMES 3
5 CO - PO/PSO MAPPING 4
6 LESSON PLAN 4
7 ACTIVITY BASED LEARNING 4
8 LECTURE NOTES 4
8.1 TRANSMISSION LINE PARAMETERS 4
8.2 TRANSMISSION LINE EQUIVALENT CIRCUIT 8
8.3 TRANSMISSION LINE EQUATIONS AND THEIR SOLUTIONS IN 8
THEIR PHASOR FORM
8.4 STANDING WAVE RATIO 12
8.5 TRANSMISSION OF FINITE LENGTH- HALF WAVE&QUARTER 13
WAVE
8.6 SMITH CHART 15
8.7 GRAPHICAL ANALYSIS OF TRANSMISSION LINES USING SMITH 22
CHART
8.8 STUB MATCHING SINGLE AND DOUBLE STUB MATCHING 22
8.10 PROBLEMS 25
9 PRACTICE QUIZ 27
10 ASSIGNMENTS 29
11 PART A QUESTIONS & ANSWERS (2 MARKS QUESTIONS) 30
12 PART B QUESTIONS 30
13 SUPPORTIVE ONLINE CERTIFICATION COURSES 31
14 REAL TIME APPLICATIONS 31
15 CONTENTS BEYOND THE SYLLABUS 32
16 PRESCRIBED TEXT BOOKS & REFERENCE BOOKS 32
17 MINI PROJECT SUGGESTION 32

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1. Course Objectives
The objectives of this course is to
1. Introduce fundamentals of static and time varying electromagnetic fields.
2. Teach problem solving in Electromagnetic fields using vector calculus.
3. Demonstrate wave concept with the help of Maxwell’s equations
4. Introduce concepts of polarization and fundamental theory of
electromagnetic waves in transmission lines and their practical
applications
5. Analyze reflection and refraction of electromagnetic waves propagated
in normal and oblique incidences
2. Prerequisites
Students should have knowledge on
1. Vector Analysis
2. Smith chart

3. Syllabus
UNIT V
Transmission Lines: Introduction, Transmission line parameters, Transmission line
equivalent circuit, Transmission line equations and their solutions in their phasor
form, input impedance, standing wave ratio, Transmission of finite length- half
wave, quarter wave transmission line, Smith chart, graphical analysis of
transmission lines using Smith chart, stub matching- single and double stub
matching, Illustrative Problems.

4. Course outcomes
1. Understand the principles of transmission lines and concept of smith chart
2. Derive the input impedance of transmission line.
3. Find the line parameters through problem solving
4. Study the applications of different lengths of transmission lines

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5. Co-PO / PSO Mapping
Machine
PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 P10 PO11 PO12 PSO1 PSO2
Tools

CO1 3 3 2 3 2 2

CO2 3 3 2 3 2 2

CO3 3 3 2 3 2 2

CO4 3 3 2 3 2 2

CO5 3 3 2 3 2 2

6. Lesson Plan
Lecture No. Weeks Topics to be covered References

1 Transmission line parameters T1

2 Transmission line equivalent circuit T1, R1

1
3 Transmission line equations and their solutions in their phasor form T1, R1

4 Standing wave ratio T1, R1

5 Transmission of finite length- half wave T1, R2

6 quarter wave transmission line T1, R1

2
7 Smith chart T1, R1

8 graphical analysis of transmission lines using Smith chart T1, R1

9 Stub matching single and double stub matching T1, R1

3
10 Problems T1, R1

7. Activity Based Learning

1. Conducting Quiz for designing VSWR
2. Designing Transmission line circuit for Different Parameters
8. Lecture Notes

8.1 Transmission line parameters:

A transmission line is used for the transmission of electrical power from
generating substation to the various distribution units. It transmits the wave of
voltage and current from one end to another. The transmission line is made

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up of a conductor having a uniform cross-section along the line. Air act as
an insulating or dielectric medium between the conductors.
For safety purpose, the distance between the line and ground is much more.
The electrical tower is used for supporting the conductors of the transmission
line.Tower are made up of steel for providing high strength to the conductor.
For transmitting high voltage, over longdistance high voltage direct current
is used in the transmission line.
A transmission line is a connector which transmits energy from one point to
another. The study of transmission line theory is helpful in the effective usage
of power and equipment.
There are basically four types of transmission lines −
 Two-wire parallel transmission lines
 Coaxial lines
 Strip type substrate transmission lines
 Waveguides
While transmitting or while receiving, the energy transfer has to be done
effectively, without the wastage of power. To achieve this, there are certain
important parameters which has to be considered.

The conventional open-wire transmission lines are not suitable for microwave
transmission, as the radiation losses would be high. At Microwave
frequencies, the transmission lines employed can be broadly classified into
three types. They are −
 Multi conductor lines
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o Co-axial lines
o Strip lines
o Micro strip lines
o Slot lines
o Coplanar lines, etc.
 Single conductor lines Waveguides
o Rectangular waveguides
o Circular waveguides
o Elliptical waveguides
o Single-ridged waveguides
o Double-ridged waveguides, etc.
 Open boundary structures
o Di-electric rods
o Open waveguides, etc.

The performance of transmission line depends on the parameters of the line.

The transmission line has mainly four parameters, resistance, inductance,
capacitance and shunt conductance. These parameters are uniformly
distributed along the line. Hence, it is also called the distributed parameter
of the transmission line.
The inductance and resistance form series impedance whereas the
capacitance and conductance form the shunt admittance.
Some critical parameters of transmission line are explained below in detail

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Line inductance – The current flow in the transmission line induces the
magnetic flux.When the current in the transmission line changes, the magnetic
flux also varies due to which emf induces in the circuit. The magnitude of
inducing emf depends on the rate of change of flux. Emf produces in the
transmission line resist the flow of current in the conductor, and this parameter
is known as the inductance of the line.
Line capacitance – In the transmission lines, air acts as a dielectric medium. This
dielectric medium constitutes the capacitor between the conductors, which
store the electrical energy, or increase the capacitance of the line. The
capacitance of the conductor is defined as the present of charge per unit of
potential difference.
Capacitance is negligible in short transmission lines whereas in long
transmission; it is the most important parameter. It affects the efficiency,
voltage regulation, power factor and stability of the system.
Shunt conductance – Air act as a dielectric medium between the conductors.
When the alternating voltage applies in a conductor, some current flow in the
dielectric medium because of dielectric imperfections. Such current is called
leakage current. Leakage current depends on the atmospheric condition and
pollution like moisture and surface deposits.
Shunt conductance is defined as the flow of leakage current between the
conductors. It is distributed uniformly along the whole length of the line. The
symbol Y represented it, and it is measured in Siemens.

The term performance includes the calculation of sending end voltage,

sending end current, sending end power factor, power loss in the lines,
efficiency of transmission, regulation and limits of power flows during steady
state and transient conditions. Voltage regulation is defined as the change in
the magnitude of the voltage between the sending and receiving ends of the
transmission line.
Admittance measures the capability of an electrical circuit or we can say it
measures the efficiency of a transmission line, to allows AC to flow through
them without any obstruction. It SI unit is Siemens and denoted by the symbol
Y.Impedance is the inverse of the admittance. Its measure the difficulty occurs
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in the transmission line when the AC flow. It is measured in ohms and
represented by the symbol z.
8.2 TRANSMISSION LINE EQUIVALENT CIRCUIT
The power transmission line is one of the major components of an electric
power system. Its major function is to transport electric energy, with minimal
losses, from the power sources to the load centers, usually separated by long
distances. The design of a transmission line depends on four electrical
parameters:
1. Series resistance
2. Series inductance
3. Shunt capacitance
4. Shunt conductance
The series resistance relies basically on the physical composition of the
conductor at a given temperature. The series inductance and shunt
capacitance are produced by the presence of magnetic and electric fields
around the conductors, and depend on their geometrical arrangement.

The shunt conductance is due to leakage currents flowing across insulators

and air. As leakage current is considerably small compared to nominal current,
it is usually neglected, and therefore, shunt conductance is normally not
considered for the transmission line modeling
8. 3 TRANSMISSION LINE EQUATIONS AND THEIR SOLUTIONS IN THEIR PHASOR
FORM: Let us take the equivalent circuit of the transmission line, for this we are
going to take the simplest form of transmission line which is two wirelines. This
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two wireline is made up of two conductors separated by a dielectric medium
usually air medium, which is shown in the below figure

If we pass a current (I) through the conductor-1, will find that there is a magnetic
field around the current-carrying wire of a conductor-1 and the magnetic field
can be illustrated using series inductor due to the current flow in the conductor-1,
there should be a voltage drop across the conductor-1, which can be illustrated
by a series of resistance and inductor. The setup of the two wireline conductor
can be made to a capacitor. The capacitor in the figure will always be loosy to
illustrate that we have added conductor G. The total setup i.e, series resistance
an inductor, parallel capacitor, and conductor make up an equivalent circuit of
a transmission line.

The inductor and resistance put together in the above figure can be called as
series impedance, which is expressed as
Z = R+jωL
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The parallel combination of capacitance and conductor n the above figure
can be expressed as
Y = G+jωc
w.k.t

V-(V + dV) = (R + jωL) dx * I

V-V-dv = (R + jωL) dx * I

Differentiating above eqaution

-dv/dx = (R + jωL) * I ………………. eq(1)

I-(I + dI) = (G + jωc)dx * V

I – I+dI = (G + jωc)dx * V

Differentiating above eqaution

-dI/dx = (G + jωc) * V … ……………. eq(2)

Differentiating eq(1) and (2) with respect to dx will get

-d2v/dx2 = (R + jωL) * dI/dx ………………. eq(3)

-d2I/dx2 = (G + jωc) * dV/dx … ……………. eq(4)

Substituting eq(1) and (2) in eq(3) and (4) will get

-d2v/dx2 = (R + jωL) (G + jωc) V ………………. eq(5)

-d2I/dx2 = (G + jωc) (R + jωL) I … ……………. eq(6)

Let P2 = (R + jωL) (G + jωc) … ……………. eq(7)

Where P – propogation constant

Substitute d/dx = P in eq(6) and (7)

-d2v/dx2 =P2V ……………….eq(8)

-d2I/dx2 = P2I … ……………. eq(9)

General solution is

V = Aepx + Be-px … ……………. eq(10)

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I = Cepx + De-px … ……………. eq(11)

Where A, B C and D are constants

Differentiating eq(10) and (11) with respect to ‘x’ will get

-dv/dx = P (Aepx – Be-px ) ………………. eq(12)

-dI/dx = P (Cepx – De-px) … ……………. eq(13)

Substitute eq(1) and (2) in eq(12) and (13) will get

-(R+jωL)*I=P(Aepx +Be-px ) ……………….eq(14)

-( G + jωc) * V = P (Cepx + De-px ) ………………. eq(15)

Substitute ‘p’ value in eq(14) and (15) will get

I = -p/ R + jωL * (Aepx + Be-px)

= √G + jωc / R + jωL * (Aepx + Be-px) ………………. eq(16)

V = -p/ G + jωc * (Cepx + De-px )

= √R + jωL / G + jωc * (Cepx + De-px ) ………………. eq(17)

Let Z0= √R + jωL / G + jωc

Where Z0is the characteristic impedenc

Substitute boundary conditions x=0, V=VS and I=IS in eq(16) and (17) will get

IS = A+B ………………. eq(18)

VS = C+D ………………. eq(19)

ISZ0= -A+B ………………. eq(20)

VS /Z0 = -C+D ………………. eq(21)

From (20) will get A and B values

A = VS -IS Z0

B =VS +IS Z0

From eq(21) will get C and D values

C = (IS – VS /Z0) /2
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D = (IS + VS /Z0) /2

Substitute A, B, C and D values in eq(10) and (11)

V = (VS -IS Z0) epx + (VS +IS Z0)e-px

= VS (epx +e-px/2) –IS Z¬0(epx -e-px/2)

= VS coshx – IS Z0 sinhx

Similarly

I = (IS -VS Z0) epx + (VS /Z0+IS / 2)e-px

=IS (epx+e-px/2) –VS /Z0 (epx -e-px/2)

=IS coshx – VS /Z0 sinhx

Thus V = VS coshx – IS Z0 sinhx

I = IS coshx – VS /Z0 sinhx

8.4STANDING WAVE RATIO

Standing-wave ratio (SWR) is a mathematical expression of the non-uniformity of
an electromagnetic field (EM field) on a transmission line such as coaxial cable.
Usually, SWR is defined as the ratio of the maximum radio-frequency (RF) voltage
to the minimum RF voltage along the line. This is also known as the voltage
standing-wave ratio (VSWR).

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The voltage standing wave ratio, VSWR is defined as the ratio of the maximum to
minimum voltage on a loss-less line.
The SWR can also be defined as the ratio of the maximum RF current to the
minimum RF current on the line (current standing-wave ratio or ISWR). For most
practical purposes, ISWR is the same as VSWR.
In practice there is a loss on any feeder or transmission line. To measure the VSWR,
forward and reverse power is detected at that point on the system and this is
converted to a figure for VSWR. In this way, the VSWR is measured at a particular
point and the voltage maxima and minima do not need to be determined along
the length of the line.

The VSWR or
voltage
standing wave ratio applies specifically to the voltage standing waves that are
set up on a feeder or transmission line. As it is easier to detect the voltage
standing waves, and in many instances voltages are more important in terms of
device breakdown, the term VSWR is often used, especially within RF design
areas.
The term power standing waves is also seen some times. However this is a
complete fallacy as the forward and reflected power are constant (assuming no
feeder losses) and the power does not rise and fall in the same way as the
voltage and current standing waveforms which are the summation of both
forward and reflected elements.
8.5TRANSMISSION OF FINITE LENGTH- HALF WAVE& QUARTER WAVE TRANSMISSION
LINE:
Standing waves at the resonant frequency points of an open- or short-circuited
transmission line produce unusual effects. When the signal frequency is such that
exactly 1/2 wave or some multiple thereof matches the line’s length, the source
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“sees” the load impedance as it is.
The following pair of illustrations shows an open-circuited line operating at 1/2
and 1 wavelength frequencies:

In either case, the line has voltage antinodes at both ends, and current nodes at
both ends. That is to say, there is maximum voltage and minimum current at
either end of the line, which corresponds to the condition of an open circuit.
The fact that this condition exists at both ends of the line tells us that the line
faithfully reproduces its terminating impedance at the source end, so that the
source “sees” an open circuit where it connects to the transmission line, just as if it
were directly open-circuited.

However, if the signal frequency is such that the line resonates at ¼ wavelength
or some multiple thereof, the source will “see” the exact opposite of the
termination impedance.
That is, if the line is open-circuited, the source will “see” a short-circuit at the point
where it connects to the line; and if the line is short-circuited, the source will “see”
an open circuit:

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A simple equation relates line impedance (Z0), load impedance (Zload), and input
impedance (Zinput) for an unmatched transmission line operating at an odd
harmonic of its fundamental frequency:

8.6 SMITH CHART

Smith chart is considered as a graphical measuring tool which is constructed
mainly for electrical engineers to solve problems related to RF transmission
lines and matching devices. At the same time, the device is also used to display
various factors like admittance, continual gain outlines, impedance, stability,
noise figure, and also shows a detailed analysis of mechanical vibrations.
The chart will be more common in use internal to the regions that have a radius
range of 1 or < 1 such as in the cases of stability performance and oscillator
design. As it was known that usage of the smith chart for resolving complicated
mathematical problems impose matching issues, but it is still an effective method
in explaining how RF frequencies perform at multiple frequency levels.
This because most RF circuit software’s consists of smith charts for displaying
results.It was originally developed to be used for solving complex maths problem
around transmission lines and matching circuits which has now been replaced by
computer software.
However, the Smith charts method of displaying data have managed to retain its
preference over the years and it remains the method of choice for
displaying how RF parameters behave at one or more frequencies with the

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alternative being tabulating the information.
Smith chart can be used to display several parameters including; impedances,
admittances, reflection coefficients, scattering parameters, noise figure circles,
constant gain contours and regions for unconditional stability, and mechanical
vibrations analysis, all at the same time.
As a result of this, most RF Analysis Software and simple impedance measuring
instruments include smith charts in the display options which makes it an
important topic for RF Engineers.

As mentioned in the introduction, the Smith Chart displays the complex reflection

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coefficient, in polar form, for a particular load impedance. Going back to basic
electricity classes, you will remember that impedance is a sum of resistance and
reactance and as such, is more often than not, a complex number, as a result of
this, the reflection coefficient is also a complex number, since it is completely
determined by the impedance ZL and the "reference" impedance

Where Zo is the impedance of the transmitter (or whatever is delivering power to the
antenna) while ZL is the impedance of the load.Hence, the Smith Chart is essentially
a graphical method of displaying the impedance of an antenna as a function of
frequency, either as a single point or a range of points.
Impedance Smith Chart contains two major elements which are the two circles/arcs
which define the shape and data represented by the Smith Chart. These circles are
known as;
1. The Constant R Circles
2. The Constant X Circles
The first set of lines referred to as Constant Resistance lines form circles, all tangent
to each other at the right hand of horizontal diameter. The constant R Circles are
essentially what you get when the Resistance part of the Impedance is held
constant, while the value of X varies. As such, all the points on a particular Constant
R circle represent the same resistance value(Fixed Resistance) . The value of the
resistance represented by each Constant R Circle is marked on the horizontal line,
at the point where the circle intersects with it. It is usually given by the diameter of
the circle.

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For example, consider a normalized impedance, ZL = R + iX, If R was equal to one

and X was equal to any real number such that, ZL = 1 +i0, ZL = 1 +i3, and ZL = 1 +i4, a
plot of the impedance on the smith chart will look like the image below.

Plotting multiple constant R Circles gives an image similar to the one below.

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This should give you an idea of how the giant circles in the smith chart is generated.
The Innermost and Outermost Constant R Circles, represent the boundaries of the
smith chart. The Innermost Circle(black) is referred to as the infinite resistance,
whilethe outermost circle is referred to as the zero resistance.
The Constant X Circles are more of arcs than circles and are all tangent to each
other on the right-hand extreme of horizontal diameter. They are generated when
the impedance has a fixed reactance but a varying value of resistance.The lines in
the upper half represent positive reactances while those in the lower half represent
negative reactances.
For example, let us consider a curve defined by ZL = R + iY, if Y = 1 and held
constant while R representing a real number, is varied from 0 to infinity is
plotted(blue line) on the Constant R Circles generated above, a plot similar to the
one in the image below is obtained.

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The Smith chart also has circumferential scaling in wavelengths and degrees. The
wavelength scale is used in distributed component problems and represents the
distance measured along the transmission line connected between the generator
or source and the load to the point under consideration. The degrees scale
represents the angle of the voltage reflection coefficient at that point.
Using a Smith chart and interpreting the results derived from it requires a good
understanding of AC circuit and transmission line theories, both of which are natural
pre-requisite for RF engineering. As an example of how smith charts, are used, we
will look at one of it’s most popular use cases which is impedance matching for
antennas and transmission lines.
In solving problems around matching, the smith chart is used to determine the value
of the component (capacitor or inductor) to use to ensure the line is perfectly
matched, that is, ensuring the reflection coefficient is zero.
For example, Let’s assume an impedance of Z = 0.5 - 0.6j. The first task to do will be
to find the 0.5 constant resistance circle on the smith chart. Since the impedance
has a negative complex value, implieing a capacitive impedance, you will need to
move counter-clockwise along the 0.5 resistance circle to find the point where it hits
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the -0.6 constant reactance arc (if it were a positive complex value, it would
represent an inductor and you would move clockwise).This then gives an idea of
the value of the components to use to match the load to the line.
Normalised scaling allows the Smith chart to be used for problems involving any
characteristic or system impedance, which is represented by the center point of the
chart. For Impedance smith charts, the most commonly used normalization
impedance is 50 ohms and it opens the graph up making tracing the impedance
easier. Once an answer is obtained through the graphical constructions described
above, it is straightforward to convert between normalised impedance (or
normalised admittance) and the corresponding unnormalized value by multiplying
by the characteristic impedance (admittance). Reflection coefficients can be read
directly from the chart as they are unit-less parameters.
Also, the value of impedances and admittances change with frequency and the
complexity of problems involving them increases with frequency. Smith charts can
however be used to solve these problems, one frequency at a time or over multiple
frequencies.
When solving the problem manually with one frequency at a time, the result is
usually represented by a point on the chart. While these are sometimes “enough”
for narrow bandwidth applications, it is usually a difficult approach for application
with Wide Bandwidth involving several frequencies. As such the smith Chart is
applied over a wide range of frequencies and the result is represented as
a Locus (connecting several points) rather than a single point, provided the
frequencies are close.
These locus of points covering a range of frequencies on the smith chart can be
used to visually represent:
1. How capacitive or inductive a Load is across the examined frequency range
2. How difficult matching is likely to be at the various frequencies
3. How well-matched a particular component is.
The accuracy of the Smith chart is reduced for problems involving a large locus of
impedances or admittances, although the scaling can be magnified for individual
areas to accommodate these.
The Smith chart may also be used for lumped element matching and analysis
problems.
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8.7 GRAPHICAL ANALYSIS OF TRANSMISSION LINES USING SMITH CHART
The graphical step-by-step procedure is
1. Identify the load reflection coefficient ΓR and the normalized load impedance
ZR on the Smith chart.
2. Draw the circle of constant reflection coefficient amplitude |Γ(d)| =|ΓR|.
3. Find the intersection of this circle with the real positive axis for the reflection
coefficient (corresponding to the transmission line location dmax).
4. A circle of constant normalized resistance will also intersect this point. Read or
interpolate the value of the normalized resistance to determine the VSWR.

8.8 STUB MATCHING SINGLE AND DOUBLE STUB MATCHING

The ability to obtain a reasonable match over a frequency band depends upon the
magnitude of the mismatch, the desired bandwidth, and the complexity of
matching circuit. But at any one frequency any impedance mismatch can be
perfectly matched to the characteristic impedance of the transmission line, as long
as it's not on the rim of the chart (perfect reflection, |Gamma| = 1). And this always
can be done with one stub that's less than a quarter-wavelength long.

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The technique is simple: move along the transmission line to rotate the mismatch to
the unity resistance (conductance) circle and insert the appropriate type and
length of stub in series (shunt) with the main line to move along this circle to the
origin. If the far end of the stub is either a short or open circuit (or generally, any
pure reactance), its input end is also a pure reactance (susceptance) so that it
doesn't affect the resistance (conductance) component of the mainline
impedance (admittance).
Since it's usually easier to add a stub in parallel with a transmission line, the example
shown below uses an admittance chart because, at the attachment point, the
resulting admittance is the sum of the stub's input susceptance and the main line
admittance. First, the mismatched point is rotated around the origin until it reaches
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the unity conductance circle. Then, the characteristic impedance and length of
the stub is chosen such that its input susceptance is equal and opposite to the main
line susceptance indicated on the unity conductance circle.
The example shows two cases: move toward the generator 39 degrees of line and
add a short-circuited stub that provides 0.8 siemens
normalized inductive susceptance, or move toward the generator 107 degrees of
line and add an open-circuited stub that provides 0.8 siemens
normalized capacitive susceptance.

There are an infinite number of possible solutions because, at one frequency, a stub
of any characteristic impedance can provide the necessary normalized
susceptance simply by adjusting its length. The differences show up when looking
over a frequency band. For example, the stub's length may be increased by an
integer multiple of half-wavelengths at a particular frequency and its input
susceptance at this frequency will not change. But over a frequency band, the
susceptance will vary considerably more than if the extra length had not been
added.

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There is one last technique we can look at which is somewhat more flexible than the
single stub matching which we just looked at. This is called double stub matching!
Suppose we have the following situation, as depicted in the figure.

8. 9 PROBLEMS:
1. Consider the characteristic impedance of a 50Ω termination and the following
impedances:
Z1 = 100 + j50Ω Z2 = 75 - j100Ω Z3 = j200Ω Z4 = 150Ω
Z5 = ∞ (an open circuit) Z6 = 0 (a short circuit) Z7 = 50Ω Z8 = 184 - j900Ω

Then, normalize and plot. The points are plotted as follows:

z1 = 2 + j z2 = 1.5 - j2 z3 = j4 z4 = 3
z5 = 8 z6 = 0 z7 = 1 z8 = 3.68 - j18
It is now possible to directly extract the reflection coefficient Γ on the Smith chart
of Figure 5. Once the impedance point is plotted (the intersection point of a
constant resistance circle and of a constant reactance circle), simply read the
rectangular coordinates projection on the horizontal and vertical axis. This will
give Γr, the real part of the reflection coefficient, and Γi, the imaginary part of the
reflection coefficient
It is also possible to take the eight cases presented in the example and extract
their corresponding Γ directly from the Smith chart of Figure 6. The numbers are:
Γ1 = 0.4 + 0.2j Γ2 = 0.51 - 0.4j Γ3 = 0.875 + 0.48j Γ4 = 0.5
Γ5 = 1 Γ6 = -1 Γ7 = 0 Γ8 = 0.96 - 0.1j

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2. Find Γ(d), given Z(d)= 25 +j100 with Z0=50 Ω

SOL: 1. Normalize the impedance
2. Find the circle of constant normalized resistance r
3. Find the arc of constant normalized reactance x
4. The intersection of the two curves indicates the reflection coefficient
in the complex plane. The chart provides directly the magnitude and
the phase angle of Γ(d)

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3. Given ZR= 25+j 100 Ω j with Z find Z(d)and Γ(d) for d = 0.18λ

9. Practice Quiz

1. The short circuit line will have a reflection coefficient of?

a) 22

b) 0

c) -1

d) 1

2. The open circuit line will have a reflection coefficient of

a) 1

b) 0

c) -1

d) None of the above

3. The short circuit impedance of the transmission line is given by

a) ZSC = -j Zo tan βl
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b) ZSC = j Zo tan βl

c) ZSC = Zo tan βl

d) ZSC = Zo tan β

4. The best stub selection for the transmission line will be?

a) Series Open

b) Series short

c) Shunt Open

d) Shunt short

5. For a matched line, the input impedance will be equal to

a) single point

b) Load impedance

c) Characteristic impedance

d) none

6. Identify the material which is not present in a transmission line setup

a) waveguides

b) cavity resonator

c) oscillator

d) none

7. The Smith chart is a polar chart which plots?

a) R vs Znorm

b) T vs Z

c) R vs Z

d) T vs Znorm

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8. The constant x-circles of Smith chart becomes smaller due to increase in the

value of 'x' from

a) 0 to π /2

b) 0 to ∞

c) 0 to 2π

d) 0 to π

9. According to Smith diagram, where should be the position of reflection

coefficient value for a unity circle with unity radius?

a) On or inside the circle

b) outside the circle

c) Both a and b

d) None of the above

10. The Smith chart is graphical technique used in the scenario of transmission

lines

a) True

b) False

10. Assignments

S.No Question BL CO

Derive the Expression for input impedance of a

1 2 6
transmission line.
Derive the expressions for propagation constant,
2 attenuation constant, phase constant, the characteristic 2 6
impedance & phase velocity.
Obtain standing wave ratios in terms of reflection
3 2 6
coefficient.
4 Explain how wave propagate through a Coaxial Cable. 2 6
5 Plot input impedance curves for open &short circuited 3 6

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lossless transmission lines.

11. Part A- Question & Answers

S.No Question & Answers BL CO

1 Define the Shorted transmission line?
Ans. when the transmission line is connected to load ZL=
1 6
0 (zero impedance) is called shorted circuited
transmission line
2 Define the matched transmission line?
Ans. when the transmission line is connected to load ZL=
1 6
Zo (characteristics impedance) is called matched
transmission line
3 Transmission lines are commonly used as?
Ans. Transmission lines are commonly used in power
1 6
distribution (at low frequencies) and in communications
(at high frequencies).
4 Various types of Transmission lines?
Ans. Various kinds of transmission lines such as the
twisted-pair and coaxial cables (thin net and thick net) 1 6
are used in computer networks such as the Ethernet and
internet.
5 What is mean by standing wave ?
Ans. The voltage standing wave ratio, VSWR is defined
1 6
as the ratio of the maximum to minimum voltage on a
loss-less line.

12. Part B- Questions

S.No Question BL CO
1 Derive the Expression for input impedance of a 1 6
transmission line.

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2 Obtain the solutions of transmission line equations. 2 6
Derive the expressions for propagation constant,
attenuation constant, phase constant, the
characteristic impedance and phase velocity
3 A Distortionless line has Z0=60 Ohms, α=mNp/m, velocity 2 6
of 0.6c. Whare c is the velocity of light in vaccum.Find
R,L,G,C and λ at 100MHz.
4 A Load of 100+j150 Ω is connected to 75 Ω lossless line. 3 6
Find(a)Γ (b)s (c) input impedance at a distance of 0.4λ
from the load. (d) Load admittance (e) locations of
Vmax and Vmin for 0.6 λ from the load. (f) Input
impedance at the Generator.
5 An antenna with impedance 40+j30 Ω is to be matched 3 6
to a 100 Ω lossless line with a short circuited stub.
Determine (i) Required stub admittance. (ii) Distance
between stub and antenna. (iii) Stub length. (iv) Find
Vmax and Vmin Points for a 0.6λ line from the load.

13. Supportive Online Certification Courses

1. http://opencourses.emu.edu.tr/course/view.php?id=26
2. https://www.youtube.com/watch?v=pGdr9WLto4A&list=PLl6m4jcR_DbOx6s2t
oprJQx1MORqPa9rG

14. Real Time Applications

S.No Application CO
1 It is used in the transmission line is used to calculate impedance 6
provided at any load

2 The chart is even employed to calculate admittance values 6

provided at any load
3 Used in the measurement of the length of the short-circuited 6
section of the Tx.line in order to offer the required amount of

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inductive reactance of capacitance
4 Employed to know the value of VSWR amongst others. 6

15. Contents Beyond the Syllabus

1. Electromagnetic interference and compatibility
2. Micro waves
3. Stub Matching
16. Prescribed Text Books & Reference Books
Text Book
1. Matthew N.O. Sadiku, “Elements of Electromagnetics”, 4th edition. Oxford
Univ. Press,2008.
2. William H. Hayt Jr. and John A. Buck, “Engineering Electromagnetics”, 7th
edition.,

References:
1. E.C. Jordan and K.G. Balmain, “Electromagnetic Waves and Radiating
Systems”, 2nd Edition, PHI, 2000.
2. John D. Krauss, “Electromagnetics”, 4th Edition,McGraw- Hill publication1999.
3. Electromagnetics, Schaum’s outline series, 2nd Edition, Tata McGraw-Hill
publications, 2006.
17. Mini Project Suggestion
1. Designing a transmission line which transmits the human voice
2. Design an FM transmitter
3. Using the ZELAND IE3D/ ZELAND Fidelity/ Ansoft HFSS and CST Sofrwares
designing a antenna

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