ELECTROMAGNETIC FIELDS AND TRANSMISSION LINES DEPT.

ECE

UNIT – IV
Transmission Lines - I:
Contents:
 Types
 Parameters
 Transmission Line Equations
 Primary & Secondary Constants
 Expressions for Characteristics Impedance, Propagation Constant, Phase and
Group Velocities
 Infinite Line Concepts
 Distortion - Condition for Distortion less Transmission and Minimum
Attenuation
 Illustrative Problems.

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Introduction:
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 up of a conductor having a uniform cross-section along
the line. Air act as an insulating or dielectric medium between the conductors.

Fig. Transmission Lines

Types of Transmission Lines

The different types of transmission lines include the following.

Open Wire Transmission Line

It consists pair of parallel conducting wires separated by a uniform distance. The two-wire
transmission lines are very simple, low cost and easy to maintain over short distances and these
lines are used up to 100 MHz Another name of an open-wire transmission line is a parallel wire
transmission line.

Coaxial Transmission Line

The two conductors placed coaxially and filled with dielectric materials such as air, gas or solid.
The frequency increases when losses in the dielectric increases, the dielectric is polyethylene.
The coaxial cables are used up to 1 GHz. It is a type of wire which carries high-frequency signals
with low losses and these cables are used in CCTV systems, digital audios, in computer network
connections, in internet connections, in television cables, etc.

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Optic Fiber Transmission Line

The first optical fiber invented by Narender Singh in 1952. It is made-up of silicon oxide or
silica, which is used to send signals over a long distance with little loss in signal and at the speed
of light. The optic fiber cables used as light guides, imaging tools, lasers for surgeries, used for
data transmission and also used in a wide variety of industries and applications.
Microstrip Transmission Lines

The microstrip transmission line is a Transverse Electromagnetic (TEM) transmission line

invented by Robert Barrett in 1950.

Wave Guides

Waveguides are used to transmit electromagnetic energy from one place to another place and
they are usually operating in dominant mode. The various passive components such as filter,
coupler, divider, horn, antennas, tee junction, etc. Waveguides are used in scientific instruments
to measure optical, acoustic ad elastic properties of materials and objects. There are two types of
waveguides are Metal waveguides and dielectric waveguides. The waveguides are used in optical
fiber communication, microwave ovens, space crafts, etc.
Applications

The applications of transmission line are

 Power transmission line

 Telephone lines
 Printed circuit board
 Cables
 Connectors (PCI, USB)

Parameters of transmission line (Primary Constants):

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.

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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

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.

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Primary & Secondary Constants:

The primary line constants are the resistance, inductance, conductance, and capacitance per unit
length of the transmission line.

However, the term “secondary line constants” is not commonly used. It is normally known as
“quaternary parameters” or “quaternary constants” used in telecommunication line analysis.
These parameters extend the analysis of transmission lines beyond the primary parameters by
including additional effects, such as radiation and shunt capacitance. Quaternary parameters are
also used to model the behavior of transmission lines at higher frequencies.

Propagation Constant Definition:

Electromagnetic waves propagate in a sinusoidal fashion. The measure of the change in

amplitude and phase per unit distance is called the propagation constant. Denoted by the
Greek letter 𝜸. The terminologies like Transmission function, Transmission constant,
Transmission parameter, Propagation coefficient, and Propagation parameter are synonymous
with this quantity. Sometimes 𝜶 and 𝜷 are collectively referred to as Propagation or
Transmission parameters.

The propagation constant can be mathematically expressed as:

γ = α + jβ

Where:

α (alpha) represents the attenuation constant, which measures the rate of amplitude decay of the
signal as it travels through the medium. It is a real number and is usually measured in Nepers per
unit length or decibels per unit length.

β (beta) represents the phase constant, which determines the phase shift experienced by the
signal as it propagates through the medium. It is an imaginary number and is usually measured in
radians per unit length.

The magnitude of the propagation constant (γ) gives the overall rate of signal decay, while the
argument or phase angle of the propagation constant (arg(γ)) gives the phase shift experienced by
the signal.

Propagation Constant of a Transmission Line:

The propagation constant for any conducting lines (like copper lines) can be calculated by
relating the primary line parameters.

Where, Z = R + iωL is the series impedance of line per unit length.

Y = G + iωC is the shunt admittance of line per unit length.

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Characteristic Impedance (Zo)

As we already discussed that primary constants are very significant in transmission lines, they
make characteristic impedance (Zo) a very significant parameter as well, because characteristic
impedance (Zo) involves all four of the primary constants in its expression.

What is Characteristic Impedance?

Characteristic impedance can be defined as the ratio of amplitude of voltage to the amplitude of
current of a unidirectional wave travelling from source to load along a uniform transmission line
in the absence of reflections.

It may also be defined as a square root of the ratio of series impedance of a line to its shunt
admittance.

Where,

Z = R + jwL (series impedance per unit length per phase)

Y = G + jwC (shunt admittance per unit length per phase)

R, L, G and C are the primary constants of a transmission line, and the above expression
confirms that characteristics of a transmission line are described by primary line constants.

Transmission Line Equations

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. These two wirelines are 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.

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Fig.equivalent_circuit_of_a_transmission_line_1

The inductor and resistance put together in the above figure can be called as series impedance,
which is expressed as

Z = R+jωL
The parallel combination of capacitance and conductor n the above figure can be expressed as

Y = G+jωc

Where l – length

Is – Sending end current

Vs – Sending end voltage

dx – element length

x – a distance of dx from sending end

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At a point, ‘p’ take current(I) and voltage(v) and at a point, ‘Q’ take I+dV and V+dV

The change in voltage for the length PQ is the

V-(V + dV) = (R + jωL) dx * I

V-V-dv = (R + jωL) dx * I
-dv/dx = (R + jωL) * I ………………. eq(1)
I-(I + dI) = (G + jωc)dx * V
I – I+dI = (G + jωc)dx * V
-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 – propagation constant

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

-d2v/dx2 = P2 V ………………. eq(8)

-d2I/dx2 = P2I … ……………. eq(9)
General solution is

V = Aepx + Be-px … ……………. eq(10)

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 (Ce px + 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 Z 0is the characteristic impedenc
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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 +I S Z0
From eq(21) will get C and D values

C = (IS – VS /Z0) /2
D = (IS + VS /Z0) /2
Substitute A, B, C and D values in eq(10) and (11)

V= (VS -I S 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 (e px -e-px/2)
=IS coshx – VS /Z0 sinhx
Thus V = VS coshx – I S Z0 sinhx
I = IS coshx – VS /Z0 sinhx
Equation of transmission line in terms of sending end parameters are derived

Phase and Group Velocities:

Phase velocity is the speed at which a point of constant phase moves through a medium. In
simple terms, it’s like tracing the path of a ruffling wave crest or trough marking a constant
phase in the wave.

In physics, phase velocity can be calculated by using the simple formula:

Where:

 ω indicates phase velocity

 vp is the angular frequency of the wave
 k is the wave number

It's worth mentioning that phase velocity depends on the medium the wave passes through. In
some media, the phase velocity might change, leading to phenomena such as refraction.

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Group Velocity

Group velocity is defined as the derivative of the wave's angular frequency with respect to its
wave number. It can be mathematically expressed as:

Relation Between Group Velocity and Phase Velocity

The Group Velocity and Phase Velocity relation can be mathematically written as-

Where,

 Vg is the group velocity.

 Vp is the phase velocity.
 k is the angular wavenumber.
The group velocity is directly proportional to phase velocity. This means-

 When group velocity increases, proportionately phase velocity will also increase.
 When phase velocity increases, proportionately group velocity will also increase.
For the amplitude of wave packet let-

 ω is the angular velocity given by ω=2πf

 k is the angular wave number given by

 t is time
 x be the position
 Vp phase velocity
 Vg be the group velocity

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The phase velocity of a wave is given by the following equation:

The above equation signifies the relationship between the phase velocity and the group velocity.

Infinite Line Concepts:

A finite line is a line having a finite length on the line. It is a line, which is terminated, in
its characteristic impedance (ZR=Z0), so the input impedance of the finite line is equal to the
characteristic impedance (Zs=Z0).
An infinite line is a line in which the length of the transmission line is infinite. A finite
line, which is terminated in its characteristic impedance, is termed as infinite line. So for an
infinite line, the input impedance is equivalent to the characteristic impedance.

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Distortion - Condition for Distortion less Transmission and Minimum

Attenuation:

It is desirable, however to know the condition on the line parameters that allows propagation
without distortion. The line having parameters satisfy this condition is termed as a distortion less
line.
The condition for a distortion less line was first investigated by Oliver Heaviside. Distortion
less condition can help in designing new lines or modifying old ones to minimize distortion.
A line, which has neither frequency distortion nor phase distortion is called a distortion less line.

Condition for a distortion less line

The condition for a distortion less line is RC=LG. Also,
a) The attenuation constant _ should be made independent of frequency. α = RG
b) The phase constant _ should be made dependent of frequency. β = ω LC
c) The velocity of propagation is independent of frequency.
V=1 / LC
For the telephone cable to be distortion less line, the inductance value should be increased
by placing lumped inductors along the line.
For a perfect line, the resistance and the leakage conductance value were neglected. The
conditions for a perfect line are R=G=0. Smooth line is one in which the load is terminated by its
characteristic impedance and no reflections occur in such a line. It is also called as flat line.

The distortion Less Iine

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If a line is to have neither frequency nor delay distortion, then attenuation constant and
velocity of propagation cannot be function of frequency.
Then the phase constant be a direct function of frequency

The above equation shows that if the the term under the second radical be reduced to equal
(RG + ω2LC)2
Then the required condition for ß is obtained. Expanding the term under the internal radical
and forcing the equality gives
R2G2- 2 ω2LCRG+ ω4L2C2+ ω2L2G2+ 2 ω2LCRG+ ω2CR2 = (RG+ ω2LC)2
This reduces to
2 ω2LCRG+ ω2L2G2+ ω2CR2=0
(LG-CR)2=0
Therefore, the condition that will make phase constant a direct form is
LG = CR
A hypothetical line might be built to fulfill this condition. The line would then have a value of ß
obtained by use of the above equation.
Already we know that the formula for the phase constant
β = ωLC
Then the velocity of propagation will be v = 1/ LC
This is the same for the all frequencies, thus eliminating the delay distortion.
May be made independent of frequency if the term under the internal radical is forced to reduce
to (RG +ω LC)2
Analysis shows that the condition for the distortion less line LG = CR , will produce the desired
result, so that it is possible to make attenuation constant and velocity independent of frequency
simultaneously. Applying the condition LG= RC to the expression for the attenuation
gives α = RG
This is the independent of frequency, thus eliminating frequency distortion on a line. To
achieve
LG = CR
Require a very large value of L, since G is small. If G is intentionally increased, attenuation are
increased, resulting in poor line efficiency.

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To reduce R raises the size and cost of the conductors above economic limits, so that the
hypothetical results cannot be achieved.
Propagation constant is as the natural logarithm of the ratio of the sending end current or voltage
to the receiving end current or voltage of the line. It gives the manner in the wave is propagated
along a line and specifies the variation of voltage and current in the line as a function of distance.
Propagation constant is a complex quantity and is expressed as γ= α + j β.
The real part is called the attenuation constant, whereas the imaginary part of propagation
constant is called the phase constant.

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UNIT-V
Transmission Lines - II:
Contents:
 SC and OC Lines
 Input Impedance Relations
 Reflection Coefficient
 VSWR
 Smith Chart - Configuration and Applications
 Illustrative Problems.

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Input Impedance Relations

 The input impedance of a transmission line is the impedance seen by any signal
entering it. It is caused by the physical dimensions of the transmission line and its
downstream circuit elements.
 If a transmission line is ideal, there is no attenuation to the signal amplitudes and the
propagation constant turns out to be purely imaginary.

 When the transmission line length is infinite, the input impedance is equal to the
characteristic impedance.

Calculating the Input Impedance

Consider a lossless, high-frequency transmission line where the voltage and currents are given by
equations 1 and 2, with the input impedance, characteristic impedance, and load impedance as
Zin, Z0, and ZL, respectively.

As the transmission line is ideal, there is no attenuation to the signal amplitudes and the
propagation constant turns out to be purely imaginary. Let’s define the output terminals with axis
point z=0 and input terminals z=-L. Our objective is to find the impedance of the circuit when
looking from Z=-L:

The input impedance is the ratio of input voltage to the input current and is given by equation 3.
By substituting equation 5 into equation 4, we can obtain the input impedance, as given in
equation 6:

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From equation 6, we can conclude that the input impedance of the transmission line depends on
the load impedance, characteristic impedance, length of the transmission line, and the phase
constant of the signals propagating through it.

It is already a known fact that the characteristic impedance Z0 is dependent on the distributed
parameters of the transmission line, such as resistance, inductance, capacitance, and conductance
(as given by equation 7), which are usually defined per unit length. Whenever any change is
made in the circuit, the input impedance changes.

The relationship between the characteristic impedance and input impedance can be deduced for
certain transmission lines. In the derivation of the input impedance equation, we have considered
the finite length of the transmission line. When the transmission line length is infinite, then the
input impedance of the transmission line is equal to the characteristic impedance. Whenever the
transmission line of finite length is terminated by a load impedance that is equal to the
characteristic impedance, there is no reflection of signals (according to equation 7). In this case,
the input impedance equals characteristic impedance.

OPEN AND SHORT-CIRCUITED LINES

As limited cases it is convenient to consider lines terminated in open circuit or short circuit,
that is with ZR = ∞or ZR =0.
First, let us consider the question at hand: What is the input impedance when the transmission
line is open- or short-circuited?

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And for the short circuit case ZR =0., so that

Zs = Z0 tanh γl

Before the open circuit case is considered, the input impedance should be
written. The input impedance of the open circuited line of length l, with ZR = ∞, is

Zoc = Z0 coth γl

By multiplying the above two equations it can be seen that

Z0 = ZocZsc

This is the same result as was obtained for a lumped network. The above equation
supplies a very valuable means of experimentally determining the value of z0 of a
line.

Also from the same two equations

Use of this equation in experimental work requires the determination of the

hyperbolic tangent of a complex angle. If
Reflection coefficient:
A reflection coefficient, sometimes called reflection parameter, defines how much energy is
reflected from the load to the source of the RF systems. A reflection coefficient is also known as
s11 parameter. By definition, a reflected coefficient is a ration of the reflected wave and the
incident wave of the electric field strength. In the literature it is presented with the capital Greek
letter gamma (Γ).

The mismatch of a load Z L to a source Z0 results in a reflection coefficient of:

Γ=(ZL-Z0)/(ZL+Z0)

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Note that the load can be a complex (real and imaginary) impedance. If you can't remember in
which order the numerator is subtracted (did we just say "Z L-Z0" or Z0 -ZL"?), you can always
figure it out by remembering that a short circuit (Z L=0) is on the left side of the Smith
chart (angle = -180 degrees) which means Γ=-1 in this case, which means that the minus sign
belongs in front of Z0.

The magnitude of the reflection coefficient is given by:

ρ=mag(Γ)
For cases where Z L is a real number,
ρ=abs((ZL-Z0)/(ZL+Z0))
Note that "abs" means "absolute value" here. VSWR can be calculated from the magnitude of the
reflection coefficient:
VSWR=(1+ρ)/(1-ρ)
For cases where Z L is real, with a little algebra you'll see there are two cases for VSWR,
calculated from load impedance:
For ZL<Z0: VSWR=Z0/ZL
For ZL>Z0: VSWR=Z L/Z0

VSWR:
VSWR is an abbreviation for Voltage Standing Wave Ratio or sometimes in literature just SWR
(Standing Wave Ratio). The value of VSWR presents the power reflected from the load to the
source. It is often used to describe how much power is lost from the source (usually a High
Frequency Amplifier) through a transmission line (usually a coaxial cable) to the load (usually an
antenna).

How to express VSWR using voltage?

By the definition, VSWR is the ratio of the highest voltage (the maximum amplitude of the
standing wave) to the lowest voltage (the minimum amplitude of the standing wave) anywhere
between source and load.

VSWR = |V(max)| / |V(min)|

V(max) = the maximum amplitude of the standing wave

Vmin) = the minimum amplitude of the standing wave

What is the ideal value of a VSWR?

The value of an ideal VSWR is 1:1 or shortly expressed as 1. In this case the reflected power
from the load to the source is zero.

How to express VSWR using an impedance?

By the definition, VSWR is the ratio of the load impedance and source impedance.

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ZL = the load impedance

Zo = the source impedance

How to express a VSWR using reflection and forward power?

By the definition VSWR is equal to

VSWR = 1 + √(Pr/Pf) / 1 – √(Pr/Pf)

where:

Pr = Reflected power
Pf = Forward power

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Smith Chart:

The Smith Chart has been in use since the 1930s as a method to solve various RF design
problems - notably impedance matching with series and shunt components - and it provides a
convenient way to find these solutions without the use of a calculator. In order to understand the
construction of the chart, you'll need to understand high school algebra and the basics of
complex numbers, as well as have a basic understanding of impedance in electronic circuits. That
said, even if you don't fully understand the derivation below, you can still use the chart to help
you with your own design. By taking the standard reflection coefficient formula and
manipulating it so that it provides us with the equations for circles of various radii, we'll be able
to construct the basic Smith Chart. That's all the Smith Chart really is: a collection of circles,
each one centered in a different place in (or outside) the plot, and each one representing
either constant resistance or constant reactance

Deriving the Smith Chart

Once we get past the derivation, there will be a few simplified images showing how those
equations can be used and combined to get the final product. Let's get started by writing the
equation for the reflection coefficient of a load impedance, given a source impedance:

The reflection coefficient is just the ratio of the complex amplitude of a reflected wave to the
amplitude of the incident wave. This is the main equation we'll be using, but there will be some
quick transformations to it. First, we'll want to simplify it a little by normalizing the equation
with respect to Zload , dividing each term on the right side:

At this point, recall that Z o , being an impedance of complex value, can be represented in the
form R + jX. Since the reflection coefficient (which is currently in polar form) can also be
represented in rectangular coordinates (we'll use A + jB for it), the above formula can be
transformed into this:

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Great! At this point we've got the equation in the form we need to start constructing the Smith
Chart. The next step - solving for the real and imaginary parts of the equation - is probably the
most difficult part of the entire derivation, and even then you only need to understand the
concept of complex conjugates to do it. Let's go ahead and split it into real and imaginary
components, first by multiplying by the complex conjugate (it helps if you separate the existing
real and imaginary parts using brackets as shown below):

At this point we can separate the real and imaginary components. After that, there will be two
final simplifications to do before we'll have the equations to draw the Smith Chart. Here are the
separated real and imaginary parts (we'll call them Equations 1 and 2):

Finally, you will want to do just a little more algebra (tedious, I know). Solving the real
component, A, for X2 , you will get Equation 3:

You can substitute this into Equation 2 to get the first of our two final equations, which allows us
to determine the circles of constant resistance (Equation 4):

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Just like the previous result, this is a circle with radius 1/X but this time there are two sets of
circles (more on that in a bit), with centers at (1,1/X)These are circles (they appear as arcs on the
diagram) of constant reactance. Now you should see how the standard Smith Chart is drawn; it
consists of constant resistance circles graphed together with the constant reactance arcs. Below
you'll find some simplified images of both equations graphed separately and combined. But first,
let's talk about how to interpret the Smith Chart and its physical relevance.

There is quite a bit of information to obtain from analyzing the equations we've derived. Here are
just a few things of note:

 At infinite R and X, both types of circles converge to the same location (typically shown
on a Smith Chart at the far right or far left side of the diagram). This is at the point (1, 0).
 Setting R = 0 will result in a circle centered at (0, 0) on your chart with a radius of 1,
which is the "boundary" of the chart.
 Approaching X = 0 results in an infinite radius; this is represented by a line crossing the
center of the chart. How do we interpret this? This is often called the real axis. In terms
of reactances, lines above the real axis in the chart (the positive arcs from the second
derived equation) represent inductive reactances, while those below (negative arcs)
represent capacitive reactances.

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 What happens if R < 0? The standard Smith Chart doesn't provide much detail about this,
but situations with R lying outside the boundary suggest oscillation in any would-be
circuit (which is pretty handy to know).
 Based on the knowledge we now have on resistance and reactance on the chart, we know
that every point represents a series combination of resistance and reactance (R + jX).
This'll help us when we want to do some plotting

Constant Resistance Circles:

Constant Reactance Arcs:

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Smith Chart:

Applications of Smith Charts:

Smith charts find applications in all areas of RF Engineering. Some of the most popular
application includes;

 Impedance calculations on any transmission line, on any load.

 Admittance calculations on any transmission line, on any load.
 Calculation of the length of a short-circuited piece of transmission line to provide a
required capacitive or inductive reactance.
 Impedance matching.
 Determining VSWR among others.

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