Digital image processing unit 1

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LCE/7.5.1/RC 01
TEACHING NOTES
Department: ELECTRONICS & COMMUNICATION ENGINEERING
Unit: II Date:
Topic name: Retarded Potential No. of marks allotted by JNTUK:
Books referred: 01. Antennas for All Applications by John D. Kraus & Ronald J. Marhefka
02. Antenna & Wave Propagation by K. D. Prasad
03. www.wikipedia.org
Retarded Potential:
The retarded potential formulae describe the scalar or vector potential for electromagnetic
fields of a time-varying current or charge distribution. The retardation between cause and effect is
thereby essential; e.g. the signal takes a finite time, corresponding to the velocity of light, to
propagate from the source point r’ of the field to the point r, where an effect is produced or
measured. Otherwise, the formulas below correspond to the same superposition principle acting,
e.g., in electrostatics for Coulomb's law.
These are the electromagnetic retarded potentials for an arbitrary source in free space
(vacuum). They satisfy the inhomogeneous wave equations for Φ and A in the Lorenz gauge.

Here, r is location, t is time, and

is the speed of light in a vacuum. tr is the "retarded time"; the time at which light must be emitted
from location r’ in order to reach location r at time t. ρ (r,t) is the electric charge density, and J (r,t) is
the current density. ε0 is the dielectric constant of free space and μ0 is the magnetic permeability of
free space. ф (r,t) is the electrical potential, and A (r,t) is the vector potential. Finally, dτ’ is the
integration measure corresponding to r’.
From ф (r,t) and A (r,t) the electromagnetic fields E (r,t) and B (r,t) can be calculated,

The corresponding advanced potentials, essentially mathematical objects, are:

Faculty/Date: HOD/Date:

Page 1 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Topic name: Electric Dipole Radiation No. of marks allotted by JNTUK
JNTUK:
03. www.wikipedia
www.wikipedia.org

The subscript a stands for advanced, and ta is the "advanced time".
Electric Dipole Radiation:
In the previous section, we examined the radiation emitted by a short electric dipole of
oscillating dipole moment,

Where

We found that, in the far field, the mean electromagnetic energy flux takes the form.

Assuming that the dipole is centered on the origin of our spherical polar coordinate system.
ssuming
The mean power radiated into the element of solid angle dΩ = sinθdθdψ, centered on the angular
ed
coordinates (θ,ψ ), is

Hence, the differential power radiated into this element of solid angle is simply

This formula completely specifies the radiation pattern of an oscillating electric dipole
(provided that the dipole is much shorter in length than the wave length of the emitted radiation Of
wave-length radiation).
course, the power radiated into a given element of solid angle is independent of dP/dΩ; otherwise
energy would not be conserved. Finally, the total radiated power is the integral of ove overall solid
angles.
Dipole Antenna:
The dipole antenna or dipole aerial is one of the most important and also one of the most
widely used types of antenna. It can be used on its own, or there are many other types of antenna
that use the dipole as the basic element within the antenna.


Page 2 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Topic name: Dipole Antenna - 1 No. of marks allotted by JNTUK:
The basic construction of a dipole is quite straightforward - a simple dipole antenna can be
constructed from a few simple pieces of wire. In this way antennas including FM dipole antennas, or
antennas for the short wave bands can easily be made. These antennas, while not having the
performance of other more complicated types of antenna can nevertheless prove very effective and
quite satisfactory in many applications.
Basic Dipole Facts:
The name dipole means two poles and the antenna does in fact consist of two "poles" or
sections. These are normally equal in length, making the antenna what is termed a centre fed
antenna. Sometimes a dipole may not be fed in the centre, although this is not normally done in
most antenna designs.
The power is applied to the dipole antenna itself through a feeder. Conversely if the dipole
antenna is used for receiving, the received signals are taken away to the receiver through a feeder.
The feeder serves to transfer the power to or from the antenna with as little loss as possible.

The most common form of dipole has an electrical length of half a wavelength. As a result
this antenna is called a half wave dipole. As before the lengths of the wires are both the same. As the
total length of the dipole is a half wavelength, this makes each section or leg of the dipole a quarter
wavelength long.


Page 3 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Current and Voltage on a Dipole:
In order that power flows into or out of an antenna that is transmitting or receiving, there
must be associated currents and voltages. The levels of current and voltage vary along the length of
the antenna, and it is found that the current distribution along a dipole is roughly sinusoidal. It falls
to zero at the end and is at a maximum in the middle. Conversely the voltage is low at the middle
and rises to a maximum at the ends. It is generally fed at the centre, at the point where the current
is at a maximum and the voltage a minimum. This provides a low impedance feed point which is
convenient to handle. High voltage feed points are far less convenient and more difficult to use.
When multiple half wavelength dipoles are used, they are similarly normally fed in the
centre. Here again the voltage is at a minimum and the current at a maximum. Theoretically any of
the current maximum nodes could be used.

Dipole Feed Impedance:
All antennas have what is termed a feed impedance. This is the impedance that is seen at
the point in the antenna where the feeder is connected. The impedance is measured in ohms, and to
ensure that the maximum amount of power is transferred between the feeder and the antenna, it is
necessary to ensure that the antenna and feeder impedances are matched, i.e. they have the same
value.
The feed impedance of a dipole antenna is dependent upon a variety of factors including the
length, the feed position, the environment and the like. A half wave centre fed dipole antenna in
free space has an impedance 73.13 ohms making it ideal to feed with 75 ohm feeder.
The feed impedance of a dipole can be changed by a variety of factors, the proximity of
other objects having a marked effect. The ground has a major effect. If the dipole antenna forms the
radiating element for a more complicated antenna, then elements of the antenna will have an effect.
Often the effect is to lower the impedance, and when used in some antennas the feed impedance of
the dipole element may fall to ten ohms or less, and methods need to be used to ensure a good
match is maintained with the feeder. One method is to use the folded dipole, outlined later on this
page.


Page 4 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Polar Diagram:
The polar diagram of a half wave dipole antenna that the direction of maximum sensitivity or
radiation is at right angles to the axis of the antenna. The radiation falls to zero along the axis of the
antenna as might be expected.

If the length of the dipole antenna is changed then the radiation pattern is altered. As the
length of the antenna is extended it can be seen that the familiar figure of eight pattern changes to
give main lobes and a few side lobes. The main lobes move progressively towards the axis of the
antenna as the length increases.
Dipole Antenna Length:
The length of a dipole is the main determining factor for the operating frequency of the
dipole antenna. Although the antenna may be an electrical half wavelength, or multiple of half
wavelengths, it is not exactly the same length as the wavelength for a signal travelling in free space.
There are a number of reasons for this and it means that an antenna will be slightly shorter than the
length calculated for a wave travelling in free space.
For a half wave dipole the length for a wave travelling in free space is calculated and this is
multiplied by a factor "A". Typically it is between 0.96 and 0.98 and is mainly dependent upon the
ratio of the length of the antenna to the thickness of the wire or tube used as the element. Its value
can be approximated from the graph:


Page 5 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Multiplication factor "A" used for calculating the length of a dipole
In order to calculate the length of a half wave dipole the simple formulae given below can be used:
Length (metres) = 150 x A / frequency in MHz
Length (inches) = 5905 x A / frequency in MHz
Using these formulae it is possible to calculate the length of a half wave dipole. Even though
calculated lengths are normally quite repeatable it is always best to make any prototype antenna
slightly longer than the calculations might indicate. This needs to be done because changes in the
thickness of wire being used etc may alter the length slightly and it is better to make it slightly too
long than too short so that it can be trimmed so that it resonates on the right frequency. It is best to
trim the antenna length in small steps because the wire or tube cannot be replaced very easily once
it has been removed.
Folded Dipole Antenna:
The standard dipole is widely used in its basic form. However under a number of
circumstances a modification of the basic dipole, known as a folded dipole provides a number of
advantages that can be used to advantage. This type of antenna is often used in the simple FM
dipole antennas that can be bought to use as temporary FM broadcast antennas. They are also used
within other larger antennas such as the Yagi.
In its basic form a dipole consists of a single wire or conductor cut in the middle to
accommodate the feeder. It is found that the feed impedance is altered by the proximity of other
objects, especially other parasitic elements that may be used in other forms of antenna. This can
cause problems with matching and because resistance losses in the antenna system can start to
become significant.
Additionally many antennas have to be able to operate over large bandwidths and a
standard dipole may be unable to fulfil this requirement adequately.


Page 6 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Topic name: Dipole Antenna – 5 & Radiation Resistance No. of marks allotted by JNTUK:
A variation of the dipole, known as a folded dipole provides a solution to these problems,
offering a wider bandwidth and a considerable increase in feed impedance. The folded dipole is
formed by taking a standard dipole and then taking a second conductor and joining the two ends. In
this way a complete loop is made as shown. If the conductors in the main dipole and the second or
"fold" conductor are the same diameter, then it is found that there is a fourfold increase in the feed
impedance. In free space, this gives a feed impedance of around 300 ohms. Additionally the antenna
has a wider bandwidth.
In a standard dipole the currents flowing along the conductors are in phase and as a result
there is no cancellation of the fields and radiation occurs. When the second conductor is added this
can be considered as an extension to the standard dipole with the ends folded back to meet each
other. As a result the currents in the new section flow in the same direction as those in the original
dipole. The currents along both the half-waves are therefore in phase and the antenna will radiate
with the same radiation patterns etc as a simple half-wave dipole.
The impedance increase can be deduced from the fact that the power supplied to a folded
dipole is evenly shared between the two sections which make up the antenna. This means that when
compared to a standard dipole the current in each conductor is reduced to a half. As the same
power is applied, the impedance has to be raised by a factor of four to retain balance in the equation
Watts = I^2 x R.
Radiation Resistance:
Radiation resistance is that part of an antenna's feedpoint resistance that is caused by the
radiation of electromagnetic waves from the antenna. The radiation resistance is determined by the
geometry of the antenna, not by the materials of which it is made. It can be viewed as the equivalent
resistance to a resistor in the same circuit.
Radiation resistance is caused by the radiation reaction of the conduction electrons in the
antenna.
When electrons are accelerated, as occurs when an AC electrical field is impressed on an
antenna, they will radiate electromagnetic waves. These waves carry energy that is taken from the
electrons. The loss of energy of the electrons appears as an effective resistance to the movement of
the electrons, analogous to the ohmic resistance caused by scattering of the electrons in the crystal
lattice of the metallic conductor.
While the energy lost by ohmic resistance is converted to heat, the energy lost by radiation
resistance is converted to electromagnetic radiation.
Power is calculated as,
P = I2R


Page 7 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Topic name: Antenna Theorems – 1 No. of marks allotted by JNTUK:
Where I is the electric current flowing into the feeds of the antenna and P is the power in
the resulting electromagnetic field. The result is that there is a virtual, effective resistance:

This effective resistance is called the radiation resistance.
Thus the radiation resistance of an antenna is a good indicator of the strength of the
electromagnetic field radiated by a transmitting antenna or being received by a receiving antenna,
since its value is directly proportional to the power of the field.
Antenna Theorems:
Antenna Reciprocity Theorem:
In classical treatises on electromagnetic fields and antennas, the antenna theorem is usually
formulated as follows:
Given two antennas A and B placed at some distance, each of them can be operated either
as a transmitting antenna or as a receiving antenna. Suppose that antenna B is kept intact, while the
performance of antenna A as a transmitter is modified so that, for a fixed amount of input power,
the signal received by antenna B changes by a factor F. Then the same modification changes also the
performance of antenna A as a receiver and does so by the same factor F.
The theorem follows from certain symmetries of Maxwell equations and its validity is easily
verified experimentally.
Adaption to Magnetic Resonance:
In magnetic resonance the signal is normally detected using what at first sight appears as an
induction coil (NMR, MRI) or a resonant cavity (ESR). However, this description presents conceptual
problems since classical induction cannot account for the evidently quantum character of MR
spectra which implies absorption and emission events involving discrete electromagnetic quanta.
Especially in the case of nuclear magnetic resonance (NMR) and imaging (MRI), where one uses
induction coils and there is no detectable radiation exiting/entering the sample area, the quantum
nature of the phenomenon is not easy to reconcile with classical Faraday induction.
Fortunately, it is clear that the interaction energy E between a magnetic moment m of a
particle and a magnetic field B is described by the same formula (E = - m.B) in both the classical and
the quantum formalism. Consequently, it is this term which plays a key role in resolving the apparent
impasse. Since the way how this works out is too long to follow here, let us see right away the result
for the radiation-less case of transmitter/receiver coils. The modified theorem, which we might also
call the MR reciprocity law, applies to all devices capable of both producing magnetic fields and
detecting MR signals.


Page 8 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Topic name: Antenna Theorems - 2 No. of marks allotted by JNTUK:
The sensitivity of a magnetic resonance assembly, used as a receiver, to nuclides present at a
point X is proportional to that assembly's efficiency, when used as a transmitter, to generate at that
same location X a radiofrequency field B1.
The efficiency of the assembly is measured by the amplitude of B1 produced using a fixed
input power P. More precisely, what matters is the component of B1 which is transversal to the main
field B0, but the orthogonality of B1 and B0 should be guaranteed by the geometry of the design.
One great merit of the theorem is that it makes superfluous the old and irrational concept of
coil filling factor. Consider, for example, a simple loop like the one in Figure-1 and suppose that we
use it as a transmitter to generate an RF magnetic field at point X. When the coil is tuned and
matched so that the whole assembly has a predefined impedance Z0 (usually 50 Ω), a fixed power P
implies a fixed current J. Suppose now that we reduce the diameter of the loop by a factor F and
retune it again to Z0. Then, at input power P, we have still the same current J but, according to the
Biot-Savart law, the generated magnetic field B1 is greater than before by the factor F. Consequently,
according to the Theorem, the sensitivity of the smaller coil to a nuclide present at X will be F times
that of the larger coil.
The sensitivity of a front-end assembly at a point X is proportional to the RF field it generates
therein when used as a transmitter and fed a pre-defined power. For the test, the input impedance
and the current must be kept constant and the assembly must be tuned to the Larmor frequency of
the observed nuclides. As always, B1 must be perpendicular to the main magnetic field B0.
The above discussion makes sense since it relates the sensitivity of an assembly (inclusive of
its tuning and matching circuitry) at a point X to its capability to produce at X a magnetic field when
used as an RF field generator rather than a sensor. This combines geometric and electric
characteristics of the assembly with the relative location of X - all features which clearly need to be
taken into account when talking, for example, about sensitivity to small voxels inside a large MRI coil
(notice that in such cases the concept of a 'filling factor' is not applicable at all). It can be shown that
the sensitivity to nuclides present at a point X is proportional to,

Where B1(X) is the RF field produced at X when the assembly is used as a transmitter and fed
the power P. One important and easy to remember corollary of the theorem is the relation between
sensitivity and the width of 90o excitation pulse. Since the latter is inversely proportional to the B1
magnitude, it follows that:
If a modification of the front-end assembly shortens by a factor F the duration of the 90-
degrees pulse, with no change in the employed RF power, the sensitivity of that assembly increases
by the same factor F.


Page 9 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Topic name: Loop Antennas - 1 No. of marks allotted by JNTUK:
Loop Antennas:
A loop antenna has a continuous conducting path leading from one conductor of a two-wire
transmission line to the other conductor. All planar loops are directional antennas with a sharp null,
and have a radiation pattern similar to the dipole antenna with E and H fields interchanged.
Small Loops:
A loop is considered a small loop if it is less than 1/4 of a wavelength in circumference. Most
directional receiving loops are about 1/10 of a wavelength. The small loop is also called the magnetic
loop because it is more sensitive to the magnetic component of the electromagnetic wave. As such,
it is less sensitive to near field electric noise when properly shielded. The received voltage of a small
loop can be greatly increased by bringing the loop into resonance with a tuning capacitor.
From considerations of symmetry, equal voltages will be induced in each limb of the loop
when a signal arrives along the loop axis. The loop output, which is the difference between the limb
voltages, is therefore zero in all cases. For signals arriving in the plane of the loop, there is a phase
difference between the limb voltages and this produces a maximum output for the small loop.
Medium Loops:
There are two special cases of loop antennas which are neither short nor long, and have
particular characteristics:
Half-wavelength loop, a half-wave dipole curved into a circle, can be mounted in the
horizontal plane as a horizontally polarized omnidirectional antenna.
Full-wavelength loop, an element of the quad antenna, which radiates on its axis (unusual
for a loop) and is polarized according to the position of the feed point.


Page 10 of 11

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LCE/7.5.1/RC 01
TEACHING NOTES
Unit: II Date:
Topic name: Loop Antennas - 2 No. of marks allotted by JNTUK:
Large Loops:
The large loop antenna is similar to a dipole, except that the ends of the dipole are
connected to form a circle, triangle (delta loop antenna) or square. Typically such a loop is some
multiple of a half or full wavelength in circumference. A circular large loop gets higher gain (about
10%) than the other forms, as gain of this antenna is directly proportional to the area enclosed by
the loop, but circles of flexible wire can be difficult to support, making squares and triangles much
more popular. Large loop antennas are less susceptible to localized noise, partly due to their lack of
a need for a ground plane. The large loop usually has its strongest signal in the plane of the loop
unless it is very large. The nulls are in the axis perpendicular to the plane of the loop.


Page 11 of 11

Digital image processing unit 1

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