ANTENNAS AND WAVE PROPAGATIONS

UNIT1
• Review of Electromagnetic Theory: Vector Potential, Solution of Wave
Equation, Retarded Case, Hertizian Dipole. Antenna Characteristics:
Radiation Pattern, Beam Solid Angle, Directivity, Gain, Input Impedance,
Polarization, Bandwidth, Reciprocity, Equivalence of Radiation Patterns,
Equivalence of Impedances, Effective Aperture, Vector Effective Length,
Antenna efficiency

Mr. P Pradeep
Assistant Professor
ECE Dept.
SNIST
1
 What is an antenna?
 Why antenna?
 Where used?
 Antenna Background
 Types of antennas

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A charged particle produces an electric field. This electric field exerts a force on other
charged particles.
A moving charged particle produces a magnetic field. This magnetic field exerts a
force on other moving charges.
An accelerating charged particle produces an electromagnetic (EM) wave.
Electromagnetic waves are electric and magnetic fields traveling through empty space
with the speed of light c.
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 Figure 2. (a) Capacitor circuit, (b) dipole, (c) dipole showing
intrinsic capacitance and charging current
Ampere's law
inconsistency of ampere's law

Electromagnetics waves are exist based on displacement current Voltage and

currents are exist based on conduction current.
𝜕𝐸
𝐽𝐷 = 𝜖
𝜕𝑡
𝐽𝐶 = 𝜎𝐸

At low frequency, particles are associated with energy.

At high frequency fields, are associated with energy.
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Figure 3. (a) Electric field E and (b) magnetic field H and TEM
field from dipole charge and current

Figure 4. (a) Electric field antenna and (b) magnetic field

antenna. 6
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Antenna (or) Radiator
An antenna is a
transducer that converts
electrical signal into an
electromagnetic wave or
vice versa.
It acts as a transition
region between a guided
wave and free space. 8
Most antennas are
reciprocal devices.
i.e. Radiates and
Receives EM waves.

An antenna radiates
electromagnetic
energy in the desired
direction (defined by
IEEE)
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Antenna’s Radiation mechanism

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EXAMPLES OF ANTENNA

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FUNCTIONS OF AN ANTENNA
1. An Antenna acts as an transducer.
2. It directs the radiated energy in desired
direction and suppresses energy in
undesired direction.
3. It is used to detect the presence of EM
waves.
4. It provides matching between the
transmitter and the free space in order to
achieve maximum power transfer. 18
 ANTENNA PARAMETERS
1) Radiation pattern 9) Equivalence of
2) Beam solid angle Radiation pattern
3) Directivity 10)Equivalence of
4) Gain impedance
5) Input Impedance 11)Physical & Effective
6) Polarization aperture
7) Beam width 12)Aperture & Antenna
8) Reciprocity Efficiency
13)Effective length 19
1.Radiation Pattern
It is a mathematical function or graphical
representation of the radiation properties of an
antenna as function of space coordinates.

Radiation Pattern usually indicate either electric field

intensity called as Field pattern or Power intensity
called power pattern.

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 Radiation Pattern
By measuring radiated energy at various angles at
constant distances from the antenna and then
plotting the energy values on a graph gives radiation
pattern.

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Often the field and power patterns are normalized with respect to their
maximum value, yielding normalized field and power patterns.

a. field pattern( in linear scale) typically represents a plot of the spatial

variation of magnitude of the electric (or magnetic) field along a
constant radius.

b. power pattern( in linear scale) typically represents a trace of the

received power at a constant radius.

c. power pattern( in dB) represents square of the magnitude of electric

or magnetic field in decibels, as a function of the angular space.

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Figure: Two-dimensional normalized field pattern( linear scale), power pattern( linear
scale), and power pattern( in dB) of a 10-element linear array with a spacing of d = 0.25λ. 24
 Lobes of an antenna
Various parts of a radiation pattern are referred to as lobes
A radiation lobe is a “portion of the radiation pattern bounded by regions of
relatively weak radiation intensity.”

1. Major Lobe: Radiation lobe contains the

direction of maximum radiation
2. Minor Lobe: A the lobes except major lobe
3. Side Lobe: The minor lobe adjacent to the
major lobe
4. Back lobe: The minor lobe opposite to the
direction of major lobe

Minor lobes usually represent radiation in undesired directions, and they should be minimized.
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 Principal Patterns

For a linearly polarized antenna, performance is

often described in terms of its principal E- and
H-plane patterns.

The E-plane (xz plane, elevation plane ∅ = 𝟎)

is defined as “the plane containing the electric-
field vector and the direction of maximum
radiation.” and,

The H-plane (xy plane, azimuthal plane; 𝛉 =

𝛑
) as “the plane containing the magnetic-field
𝟐
vector and the direction of maximum radiation.”
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Types of Radiation Pattern
I. Isotropic antenna: It radiates equally in all
directions. Its radiation pattern is spherical (not
exists in real time).
II. Directional antenna: It radiates in particular
direction
III. Omni-directional antenna: Non directional in
horizontal plane and directional in vertical plane
or Major lobe is in one plane.

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 3D Radiation Pattern

Isotropic Omni-Directional Directional

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 2.BEAM WIDTH
It is the angular width of the
radiation pattern of an antenna
1.Half Power Beam width
(HPBW):
Angular difference between
the points where radiation
intensity is 0.707 of maximum
value. Also called as 3 dB
beam width.
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. 2. Beam width between
First Nulls (FNBW):
Angular difference
between two nulls
enclosing the main
beam. Null indicates
zero radiation

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 Radiating Near and Far Fields

• The field patterns generated by a radiating

antenna vary with distance and are associated
with (i) radiating energy and (ii) reactive energy.
• The space surrounding an antenna is usually
subdivided into three regions: (a) reactive near-
field, (b) radiating near-field (Fresnel) and (c)
far-field (Fraunhofer) regions as shown in
Figure

• The boundaries of these regions are not defined

precisely but are only approximations.
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Reactive near-field region is defined as “that portion of the near-field region
immediately surrounding the antenna wherein the reactive field predominates.”

Reactive near field 𝑅 ≤ 𝑅1

where λ is the wavelength and D is the largest dimension of the antenna.

Radiating near-field (Fresnel) region is defined as “that region of the field of an

antenna between the reactive near-field region and the far-field region wherein
radiation fields predominate and wherein the angular field distribution is dependent
upon the distance from the antenna.
Radiating near field 𝑅1 < 𝑅 ≤ 𝑅2

Far-field (Fraunhofer) region is defined as “that region of the field of an antenna

where the angular field distribution is essentially independent of the distance from
the antenna.
Radiating far-field 𝑅 > 𝑅2 32
 3. Angle and Beam solid angle

The measure of a plane angle is a radian.

One radian is defined as the plane angle with its vertex at the center of a circle
of radius r that is subtended by an arc whose length is r.
Measurement of angle between two point in two
dimension

θ = arc length/radius
= L / r (radian)
For complete circle,
arc length = 2πr
Then, Angle θ = 2πr /r
θ = 2π radian
θ = 2π radian
 Beam solid angle

The measure of a solid angle is a steradian.

One steradian is defined as the solid angle with its vertex at

the center of a sphere of radius r that is subtended by a
spherical surface area equal to that of a square with each
side of length r.
It is the amount of area per square
of radius.

Solid angle φ = area / radius²

For complete sphere,
surface area = 4πr²
φ = 4π steradian Solid angle φ = 4πr²/ r²
φ = 4π steradian
Radian: A plane angle is taken in 2D
Steradian: A solid angle taken in 3D
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Solid angle (dΩ)
For a spherical coordinate system, an incremental area 𝑑𝐴 𝑖𝑠

Longitude = length in 𝜙 direction = 𝑟𝑠𝑖𝑛𝜃𝑑𝜙

Latitude = length in 𝜃 direction = 𝑟 𝑑𝜃

Therefore, 𝑑𝐴 = 𝑟 2 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙
𝑑Ω- Solid angle in Steradian = 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙

𝑑𝐴 = 𝑟 2 𝑑Ω
Area of sphere = 4𝜋𝑟 2
𝑑Ω=4𝜋 Steradian
𝑑Ω
1 Steradian =
4𝜋
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 𝒔𝒐𝒍𝒊𝒅 𝒂𝒏𝒈𝒍𝒆 𝒐𝒇 𝒔𝒑𝒉𝒆𝒓𝒆
• 1 steradian = 1 sr =
4𝜋
180 2
=1 𝑟𝑎𝑑 =
2
𝑑𝑒𝑔 2 = 3282.8064
𝜋
square degree

Therefore,
4𝜋 steradians = 3282.8064 x 4𝜋 =
41252.96 square degree Solid angle in a
sphere

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Beam angle or Beam solid angle (Ω𝐴 )
The beam solid angle Ω𝐴 is defined as the solid angle through which
all the power of the antenna would flow if its radiation intensity is
constant (and equal to the maximum value of U) for all angles within
Ω𝐴 .

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RADIATION POWER DENSITY
Electromagnetic waves are used to transport information through a wireless medium
or a guiding structure, from one point to the other.
It is then natural to assume that power and energy are associated with
electromagnetic fields.

The quantity used to describe the power associated with an electromagnetic wave is
the instantaneous Poynting vector defined as

𝒲 =ℰ×ℋ

𝒲= instantaneous Poynting vector (W/m2)

ℰ= instantaneous electric-field intensity (V/m)
ℋ= instantaneous magnetic-field intensity (A/m)

Note that script letters are used to denote instantaneous fields and quantities
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Since the Poynting vector is a power density, the total power crossing a closed surface
can be obtained by integrating the normal component of the Poynting vector
over the entire surface. In equation form
.

𝒫 = ඾ 𝒲. d𝒔
𝑆

The average power radiated by an antenna

(radiated power) can be written as

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Radiation intensity
• Radiation intensity in a given direction is defined as “the power
radiated from an antenna per unit solid angle.” The radiation intensity
is a far-field parameter, and it can be obtained by simply multiplying
the radiation density by the square of the distance. In mathematical
form it is expressed as
𝑈 = 𝑟 2 𝑊𝑟𝑎𝑑
𝑈 = radiation intensity (W/unit solid angle)
𝑊𝑟𝑎𝑑 = radiation density (W/m2)

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The total power is obtained by integrating the radiation intensity, over the entire
solid angle of 4π

For anisotropic source U will be independent of the angles θ and φ, as was the
case for Wrad.

the radiation intensity of an isotropic source as

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DIRECTIVITY
Directivity of an antenna defined as “the ratio of the radiation intensity in a
given direction from the antenna to the radiation intensity averaged over all
directions.
The average radiation intensity is equal to the total power radiated by the
antenna divided by 4π

If the directions not specified, the direction of maximum radiation intensity is

implied
𝑃 𝜃, ∅ 𝑚𝑎𝑥
𝐷=
𝑃 𝜃, ∅ 𝑎𝑣𝑔

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 ANTENNA EFFICIENCY

The total antenna efficiency 𝑒0 is used to take into account losses at the input
terminals and within the structure of the antenna. Such losses may be due
1. reflections because of the mismatch between the transmission line and the antenna
2. I 2R losses (conduction and dielectric)

𝑒0= 𝑒𝑟 𝑒𝑐 𝑒𝑑

𝑒0 = total efficiency (dimensionless)

𝑒𝑟 = reflection(mismatch) efficiency = (1 − |Γ|2 )
(dimensionless)
𝑒𝑐 = conduction efficiency (dimensionless)
𝑒𝑑 = dielectric efficiency (dimensionless) 49
50
GAIN
Another useful measure describing the performance of an antenna is the gain.

Although the gain of the antenna is closely related to the directivity, it is a

measure that takes into account the efficiency of the antenna as well as its
directional capabilities

Gain of an antenna (in a given direction) is defined as “the ratio of the intensity,
in a given direction, to the radiation intensity that would be obtained if the power
accepted by the antenna” were radiated isotropically.

radiation intensity 𝒰(𝜃,∅))

Gain = 4π = 4π (dimensionless)
total input (accepted) power 𝑃𝑖𝑛
INPUT IMPEDANCE
Input impedance is defined as “the impedance presented by an antenna at its
terminals or the ratio of the voltage to current at a pair of terminals or the ratio
of the appropriate components of the electric to magnetic fields at a point.”

In general the resistive part of consists of two components; that is

where
Rr = radiation resistance of the antenna
RL = loss resistance of the antenna

The power that is delivered to the antenna, part is radiated through the
mechanism provided by the radiation resistance and the other is dissipated as
heat which influences part of the overall efficiency of the antenna. 52
 Classification of Polarization

Linear : Two orthogonal plane waves with same phase but

possibly different amplitudes.

Circular: Two orthogonal plane waves with 90 deg phase

shift but same amplitudes.

Elliptical: Possibly any degree phase shift with

different amplitudes.

https://www.edmundoptics.com/knowledge-
center/application-notes/optics/introduction-to-polarization/

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 Linear Polarization

Linear Polarization A time-harmonic wave is

linearly polarized at a given point in space if the
electric-field (or magnetic-field) vector at that
point is always oriented along the same straight
line at every instant of time.
This is accomplished if the field vector (electric or
magnetic) possesses:
a. Only one component, or
b. Two orthogonal linear components that are in
time phase or 180◦ (or multiples
of 180◦) out-of-phase.

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 Circular Polarization
Circular Polarization A time-harmonic wave is circularly polarized at a given point in
space if the electric (or magnetic) field vector at that point traces a circle as a function
of time.
The necessary and sufficient conditions to accomplish this are if the field vector
(electric or magnetic) possesses all of the following:
a. The field must have two orthogonal linear components, and
b. The two components must have the same magnitude, and
c. The two components must have a time-phase difference of odd multiples of 90◦

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 Elliptical Polarization
Elliptical Polarization A time-harmonic wave is elliptically polarized if the tip
of the field vector (electric or magnetic) traces an elliptical locus in space.
A wave is elliptically polarized if it is not linearly or circularly polarized
The necessary and sufficient conditions to accomplish this are
if the field vector (electric or magnetic) possesses all of the
following:

a. The field must have two orthogonal linear components, and

b. The two components can be of the same or different
magnitude.
c. (1) If the two components are not of the same magnitude, the
time-phase difference between the two components must not
be 0◦ or multiples of 180◦ (because it will then be linear).
(2) If the two components are of the same magnitude, the time-
phase difference between the two components must not be odd
multiples of90◦ (because it will thenbe circular).
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 Transmitter Receiver
Horizontally polarized Horizontally polarized

Vertically polarized Vertically polarized

Horizontally polarized Vertically polarized

Horizontally polarized Circularly polarized

Vertically polarized Circularly polarized

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Antenna Bandwidth
• Antenna bandwidth can be defined as a range of frequencies over
which the antenna maintains required characteristics like gain, FBR,
SWR, Radiation pattern, polarization and impedance etc. i.e. without
any considerable changes.
• Antenna bandwidth classified into two types
1. Pattern bandwidth
2. impedance bandwidth
• Associated with pattern bandwidth are gain, side lobe level, beam
width, polarization, and beam direction
• input impedance and radiation efficiency are related to impedance
bandwidth.
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Front to Back Ratio
• The ratio of radiated power in desired
direction to the radiated power in
opposite direction is called front to back
ratio.

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Effective Aperture (Area) (or) Capture Area
The effective area (or aperture) is the area over which the
antenna would "collect" all the power or energy of a plane
wave.

Received power / incident

Effective aperture =
power density

For a lossless antenna,

physical Aperture = Effective aperture 60
 Physical Aperture

It is the measure of physical

size of the receiving antenna.

Physical aperture = A x B

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 Aperture Efficiency

It is the ratio of effective aperture to the

physical aperture.

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 Vector Effective length or Height
The effective length of an antenna is a quantity that is used to determine the
voltage induced on the open-circuit terminals of the antenna when a wave
impinges upon it

It should be noted that it is also referred to as the effective height

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 Reciprocity

It states that the receive and transmit properties of

an antenna are identical.

If you know the radiation pattern in the transmit

mode, then you also know the pattern in the
receive mode.

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If a current I1 at terminals of antenna 1 induces an EMF E2
at open terminals of antenna 2 and a current I2 at terminals
of antenna 2 induces an EMF E1 at open terminals of
antenna 2, then,

E1 = E2 (if I1 =I2)

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• Applying KVL to Loop 2

• Applying KVL to Loop1

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• By symmetry .
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•.

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Applications of Reciprocity Theorem
• Reciprocity theorem is used to
1. find equality of Radiation patterns
2. Find Equality of Effective length
3. Find Equality of antenna Impedance.

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PROPERTIES OF ANTENNA

1. It has identical impedance when used for transmitting and

receiving purposes. This property is called equality of impedances.

2. It has identical directional characteristics/patterns when it is used

for transmitting and receiving purposes. This property is called
equality of directional patterns.

3. It has the same effective length when it is used for transmitting

and receiving purposes. This property is called equality of effective
lengths.

These properties can be proved using Reciprocity theorem.

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Equivalence of Radiation Pattern
For Linear, passive and isotropic medium, the radiation
patterns of the transmitting and receiving antennas are
identical.
Directivity depends on the shape of Power pattern.

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• Proof:
• In the above figure Antenna (1) is test antenna (Transmitting) and
Antenna 2 is exploring Antenna(Receiving).
• Pattern may be field pattern or Power pattern. In the present case
field pattern is considered.
• Test antenna (Antenna:1) is at the center of the observation circle
and Receiving antenna (Antenna :2 ) is moved on the observation
circle.
• Antenna number 2 is perpendicular to the radius vector and
parallel to electric Vector.

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• Case :I
• If voltage V is applied to the transmitting antenna No :1 and resulting
current I at the terminals of the receiving antenna is measured .
• Ratio is calculated.
• Case: 2
• Antenna2 is kept at the center of Observation circle. Antenna 1 is moved
on the observation circle. If voltage V is applied to the Receiving antenna
No :2 and resulting current I at the terminals of the antenna number 1 is
measured.
• Ratio is calculated.

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• Are same for every position of test

antenna and Receiving antenna.

Conclusion :
Radiation pattern of Test antenna (Antenna :1) is identical with the
radiation pattern obtained when antenna number 2 is transmitting and
Antenna number 1 is Receiving.

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Equivalence of Antenna Impedance

For Linear, passive and isotropic medium, the

antenna impedance of the transmitting and
receiving antennas are identical.

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 Reciprocity therem-Application-3.Equivalence
of Impedances
• During transmission one point of
antenna length is excited.
• During reception entire length of
antenna is excited.
• Excited lengths are different during
transmission and reception. Hence d is the distance between antennas, it is very large
current distribution during transmission
and reception is also different.

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• Proof
• Consider two antennas with wide separation. So that
mutual inductance can be neglected.
• Transmission Antenna-1
• Self impedance of Antenna number1

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Reciprocity therem-Application-3.Equivalence
of Impedances
• Under short circuit
condition

• Note : is voltage
source and is Self
impedance and their ratio
gives short circuit current. 80
 Under open circuit condition =0

𝐸O𝐶 = 𝑧12 𝐼2

𝐸OC 𝐸O𝐶 = 𝑧12 𝐼2

𝑧𝐿 =
𝐼𝑆𝐶

𝑧𝐿 = 𝑍11

Antenna circuit is same. Hence impedance is same during

transmission and Reception. 81