Antennas and Wave
Propagation
Lectures Notes
Written by
Dr. Saad Saffah
Fundamentals of Antennas and Wave Propagation
Contains
1-1 INTRODUCTION AND DEFINITIONS:
1-1-1 THE ELECTROMAGNETIC SPECTRUM
1-1-2 THE RADIO TRANSMITTER
1-1-3 THE RADIO RECEIVER
1-1-4 WIRELESS COMMUNICATION SYSTEMS
1.2 MAXWELL’S EQUATIONS AND THE FUNDAMENTAL OF RADIATION:
1-2-1 UNIFORM PLANE WAVE (UPW) SOLUTION
1-2-2 PROPAGATION IN LOSSLESS, CHARGE-FREE REGION
1-2-3 POYNTING VECTOR
2-1 RADIATED FIELD OF A HERTZIAN DIPOLE
2-1 BASIC ANTENNA PARAMETERS:
2-1-1 DIRECTIVITY AND GAIN:
2-1-2 RADIATION RESISTANCE :
2-1-3 EFFECTIVE AREA OF AN ANTENNA:
2-1-4 EFFECTIVE LENGTH/HEIGHT OF THE ANTENNA:
2-1-5 ANTENNA EQUIVALENT CIRCUIT:
3-1 HALF WAVE DIPOLE ANTENNA:
3-2 QUARTER WAVE MONOPOLE ANTENNA:
4-1 SMALL LOOP ANTENNAS:
5-1 INTRODUCTION TO ANTENNA ARRAYS:
5-2 ARRAY OF IDENTICAL ELEMENTS:
5-3 TWO-ELEMENT ARRAY:
5-3-1 CASE-1:
5-3-2 CASE 2:
5-4 UNIFORM ONE-DIMENSIONAL ARRAY
5-4-1 BROAD SIDE CASE:
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Fundamentals of Antennas and Wave Propagation
5-4-2 END FIRE CASE:
5-5 ARRAY PATTERN SYNTHESIS:
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Fundamentals of Antennas and Wave Propagation
1-1 INTRODUCTION AND DEFINITIONS:
There are more “radios” being built than ever before!
1. Telephony
1. Cellular
2. Public Cabinets System (PCS)
3. Global Satellite Systems
4. Microwave Links
2. Broadcasting
1. AM radio, FM radio, VHF/UHF TV
2. Satellite Links
3. Direct Broadcasting
3. Networking
1. LAN
2. WiFi
3. WiMAX
4.
LiFi
4. Radar and Navigation
1. Global Positioning System (GPS)
2. Radar detection, tracking and imaging
3. Radio Frequency Identification (RFID)
Q: Just what is a radio?
A: A device that transfers information to a distant site, by means of unbounded
electromagnetic propagation.
A radio system has three sections, with antennas serving as couplers between each
section:
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Fundamentals of Antennas and Wave Propagation
1-1-1 THE ELECTROMAGNETIC SPECTRUM
We can propagate energy anywhere within the electromagnetic spectrum, but we
typically use frequencies less than, say, 40 GHz.
Q: Why don’t we use frequencies greater than 40 GHz ?
A: Two reasons:
1. The difficulty in making electronic components.
2. The Earth’s atmosphere rapidly attenuates the propagating wave!
1-1-2 THE RADIO TRANSMITTER
There are five main components of a transmitter:
1) The signal a(t) – This is the information we are trying to transmit. It may be in
either digital or analog form. It also may have been encoded to remove
redundancy, in a process known as source coding.
2) RF source – Generates electromagnetic energy at RF/microwave frequencies that
are suitable for electromagnetic propagation (subject to FCC restrictions !).
3) Modulator – Places signal a(t) (i.e., the information) onto the RF signal, known as
the carrier. Accomplished by modulating some parameter of the carrier signal –
e.g., magnitude, phase, frequency, or some combination thereof. In general, this
process is called channel coding. Its goal is to maximize the rate at which
information is sent, while minimizing the effect of unknown channel parameters.
4) Power Amplifier – Increases the power (i.e., energy flow) of the modulated carrier
signal, without (hopefully) distorting it.
5) Antenna – Acts as the coupling mechanism between the bounded e.m. wave of a
transmission line and the unbounded propagating wave in space. Often, an
antenna is required to launch the unbounded wave in a specific direction.
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Fundamentals of Antennas and Wave Propagation
1-1-3 THE RADIO RECEIVER
There are eight basic components in a radio receiver:
1) Antenna – Couples the incoming EM propagating wave into the receiver.
2) Low-Noise Amplifier – Boosts the power of the initial signal above the receiver
noise.
3) Pre-selector Filter – Allows only the frequency band of interest to pass into the
receiver (e.g., for FM radio 88-108 MHz).
4) Local Oscillator/Mixer – Translates the signal from its propagation frequency to a
lower, fixed intermediate frequency (IF).
5) IF Amplifier – A high-gain amplifier that greatly increases signal power (i.e., to a
detectable level).
6) IF Filter - Allows only the signal of interest to pass. Bandwidth is typically that of
the desired signal. (e.g., 200 kHz for FM radio, 20 kHz for AM radio).
7) Detector/Demodulator – Extracts the signal information from the IF signal.
8) The Recovered Signal â(t)- The receiver’s at what the original signal was.
1-1-4 WIRELESS COMMUNICATION SYSTEMS
In wireless communication systems, signals are radiated in space as an
electromagnetic wave by using a transmitting antenna and a fraction of this radiated
power is intercepted by using a receiving antenna. Thus, an antenna is a device used
for radiating or receiver radio waves.
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Fundamentals of Antennas and Wave Propagation
An antenna can also be thought of as a transitional structure between free
space and a guiding device (such as transmission line or waveguide). Usually antennas
are metallic structures, but dielectric antennas (lens antennas) are also used now a
day. In our discussion, we shall consider only metallic antennas. Here we shall restrict
our discussion to some very commonly used antenna structures. Some of the most
commonly used antenna structures are shown.
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Fundamentals of Antennas and Wave Propagation
A radio antenna: may be defined as the structure associated with the region of
transition between a guided wave and a free-space wave, or vice versa.
To be more explicit, the region of transition between the guided wave and the freespace wave may be defined as an antenna.
Electromagnetic wave: An electromagnetic wave, as the name implies, consists of a
combination of the properties of both electric and magnetic fields.
It is conceptually easy to generate an electromagnetic wave. All you need to do is to
cause a changing current in a conductor and a wave will propagate outward to the
end of the universe.
A transmission line: is a device for transmitting or guiding radio-frequency energy
from one point to another. Usually it is desirable to transmit the energy with a
minimum of attenuation, heat and radiation losses being as small as possible.
The term transmission line (or transmission system) include not only coaxial and 2wire transmission lines but also hollow pipes, or waveguides.
The wave transmitted along the line: is 1-dimensional in that it does not spread out
into space but follows along the line.
A generator connected to an infinite, lossless transmission line produces a uniform
traveling wave along the line. If the line is short-circuited, the outgoing traveling wave
is reflected, producing a standing wave on the line due to the interference between
the outgoing and reflected waves.
A guided wave traveling along a transmission line which opens out, as in Fig. 2-1, will
radiate as a free-space wave. The guided wave is a plane wave while the free-space
wave is a spherically expanding wave.
Resonant circuit: The energy concentrations in such a wave oscillate from entirely
electric to entirely magnetic and back twice per cycle. Such energy behavior is
characteristic of a resonant circuit, or resonator.
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Fundamentals of Antennas and Wave Propagation
1.2 MAXWELL’S EQUATIONS AND THE FUNDAMENTAL OF RADIATION:
Radiation is the process of emitting energy from a source. Electromagnetic
radiation can be at all frequencies except zero (DC), but radiation at various
frequencies may take different forms. The energy associated with the radiation
depends on the frequency.
Time varying currents radiate electromagnetic waves. A time varying current
generates time varying electric and magnetic fields. When such fields exist, power is
generated and propagated.
In regions of free space (i.e. the vacuum), where no electric charges, no electric
currents and no matter of any kind are present, Maxwell’s equations (in differential
form) are:
We can de-couple Maxwell’s Equations e.g. by applying the curl operator to
equations (3) and (4):
These are three-dimensional de-coupled wave equations for 𝐸̅ and 𝐵̅
Note that they have exactly the same structure – both are linear, homogeneous, 2nd
order differential equations.
Remember that each of the above equations is explicitly dependent on space and
time, i.e. 𝐸̅ = 𝐸̅ (𝑟̅ , 𝑡) and 𝐵̅ = 𝐵̅(𝑟̅ , 𝑡)
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Fundamentals of Antennas and Wave Propagation
Thus, Maxwell’s equations implies that empty space (the vacuum: which is not empty,
at the microscopic scale) supports the propagation of {macroscopic} electromagnetic
waves, which propagate at the speed of light:
𝑐=
1
√ 𝜇 𝑜 𝜀𝑜
= 3 × 108 𝑚/𝑠 .
EM waves have associated with them a frequency f and wavelength λ, related to
each other via = 𝑓𝜆 .
1-2-1 UNIFORM PLANE WAVE (UPW) SOLUTION
Now, let’s try to solve the wave equation. We suppose that we have
a). Electrical filed only have one components (say Ex component)
b). This components is constant in two directions (say x-y plane), only change along zaxis (propagation direction).
So the proposed solution is


E  E x z a x
How do we know this is a solution?
Just plug in the wave equation, if we can find an Ex(z) which satisfies the wave
equation, we have a solution.
d 2 Ex
dz 2
  2 Ex  0
the solution is




Ex z   E 0ez  E 0ez a x
If we translate this solution back into instantaneous solution, it is:




Ex z   E 0ez e j t z   E 0ez e j t z  a x
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Fundamentals of Antennas and Wave Propagation
The first term characterizes the forward-propagating UPW, the second term
characterizes the backward-propagating one.
By the help of the Maxwell equations, we can also figure out other field such D, B. and
H.


  Es   jH s
Therefore, the H field can be calculated as


1
Hs  
  Es
j

 E0 z E0 z  
H s  
e 
e a y
j
 j

Having E and H solution available, the ratio between the E and H field can be taken.
Let’s take a forward-propagation UPW as examples:

E
H


j


j
  j
Since E and H fields have unit (V/m) and (A/m). The ratio yields a unit of (). This ratio
is referred as intrinsic impedance (η) of the materials.


1
H s  a p  Es




Es  a p  H s
Now, we can see, the propagation characteristics for UPW is ultimately determined
by  
j  j     j .
1-2-2 PROPAGATION IN LOSSLESS, CHARGE-FREE REGION
In a charge free region with zero loss, the propagation constant

=0, so
j   j     j

j 0  j   j
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Fundamentals of Antennas and Wave Propagation
   







E x z   E 0e j t  z   E 0e j t  z  a x
up 

1



A special example will be vacuum, where  = o ,  = o.
o
 120
o
o 
m / s 
u p  3  108
1-2-3 POYNTING VECTOR
 E  H  dS   J  E dv  t  2   E



 

2
2
 1
dv     H dv


t 2 
1
The cross product in the left-hand side can be written as following and known as
instantaneous Poynting vector.
  
P  EH
UPW power transmission, apply Poynting theorem to UPW. WE can easily show that
the time-averaged power density in a plane wave is




1
Pave  Re Es  H s*
2

The power through a surface is then


P   Pave  dS .
If we have a plane wave traveling in lossy media, the spatial part of wave is


Es  Ex 0ez e jz e j ax
The intrinsic impedance can be complex and can be written in the form of
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Fundamentals of Antennas and Wave Propagation
  e
j
The magnetic field is then


E
1
 j 
H s  a p  E s  x 0 e z e  jz e j e  a y


The Poynting vector can be expressed as:
2

1 Exo   2z

Pave 
e
cos az
2 
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Fundamentals of Antennas and Wave Propagation
1-3 POLARIZATION
The polarization of a UPW describes the shape and locus of the tip of E-vector at a
given point in space as function of time. If we have a UPW traveling along +z direction,
the E-field could have two components along x-axis and y-axis. Mathematically, it can
be written as:



j
E t , z   E xo e jx e jt e jkz a x  E yo e y e jt e jkz a y


j t  kz  y  
E t , z   E xo e j t  kzx a x  E yo e
ay
If we observe the E-field at a fix point, say… z = 0 , we have:


j ( t  y ) 
E t ,0  E xo e j ( t x ) a x  E yo e
ay




 E xo cos t   x a x  E yo cos  t   y a y
In an x-y plane, the above equation represent parametric equations describing various
shape of circle, ellipse, or straight line. These all depends on the phase in two axis and
amplitudes. If we choose a proper reference point for the time, we can set x = 0, and
 = y - x , so



E t ,0  E xo e j ( t ) a x  E yo e j ( t  ) a y
1-3-1 LINEAR POLARIZATION
If the phases x and y are in phase or completely out of phase
x   y
x   y  


in  phase


out  of  phase



E z , t   E xo a x  E yo a y e j ( t  kz)



E z , t   E xo a x  E yo a y e j ( t  kz)
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Fundamentals of Antennas and Wave Propagation
If we observe the E-field at a fix point, say… z = 0 , we have:





E 0, t   E xo a x  E yo a y e j ( t )
 E yo
 Ex
o

  tan 1  

E 0, t   E x2o  E y2o




1/ 2 e j( t )
The  is defined as inclination angle shown below. Since  is independent of both z
and t, E(z,t) maintains a direction along the line making an angle  with the x-axis.
1-3-2 CIRCULAR POLARIZATION
If Exo = Eyo  a, and the phase different between two E-field components are
   y  x  

2
Here we have two situations:
 Left-handed circular (LHC) polarization    y   x  
15

2
Fundamentals of Antennas and Wave Propagation
 Right-handed circular (RHC) polarization    y   x  

2
For LHC polarization, at z=0 (observation point)

j ( t  2 ) 
j ( t ) 


E 0, t  ae
a x  ae
ay

E z, t   a 2
 ae j ( t  2 ) 


E
0
,
t


y
1
  tan 1   a sin(  t )     t
  tan 1 
  tan 
 ae j ( t ) 
 a cos( t ) 
 E x 0, t  


From above equations, we see that the amplitude of the total E-field is a
constant, however, the inclination angle linearly decreases. If you face the incoming
EM field, you find that E-field is Left-hand-circulated.
For RHC polarization, at z=0 (observation point)
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Fundamentals of Antennas and Wave Propagation


j ( t  2 ) 
E 0, t   ae j ( t ) a x  ae
ay

E 0, t   a 2
 ae j ( t  2 ) 


E
0
,
t


y
1 
1  a sin(  t ) 

  tan 1 

tan

tan

 a cos( t )     t
j ( t )


E
0
,
t


ae


 x



From above equations, we see that the amplitude of the total E-field is a constant,
however, the inclination angle linearly increases. If you face the incoming EM field,
you find that E-field is Right-hand-circulated.
1-3-3 ELLIPTICAL POLARIZATION
In most general cases, we have Ex0  Ey0, and   0, , or  /2. The wave is said to be
elliptically polarized. The shape and handedness (RH or LH) are determined by the
values of the ratio (Ey0 / Ex0).
Illustrated by the figure below, the major and minor axes of the ellipse do not
necessarily overlap with the x and y-axis. We then define the rotation angle  that is
between the major axis  and the x-axis. If we define an auxiliary angel 0:
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Fundamentals of Antennas and Wave Propagation
tan 0 
E xo
E yo
We can prove that the rotation angle is:
tan 2  tan 2 0 cos
 /2   /2
tan 2   sin 2 0 sin 
 /4   /4
Also, we have
 If  > 0, then sin > 0, this leads to left-handed rotation
 If  < 0, then sin < 0, this leads to right-handed rotation
Example: Determine the polarization state of a plane wave with E-field.



j t  kz 
 j 4a y e j t kz 
1- E z , t   j3a x e









 j  t kz 6 
 j  t kz 4 
 j 4a y e
2- E z , t   3a x e

 j  t  kz 4 
 j  t  kz 4 
 j 4a y e
3- E  z , t   4a x e
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Fundamentals of Antennas and Wave Propagation
Chapter 2: RADIO WAVE PROPAGATION
2.1- ELECTROMAGNETIC FIELDS
There are two basic fields associated with every antenna; an INDUCTION FIELD and a
RADIATION FIELD.
2.1.1- INDUCTION FIELD
Figure 2-1, a low-frequency generator connected to an antenna, will help you
understand how the induction field is produced. Let us follow the generator through
one cycle of operation.
1- Initially, you can consider that the generator output is zero and that no fields exist
about the antenna, as shown in view (A).
2- Now assume that the generator produces a slight potential and has the
instantaneous polarity shown in view (B). Because of this slight potential, the
antenna capacitance acts as a short, allowing a large flow of current (I) through
the antenna in the direction shown. This current flow, in turn, produces a large
magnetic field about the antenna. Since the flow of current at each end of the
antenna is minimum, the corresponding magnetic fields at each end of the
antenna are also minimum.
3- As time passes, charges, which oppose antenna current and produce an
electrostatic field (E field), collect at each end of the antenna. Eventually, the
antenna capacitance becomes fully charged and stops current flow through the
antenna. Under this condition, the electrostatic field is maximum, and the
magnetic field (H field) is fully collapsed, as shown in view (C).
4- As the generator potential decreases back to zero, the potential of the antenna
begins to discharge. During the discharging process, the electrostatic field
collapses and the direction of current flow reverses, as shown in view (D).
5- When the current again begins to flow, an associated magnetic field is generated.
Eventually, the electrostatic field completely collapses, the generator potential
reverses, and current is maximum, as shown in view (E).
6- As charges collect at each end of the antenna, an electrostatic field is produced
and current flow decreases. This causes the magnetic field to begin collapsing. The
collapsing magnetic field produces more current flow, a greater accumulation of
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Fundamentals of Antennas and Wave Propagation
charge, and a greater electrostatic field. The antenna gradually reaches the
condition shown in view (F), where current is zero and the collected charges are
maximum.
7- As the generator potential again decreases
toward zero, the antenna begins to
discharge and the electrostatic field begins
to collapse. When the generator potential
reaches zero, discharge current is
maximum and the associated magnetic
field is maximum. A brief time later,
generator potential reverses, and the
condition shown in view B recurs.
NOTE: The electric field (E field) and the
electrostatic field (E field) are the same. They
will be used interchangeably throughout this
text.
The graph shown in figure 2-2 shows the
relationship between the magnetic field (HField) and the electric field (E-Field) plotted
against time. Note that the two fields are 90
degrees out of phase with each other.
If you compare the graph in figure 2-2 with
figure 2-1, you will notice that the two fields
around the antenna are displaced 90 degrees
from each other in space. (The H field exists in
a plane perpendicular to the antenna. The E
field exists in a plane parallel with the antenna,
as shown in figure 2-1
Figure 2-1.—Induction field about an antenna.
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Fundamentals of Antennas and Wave Propagation
Figure 2-2.—Phase relationship of induction field components.
All the energy supplied to the induction field is returned to the antenna by the
collapsing E and H fields. No energy from the induction field is radiated from the
antenna. Therefore, the induction field is considered a local field and plays no part in
the transmission of electromagnetic energy. The induction field represents only the
stored energy in the antenna and is responsible only for the resonant effects that the
antenna reflects to the generator.
2.1.2- RADIATION FIELDS
The E and H fields that are set up in the transfer of energy through space are known
collectively as the radiation field. The radiation field decreases as the distance from
the antenna is increased. Because the decrease is linear, the radiation field reaches
great distances from the antenna.
Let us look at a half-wave antenna to illustrate how this radiation actually takes place.
Simply stated, a half-wave antenna is one that has an electrical length equal to half
the wavelength of the signal being transmitted. Assume, for example, that a
transmitter is operating at 30 megahertz. If a half-wave antenna is used with the
transmitter, the antenna's electrical length would have to be at least 16 feet long. (The
formula used to compute the electrical length of an antenna would be explained
later).
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Fundamentals of Antennas and Wave Propagation
Figure 2-3 is a simple picture of an E field detaching itself from an antenna. (The H
field will not be considered, although it is present.)
1- In view (A), the voltage is maximum and the electric field has maximum
intensity. The lines of force begin at the end of the antenna that is positively
charged and extend to the end of the antenna that is negatively charged. Note
that the outer E lines are stretched away from the inner lines. This is because
of the repelling force that takes place between lines of force in the same
direction.
2- As the voltage drops view (B), the separated charges come together, and the ends
of the lines move toward the center of the antenna. But, since lines of force in the
same direction repel each other, the centers of the lines are still being held out.
3- As the voltage approaches zero view (B), some of the lines collapse back into the
antenna. At the same time, the ends of other lines begin to come together to form
a complete loop. Notice the direction of these lines of force next to the antenna
in view (C). At this point, the voltage on the antenna is zero.
4- As the charge starts to build up in the opposite direction, view (D), electric lines of
force again begin at the positive end of the antenna and stretch to the negative
end of the antenna. These lines of force, being in the same direction as the sides
of the closed loops next to the antenna, repel the closed loops and force them out
into space at the speed of light. As these loops travel through space, they
generate a magnetic field in phase with them.
Figure 2-3.—Radiation from an antenna.
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Fundamentals of Antennas and Wave Propagation
Figure 2-4 shows a comparison between the induction field and the radiation field.
Figure 2-4.—E and H components of induction and radiation fields.
Q1. Which two composite fields (composed of E and H fields) are associated with
every antenna?
Q2. What composite field (composed of E and H fields) is found stored in the antenna?
Q3. What composite field (composed of E and H fields) is propagated into free space?
2.2- RADIO WAVES
An energy wave generated by a transmitter is called a RADIO WAVE. The radio wave
radiated into space by the transmitting antenna is a very complex form of energy
containing both electric and magnetic fields. Because of this combination of fields,
radio waves are also referred to as ELECTROMAGNETIC RADIATION.
NOTE: The term radio wave is not limited to communications equipment alone. The
term applies to all equipment that generate signals in the form of electromagnetic
energy.
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Fundamentals of Antennas and Wave Propagation
2.2.1- COMPONENTS OF RADIO WAVES
The basic shape of the wave generated by a transmitter is that of a sine wave. The
wave radiated out into space, however, may or may not retain the characteristics of
the sine wave.
The frequencies falling between 3000 hertz (3 kHz) and 300,000,000,000 hertz (300
GHz) are called RADIO FREQUENCIES (abbreviated RF) since they are commonly used
in radio communications. This part of the radio frequency spectrum is divided into
bands, each band being 10 times higher in frequency than the one immediately below
it. This arrangement serves as a convenient way to remember the range of each band.
The RF bands are shown in table 2-1. The usable radio-frequency range is roughly 10
kilohertz to 100 gigahertz.
Table 2-1.—Radio Frequency Bands
DESCRIPTION
Very low
Low
Medium
High
Very high
Ultrahigh
Super high
Extremely high
ABBREVIATION
VLF
LF
MF
HF
VHF
UHF
SHF
EHF
FREQUENCY
3 to 30 KHz
30 to 300 KHz
300 to 3000 KHz
3 to 30 MHz
30 to 300 MHz
300 to 3000 MHz
3 to 30 GHz
30 to 300 GHz
Any frequency that is a whole number multiple of a smaller basic frequency is known
as a HARMONIC of that basic frequency. The basic frequency itself is called the first
harmonic or, more commonly, the FUNDAMENTAL FREQUENCY. A frequency that is
twice as great as the fundamental frequency is called the second harmonic; a
frequency three times as great is the third harmonic; and so on. For example:
First harmonic (Fundamental frequency)
3000 kHz
Second harmonic
6000 kHz
Third harmonic
9000 kHz
The PERIOD of a radio wave is simply the amount of time required for the completion
of one full cycle. If a sine wave has a frequency of 2 hertz, each cycle has a duration,
or period, of one-half second. If the frequency is 10 hertz, the period of each cycle is
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Fundamentals of Antennas and Wave Propagation
one-tenth of a second. Since the frequency of a radio wave is the number of cycles
that are completed in one second, you should be able to see that as the frequency of
a radio wave increases, its period decreases.
Q4. What is the term used to describe the basic frequency of a radio wave?
Q5. What is the term used to describe a whole number multiple of the basic frequency
of a radio wave?
2.3- WAVELENGTH-TO-FREQUENCY CONVERSIONS
As discussed earlier, a radio wave travels 300,000,000 meters a second (speed of
light); therefore, a radio wave of 1 hertz would have traveled a distance (or
wavelength) of 300,000,000 meters. Obviously then, if the frequency of the wave is
increased to 2 hertz, the wavelength will be cut in half to 150,000,000 meters. This
illustrates the principle that the HIGHER THE FREQUENCY, the SHORTER THE
WAVELENGTH.
Wavelength-to-frequency conversions of radio waves are really quite simple because
wavelength and frequency are reciprocals: Either one divided into the velocity of a
radio wave yields the other. Remember, the formula for wavelength is:
Q6. It is known that WWV operates on a frequency of 10 megahertz. What is the
wavelength of WWV?
Q7. A station is known to operate at 60-meters. What is the frequency of the unknown
station?
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Fundamentals of Antennas and Wave Propagation
2.4- POLARIZATION
For maximum absorption of energy from the electromagnetic fields, the receiving
antenna must be located in the plane of polarization. This places the conductor of the
antenna at right angles to the magnetic lines of force moving through the antenna and
parallel to the electric lines, causing maximum induction.
Normally, the plane of polarization of a radio wave is the plane in which the E field
propagates with respect to the Earth. If the E field component of the radiated wave
travels in a plane perpendicular to the Earth's surface (vertical), the radiation is said
to be VERTICALLY POLARIZED, as shown in figure 2-5, view A. If the E field propagates
in a plane parallel to the Earth's surface (horizontal), the radiation is said to be
HORIZONTALLY POLARIZED, as shown in view B.
Figure 2-5.—Vertical and horizontal polarization.
The position of the antenna in space is important because it affects the polarization
of the electromagnetic wave. When the transmitting antenna is close to the ground,
vertically polarized waves cause a greater signal strength along the Earth's surface. On
the other hand, antennas high above the ground should be horizontally polarized to
get the greatest possible signal strength to the Earth's surface.
If you know the directions of the E and H components, you can use the "right-hand
rule" (see figure 2-6) to determine the direction of wave propagation. This rule states
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Fundamentals of Antennas and Wave Propagation
that if the thumb, forefinger, and middle finger of the right hand are extended so they
are mutually perpendicular, the middle finger will point in the direction of wave
propagation if the thumb points in the direction of the E field and the forefinger points
in the direction of the H field. Since both the E and H fields reverse directions
simultaneously, propagation of a particular wavefront is always in the same direction
(away from the antenna).
Figure 2-6.—Right-hand rule for propagation.
Q8. If a transmitting antenna is placed close to the ground, how should the antenna
be polarized to give the greatest signal strength?
Q9. In the right-hand rule for propagation, the thumb points in the direction of the E
field and the forefinger points in the direction of the H field. In what direction does the
middle finger point?
2.5- ATMOSPHERIC PROPAGATION
Within the atmosphere, radio waves can be reflected, refracted, and diffracted like
light and heat waves.
2.5.1- REFLECTION
Radio waves may be reflected from various substances or objects they meet during
travel between the transmitting and receiving sites. The amount of reflection depends
on the reflecting material. Smooth metal surfaces of good electrical conductivity are
efficient reflectors of radio waves. The surface of the Earth itself is a fairly good
reflector. The size of the area required for reflection to take place depends on the
27
Fundamentals of Antennas and Wave Propagation
wavelength of the radio wave and the angle at which the wave strikes the reflecting
substance.
Figure 2-7 shows two radio waves being reflected from the Earth's surface. Notice that
the positive and negative alternations of radio waves (A) and (B) are in phase with
each other in their paths toward the Earth's surface.
Figure 2-7.—Phase shift of reflected radio waves.
2.5.2- REFRACTION
Another phenomenon common to most radio waves is the bending of the waves as
they move from one medium into another in which the velocity of propagation is
different. This bending of the waves is called refraction.
As an example, the radio wave shown in figure 2-8 is traveling through the Earth's
atmosphere at a constant speed. As the wave enters the dense layer of electrically
charged ions, the part of the wave that enters the new medium first travels faster than
the parts of the wave that have not yet entered the new medium. This abrupt increase
in velocity of the upper part of the wave causes the wave to bend back toward the
Earth. This bending, or change of direction, is always toward the medium that has the
lower velocity of propagation.
Figure 2-8.—Radio wave refraction.
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Fundamentals of Antennas and Wave Propagation
2.5.3- DIFFRACTION
A radio wave that meets an obstacle has a natural tendency to bend around the
obstacle as illustrated in figure 2-9. The bending, called diffraction, results in a change
of direction of part of the wave energy from the normal line-of-sight path. This change
makes it possible to receive energy around the edges of an obstacle as shown in view
A or at some distances below the highest point of an obstruction, as shown in view B.
Although diffracted RF energy usually is weak, it can still be detected by a suitable
receiver.
Figure 2-9.—Diffraction around an object.
Q10. What is one of the major reasons for the fading of radio waves that have been
reflected from a surface?
2.6- THE EFFECT OF THE EARTH'S ATMOSPHERE ON RADIO WAVES
This discussion of electromagnetic wave propagation is concerned mainly with the
properties and effects of the medium located between the transmitting antenna and
the receiving antenna. While radio waves traveling in free space have little outside
influence affecting them, radio waves traveling within the Earth's atmosphere are
affected by varying conditions. Atmospheric conditions vary with changes in height,
geographical location, and even with changes in time (day, night, season, and year). A
knowledge of the composition of the Earth's atmosphere is extremely important for
understanding wave propagation.
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Fundamentals of Antennas and Wave Propagation
The Earth's atmosphere is divided into three separate regions, or layers. They are the
TROPOSPHERE, the STRATOSPHERE, and the IONOSPHERE. The layers of the
atmosphere are illustrated in figure 2-10.
Figure 2-10.—Layers of the earth's atmosphere.
2.6.1- TROPOSPHERE
The troposphere is the portion of the Earth's atmosphere that extends from the
surface of the Earth to a height of about 3.7 miles (6 km) at the North Pole or the
South Pole and 11.2 miles (18 km) at the equator. Virtually all weather phenomena
take place in the troposphere. The temperature in this region decreases rapidly with
altitude, clouds form, and there may be much turbulence because of variations in
temperature, density, and pressure. These conditions have a great effect on the
propagation of radio waves, which will be explained later in this chapter.
2.6.2- STRATOSPHERE
The stratosphere is located between the troposphere and the ionosphere. The
temperature throughout this region is considered to be almost constant and there is
little water vapor present. The stratosphere has relatively little effect on radio waves
because it is a relatively calm region with little or no temperature changes.
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Fundamentals of Antennas and Wave Propagation
2.6.3- IONOSPHERE
The ionosphere extends upward from about 31.1 miles (50 km) to a height of about
250 miles (402 km). It contains four cloud-like layers of electrically charged ions, which
enable radio waves to be propagated to great distances around the Earth. This is the
most important region of the atmosphere for long distance point-to-point
communications. This region will be discussed in detail a little later in this chapter.
Q11. What are the three layers of the atmosphere?
Q12. Which layer of the atmosphere has relatively little effect on radio waves?
2.7- RADIO WAVE TRANSMISSION
Electromagnetic (radio) energy travels from a transmitting antenna to a receiving
antenna in two principal ways. One way is by GROUND WAVES and the other is by
SKY WAVES. Ground waves are radio waves that travel near the surface of the Earth
(surface and space waves). Sky waves are radio waves that are reflected back to Earth
from the ionosphere. (See figure 2-11.)
Figure 2-11.—Ground waves and sky waves.
2.7.1- GROUND WAVES
The ground wave is actually composed of two separate component waves. These are
known as the SURFACE WAVE and the SPACE WAVE (fig. 2-11). The determining factor
in whether a ground wave component is classified as a space wave or a surface wave
is simple. A surface wave travels along the surface of the Earth. A space wave travels
over the surface.
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Fundamentals of Antennas and Wave Propagation
2.7.1.1- SURFACE WAVE.—The surface wave reaches the receiving site by traveling
along the surface of the ground as shown in figure 2-12. A surface wave can follow the
contours of the Earth because of the process of diffraction. When a surface wave
meets an object and the dimensions of the object do not exceed its wavelength, the
wave tends to curve or bend around the object. The smaller the object, the more
pronounced the diffractive action will be.
Figure 2-12.—Surface wave propagation.
As a surface wave passes over the ground, the wave induces a voltage in the Earth.
The induced voltage takes energy away from the surface wave, thereby weakening, or
attenuating, the wave as it moves away from the transmitting antenna. To reduce the
attenuation, the amount of induced voltage must be reduced. This is done by using
vertically polarized waves that minimize the extent to which the electric field of the
wave is in contact with the Earth. When a surface wave is horizontally polarized, the
electric field of the wave is parallel with the surface of the Earth and, therefore, is
constantly in contact with it.
Another major factor in the attenuation of surface waves is frequency. Recall from
earlier discussions on wavelength that the higher the frequency of a radio wave, the
shorter its wavelength will be. These high frequencies, with their shorter wavelengths,
are not normally diffracted but are absorbed by the Earth at points relatively close to
the transmitting site. You can assume, therefore, that as the frequency of a surface
wave is increased, the more rapidly the surface wave will be absorbed, or attenuated,
by the Earth.
2.7.1.2- SPACE WAVE.—The space wave follows two distinct paths from the
transmitting antenna to the receiving antenna—one through the air directly to the
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Fundamentals of Antennas and Wave Propagation
receiving antenna, the other reflected from the ground to the receiving antenna. This
is illustrated in figure 2-13. The primary path of the space wave is directly from the
transmitting antenna to the receiving antenna. Therefore, the receiving antenna must
be located within the radio horizon of the transmitting antenna.
Figure 2-13.—Space wave propagation.
Space waves suffer little ground attenuation, they nevertheless are susceptible to
fading. This is because space waves actually follow two paths of different lengths
(direct path and ground reflected path) to the receiving site and, therefore, may arrive
in or out of phase. If these two component waves are received in phase, the result is
a reinforced or stronger signal. Likewise, if they are received out of phase, they tend
to cancel one another, which results in a weak or fading signal.
Q13. What is the determining factor in classifying whether a radio wave is a ground
wave or a space wave?
Q14. What is the best type of surface or terrain to use for radio wave transmission?
Q15. What is the primary difference between the radio horizon and the natural
horizon?
Q16. What three factors must be considered in the transmission of a surface wave to
reduce attenuation?
2.7.2- SKY WAVE
The sky wave, often called the ionospheric wave, is radiated in an upward direction
and returned to Earth at some distant location because of refraction from the
ionosphere. This form of propagation is relatively unaffected by the Earth's surface
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Fundamentals of Antennas and Wave Propagation
and can propagate signals over great distances. Usually the high frequency (HF) band
is used for sky wave propagation. The following in-depth study of the ionosphere and
its effect on sky waves will help you to better understand the nature of sky wave
propagation.
2.8- STRUCTURE OF THE IONOSPHERE
As we stated earlier, the ionosphere is the region of the atmosphere that extends from
about 30 miles above the surface of the Earth to about 250 miles. It is appropriately
named the ionosphere because it consists of several layers of electrically charged gas
atoms called ions. The ions are formed by a process called ionization.
2.8.1- FOUR DISTINCT LAYERS
The ionosphere is composed of three layers designated D, E, and F, from lowest level
to highest level as shown in figure 2-14. The F layer is further divided into two layers
designated F1 (the lower layer) and F2 (the higher layer). The presence or absence of
these layers in the ionosphere and their height above the Earth varies with the
position of the sun. At high noon, radiation in the ionosphere directly above a given
point is greatest. At night, it is minimum. When the radiation is removed, many of the
particles that were ionized recombine. The time interval between these conditions
finds the position and number of the ionized layers within the ionosphere changing.
Since the position of the sun varies daily, monthly, and yearly, with respect to a
specified point on Earth, the exact position and number of layers present are
extremely difficult to determine. However, the following general statements can be
made:
Figure 2-14.—Layers of the ionosphere.
34
Fundamentals of Antennas and Wave Propagation
a) The D layer ranges from about 30 to 55 miles. Ionization in the D layer is low
because it is the lowest region of the ionosphere. This layer has the ability to
refract signals of low frequencies. High frequencies pass right through it and are
attenuated. After sunset, the D layer disappears because of the rapid
recombination of ions.
b) The E layer limits are from about 55 to 90 miles. This layer is also known as the
Kennelly- Heaviside layer, because these two men were the first to propose its
existence. This layer has the ability to refract signals as high as 20 megahertz.
For this reason, it is valuable for communications in ranges up to about 1500
miles.
c) The F layer exists from about 90 to 240 miles. During the daylight hours, the F
layer separates into two layers, the F1 and F2 layers. The F layers are
responsible for high frequency, long distance transmission.
Q17. What causes ionization to occur in the ionosphere?
Q18. How are the four distinct layers of the ionosphere designated?
Q19. What is the height of the individual layers of the ionosphere?
2.8.2- REFRACTION IN THE IONOSPHERE
When a radio wave is transmitted into an ionized layer, refraction, or bending of the
wave, occurs. As we discussed earlier, refraction is caused by an abrupt change in the
velocity of the upper part of a radio wave as it strikes or enters a new medium. The
amount of refraction that occurs depends on three main factors:
(1) the density of ionization of the layer,
(2) the frequency of the radio wave,
(3) the angle at which the wave enters the layer.
2.8.2.1- DENSITY OF LAYER
Figure 2-15 illustrates the relationship between radio waves and ionization density.
Each ionized layer has a central region of relatively dense ionization, which tapers off
in intensity both above and below the maximum region. As a radio wave enters a
region of INCREASING ionization, the increase in velocity of the upper part of the wave
35
Fundamentals of Antennas and Wave Propagation
causes it to be bent back TOWARD the Earth. While the wave is in the highly dense
center portion of the layer, however, refraction occurs more slowly because the
density of ionization is almost uniform. As the wave enters into the upper part of the
layer of DECREASING ionization, the velocity of the upper part of the wave decreases,
and the wave is bent AWAY from the Earth.
Figure 2-15.—Effects of ionospheric density on radio waves.
If a wave strikes a thin, very highly ionized layer, the wave may be bent back so rapidly
that it will appear to have been reflected instead of refracted back to Earth. To reflect
a radio wave, the highly ionized layer must be approximately no thicker than one
wavelength of the radio wave. Since the ionized layers are often several miles thick,
ionospheric reflection is more likely to occur at long wavelengths (low frequencies).
2.8.2.2- FREQUENCY
For any given time, each ionospheric layer has a maximum frequency at which radio
waves can be transmitted vertically and refracted back to Earth. This frequency is
known as the CRITICAL FREQUENCY. Radio waves transmitted at frequencies higher
than the critical frequency of a given layer will pass through the layer and be lost in
space; but if these same waves enter an upper layer with a higher critical frequency,
they will be refracted back to Earth. Radio waves of frequencies lower than the critical
frequency will also be refracted back to Earth unless they are absorbed or have been
refracted from a lower layer.
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Fundamentals of Antennas and Wave Propagation
Figure 2-16 shows three separate waves of different frequencies entering an
ionospheric layer at the same angle. Notice that the 5-megahertz wave is refracted
quite sharply. The 20-megahertz wave is refracted less sharply and returned to Earth
at a greater distance. The 100-megahertz wave is obviously greater than the critical
frequency for that ionized layer and, therefore, is not refracted but is passed into
space.
Figure 2-16.—Frequency versus refraction and distance.
2.8.2.3- ANGLE OF INCIDENCE
The rate at which a wave of a given frequency is refracted by an ionized layer depends
on the angle at which the wave enters the layer. Figure 2-17 shows three radio waves
of the same frequency entering a layer at different angles. The angle at which (wave
A) strikes the layer is too nearly vertical for the wave to be refracted to Earth. As the
wave enters the layer, it is bent slightly but passes through the layer and is lost. When
the wave is reduced to an angle that is less than vertical (wave B), it strikes the layer
and is refracted back to Earth. The angle made by (wave B) is called the CRITICAL
ANGLE for that particular frequency. Any wave that leaves the antenna at an angle
greater than the critical angle will penetrate the ionospheric layer for that frequency
and then be lost in space. (Wave C) strikes the ionosphere at the smallest angle at
which the wave can be refracted and still return to Earth. At any smaller angle, the
wave will be refracted but will not return to Earth.
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Fundamentals of Antennas and Wave Propagation
Figure 2-17.—Different incident angles of radio waves.
Q20. What factor determines whether a radio wave is reflected or refracted by the
ionosphere?
Q21. There is a maximum frequency at which vertically transmitted radio waves can
be refracted back to Earth. What is this maximum frequency called?
Q22. What three main factors determine the amount of refraction in the ionosphere?
2.9- PROPAGATION PATHS
The path that a refracted wave follows to the receiver depends on the angle at which
the wave strikes the ionosphere. You should remember, however, that the RF energy
radiated by a transmitting antenna spreads out with distance. The energy therefore
strikes the ionosphere at many different angles rather than a single angle.
After the RF energy of a given frequency enters an ionospheric region, the paths that
this energy might follow are many. It may reach the receiving antenna via two or more
paths through a single layer. It may also, reach the receiving antenna over a path
involving more than one layer, by multiple hops between the ionosphere and Earth,
or by any combination of these paths.
Figure 2-20 shows how radio waves may reach a receiver via several paths through
one layer. The various angles at which RF energy strikes the layer are represented by
dark lines and designated as rays 1 through 6.
38
Fundamentals of Antennas and Wave Propagation
Figure 2-18.—Ray paths for a fixed frequency with varying angles of incidence.
When the angle is relatively low with respect to the horizon (ray 1), there is only slight
penetration of the layer and the propagation path is long. When the angle of incidence
is increased (rays 2 and 3), the rays penetrate deeper into the layer but the range of
these rays decreases. When a certain angle is reached (ray 3), the penetration of the
layer and rate of refraction are such that the ray is first returned to Earth at a minimal
distance from the transmitter. Notice, however, that ray 3 still manages to reach the
receiving site on its second refraction (called a hop) from the ionospheric layer.
As the angle is increased still more (rays 4 and 5), the RF energy penetrates the central
area of maximum ionization of the layer. These rays are refracted rather slowly and
are eventually returned to Earth at great distances. As the angle approaches vertical
incidence (ray 6), the ray is not returned at all, but passes on through the layer.
2.9.1- ABSORPTION IN THE IONOSPHERE
Many factors affect a radio wave in its path between the transmitting and receiving
sites. The factor that has the greatest adverse effect on radio waves is ABSORPTION.
Absorption results in the loss of energy of a radio wave and has a pronounced effect
on both the strength of received signals and the ability to communicate over long
distances.
2.9.2- FADING
The most troublesome and frustrating problem in receiving radio signals is variations
in signal strength, most commonly known as FADING. There are several conditions
that can produce fading.
39
Fundamentals of Antennas and Wave Propagation
Fading also results from absorption of the RF energy in the ionosphere. Absorption
fading occurs for a longer period than other types of fading, since absorption takes
place slowly.
Usually, however, fading on ionospheric circuits is mainly a result of multipath
propagation.
2.9.2.1- MULTIPATH FADING
MULTIPATH is simply a term used to describe the multiple paths a radio wave may
follow between transmitter and receiver. Such propagation paths include the ground
wave, ionospheric refraction, reradiating by the ionospheric layers, reflection from the
Earth's surface or from more than one ionospheric layer, etc. Figure 2-19 shows a few
of the paths that a signal can travel between two sites in a typical circuit. One path,
XYZ, is the basic ground wave. Another path, XEA, refracts the wave at the E layer and
passes it on to the receiver at A. Still another path, XFZFA, results from a greater angle
of incidence and two refractions from the F layer. At point Z, the received signal is a
combination of the ground wave and the sky wave. These two signals having traveled
different paths arrive at point Z at different times. Thus, the arriving waves may or
may not be in phase with each other. Radio waves that are received in phase reinforce
each other and produce a stronger signal at the receiving site. Conversely, those that
are received out of phase produce a weak or fading signal. Small alternations in the
transmission path may change the phase relationship of the two signals, causing
periodic fading. This condition occurs at point A. At this point, the double-hop F layer
signal may be in or out of phase with the signal arriving from the E layer.
Figure 2-19.—Multipath transmission.
Multipath fading may be minimized by practices called SPACE DIVERSITY and
FREQUENCY DIVERSITY. In space diversity, two or more receiving antennas are spaced
40
Fundamentals of Antennas and Wave Propagation
some distance apart. Fading does not occur simultaneously at both antennas;
therefore, enough output is usually available from one of the antennas to provide a
useful signal. In frequency diversity, two transmitters and two receivers are used, each
pair tuned to a different frequency, with the same information being transmitted
simultaneously over both frequencies. One of the two receivers will usually provide a
useful signal.
2.9.2.2- SELECTIVE FADING
Fading resulting from multipath propagation is variable with frequency since each
frequency arrives at the receiving point via a different radio path. When a wide band
of frequencies is transmitted simultaneously, each frequency will vary in the amount
of fading. This variation is called SELECTIVE FADING. When selective fading occurs, all
frequencies of the transmitted signal do not retain their original phases and relative
amplitudes. This fading causes severe distortion of the signal and limits the total signal
transmitted.
Q23. What is the skip zone of a radio wave?
Q24. Where does the greatest amount of ionospheric absorption occur in the
ionosphere?
Q25. What is meant by the term "multipath"?
Q26. When a wide band of frequencies is transmitted simultaneously, each frequency
will vary in the amount of fading. What is this variable fading called?
2.10- WEATHER VERSUS PROPAGATION
Weather is an additional factor that affects the propagation of radio waves. In this
section, we will explain how and to what extent the various weather phenomena
affect wave propagation.
Wind, air temperature, and water content of the atmosphere can combine in many
ways.
2.10.1- RAIN
Attenuation because of raindrops is greater than attenuation because of other forms
of precipitation. Attenuation may be caused by absorption, in which the raindrop,
acting as a poor dielectric, absorbs power from the radio wave and dissipates the
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Fundamentals of Antennas and Wave Propagation
power by heat loss or by scattering (fig. 2-24). Raindrops cause greater attenuation by
scattering than by absorption at frequencies above 100 megahertz. At frequencies
above 6 gigahertz, attenuation by raindrop scatter is even greater.
Figure 2-24.—RF energy losses from scattering.
2.10.2- FOG
In the discussion of attenuation, fog may be considered as another form of rain. Since
fog remains suspended in the atmosphere, the attenuation is determined by the
quantity of water per unit volume and by the size of the droplets. Attenuation because
of fog is of minor importance at frequencies lower than 2 gigahertz. However, fog can
cause serious attenuation by absorption, at frequencies above 2 gigahertz.
2.10.3- SNOW
The scattering effect because of snow is difficult to compute because of irregular sizes
and shapes of the flakes. While information on the attenuating effect of snow is
limited, scientists assume that attenuation from snow is less than from rain falling at
an equal rate. This assumption is borne out by the fact that the density of rain is eight
times the density of snow. As a result, rain falling at 1 inch per hour would have more
water per cubic inch than snow falling at the same rate.
Q27. How do raindrops affect radio waves?
Q28. How does fog affect radio waves at frequencies above 2 gigahertz?
42
Fundamentals of Antennas and Wave Propagation
ANSWERS TO QUESTIONS Q1. THROUGH Q28.
A1. Induction field and radiation field.
A2. Induction field.
A3. Radiation field.
A4. Fundamental frequency.
A5. Harmonic frequency or harmonics.
A6. 30 meters.
A7. 5 megahertz.
A8. Vertically polarized.
A9. Direction of wave propagation.
A10. Shifting in the phase relationships of the wave.
A11. Troposphere, stratosphere, and ionosphere.
A12. Stratosphere.
A13. Whether the component of the wave is travelling along the surface or over the
surface of the earth.
A14. Radio horizon is about 1/3 farther.
A15. Sea water.
A16. (a) electrical properties of the terrain (b) frequency (c) polarization of the
antenna
A17. High-energy ultraviolet light waves from the sun.
A18. D, E, F1, and F2 layers.
A19. D layer is 30-55 miles, E layer 55-90 miles, and F layers are 90-240 miles.
A20. Thickness of ionized layer.
A21. Critical frequency.
A22. (a) density of ionization of the layer (b) frequency (c) angle at which it enters the
layer
A23. A zone of silence between the ground wave and sky wave where there is no
reception.
43
Fundamentals of Antennas and Wave Propagation
A24. Where ionization density is greatest.
A25. A term used to describe the multiple pattern a radio wave may follow.
A26. Selective fading.
A27. They can cause attenuation by scattering.
A28. It can cause attenuation by absorption.
44
Fundamentals of Antennas and Wave Propagation
2-1 BASIC ANTENNA PARAMETERS:
A radio antenna may be defined as the structure associated with the region of
transition between a guided wave and a free-space wave, or vice versa. Antennas
convert electrons to photons, or vice versa.
An antenna does not radiate uniformly in all directions. We consider an
antenna called an isotropic radiator having equal radiation in all directions. The
relative distribution of radiated power as a function of direction in space is called the
radiation pattern of the antenna.
The two-wire transmission line in Fig. 2- 1.a is connected to a radio-frequency
generator (or transmitter). Along the uniform part of the line, energy is guided as a
plane Transverse Electro-Magnetic Mode (TEM) wave with little loss. The spacing
between wires is assumed to be a small fraction of a wavelength. Further, on, the
transmission line opens out in a tapered transition. As the separation approaches the
order of a wavelength or more, the wave tends to be radiated so that the opened-out
line acts like an antenna, which launches a free-space wave. The currents on the
transmission line flow out on the antenna and end there, but the fields associated
with them keep on going.
The transmitting antenna in Fig. 2-1.a is a region of transition from a guided
wave on a transmission line to a free-space wave. The receiving antenna (Fig. 2-1.b)
is a region of transition from a space wave to a guided wave on a transmission line.
Thus, an antenna is a transition device, or transducer, between a guided wave and a
free-space wave, or vice-versa. The antenna is a device, which interfaces a circuit and
space.
45
Fundamentals of Antennas and Wave Propagation
Figure 2-1 (a) Radio (or wireless) communication link with transmitting antenna and (b) receiving
antenna. The receiving antenna is remote from the transmitting antenna so that the spherical
wave radiated by the transmitting antenna arrives as an essentially plane wave at the receiving
antenna.
2-1-1 PATTERNS
Figure 2-2 shows a three-dimensional field pattern with pattern radius r (from
origin to pattern boundary at the dot) proportional to the field intensity in the
direction  and . The pattern has its main lobe (maximum radiation) in the z direction
( = 0) with minor lobes (side and back) in other directions.
A typical radiation patter plot is shown in fig (2-2).
Figure 2-2 Three-dimensional field pattern of a directional antenna with maximum radiation in zdirection at = 0o.
46
Fundamentals of Antennas and Wave Propagation
Any field pattern can be presented in three-dimensional spherical coordinates,
as in Fig. 2- 2, or by plane cuts through the main-lobe axis.
The main lobe contains the direction of maximum radiation. However, in some
antennas, more than one major lobe may exist. Lobe other than major lobe are called
minor lobes. Minor lobes can be further represent radiation in the considered
direction and require to be minimized.
Figures 2-3.a and 2-3.b are principal plane field and power patterns in polar
coordinates. The same pattern is presented in Fig. 2-3.c in rectangular coordinates on
a logarithmic, or decibel, scale, which gives the minor lobe levels in more detail. The
angular beamwidth at the half-power level or half-power beamwidth (HPBW) (or 3dB beamwidth (-3dB=10*log(0.5))) and the beamwidth between first nulls (FNBW)
as shown in Fig. 2-3, are important pattern parameters.
Dividing a field component by its maximum value, we obtain a normalized or
relative field pattern, which is a dimensionless number with maximum value of unity.
Thus, the normalized field pattern (Fig. 2-3.a) for the electric field is given by:
The half-power level occurs at those angles  and  for which 𝐸𝜃 (𝜃, 𝜙)𝑛 =
1/√2 = 0.707
At distances that are large compared to the size of the antenna and large
compared to the wavelength, the shape of the field pattern is independent of
distance. Usually the patterns of interest are for this far-field condition.
Patterns may also be expressed in terms of the power per unit area [or Poynting
vector 𝑆(𝜃, 𝜙). Normalizing this power with respect to its maximum value yields a
normalized power pattern as a function of angle, which is a dimensionless number
with a maximum value of unity. Thus, the normalized power pattern (Fig. 2-3.b) is
given by:
47
Fundamentals of Antennas and Wave Propagation
Figure 2-3 Two-dimensional field, power and decibel plots of the 3-D antenna pattern of
Fig. 2-2. Taking a slice through the middle of the 3-dimensional pattern of Figure 2-2 results in the
2-dimensional pattern at (a). It is a field pattern (proportional to the electric field E in V/m) with
normalized relative field En() = 1 at  = 0o. The half-power beam width (HPBW) = 40° is measured
at the E = 0.707 level.
The pattern at (b) is a power plot of (a) (proportional to E2) with relative power Pn = 1 at  = 0o and
with HPBW = 40° as before and measured at the Pn = 0.5 level.
A decibel (dB) plot of (a) is shown at (c) with HPBW = 40° as before and measured at the -3dB
level. The first side lobes are shown at the -9dB and second side lobes at -13dB. Decibel plots are
useful for showing minor lobe levels.
48
Fundamentals of Antennas and Wave Propagation
Although the radiation pattern characteristics of an antenna involve threedimensional vector fields for a full representation, several simple single-valued scalar
quantities can provide the information required for many engineering applications.
These are:
• Half-power beamwidth, HPBW
• Beam area, A
• Beam efficiency, M
49
Fundamentals of Antennas and Wave Propagation
• Directivity D or gain G
• Effective aperture Ae
2-1-2 BEAM AREA (OR BEAM SOLID ANGLE) A
In polar two-dimensional coordinates, an incremental area dA on the surface
of a sphere is the product of the length 𝑟𝑑𝜃 in the  direction (latitude) and 𝑟𝑠𝑖𝑛𝜃𝑑𝜙
in the  direction (longitude), as shown in Fig. 2-4.
Thus,
Figure 2-4 (a) Polar coordinates showing incremental solid angle dA = r2d on the surface of a
sphere of radius r where d= solid angle subtended by the area dA. (b)Antenna power pattern
and its equivalent solid angle or beam area A.
The area of the strip of width r dθ extending around the sphere at a constant angle θ
is given by (2π r sinθ) (r dθ). Integrating this for θ values from 0 to π yields the area of
the sphere. Thus,
50
Fundamentals of Antennas and Wave Propagation
The beam area A is the solid angle through which all of the power radiated by
the antenna would stream if P (θ, Ø) maintained its maximum value over A and was
zero elsewhere. Thus the power radiated= P (θ, Ø)A watts. The beam area of an
antenna can often be described approximately in terms of the angles subtended by
the half-power points of the main lobe in the two principal planes. Thus,
51
Fundamentals of Antennas and Wave Propagation
52
Fundamentals of Antennas and Wave Propagation
2-1-3 DIRECTIVITY AND GAIN:
We have already mentioned that an antenna does not radiate uniformly in all
directions. Directivity function D  ,   describes the variation of the radiation
intensity. The directivity function D  ,   is defined by
D  ,   =
Power radiated per unit solid angle
Average power radiated per unit solid angle
If Pr is the radiated power, the
dPr
gives the amount of power radiated per unit solid
d
angle. Had this power beam uniformly radiated in all directions then average power
radiated per unit solid angle is
Pr
.
4
dPr
dPr
D  ,    d   4 d  ............................. (1.27)
Pr
Pr
4
The maximum of directivity function is called the directivity.
In defining directivity function, total radiated power is taken as the reference.
Another parameter called the gain of an antenna is defined in the similar manner,
which takes into account the total input power, rather than the total radiated power
is used as the reference. The amount of power given as input to the antenna is not
fully radiated.
Pr   Pin …………………………………… (1.28)
where  is the radiation efficiency of the antenna.
The gain of the antenna is defined as
G  ,    4
Radiated power per unit solid angle
input power
G  ,     D  , 
The maximum gain function is termed as gain of the antenna.
Another parameter that incorporates the gain is effective isotropic radiated power or
EIRP, which is defined as the product of the input power and maximum gain or simply
the gain. An antenna with a gain of 100 and input power of 1 W is equally effective as
an antenna having a gain of 50 and input power 2 W.
53
Fundamentals of Antennas and Wave Propagation
2-1-4 RADIATION RESISTANCE:
The radiation resistance of an antenna is defined as the equivalent resistance that
would dissipate the same amount of power as is radiated by the antenna. For the
elementary current element, we have discussed so far. From equation (1.26), we find
that radiated power density
I 0  dl  k02 sin 2 
2
Sav = Pav 
2
32 2 r 2

ar
 Radiated power
P𝑟 = ∫ 𝑆𝑎𝑣 . 𝑟 2 𝑑Ω
𝑠
I 0 dl 2 k02  2
Pr 
sin 2 r 2 sin  d d
2 2


32 r  0  0
2
2
I 0 dl 2 k02 2

d  sin 3 d
2

32
 0
 0
2
I 0  k0 dl 
2
P
12
2
................................... (1.29)
Further,

dPr  Pav .r 2 sin  d d a r

 Pav . a r r 2 d 
I 0  k0 dl  sin 2 
dPr


……………… (1.30)
d
32 2
2
2
From (1.29) and (1.30)
D  ,    1.5sin 2 
Directivity D  D  ,   max , which occurs at  

.
2
If Rr is the radiation resistance of the elementary dipole antenna, then
1 2
I Rr  Pr
2
Substituting Pr from (1.29), we get
54
Fundamentals of Antennas and Wave Propagation
  dl 
Rr  0  2 
6  0 
2
Substituting 𝜂𝑜 = 120𝜋
480 3  dl 
Rr 
 
6  0 
2
2
 dl 
Rr  80   ………………….. (1.32)
 0 
2
For such an elementary dipole antenna the principal E and H plane pattern are shown
in Fig 1.4(a) and (b).
Fig 1.4 (a) Principal E plane pattern of an elementary Dipole. (b) Principal H
plane pattern of an elementary
Dipole.
The bandwidth (3 dB beam width) can be found to be 900 in the E plane.
2-1-5 EFFECTIVE AREA OF AN ANTENNA:
An antenna operating as a receiving antenna extracts power from an incident
electromagnetic wave. The incident wave on a receiving antenna may be assumed to
be a uniform plane wave being intercepted by the antenna. This is illustrated in Fig
1.5.
The incident electric field sets up currents in the antenna and delivers power to
any load connected to the antenna. The induced current also re-radiates fields known
as scattered field. The total electric field outside the antenna will be sum of the
incident and scattered fields and for perfectly conducing antenna the total tangential
electric field component must vanish on the antenna surface.
55
Fundamentals of Antennas and Wave Propagation
Fig 1.5: Plane wave intercepted by an antenna
Let Pinc represents the power density of the incident wave at the location of the
receiving antenna and PL represents the maximum average power delivered to the
load under matched conditions with the receiving antenna properly oriented with
respect to the polarization of the incident wave.
We can write,
PL  Aem Pinc ................................ (1.33)
where Pinc 
E2
and the term Aem is called the maximum effective aperture of the
20
antenna.
Aem is related to the directivity of the antenna D as,
D
4
2
Aem ................................ (1.34)
If the antenna is lossy then some amount of the power intercepted by the antenna
will be dissipated in the antenna.
From eqn. (1.28), we have G   D
Therefore, from (1.34),
G
4

2
 Aem  
4
2
Ae ....................................................(1.35)
Ae   Aem is called the effective aperture of the antenna (in m2).
So effective area or aperture Ae of an antenna is defined as that equivalent area which
when intercepted by the incident power density Pin gives the same amount of
received power PR which is available at the antenna output terminals.
If the antenna has a physical aperture A then aperture efficiency a 
56
Ae
A
Fundamentals of Antennas and Wave Propagation
2-1-6 EFFECTIVE LENGTH/HEIGHT OF THE ANTENNA:
When a receiving antenna intercepts incident electromagnetic waves, a voltage is
induced across the antenna terminals. The effective length he of a receiving antenna
is defined as the ratio of the open circuit terminal voltage to the incident electric field
strength in the direction of antennas polarization.
ℎ𝑒 =
𝑉𝑜𝑐
𝐸
[m] ……………………………….. (1.36)
where Voc = open circuit voltage
E = electric field strength
Effective length he is also referred to as effective height.
2-1-7 ANTENNA EQUIVALENT CIRCUIT:
To a generator feeding a transmitting antenna, the antenna appears as a lead.
In the same manner, the receiver circuitry connected to a receiving antenna's output
terminal will appear as load impedance.
Both transmitting and receiving antennas can be represented by equivalent
circuits as shown by figure 1.6(a) and figure 1.6(b).
Fig 1.6 (a): Equivalent circuit of a Tx antenna
Vg = open circuit voltage of the generator
Zg = generator impedance
Z0 = Characteristics impedance of the transmission line connecting generator to the
antenna.
Pinc = Incident power to the antenna terminal
57
Fundamentals of Antennas and Wave Propagation
Prefl = Power reflected from the antenna terminal.
Pin = Input power to the antenna
XA = Antenna reactance
Rl = Loss resistance of the antenna
Rr = Radiation resistance
Z A   Rl  Rr   jX A  RA  jX A antenna impedance.
Fig 1.6 (b): Equivalent circuit of receiving antenna.
he = effective length
E = incident field strength
Voc = h0 E open circuit voltage
Zload = Input impedance of the receiver.
Re, Rr and XA as defined earlier.
From equation (1.7) to (1.9), we have seen that solution for E and H can be obtained
provided solution of A is unknown for a given J. Further, while computation of
radiated fields for a Hertzian dipole, in equation (1.23a) and (1.23b) we have
neglected the higher order terms of
1
1
and retained only those terms having
n
r
r
variation. In fact, once A is known the radiation field components can be completed
for the far field region as:
Er
0......................................................(7.37a )
E
 j A ............................................(7.37b)
E
 j A .............................................(7.37c)
58
Fundamentals of Antennas and Wave Propagation
Hr
0...............................................................(7.37 d )
H  
E

j
A ...........................................(7.37e)
 
E
j
H    
A ............................................(7.37 f )

 

The relationship stated above equation (1.37a) - (1.37f) may be verified for a Hertzian
dipole using equations (1.22), (1.24a) and (1.24b).
3-1 HALF WAVE DIPOLE ANTENNA:
Let us consider linear antennas of finite length and having negligible diameter. For
such antennas, when fed at the center, a reasonably good approximation of the
current is given by,
Fig 1.7: Current distribution on a center fed dipole antenna

 l

 I 0 sin  k   z '  


 2
I  z '  
 I sin  k  l  z '  

 2
 0

 

0  z'

l
2
l
 z' 0
2
This distribution assumes that the current vanishes at the two end points i.e. z '  l / 2
The plots of current distribution are shown in the figure 1.7 for different 'l'.
For a half wave dipole, i.e. l   / 2 , the current distribution expressed as
I  I 0 cos k0 z '

0
4
 z'
0
4
………………………….. (1.39)
59
Fundamentals of Antennas and Wave Propagation
(b)
(a)
Fig 1.8(a): Half wave dipole, (b): Far field approximation for half wave dipole
From equation (1.21) we can write

dA  a 3
0 I  z ' dz ' e jk R
……………………… (1.40)
4
R
0
From Fig 1.8(b), for the far field calculation, R  r  z 'cos for the phase variation and
R  r for amplitude term.

 dA  a3
0 I  z ' dz ' e jk0R jk0 z 'cos
e
..................... (1.41)
4
r
Substituting I  z '  I 0 cos k0 z ' from (1.39) to (1.41) we get (for lemda/2 only)

dA  a3
0 I 0 e jk rr
cos k0 z ' e jk z 'cos dz ' ............................ (1.42)
4
r
0
0
Therefore, the vector potential for the half wave dipole can be written as:

A = a3

A  a3
0 I 0 e jk r
4 r
0
0 I 0 e
4
r
 jk0 rr
/4


cos k0 z ' e jk0 z 'cos dz '
 /4


2 cos  cos  
2
 ................................ (7.43)
2
k0 sin 
From (1.37b),
 I  e jk0r cos  / 2cos  

E    j    sin  0 0 
2  r
k0 sin 2 

60
Fundamentals of Antennas and Wave Propagation
 jI 0
 j
 k0 e jk r cos  / 2 cos  
2 k0 r
k0 sin 2 
0
 / 2 cos   …………………. (1.44)
I 00 e jk0r
cos
2 r
sin 
Similarly from (1.37c)
E  0 ........................................................................ (1.45)
and from (1.37e) and (1.37f)
H 
jI 0 e jk0r cos  / 2cos  
................................ (1.46)
2 r
sin 
and H  0 …………………………………………. (1.47)
The radiated power can be computed as
2 
Pr 
1
  2 E  H r
2
sin  d d
0 0
2



cos   
2
2   cos 
I 
2
  sin  d d
 0 2 0  
8
sin 

0 0 


 36.565 I 0 ……………………………………. (1.48)
2
Therefore, the radiation resistance of the half wave dipole antenna is 36.565  2  =
73.13 
Further, using Eq(1.27) the directivity function for the dipole antenna can be written
 cos  / 2 cos   
D  ,    1.64 
 …………………. (1.49)
sin 


2
as
Thus, directivity of such dipole antenna is 1.64 as compared to 1.5 for an elementary
dipole. The half power beam width in the E-plane can be found to be 780 as compared
to 900 for a horizon dipole.
3-2 QUARTER WAVE MONOPOLE ANTENNA:
A quarter wave monopole antenna is half of a dipole antenna placed over a grounded
plane. The geometry of such antennas is shown in Fig 1.9(a) and equivalent half wave
dipole is shown in fig 1.9(b).
61
Fundamentals of Antennas and Wave Propagation
Fig 1.9 (a): Quarter wave monopole, (b) Equivalent Half wave dipole
If the ground plane is perfectly conducting, the monopole antenna shown in Fig 1.9(a)
will be equivalent to a half wave dipole shown in Fig 1.9(b) taking image into account.
The radiation pattern above the grounded plane ( in the upper hemisphere) will be
same as that of a half wave dipole, however, the total radiated power will be half of
that of a dipole since the field will be radiated only in the upper hemisphere.
An ideal quarter wave antenna mounted over a perfectly conducting ground plane
has radiation resistance 36.56, half that of a dipole antenna, radiating in free space.
The directivity of such antennas become double of that of dipole antennas.
Quarter wave monopole antennas are often used as vehicle mounted antennas, the
vehicle providing required ground plane for the antenna. For quarter-wave antennas
mounted above earth, the poor conductivity of the soil results in excessive power loss
from the induced amount in the soil.
The effect of poor ground conductivity is taken care of by installing a ground screen
consisting of radial wires extending outward from the antenna base for a distance of
Such arrangement is shown in Fig 1.10.
radial wires of length
buried below grounded
surface
Fig 1.10: Grounded screen for improving performance of monopole antennas operating near earth
surface.
62
Fundamentals of Antennas and Wave Propagation
4- FRIIS TRANSMISSION FORMULA.
The usefulness of the aperture concept will now be illustrated by using it to
derive the important Friis transmission formula published in 1946 by Harald T. Friis
of the Bell Telephone Laboratories.
The Friis Transmission Equation is used to calculate the power received from
one antenna (with gain Gt), when transmitted from another antenna (with gain Gr),
separated by a distance R, and operating at frequency f or wavelength lambda.
Derivation of Friis Transmission Formula
To begin the derivation of the Friis Equation, consider two antennas in free space
(no obstructions nearby) separated by a distance R:
Tx
R
Rx
Pt
Gt
f
Pr
Gr
Figure 4-1. Transmit (Tx) and Receive (Rx) Antennas separated by R.
Assume that 𝑃𝑇 [Watts] of total power are delivered to the transmit antenna. For
the moment, assume that the transmit antenna is omnidirectional, lossless, and that the
receive antenna is in the far field of the transmit antenna. Then the power density S (in
Watts per square meter) of the plane wave incident on the receive antenna a distance R
from the transmit antenna is given by:
𝑃𝑇
𝑆=
4𝜋𝑅 2
If the transmit antenna has an antenna gain in the direction of the receive antenna
given by 𝐺𝑇 , then the power density equation above becomes:
𝑃𝑇
𝑆=
𝐺
4𝜋𝑅2 𝑇
The gain term factors in the directionality and losses of a real antenna.
Assume now that the receive antenna has an effective aperture given by 𝐴𝐸𝑅 .
Then the power received by this antenna (𝑃𝑅 ) is given by:
𝑃𝑇
𝑃𝑅 =
𝐺 𝐴
4𝜋𝑅2 𝑇 𝐸𝑅
Since the effective aperture for any antenna can also be expressed as:
𝜆2
𝐴𝑒 =
𝐺
4𝜋
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Fundamentals of Antennas and Wave Propagation
The resulting received power can be written as:
𝑃𝑇 𝐺𝑇 𝐺𝑅 𝜆2
𝑃𝑅 =
(4𝜋𝑅)2
[𝐄𝐪𝐮𝐚𝐭𝐢𝐨𝐧 𝟏]
This is known as the Friis Transmission Formula. It relates the free space path
loss, antenna gains and wavelength to the received and transmit powers. This is one of
the fundamental equations in antenna theory, and should be remembered (as well as
the derivation above).
Another useful form of the Friis Transmission Equation is given in Equation [2].
Since wavelength and frequency f are related by the speed of light c (see intro to
frequency page), we have the Friis Transmission Formula in terms of frequency:
𝑃𝑇 𝐺𝑇 𝐺𝑅 𝑐 2
𝑃𝑅 =
(4𝜋𝑅𝑓)2
[𝐄𝐪𝐮𝐚𝐭𝐢𝐨𝐧 𝟐]
Equation [2] shows that more power is lost at higher frequencies. This
is a fundamental result of the Friis Transmission Equation.
This means that for antennas with specified gains, the energy transfer
will be highest at lower frequencies. The difference between the power
received and the power transmitted is known as path loss. Said in a different
way, Friis Transmission Equation says that the path loss is higher for higher
frequencies.
The importance of this result from the Friis Transmission Formula
cannot be overstated. This is why mobile phones generally operate at less than
2 GHz. There may be more frequency spectrum available at higher frequencies,
but the associated path loss will not enable quality reception. As a further
consequence of Friss Transmission Equation, suppose you are asked about 60
GHz antennas. Noting that this frequency is very high, you might state that the
path loss will be too high for long range communication - and you are
absolutely correct. At very high frequencies (60 GHz is sometimes referred to
as the mm (millimeter wave) region), the path loss is very high, so only pointto-point communication is possible. This occurs when the receiver and
transmitter are in the same room, and facing each other.
As a further corollary of Friis Transmission Formula, do you think the
mobile phone operators are happy about the new LTE (4G) band that operates
at 700MHz? The answer is yes: this is a lower frequency than antennas
traditionally operate at, but from Equation [2], we note that the path loss will
therefore be lower as well. Hence, they can "cover more ground" with this
frequency spectrum, and a Verizon Wireless executive recently called this
"high quality spectrum", precisely for this reason.
64
Fundamentals of Antennas and Wave Propagation
Side Note: On the other hand, the cell phone makers will have to fit an
antenna with a larger wavelength in a compact device (lower frequency =
larger wavelength), so the antenna designer's job got a little more complicated!
Finally, if the antennas are not polarization matched, the above received power
could be multiplied by the Polarization Loss Factor (PLF) to properly account for this
mismatch. Equation [2] above can be altered to produce a generalized Friis
Transmission Formula, which includes polarization mismatch:
𝑃𝑇 𝐺𝑇 𝐺𝑅 𝑐 2
𝑃𝑅 = (𝑃𝐿𝐹) ∙
(4𝜋𝑅𝑓)2
[𝐄𝐪𝐮𝐚𝐭𝐢𝐨𝐧 𝟑]
The decibel system (and the properties of logarithms) makes the mathematics
involved in some equations much simpler. For instance, recall the Friis Transmission
Formula:
𝑃𝑇 𝐺𝑇 𝐺𝑅 𝜆2
𝑃𝑅 =
(4𝜋𝑅)2
To convert this equation from linear units in Watts to decibels, we take the
logarithm of both sides and multiply by 10:
𝑃𝑇 𝐺𝑇 𝐺𝑅 𝜆2
10 ∙ 𝑙𝑜𝑔10 (𝑃𝑅 ) = 10 ∙ 𝑙𝑜𝑔10 (
)
(4𝜋𝑅)2
A nice property of logarithms is that for two numbers A and B (both positive),
the following result is always true:
𝑙𝑜𝑔10 (𝐴𝐵) = 𝑙𝑜𝑔10 (𝐴) + 𝑙𝑜𝑔10 (𝐵)
Equation (1) then becomes:
10 ∙ 𝑙𝑜𝑔10 (𝑃𝑅 )
= 10 ∙ 𝑙𝑜𝑔10 (𝑃𝑇 ) + 10 ∙ 𝑙𝑜𝑔10 (𝐺𝑇 ) + 10 ∙ 𝑙𝑜𝑔10 (𝐺𝑅 ) + 10
𝜆 2
∙ 𝑙𝑜𝑔10 (
)
4𝜋𝑅
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Fundamentals of Antennas and Wave Propagation
Using the definition of decibels, the above equation becomes a simple addition
equation in dB:
𝜆 2
[𝑃𝑅 ]𝑑𝐵 = [𝑃𝑇 ]𝑑𝐵 + [𝐺𝑇 ]𝑑𝐵 + [𝐺𝑅 ]𝑑𝐵 + [(
) ]
4𝜋𝑅 𝑑𝐵
The above representation is easier to work with, which is kind of cool.
Examples
Ex1:- What is the maximum power received at a distance of 0.5 km over a free-space
1GHz circuit consisting of a transmitting antenna with a 25-dB gain and a receiving
antenna with a 20-dB gain? The gain is with respect to a lossless isotropic source. The
transmitting antenna input is 150W.
Solution:
With respect to a lossless isotropic source and at maximum power D = G
𝑐 3 × 108
𝜆= =
= 0.3 𝑚
𝑓
109
𝐺𝑇 = 25|𝑑𝐵 = 1025/10 = 102.5 = 316
𝐺𝑟 = 20|𝑑𝐵 = 1020/10 = 102 = 100
Pt Gt Gr  2
316  0.32  100
Pr 
 150
 0.0108 W  10.8 mW
(4 r ) 2
(4 ) 2 500 2
(ans.)
Ex2:- Two spacecraft are separated by 100 Mm. Each has an antenna with D = 1000
operating at 2.5 GHz. If craft A's receiver requires 20 dB over 1 pW, what transmitter
power is required on craft B to achieve this signal level?
Solution:
With respect to a lossless isotropic source and at maximum power D = G
  c / f  3 108 / 2.5 109  0.12 m,
Aet  Aer 
D2
4
Pr (required)  100 1012  1010 W
Pt  Pr
16
2
r 2 2
(4 )2 r 2 2
r 2 (4 )2
10 10 (4 )

P

P

10
 10966 W  11 kW (ans.)
r
r
Aet 2
D2 4
D2 2
106 0.122
66
Fundamentals of Antennas and Wave Propagation
4- ARRAY ANTENNA
4-1 INTRODUCTION TO ANTENNA ARRAYS:
An antenna array is an assembly of radiating elements. Radiation pattern of a single
element is relatively wide; each element provides low values of directivity. However,
in many applications we require antennas with very high directive characteristics. The
directive characteristics of the antennas can be improved by increasing the electrical
size of the antenna. One way to increase the dimension of the antenna without
necessarily increasing the size of the individual elements is to form an array of
antenna elements. The total field of the array is determined by the vector addition of
the fields radiated by the individual elements. The elements of the array need not be
identical, but it is often convenient and simpler to design such arrays when the
individual elements are considered to be identical. Therefore, here we will consider
an array with identical elements. In designing arrays, we have several controls such
as geometrical configuration of the overall array, distance between the elements,
excitation (amplitude and phase) and pattern of individual elements.
5-2 ARRAY OF IDENTICAL ELEMENTS:
In this section, we establish the basic methodology for analyzing an array of identical
elements.
Fig. 1.12: A general N-element array.
As shown in fig 1.12, let us consider an array of N identical elements. The
position vector of the ith element is given by ri . The excitation of ith element is given
by:
Ci e ji
where Ci and  i are respectively the relative amplitudes and phases.
67
Fundamentals of Antennas and Wave Propagation
Let the electric field radiated by an element, when placed at the origin and with an
unity excitation is given by
e jk0r
E  r   f  ,  
……………………..(1.54)
4 r

The distance from the ith element to the far field point of interest is Ri  r  a r .ri
for phase variation and Ri  r for amplitude variation.
The total electric field at the point P is given by

N
E  r    Ci e ji f  , 
e
i 1
 f  , 
e
 jk0r N
r


 jk0  r  ar .ri 
Ci e

i 1


r



j i  k0 a r .ri 


………………………. (1.55)
As can be seen from (1.56), the total radiation field is given by the product of the
N
radiation field of the reference element and the term
C e
i 1
N
The term F  ,    Ci e


i 1
j



  i  k0 a r .r  .



j i  k0 ar .ri 

i

……………………….... (1.56)
is called the array factor of the antenna array.
If we consider isotropic elements then f  ,   1 ; hence the radiation pattern
of the array depends only on the array factor F  ,   . Further, it is worth mentioning
here that while discussing the properties of array we are neglecting the effect of
radiation of one element on the source distribution of the other, i.e., we assume that
mutual coupling effect among the elements of the array are neglected. Such effects
are included when very accurate characterization of arrays is required.
4-3 TWO-ELEMENT ARRAY:
In equation (1.57), we derived the expression for the array factor for an Nelement array. To simplify our discussion, let us consider a two-element array.
Further, we consider the elements are to be isotropic point sources. The array
configuration under consideration is shown in Fig. 1.13.
68
Fundamentals of Antennas and Wave Propagation
Fig 1.13: two-element array of isotropic point sources.
For this array, from (1.57) the array factor is given by
F  , 
d 
j1 jk0 2 a X .ar
 C1e e
 F  ,   C1e e
j1
j
 
j2
 C2e e
k0 d
sin cos
2

jk0 d   a X .ar
2

 C2e j2 e
 jk0 d sin cos
2
We now consider some specific cases.
4-3-1 CASE-1:
Point sources have same amplitude and phase.
For this case we consider C1  C2  1 & 1   2  0
 F  ,   e


j  d  sin cos
 

e

 j  d  sin cos

 d

 2cos     sin  cos  
 
Let us plot the array pattern on x-y plane i.e.  


2
. Fig 1.14 (a) – Fig 1.14(d) show
the nature of variation of the array factor as a function of
69
d

.
Fundamentals of Antennas and Wave Propagation
It can be seen that for
d

 0.5 , the maximum radiation take place in a direction
perpendicular to array axis( broad side direction) and no radiation along the axis of
the array (end fix) for
(a)
(c)
d

d

d

 0.5 the radiation increases along the array axis.
 0.25
(b)
 0.50
(d)
d

d

 0.75
 1.00
Fig 1.14: Plot of F  / 2,   vs  for different values of d /  , the elements excited in
the same phase.
4-3-2 CASE 2:
Point sources have equal amplitude and opposite phase.
For this case let C1  C2  1 and 1   / 2
70
 2   / 2
Fundamentals of Antennas and Wave Propagation
F  ,    e

 d
j     sin  cos   
2
 
e

 d 
 j     sin  cos   
2
 

 d
 2 cos   sin  cos    ..................................(7.57)
2
 
Once again, we plot array pattern on the x-y plane, i.e.  

2
. The same is shown in
Fig 1.15(a) to Fig 1.15(d).
It can be seen from Fig 1.14(b) and Fig 1.15(b), that for d   / 2 spacing,
broadside pattern is obtained for elements having same phase while end side pattern
is obtained when the elements are excited in the opposite phase.
(a)
(c)
d

d

 0.25
(b)
 0.50
(d)
d

d

 0.75
 1.00
Fig 1.15: Plot of F  / 2,   vs  for different values of d /  , the elements excited in
different phase.
71
Fundamentals of Antennas and Wave Propagation
4-4 UNIFORM ONE-DIMENSIONAL ARRAY
So far, we have considered the behavior of arrays having only two elements.
Let us now consider a uniform array having N +1 point sources. Each antenna element
is assumed to have same amplitude Ci  I 0 and a progressive phase shift of  d
between two elements where‘d’ is the separation between the elements. Thus, with
reference to the Fig 1.1b, the ith element has a phase i  id .
Fig1.16: Uniform linear array
N
 F  ,    I 0  e
  

j   id  k0id a x . a r 


i 0
N
 I0  e
j  id  k0id cos  
...................................(7.58)
i 0
Where cos   sin  cos 
Using the relation
1  wN 1
wi 
…………………………………….(1.59)

1 w
i 0
N
For a G.P., from (1.61) we can write
F  ,    I 0
1  e
 I 0e
N 1 d  k0 d cos  
1  e
 d  k0 d cos  
N
j  d  k0 d cos  
2
 N  1 

sin 
  d  k0 d cos    …………………….. (1.60)
 2 

  d  k0 d cos   
sin 

2


If we define u  k0 d cos 
and u0   d , then from (1.63) we can write array field pattern F to be
  N  1

sin 
 u  u0  
 2
 ………………………..(1.61)
F U   I 0
sin  u  u0 
The function defined by equation (1.64) is a periodic function whose peak value
occurs at u  u0 and when ever
 u  u0   m is an integer. The peak value is
 N  1 I 0 .
2
72
Fundamentals of Antennas and Wave Propagation
Since cos  lies in the range 1  cos   1 , the corresponding range of u, k0 d  u  k0 d
is the physical space or visible region. The plot of array factors F  u  as a function of
u is shown in Fig 1.17.
As we can see from Fig 1.17, along with the major lobe, in the visible space there are
several smaller maxima. These smaller maxima corresponds to ride lobes.
4-4-1 BROAD SIDE CASE:
If   0 , i.e., all the elements are in the same phase, then the maximum occurs at u =
0 i.e., cos   0 .
i.e.,  

2
. Thus, the maximum radiation occurs broad side to array axis. If we
the y plane for which    / 2 .
cos   sin  cos   0     / 2 . i.e., maximum radiation is along y-axis.
consider
the
pattern
in
Then
4-4-2 END FIRE CASE:
If u0 is chosen to be k0 d , then the beam maximum is formed along u  k0 d , i.e.,
cos   1, i.e.,  =0 maximum of the array pattern is formed along the array axis.
73