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Fundamentals of Antennas and Radiating systems
Introduction:
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
receiveing radio waves. 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 are also used now a days.
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 in fig 7.1.
Fundamentals 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. At relatively lower frequencies it is in the form of electromagnetic
waves, in the visible domain the emission is in the form of light and at still higher
frequencies it may be in the form of ultra violet or X-ray radiation. 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. Although, theoritically any structure carrying time varying current can
radiate electromagnetic waves, all structures are not equally efficient in doing that. While
in many applications we try to reduce the radiation, when radiation is intended, launching
of waves into space is accomplished with the aid of specially designed structures called
antennas. If the time varying current density established on an antenna structure is
known, the radiated fields can be calculated without great difficulty. A more difficult
problem is determination of current density on an antenna such that the resultant field
will satisfy the required boundary conditions on the antenna surface.
In many practical antenna structures it is often possible to estimate the current
distribution with sufficient accuracy to obtain good approximation of radiated fields.
However, in order to calculate the impedance properties of an antenna, the current
distribution is required to be known with higher accuracy.
As we have mentioned that if the time varying current density
on the antenna is
known, the radiated and fields can be determined. However, it is often advantageous
to compute the radiated fields in an indirect manner, by introducing potential functions.
We illustrate the procedure below.
In a region not containing free charges, for time harmonic case we can write Maxwell's
equations as:
................................(7.1a)
.....................................(7.1b)
................................................(7.1c)
...............................................(7.1d)
From equations (7.1a) and (7.1b)
where
Using
, we can write
.....................................(7.2)
By solving equation (7.2) electric field
can be determined for a specified current
density and can be computed from by using (7.1b). However, as mentioned such
direct computations are often difficult. Simplification can be obtained by introducing
potential functions.
Since
, we can write
.....................................................(7.3)
because
From (7.1) and (7.3)
or,
and A is the vector potential.
Fig 7.1
A curl free vector function can be expressed as the gradient of a scalar function.
Therefore we assume,
............................................(7.4)
From equation (7.3) and (7.1a),
or,
Substituting
from (7.4)
or,
........................(7.5)
So far we have defined
and free to specify divergence of
. If we choose
...................................................................(7.6)
which is known as Lorentz condition, we can simplify equation (7.5) as
........................(7.7)
Equation (7.7) can be solved to determine
and when
is known we can find,
.....................................(7.8)
From equation (7.4) and (7.6) we can write.
........................(7.9)
From equations (7.7) - (7.9) we find that both
potential
and
can be computed when the vector
is known.
Radiated field of an Herzian dipole: In the previous section we have outlined the
procedure for computing the electric magnetic field distribution of a known current
density . In this section we consider the radiation from a short current filament. We
consider an ideal short linear element (the length of the element dl << operating
wavelength) with current considered uniform over its length. More complex antennas can
be considered to be made up of a large number of such differential antennas with proper
magnitude and phase of their current. For current element under consideration, by
continuity, equal and opposite time varying charges must exist on both ends of dl/2 so
that such elements are also called a Herzian dipole.
As shown in the Fig 7.2, the current element is located at the origin and oriented in the zdirection. We consider the time harmonic case where the current varies sinusoidally with
time and I represents the current phasor. For a z-directed current density located in free
space, from equation (7.7) we can write
................................(7.10)
In the source free region the wave equation reduces to
............................(7.11)
Since the current element is infinitesimally small, it can be regarded as a point source so
that away from the source
be written as:
will be a function of r only. Therefore equation(7.11) can
..............(7.12)
Fig 7.2
If we substitute
, then
or,
Therefore, Eqn(7.12) can be written as
or,
...................................(7.13)
Equation (7.13) has solution of the form
and
where C1 and C2 are
constants. Out of these two solutions,
represents a wave solution which represents
an outward travelling wave. Therefore we consider this solution.
Hence,
....................................(7.14)
In the static case,
and
and Eq(7.14) simplifies to
...............................................................(7.15)
For
, eqn(7.10) simplifies to
.......................................................(7.16)
(7.16) is recognized to be as the Poisson's equation and therefore the solution can be
written as,
..................................................(7.17)
Fundamentals of Antennas and Radiating systems
Both (7.15) and (7.17) represents the solutions of the equation (7.10) for k = 0.
From (7.14) and (7.15) we observe that time varying soln (7.14) is obtained by
multiplying static case solution (7.15) by multiplying the factor
manner we can write the time varying solution from (7.17) as,
. In an analogus
.......................................(7.18)
If the current densities were in x or y direction similar expression could be obtained.
Therefore, in general we can write:
........................................(7.19)
If the source is placed at a position (x', y', z') instead of origin, the vector potential at a
point (x, y, z) can be written as
.........(7.20)
where
Returning back to our problem of computation of radiated field for the current element I
dl, we observe that the current I is assumed to be constant over the length of the dipole
and
can be replaced with
can write
. Further, dl being very small,
and hence, we
............................(7.21)
Converting to spherical polar coordinates
............................(7.22)
From (7.22) using (7.8) and (7.9) we obtain
.....................................(7.23a)
and
..................(7.23b)
when the distance is very large compared to , amplitude variations corresponding to
1/r are important and 1/rn terms
can be neglected. Retaining the terms containing 1/r
variations, which constitute the radiated field or the field in the far zone, the electric and
magnetic field components can be written as:
...................................(7.24a)
.......................................(7.24b)
We find that the radiated field has transverse components only and they satisfy the
relation
.....................................................(7.25a)
.......................................................(7.25b)
The Poynting vector of the radiation field
.......(7.26)
is directed radially outward.
The terms varying as 1/r2 and 1/r3 in (7.23a) and (7.23b) constitutes near zone reactive
field for
. These fields do not contribute to the radiated power, rather they represent
stored electric and magnectic energy in space in the vicinity of the antenna and account
for the reactive part of the impedance seen looking into the antenna terminals. Therefore,
in antenna impedance calculation, the near fields are to be taken into account.
Basic Antenna Parameters:
An antenna does not radiate uniformely in all directions. For the sake of a reference, we
consider a hypothetical antenna called an isotropic radiator having equal radiation in all
directions. A directional antenna is one which can rediate or receive electromagnetic
waves more effectively in some directions than in others. The relative distribution of
radiated power as a function of direction in space (i.e., as function of and ) is called
the radiation pattern of the antenna. Instead of 3D surface, it is common practice to show
planar cross section radiation pattern. E-plane and H-plane patterns give two most
important views. The E-plane pattern is a view obtained from a section containing
maximum value of the radiated field and electric field lies in the plane of the section.
Similarly when such a section is taken such that the plane of the section contains H field
and the direction of maximum radiation.
A typical radiation pattern plot is shown in fig(7.3)
Fig 7.3(a) shows a typical radiation pattern plot in polar coordinates and Fig 7.3(b) shows
the same in rectangle coordinates.
Fig 7.3 (a): Typical radiation pattern in Polar coordinates
Fig 7.3 (b): Typical radiation pattern in rectangular coordinates
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 categorized as side lobes and back lobes. Minor lobes
represent radiation in the undesired direction and require to be minimized.
HPBW or half power beamwidth refers to the angular width between the points at which
the radiated power per unit area is one half of the maximum.
Similarly FNBW (First null beam width) refers to the angular width between the first two
nulls as shown in Fig 7.3. By the term beam width we usually refer to 3 dB beamwidth or
HPBW.
Directivity and gain:
We have already mentioned that an antenna does not radiate uniformly in all directions.
Directivity function
describes the variation of the radiation intensity. The
directivity function
is defined as,
=
If Pr is the radiated power, the
gives the amount of power radiated per unit solid
angle. Had this power been uniformly radiated in all directions then average power
radiated per unit solid angle is
.
.............................(7.27)
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.
.......................................................(7.28)
where
is the radiation efficiency of the antenna.
The gain of the antenna is defined as
The maximum gain function is termed as gain of the antenna.
Another parameter which 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.
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 (7.26) we find that radiated
power density
Radiated power
....................................(7.29)
Further,
......................(7.30)
From (7.29) and (7.30)
....................................(7.31)
Directivity
= 1.5 which occurs at
If Rr is the radiation resistance of the elementary dipole antenna, then
Substituting Pr from (7.29) we get
Substituting
..............................................(7.32)
For such an elementary dipole antenna the principal E and H plane pattern are shown in
Fig 7.4(a) and (b).
Fig 7.4 (a): Principal E plane pattern of an elementary dipole
Fig 7.4 (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.
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 7.5. 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.
Let Pinc represents the power density of the incident wave at the location of the receiving
antenna and PL represents the maximum average power deleivered to the load under
matched conditions with the receiving antenna properly oriented with respect to the
polarization of the incident wave.
We can write,
.................................(7.33)
where
and the term Aem is called the maximum effective aperture of the
antenna. Aem is related to the directivity of the antenna D as,
Fig 7.5: Plane wave intercepted by an antenna
If the antenna is lossy then some amount of the power intercepted by the antenna will be
dissipated in the antenna.
From eq(7.28) we find that
Therefore, from (7.34),
....................................................(7.35)
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
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.
.............................................(7.36)
where Voc = open circuit voltage
E = electric field strength
Effective length he is also referred to as effective height.
Antenna Equivalent Circuit:
To a generator feeding a transmitting antenna, the antenna appears as a load. In the same
manner, the receiver circuitry connected to a receiving antenna's output terminal will
apear as a load impedance. Both transmitting and receiving antennas can be represented
by equivalent circuits as shown by figure 7.6(a) and figure 7.6(b).
Fig 7.6 (a) : Equivalent
circuit of a Tx antenna
Vg = open circuit voltage of the generator
Zg = antenna impedance
Z0 = Characteristics impedance of the transmission line connecting generator to the
antenna
Pinc = Incident power to the antenna terminal
Prefl = Power reflected from the antenna terminal
Pin = Input power to the antenna
XA = Antenna reactance
Re = Loss resistance of the antenna
Rr = Radiation resistance
antenna impedance.
Fig 7.6 (b):
Equivalent circuit of receiving antenna
he = effective length
E = incident field strength
Voc = heE open circuit voltage
Zload = Input impedance of the receiver.
Re, Rr and XA as defined earlier.
Computation of far field radiation :
From equation (7.7) to (7.9) we have seen that solution for
and
can be obtained
provided solution of is unknown for a given . Further while computation of radiated
fields for a Herzian dipole, in equation (7.23a) and (7.23b) we have neglected the higher
order terms of
and retained only those terms having variation. In fact, once is
known the radiation field components can be completed for the far field region as:
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,
The relationship stated above equation (7.37a) - (7.37f) may be verified for a Herzian
dipole using equations (7.22), (7.24a) and (7.24b).
Fig 7.7: Current
distribution on a center fed dipole antenna
...(7.38)
This distribution assumes that the current vanishes at the two end points i.e.,
The plot of current distribution are shown in the figure 7.7 for different 'l'.
For a half wave dipole, i.e.,
, the current distribution expressed as
.....................................(7.39)
Fig 7.8(a): Half wave dipole
Fig 7.8(b): Farfield approximation for half wave dipole
.
From equation (7.21) we can write
...............................................(7.40)
From Fig 7.8(b), for the far field calculation,
for amplitude term.
for the phase variation and
.....................(7.41)
Substituting
from (7.39) to (7.41) we get
............................(7.42)
Therefore the vector potential for the halfwave dipole can be written as:
.............................(7.43)
From (7.37b),
..............................(7.44)
Similarly from (7.37c)
.........................................................................(7.45)
and from (7.37e) and (7.37f)
................................(7.46)
and
...............................................................(7.47)
The radiated power can be computed as
.......................................................(7.48)
Therefore the radiation resistance of the half wave dipole antenna is
=
Further, using Eqn(7.27) the directivity function for the dipole antenna can be written as
...........................(7.49)
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 Hertzian dipole.
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 7.9(a) and equivalent half wave
dipole is shown in fig 7.9(b).
Fig 7.9 (a): Quarter wave monopole
(b) Equivalent Half wave dipole
If the ground plane is perfectly conducting, the monopole antenna shown in Fig 7.9(a)
will be equivalent to a half wave dipole shown in Fig 7.9(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 an arrangement is shown in Fig 7.10.
radial wires of length
buried below grounded surface
Fig 7.10: Grounded scree for improving performance of monopole antennas
operating near earth surface
Small Loop Antennas:
Loop antennas may take many different forms such as circle, square, rectangle etc. Loop
antennas are generally classified into two categories viz, electrically small and
electrically large antennas. Electrically small antennas are those whose overall length is
less than one tenth is number of terms in the loop times the circumference of the loop.
Here we shall keep our discussion confined to small loop antennas only.
Small loops are usually not used as transmitting antennas as they have radiation
resistance smaller compared to short dipoles. However many unintentional sources of
radiation such as transformers, inductor, printed circuit boards etc essentially behave as
small loop antennas. A small loop of current is also called a magnetic dipole and its
magnetic dipole moment is equal to the product of the area with the current it carries.
Thus for these types of small current loops, the shape of the loop is not important. For a
given current, it is the area of the loop that determines the magnitude of the radiated
fields.
Fig. 7.11 shows a small current loop of radius
placed on the xy plane with its axis
oriented in the z direction. The loop carries a current I0.
Fig 7.11: Small Current loop
For
, the loop may be treated as a point source. As shown in Fig 7.11, the
elementary current element placed at
has a vector orientation
. For this current element the vector potential by equation (7.21)
................................(7.50)
where
Since
in the far field,
for amplitude variation, we assume
expansion after neglecting
and for phase variation [By Bionomial
w.r.to 1]
{ By approximating
very small as
since
)
...................................................(7.51)
Using (7.37c) and (7.37e), the radiated field components can be written as
.........................................(7.52a)
......................................(7.52b)
where
is the dipole moment of the loop.
is
The field radiated by a small loop antenna is dual of that small dipole antenna, i.e., a
short current filament, the role of electric and magnetic fields are interchanged.
From (7.52) ,
therefore the radiated power
The radiation resistance of a loop antenna can be found be
If the antenna consists of N number of turns; the radiation resistance increases by a factor
of N2.
Small loop antennas are often used as receiving antennas.
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