THE RETARDED POTENTIALS
Non homogeneous Wave Equations
Maxwell's equations in complete generality are
In our development we w
ill use the following vector iden

tities
where A and V can be any functions but in particular will be the magnetic vector
potential and electric scalar potential respectively. Because in (3) the magnetic fi
eld has no
divergence, the identity in (6) allows us to again define the vector potential A as we had for
quasi

statics in Section 5

4:
Radiation from an infinitesimal dipole (current element).
Definition:
The infinitesimal
dipole is a dipole whose length
dl
is much smaller than the wavelength
of the excited wave, i.e.
dl
<< λ
(
dl
<
λ
/50). The infinitesimal dipole is equivalent to a current
element
Idl
,
A current element is best illustrated by a very short (
compared to
λ
) piece of infinitesimally thin wire
with current
I
. Since the current element is very short, the current is assumed constant along
dl
. The
ideal current element is practically unrealizable, but a very good approximation of it is the short top

hat antenna. To realize a uniform current distribution along the wire, capacitive plates are used to
provide enough charge storage at the end of the wire, so that current is not zero there.
The magnetic vector potential (VP)
A
due to a
linear
source is
Equation (3.2) gives the field due to an electric current element (infinitesimal dipole) expressed via
the magnetic VP
A
.
The field radiated by any complex
antenna in linear medium can be
represented as
a superposition of the fields
due to the current elements on the antenna surface
.
The
A
vector will now be represented with its spherical components. In antenna theory, the preferred
coordinate system is the spherical one. This is mostly because the
far f
ield
radiation is of most
significant interest, i.e. the field is analyzed so very far from the source, that it is assumed to
propagate only radially away from the source. The transformation from rectangular to spherical
coordinates is given by:
Note:
1) Equations (3.7) and (3.9) show that the EM field generated by the, current element is rather
complicated unlike the VP
A
. The advantage of using the VP instead of the field vectors is obvious
even in this simplest e
xample.
2) The field vectors contain terms, which depend on the distance from the source as (1/
r
), (1/
r
2
) and
(1/
r
3
), and, therefore, some of them can be neglected at large distances from the dipole.
3) The longitudinal ( r
) components of the field vector
s decrease fast as the field propagates away
from the source (as 1/
r
2
and 1/
r
3
only).
4) The non

zero transverse field components,
E
and
H
, are orthogonal to each other, and they have
terms, which depend on the distance as 1/
r
. These terms differ by a fa
ctor of
. They represent the
so

called
far
field
. The concept of far field will be re

visited later, when the radiation zones are
defined.
Antenna effective area
In telecommunications, antenna effective area or effective aperture is the functionally
equivalent area from which an antenna directed toward the source of the received signal gathers or
absorbs the energy of an incident electromagnetic wave.
Note 1: Antenna effective area is usually expressed in square meters.
Note 2: In the case of paraboli
c and horn

parabolic antennas, the antenna effective area is about 0.35
to 0.55 of the geometric area of the antenna aperture.
A
eff
= p
0
/P
Where
P
0
is the power absorbed by the antenna in watts, and
P
is the power density incident on the
antenna in watt
s per square meter. It is assumed that the antenna is terminated with a matched load to
absorb the maximum power
Relationship to antenna gain
The effective area is related to the antenna gain by
A
eff
= (λ
2
/4
π
)G
where G is the antenna gain (not in decibels
) and λ is the wavelength. Note that G is the antenna gain
with respect to the isotropic radiator. This formula can be derived as a consequence of
electromagnetic reciprocity which relates the transmit properties of an antenna to the receiving
properties.
It may not hold if the antenna is made with certain non

reciprocal materials. Like the
antenna gain, the effective area varies with direction. If no direction is specified, the direction that
produces the maximum gain and maximum effective area is assumed.
Relationship to physical area
Simply increasing the size of antenna does not guarantee an increase in effective area;
however, other factors being equal, antennas with higher maximum effective area are generally
larger.
In the case of wire antennas, the
re is no simple relationship between physical area and effective area.
In the case of aperture antennas (for example, horns and parabolic reflectors) considered in their
direction of maximum radiation, the aperture efficiency is the ratio of effective area
to physical area
A
eff
=
e
ap
A
phys
where eap is the aperture efficiency, Aphys is the physical size of the aperture, and Aeff is the
effective aperture.
Note 2 in the definition section above, derived from the Federal Standard, implies that the aperture
ef
ficiency is 0.35 to 0.55, which is true for simple designs. However, carefully designed and
constructed reflector antennas can easily have efficiencies in the 0.65 to 0.75 range; and values as
high as 0.85 have been reported in the literature. Very high ap
erture efficiency is not always
desirable, since such antennas tend to have high side lobe levels.
Factors limiting the aperture efficiency are non uniform illumination of the aperture, phase variations
of the aperture field (typically due to surface erro
rs in a reflector and high flare angle in horns), and
scattering from obstructions. The incident wavefront may also not be completely phase coherent due
to variations in the propagating medium; this results in an increase in the effective area of an antenn
a
not resulting in a commensurate increase in signal power, an effect known as 'aperture loss'
Radiated Power
The time average power density, which is the average Poynting vector, can be written in the
following form:
The 1 / 2 in (1

9) appears
because the electric and magnetic ﬁelds represent the peak values.
The radiated power Pr , through a closed surface S surrounding the antenna, can be obtained by
integrating the normal component of the average Poynting vector over the entire surface. On th
e
basis of the above deﬁnition we have
The surrounding surface can be of arbitrary shape, and without losing generality a sphere is used.
Radiation Intensity
Radiation intensity in a given direction is the power radiated per unit solid angle in the
above

mentioned direction. It is expressed in watts per unit solid angle and is related to the far ﬁeld of the
antenna. In a spherical coordinate system (r, θ, φ), radiation intensity is deﬁned as U
where r denotes the distance between the antenna
and the observation point and ˆ r is the
corresponding radial unit vector.
U(θ,φ) can be expressed only by the electric or the magnetic far

zone ﬁeld:
where η is the intrinsic impedance of the medium. (For the free space η = 120π Ohms).
The total powe
r Pr radiated is obtained by taking the integral of the radiation intensity over all angles
around the antenna. Thus, Pr is
Another parameter related to U(θ,φ) is the average radiation intensity, deﬁned as the radiation
intensity of an isotropic s
ource radiating the same power as that of the actual antenna. Thus,
where
is the element of solid angle. It is obvious that an isotropic source (antenna) is not realizable;
however, it is often used as a reference element for many antennas.
Directivity
The directivity of an antenna in a given direction is deﬁned as the ratio of the radia

tion intensity in
the above

mentioned direction to the average radiation intensity. If the direction is not speciﬁed, then
it is implied in the direction of
maximum radiation. By using (1

12) and (1

13), directivity D is
Antenna Efﬁciency
It is well known that only one part of the input power at the input terminals of an antenna is radiated.
Various reasons, like the mismatch between the transmissi
on line and the antenna or conduction and
dielectric losses of the antenna itself, reduce the power radiated. The total efﬁciency of an antenna
can be expressed by
Gain
The performance of an antenna can also be described by the gain. The gain is
related to directivity. It
is an index that takes the directional properties and the efﬁciency of the antenna into account. The
gain G(θ, φ) is deﬁned by
When the direction is not speciﬁed, the gain is taken in the direction of maximum radiation.
Expr
ession (1

21) counts the losses of the antenna itself and can be written as
Antenna Patterns
One of the main characteristics of an antenna is its radiation pattern. It presents graphically the
radiation properties and can be measured by moving a p
robe antenna around the antenna under test
at a constant distance in the far ﬁeld (see Fig. 1.2a). The response, as a function of the angular
coordinates, constitutes the radiation pattern. Depending on probe type and orientation, the
appropriate componen
t of an electric or a magnetic ﬁeld can be measured. If a probe is moved over
the spherical surface, its terminal voltage will present the 3

D radiation pattern. A pattern taken on
one plane is known as a ‘plane pattern’. The pattern that contains the elec
tric ﬁeld vector is the E

plane pattern while the pattern that contains the magnetic ﬁeld vector is the H

plane. The above two
are referred to as the ‘principal plane patterns’. As an example, Fig. 1.2b presents the 3

D radiation
pattern of a uniform linea
r array of three collinear vertical electric dipoles with equal mutual
distance of 0.75λ. Figures 1.2c and d show the E

and H

plane patterns of the
array.
A plane pattern can be depicted as a polar or as a rectangular plot. The units of the patterns ar
e either
linear (Fig. 1.3a) or logarithmic (dB) (Fig. 1.3b). The lobe in the direction of maximum radiation is
called the ‘main lobe’ or the ‘main beam’. Any lobe of the pattern other than the main lobe is a
‘minor lobe’. Usually, a minor lobe is also cal
led a ‘side lobe’. A side lobe in a pattern is, in general,
any lobe other than that of the intended one. Since the intended lobe is usually the main lobe, it is
obvious that minor and side lobes are the same.
A measure of the characteris
tics of the pattern is given by certain quantities:
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