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We have seen that standard antennas emit more radiation in some directions
than in others. Indeed, it is topologically impossible for an antenna
to emit transverse waves uniformly in all directions
(for the same reason that it is impossible to comb the hair on a sphere
in such a manner
that there is no parting). One of the aims of antenna engineering
is to design antennas which transmit most of their radiation in a
particular direction. By a reciprocity argument, such an antenna, when
used as a receiver, is preferentially sensitive to radiation incident
from the same direction.
The directivity or gain of an antenna is defined as the ratio
of the maximum value of the power radiated per unit solid angle, to
the average power radiated per unit solid angle:

(982) 
Thus, the directivity measures how much more intensely the antenna
radiates in its preferred direction than a mythical ``isotropic
radiator'' would when fed with the same total power. For a Hertzian
dipole the gain is . For a halfwave antenna the gain is
. To achieve a directivity which is significantly greater than
unity, the antenna size needs to be much
larger than the wavelength. This is
usually achieved using a phased array of halfwave or fullwave antennas.
Antennas can be used to receive, as well
as emit, electromagnetic radiation. The incoming wave
induces a voltage in the antenna which can be detected in an electrical
circuit
connected to the antenna. In fact, this process is equivalent to the emission
of electromagnetic waves by the antenna viewed in reverse.
In the theory of electrical circuits, a receiving antenna is represented
as an e.m.f connected
in series with a resistor. The e.m.f.,
, represents
the voltage induced in the antenna by the incoming wave. The resistor,
, represents the power reradiated by the antenna (here,
the real resistance
of the antenna is neglected). Let us represent the detector circuit as a single
load resistor connected in series with the antenna. The question is:
how can we choose so that the maximum power is extracted from the
wave and transmitted to the load resistor? According to Ohm's law:

(983) 
where
is the current induced in the circuit.
The power input to the circuit is

(984) 
The power transferred to the load is

(985) 
The power reradiated by the antenna is

(986) 
Note that
.
The maximum power transfer to the load occurs when

(987) 
Thus, the maximum transfer rate corresponds to

(988) 
In other words, the resistance of the load circuit must match the radiation
resistance of the antenna.
For this optimum case,

(989) 
So, even in the optimum case one half
of the power absorbed by the antenna is immediately
reradiated. If
then more than one half of the
absorbed power is reradiated. Clearly, an antenna which
is receiving electromagnetic radiation is also emitting it.
This is how the BBC catch people who do not pay their television license fee in
England. They have vans which can detect the radiation emitted by
a TV aerial whilst it is in use (they can even tell which channel you are watching!).
For a Hertzian dipole antenna interacting with an incoming wave whose electric
field has an amplitude we expect

(990) 
Here, we have used the fact that the wavelength of the radiation is much
longer
than the length of the antenna, and
that the relevant e.m.f. develops between the
two ends and the centre of the antenna.
We have also assumed that the antenna is
properly aligned (i.e.,
the radiation is incident perpendicular to the axis of the
antenna). The Poynting flux of the incoming wave is

(991) 
whereas the power transferred to a properly matched detector circuit is

(992) 
Consider an idealized antenna in which all
incoming radiation incident on some area is absorbed and then
magically transferred to the detector circuit with no reradiation.
Suppose that the power absorbed from the idealized antenna
matches that absorbed from
the
real antenna. This implies that

(993) 
The quantity is called the effective area
of the antenna; it is
the area of the idealized antenna which absorbs as much net power from the incoming
wave as the actual antenna. Alternatively, is the area of the
incoming wavefront which is captured by the receiving antenna and fed to
its load circuit.
Thus,

(994) 
giving

(995) 
It is clear that the effective area of a Hertzian dipole antenna is of
order the wavelength squared of the incoming radiation.
We can generalize from this analysis of a special case. The
directivity of a Hertzian dipole is . Thus, the effective area of
the isotropic radiator (the mythical reference antenna against which
directivities are measured) is

(996) 
or

(997) 
where
. Here, we have used the formal definition
of the effective area of an antenna: is that area which, when
multiplied by the timeaveraged Poynting flux of the incoming wave,
equals the maximum power received by the antenna (when its orientation
is optimal).
Clearly, the effective area of
an isotropic radiator is the same as the area of a circle whose radius is
the reduced wavelength
.
We can take yet one more step and conclude that the effective area of
any antenna of directivity is

(998) 
Of course, to realize this full capture area the antenna must be
orientated properly.
Let us calculated the coupling or insertion loss of an
antennatoantenna communications link. Suppose that a generator
delivers the power to a transmitting antenna, which is
aimed at a receiving antenna a distance away. The receiving
antenna (properly aimed) then captures and delivers the power
to its load circuit. From the definition of
directivity, the transmitting antenna produces the timeaveraged Poynting flux

(999) 
at the receiving antenna. The received power is

(1000) 
Here, is the gain of the transmitting antenna, and is
the gain of the receiving antenna.
Thus,

(1001) 
where and are the effective areas of the transmitting
and receiving antennas, respectively.
This result is known as the Friis transmission formula. Note that
it depends on the product of the gains of the two antennas.
Thus, a properly aligned communications link has the same insertion
loss operating in either direction.
A thin wire linear antenna might appear to be essentially one dimensional.
However, the concept of an effective area shows that it possesses a
second dimension determined by the wavelength. For instance, for a
halfwave antenna, the gain of which is , the effective area
is

(1002) 
Thus, we can visualize the capture area as a rectangle which
is the physical length of the antenna in one direction, and
approximately one quarter of the wavelength in the other.
Next: Antenna arrays
Up: Radiation and scattering
Previous: Basic antenna theory
Richard Fitzpatrick
20020518