Antenna Patterns

The most important characteristic of an antenna is its ability
to absorb radio waves incident upon it. This is usually described in
terms of its effective aperture. The effective aperture of an antenna is
defined as

The units are

The effective area is a function of the direction of the incident wave,
because the antenna works better in some directions than in others. Hence

This directional property of the antenna is often described in the form of a

The other common way to specify the directive property of an antenna is the

A typical power pattern is shown in Figure 3.9. The
power pattern has a primary maximum, called the **main lobe** and several
subsidiary maxima, called *side lobes*. The points at which the main
lobe falls to half its central value are called the Half Power points and
the angular distance between these points is called the **Half Power
Beamwidth (HPBW)**. The minima of the power pattern are called **nulls**.
For radio astronomical applications one generally wants the HPBW to be small
(so that the nearby sources are not confused with one another), and the
sidelobes to be low (to minimize stray radiation). From simple diffraction
theory it can be shown that the HPBW of a reflecting telescope is given by

where D is the physical dimension of the telescope. and D must be measured in the same units and is in radians. So the larger the telescope, the better the resolution. For example, the HPBW of a 700 foot telescope at 2380 MHz is about 2 arcmin. This is very poor resolution - an optical telescope ( ), a few inches in diameter has a resolution of a few arc seconds. However, the resolution of single dish radio telescopes, unlike optical telescopes, is not limited by atmospheric turbulence. Figure 3.10 shows the power pattern of the (pre-upgrade) Arecibo telescope at 2380 MHz. Although the telescope is 1000 ft in diameter, only a 700 ft diameter aperture is used at any given time, and the HPBW is about 2 arc min. There are two sidelobe rings, which are not quite azimuthally symmetric.

There are two other patterns which are sometimes used to describe antennas. The first is the directivity . The directivity is defined as:

This is the `transmitting' pattern of the antenna, and from reciprocity should be the same as the recieving power pattern to within a constant factor. We will shortly work out the value of this constant. The other pattern is the gain . The gain is defined as:

The gain is the same as the directivity, except for an efficiency factor. Finally a figure of merit for reflector antennas is the aperture efficiency, . The aperture efficiency is defined as:

(3.4.4) |

where is the geometric cross-sectional area of the main reflector. As we shall prove below, the aperture efficiency is at most 1.0.

Consider observing a sky brightness distribution
with a telescope with a power pattern like that shown in
Figure 3.9. The power available at the antenna terminals is
the integral of the brightness in a given direction times the effective
area in that direction (Figure 3.11).

(3.4.5) |

where the available power is a function of , the direction in which the telescope is pointed. The factor of is to account for the fact that only one polarization is absorbed by the antenna. In two dimensions, the expression for is:

(3.4.6) |

in temperature units, this becomes:

(3.4.7) |

So the antenna temperature is a weighted average of the sky temperature,
the weighting function being the power pattern of the antenna. Only if the
power pattern is a single infinitely sharp spike is the antenna temperature
the same as the sky temperature. For all real telescopes, however, the
antenna temperature is a smoothed version of the sky temperature. Supposing
that you are making a sky survey for sources. Then a large increase in the
antenna temperature could mean either that there is a source in the
main beam, or that a collection of faint sources have combined to give
a large total power. From the statistics of the distribution of sources
in the sky (presuming you know it) and the power pattern, one could compute
the probability of the latter event. This gives a lower limit to the
weakest detectable source, below this limit,(called the
**confusion limit**), one can no longer be confident that increases
in the antenna temperature correspond to a single source in the main beam.
The confusion limit is an important parameter of any given telescope,
it is a function of the frequency and the assumed distribution of sources.

Now consider an antenna terminated in a resistor, with the entire
system being placed in a black box at temperature . After thermal
equilibrium has been reached, the power flowing from the resistor to
the antenna is:

The power flow from the antenna to the resistor is (from equation (3.4.9) and using the fact that the sky temperature is the same everywhere)

In thermal equilibrium the net power flow has to be zero, hence

i.e. the maximum effective aperture is determined by the

so

The max. effective aperture scales like the geometric area of the reflector, as expected. Also from equation 3.4.10

Comparing this with equation (3.4.1) gives the constant that relates the effective area to the directivity

As an application for all these formulae, consider the standard communications problem of sending information from antenna 1 (gain , input power ) to antenna 2 (directivity ), at distance R away. What is the power available at the terminals of antenna 2?

The flux density at antenna 2 is given by:

. i.e., the power falls off like , but is not isotropically distributed. (The gain tells you how collimated the emission from antenna 1 is). The power available at the terminals of antenna 2 is:

substituting for the effective aperture from equation (3.4.12)

This is called the **Friis transmission equation**. In Radar astronomy,
there is a very similar expression for the power available at an antenna
after bouncing off an unresolved target (the **radar range equation**).
The major difference is that the signal has to make a round trip, (and the
target reradiates power falling on it isotropically), so the received power
falls like the fourth power of the distance to the target.