### 8.1 The Link Gain Equation.

Up until now we've just considered the behaviour of transmitting and receiving antennas in isolation. We can now examine what happens when we use a pair of antennas to send electromagnetic power from place to place.

Consider a transmitter, *TX*, which is radiating a total power, *P*. All of this power will pass through a sphere of radius, *r*, centred on the transmitter. The total surface area of such a sphere will be . As a result, if the *TX* antenna is isotropic (omnidirectional), an area, , laid on the sphere will intercept a power

We can therefore say that using a transmitter antenna whose gain is aimed at the collecting area will cause it to receive a power

From the basic properties of antennas we can say that the gain of a receiving antenna whose effective area is will be

Combining this with expression 8.2 we can get the result

This equation is called the *Link Gain Equation*. It allows us to work out how much power we will receive using a pair of antennas in a given situation. In some books you may see it referred to as the *Friis Formula* named after the first person to work it out. From it we can define the *Link Gain*

Note that — as is often true in engineering — calling this a ‘gain’ is optimistic. Usually, the received power is only a small fraction of that transmitted and . In reality, the above equation is only exactly correct in empty space. A terrestrial communications or signal link will probably look more like the situation illustrated in figure 8.2.

The main factors which affect the link's behaviour are as follows:-
- 1) The radiation must pass through the atmosphere. This may absorb or scatter some of the field, attenuating the level reaching the receiver.
- 2) The Earth is curved and tends to be covered with irregular features. This may cause the direct
*line of sight* to be blocked. It may also provide surfaces which can reflect power into the receiver antenna.
- 3) Some of the transmitted power may initially move in a direction towards space, but be reflected by the
*ionosphere* back towards the receiver.
- 4) Some external sources may radiate unwanted
*noise* power onto the receiver.

The effects of the ground and ionosphere vary a lot from place to place and time to time. For that reason they can only be analysed on a case-by-case basis, so we can indicate their effect by turning the above equality into an approximation. The effects of atmospheric attenuation can be described in terms of an atmospheric *attenuation coefficient*, . We can use this to write a modified form of the link gain equation

The value of the attenuation coefficient depends upon the signal frequency and, at high frequencies, on the weather. For frequencies below about 1 GHz it is small enough that it can often be ignored. Between 1 GHz and 100 GHz it rises and may produce a loss of around 10 dB/km. This value is higher at *line frequencies* (e.g. at 60 GHz, 120 GHz, etc) and can exceed 100 dB/km at some frequencies in a downpour!

There is more detailed information on how atmospheric attenuation depends upon the signal frequency, etc, on page 3 of this section.