### 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.

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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.