For communications systems to operate reliably a good signal needs to be received. Too weak a signal will cause loss of data and too strong a signal can result in overloading of the receiver or interference to other communications systems. Environmental factors will (normally) reduce the received signal below the level resulting from a simple line of sight computation. Note that this posting only considers propagation in the far field – antennas more than a few wavelengths apart and in an uncluttered environment.

The transmitting isotropic antenna

This is an antenna which radiates equally in all directions, in effect distributing the power over the surface of a sphere with the antenna at the centre. This is an important definition because all antennas are characterised in comparison with an isotropic antenna. A high gain antenna concentrates the power in a preferred direction at the expense of power being radiated in other directions. A useful analogy is squeezing a balloon where it shrinks where it is squeezed and pops out further in other areas.

Antenna gains are usually expressed in dB referenced to an isotropic antenna and these figures are denoted as dBi (dB relative to an isotropic antenna). An example of this is a dipole antenna which shows a gain of 2.2dBi in its preferred radiation directions.

An isotropic antenna is defined as being 100% efficient. This means all the power which is put into the antenna is transmitted onto its coverage area. This is known as radiation efficiency and all antennas are less than 100% efficient. Factors which can impact radiation efficiency include the impedance matching to the signal source (mismatches result in power being reflected to the source) and resistive losses of the surfaces of the antenna.

As the surface area of a sphere is equal to

the power density at a distance r from the antenna can be computed as

where Pt is the transmitted power.

From this it can be seen that the power density reduces by the square of the distance.

The receiving isotropic antenna.

Antennas receive as well as transmit and the amount of power they receive is a function of the area of the signal they intercept. The effective area of an isotropic antenna is defined as

where Λ is the wavelength of the signal being received.

Isotropic antennas receive equally well in all directions and are defined as being 100% efficient. Real world antennas have efficiency losses and can increase their sensitivity in some directions at the expense of others. This figure is exactly the same as the transmit case as antennas are reciprocal in nature.

Path loss

Knowing the power density of a transmitting isotropic antenna and the effective area of a receiving antenna it is possible to work out the power transferred by two antennas.

and the path loss is

As wavelength isn’t normally used we convert to frequency

where c is the speed of light (300,000,000 m/s)

Converting everything to dB and separating the terms we end up with

As antenna gains are expressed in dBi and we can easily add the effect of them to the pathloss computation:

This is a useful result as it allows us to quickly work out the loss between two antennas under optimal conditions. If you only have the natural logs rather than base 10 then you can convert the base using the substitution below:

Because the antennas are receiving signals on the same frequency from two sources there exists a possibility of signal cancellation at certain frequencies. This would result in antennas failing to transmit (and receive) in certain areas as the signals cancelled at specific frequencies. The cancellation would depend on the relative distances from the amplifiers and the velocity factors of the intervening feeders.  The phasing of the signals would change with time and temperature making it near to impossible to predict the behaviour.

The solution was simple. Ensure that the signals at the antennas coming from the two sources are at different levels and therefore cannot completely cancel. This limits the variation in antenna gain across the band to a few dB. In this case a minimum distribution network design difference in power level of 6dB was adopted which results in a gain variation of 2.5dB across the band.

If the RF over fibre amplifiers are matched to within 2dB then this ripple could be about 4dB if the distribution network is set to 6dB difference. Obviously the coverage contour from the antenna would have to be at least at least 4dB above that level required for satisfactory performance to ensure that there are no dead spots.

The table below shows the worst case ripple resulting from the antenna being driven from two sources.