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At frequencies below 30 MHz or so, radio transmitters can normally be received over great distances because certain layers of the ionosphere reflect radio signals with a certain frequency. These reflections normally do not take place at higher frequencies, so the maximum distance that can be covered is, in principle, limited to the visible horizon. How this theoretical distance can be calculated is explained here.

The accompanying figure indicates the various distances required. M is the centre point of the Earth, r is the radius, H is the height at which the antenna is placed, s is the length of the signal path between antenna and horizon and D is the distance across the Earth’s curved surface.

Because in practice H will be much smaller than r, s will be approximately equal to D. The signal path s between antenna and horizon is perpendicular to the radius of the Earth. This means that we can apply Pythagoras’ Theorem to find the relationship between antenna height and distance to the horizon. According to the Theorem:

r^2 + s^2 = (r + H)^2
= r^2 + H^2 + 2rH

Collecting terms results in:    s^2 = H^2 + 2rH

Because H^2 is much smaller than 2rH it can be left out.

So it follows that: s^2 = 2rH    or:   s = √(2rH)

The average radius of the Earth is 6,371 km. √(2r) is therefore about 113. The formula can now be simplified to: s = 113 √(H) where s and H have to be expressed in kilometers.

An example: a VHF FM antenna is positioned at a height of 15 m. the maximum distance at which a line-of-sight connection is possible amounts to: 113 √(0.015) = 13.8 km.

In practice these distances turn out to be larger than those computed using the formula. This has to do with the propagation of electromagnetic fields. It appears that the wave is subject to reflection and does curve a little with the surface of the Earth. This is readily observed with so-called temperature inversion layers. The weather circumstances are such that hundreds of kilometres can be covered without problems using signal frequencies in the VHF
range. But even without these special weather conditions the distances that can be covered appear to be larger than predicted by theory, as already mentioned. With the antenna height of 15 m assumed earlier, the distance that can be covered appears to be of the order of 40 km, instead of the calculated 13.8 km. How the propagation of electromagnetic waves actually works is a complicated matter covered in many excellent books and publications. However, it is known that at frequencies in the GHz range the distance that can be covered becomes progressively smaller as the frequency increases. This is also the reason why parabolic antennas for SHF frequencies are positioned as high as is practicable. The amount of transmitter power plays a secondary role in all this.

What does matter however, is the height of the receiving antenna. The same formula can be used for this antenna (that is s = 113 √(H)). The theoretical total distance that can be covered is then the sum of both distances to the horizon.

Author: Gert Baars

 (Elektor Electronics Magazine – 07-08/2006)

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