The Inverse Square Law Of RF Signal Strength

A Communicator Reprise: September 2010

The Inverse Square Law when referring to RF signal strength can be expressed from both the transmitted signal and the receive signal end in the following ways;

With an identical antenna at both the transmit site and the receiving station, regardless of whether the antenna is an omni-directional or highly directional antenna, if it took 1 watt of Effective Radiated Power to talk to a station 1 mile/kilometre away, then applying the I.S. Law would mean you need 4 watts for 2 miles, 9 watts for 3 miles, 16 watts for 4 miles, 25 watts for 5 miles... 100 watts for 10 miles/kilometres away.  So if you had 1000 watt ERP output to a no-gain antenna, that would only give you 31.625 miles – line of sight communications.  But that does not take into consideration any propagation, reflections, or the fact that the receiver is generally a lot more sensitive to weak signals… This I.S. Law is saying, for a given amount of signal strength to remain the same at each of the distances above, that is what you’d have to increase the ERP [by distance squared] to accomplish this.

At the receive end, you can look at it this way;  For a given [fixed] ERP at the transmit end, let’s say for example, 100 Watts; The strength of the signal at the receive antenna would be ¼ of that at  2 miles distant [25 watts] , 1/9th  of that at 3 miles [11.1 watts], 1/16th [6.25 watts]  at 4 miles, 1/25th at 5 miles [4 watts], 1/100th at 10 miles [1 watt], 1/1000th at 31.625 miles [100 milliwatts]… ad infinitum…

And that doesn’t take into account, atmospheric and reflective/refractive absorption,  and other losses, such as coax cable losses… this is just in AIR!

So with this in mind, three things come to light.

• The difference between a transceiver with 100 watts output isn’t going to increase the signal strength much more than a 200 watt transceiver and not really that much more for 1000 watts, when you have to consider dollars spent to get the power increase through amplification. Ok 1000 watts and 100 watts will yield a S-unit increase of maybe 10 units… or 10 db over S9… but at what point over S3 [or the noise floor] could you understand the conversation?
• Antenna gain, even just 3db is enough to almost double the signal strength – at both ends. So you’d be better off spending the money on better gain antennas, than power amplifiers.  Consider antennas with 3, 6, 10, or 20dbd with a 100 watt transceiver.  What is your ERP at each of those decibel gain figures?
• It’s absolutely amazing that taking into account this Law, we can hear, under ideal conditions, a 5 watt HF signal half way around the world!