## Friis Equation

A basic equation relating to propagation in free space is the Friis equation (Ref.1). This gives the receive power in terms of transmit power, antenna gains, frequency of operation and separation distance. Basically it is saying that a point source of power is spread out over the surface of an ever expanding sphere of radius R. The original equation was written with antenna apertures, but this can be modified for antenna gains (Ref.2). Pt & Pr are the transmit & receive powers, lambda is the wavelength and Gt & Gr are the transmit & receive isotropic antenna gains:

This equation can be simplified so that we can use dBm for power and dB for gain & attenuation.

0dBm = 10*log(PmW/1mW)

Taking 10*log of both sides of the Friis equation, we can solve for the receive power Pr in dBm (Pt in dBm, Gr & Gt in dB):

Consider the distance R in Km (10^3m) and frequency f in MHz (10^6 Hz):

c = speed of light = 3*10^8m/sec

The free space path loss can be represented by:

Note for completeness we have to modify Pt & Pr to take into account the transmit & receive feeder losses Lt & Lr which in some cases can be considerable.

## Example

Let’s look at a practical example. At this point we are only considering Free Space loss as there may be other losses involved. We use a ScicosLab script to solve for receive power. Consider the receive power level of a Marine AIS Class B transmitter at 10 nautical miles. The Tx & Rx antennas have gains of 3dBi and are mounted with an RG-58/U feeder lengths of 20m at both ends.

Pt = 2W = 2000mW = +33dBm

Freq = Ch87B/88B = 161.975/162.025MHz = 162MHz approx.

Gt = Gr = 3dBi

Lt = Lr = 20m RG58/U = 8.1dB/100ft@200MHz = 5.3dB/20m approx.

R = 10 Nautical Miles = 18.52Km

The Receive Level is -73.6dBm with a free space loss of 102dB.

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## References

#1. – “Friis Transmission Equation”.

#2. – “HF High Frequency Radio Telecommunications Learn by Simulation”, EBook

https://www.clarktelecommunications.com/simulation.htm