Friis Transmission Formula

This page is worth reading a couple times and should be fully understood. Consider two antennas in free space (no obstructions nearby) separated by a distance R: Figure 1. Transmit (Tx) and Receive (Rx) Antennas separated by R. Assume that Watts of total power are delivered to the transmit antenna. For the moment, assume that the transmit antenna is omnidirectional, lossless, and that the receive antenna is in the far field of the transmit antenna. Then the power p of the plane wave incident on the receive antenna a distance R from the transmit antenna is given by: If the transmit antenna has a gain in the direction of the receive antenna given by , then the power equation above becomes: The gain term factors in the directionality and losses of a real antenna. Assume now that the receive antenna has an effective aperture given by . Then the power received by this antenna () is given by: Since the effective aperture for any antenna can also be expressed as: The resulting received power can be written as: This is known as the Friis Transmission Formula. It relates the free space path loss, antenna gains and wavelength to the received and transmit powers. This is one of the fundamental equations in antenna theory, and should be remembered (as well as the derivation above). Finally, if the antennas are not polarization matched, the above received power could be multiplied by the Polarization Loss Factor (PLF) to properly account for this mismatch. See also decibel math.