Friis transmission equation

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The Friis transmission equation is used in telecommunications engineering, and gives the power received by one antenna under idealized conditions given another antenna some distance away transmitting a known amount of power. The formula was derived in 1945 by Danish-American radio engineer Harald T. Friis at Bell Labs.

Basic form of equation

In its simplest form, the Friis transmission equation is as follows. Given two antennas, the ratio of power available at the input of the receiving antenna, ${\displaystyle P_{r}}$, to output power to the transmitting antenna, ${\displaystyle P_{t}}$, is given by

${\displaystyle {\frac {P_{r}}{P_{t}}}=G_{t}G_{r}\left({\frac {\lambda }{4\pi R}}\right)^{2}}$

where ${\displaystyle G_{t}}$ and ${\displaystyle G_{r}}$ are the antenna gains (with respect to an isotropic radiator) of the transmitting and receiving antennas respectively, ${\displaystyle \lambda }$ is the wavelength, and ${\displaystyle R}$ is the distance between the antennas. The inverse of the factor in parentheses is the so-called free-space path loss. To use the equation as written, the antenna gain may not be in units of decibels, and the wavelength and distance units must be the same. If the gain has units of dB, the equation is slightly modified to:

${\displaystyle P_{r}=P_{t}+G_{t}+G_{r}+20\log _{10}\left({\frac {\lambda }{4\pi R}}\right)}$ (Gain has units of dB, and power has units of dBm or dBW)

This simple form applies only under the following ideal conditions:

The ideal conditions are almost never achieved in ordinary terrestrial communications, due to obstructions, reflections from buildings, and most importantly reflections from the ground. One situation where the equation is reasonably accurate is in satellite communications when there is negligible atmospheric absorption; another situation is in anechoic chambers specifically designed to minimize reflections.

Modifications to the basic equation

The effects of impedance mismatch, misalignment of the antenna pointing and polarization, and absorption can be included by adding additional factors; for example:

${\displaystyle {\frac {P_{r}}{P_{t}}}=G_{t}(\theta _{t},\phi _{t})G_{r}(\theta _{r},\phi _{r})\left({\frac {\lambda }{4\pi R}}\right)^{2}(1-|\Gamma _{t}|^{2})(1-|\Gamma _{r}|^{2})|\mathbf {a} _{t}\cdot \mathbf {a} _{r}^{*}|^{2}e^{-\alpha R}}$

where

Empirical adjustments are also sometimes made to the basic Friis equation. For example, in urban situations where there are strong multipath effects and there is frequently not a clear line-of-sight available, a formula of the following 'general' form can be used to estimate the 'average' ratio of the received to transmitted power:

${\displaystyle {\frac {P_{r}}{P_{t}}}\propto G_{t}G_{r}\left({\frac {\lambda }{R}}\right)^{n}}$

where ${\displaystyle n}$ is experimentally determined, and is typically in the range of 3 to 5, and ${\displaystyle G_{t}}$ and ${\displaystyle G_{r}}$ are taken to be the mean effective gain of the antennas. However, to get useful results further adjustments are usually necessary resulting in much more complex relations, such the Hata Model for Urban Areas.