Friis transmission equation

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 tramitting a known amount of power. The formula was derived by Harald T. Friis.

Basic form of Friis Transmission Equation

In its simplest form, the Friis transmission equation is as follows. Given two antennas, the ratio of power received by the receiving antenna, $P\_r$, to power input to the transmitting antenna, $P\_t$, is given by

:$frac\{P\_r\}\{P\_t\}\; =\; G\_t\; G\_r\; left(\; frac\{lambda\}\{4\; pi\; R\}\; ight)^2$

where $G\_t$ and $G\_r$ are the antenna gain of the transmitting and receiving antennas, respectively, $lambda$ is the wavelength, and $R$ is the distance. The antenna gains are with respect to isotropic (and "not" in decibels), and the wavelength and distance units must be the same. This simple form applies only under the following ideal conditions:

*The antennas are in unobstructed free space, with no multipath.
*$P\_r$ is understood to be the available power at the receive antenna terminals. There is loss introduced by both the cable running to the antenna and the connectors. Furthermore, the power at the output of the antenna will only be fully delivered into the transmission line if the antenna and transmission line are conjugate matched (see impedance match).
*$P\_t$ is understood to be the power delivered to the transmit antenna. There is loss introduced by both the cable running to the antenna and the connectors. Furthermore, the power at the input of the antenna will only be fully delivered into freespace if the antenna and transmission line are conjugate matched.
*The antennas are correctly aligned and polarized.
*The bandwidth is narrow enough that a single value for the wavelength can be assumed.

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:

:$frac\{P\_r\}\{P\_t\}\; =\; G\_t(\; heta\_t,phi\_t)\; G\_r(\; heta\_r,phi\_r)\; left(\; frac\{lambda\}\{4\; pi\; R\}\; ight)^2\; (1-|Gamma\_t|^2)(1-|Gamma\_r|^2)\; |mathbf\{a\}\_t\; cdot\; mathbf\{a\}\_r^*|^2\; e^\{-alpha\; R\}$

where

*$G\_t(\; heta\_t,phi\_t)$ is the gain of the transmit antenna in the direction $(\; heta\_t,phi\_t)$ in which it "sees" the receive antenna.
*$G\_r(\; heta\_r,phi\_r)$ is the gain of the receive antenna in the direction $(\; heta\_r,phi\_r)$ in which it "sees" the transmit antenna.
*$Gamma\_t$ and $Gamma\_r$ are the reflection coefficients of the transmit and receive antennas, respectively
*$mathbf\{a\}\_t$ and $mathbf\{a\}\_r$ are the polarization vectors of the transmit and receive antennas, respectively, taken in the appropriate directions.
*$alpha$ is the absorption coefficient of the intervening medium.

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:

:$frac\{P\_r\}\{P\_t\}\; =\; G\_t\; G\_r\; left(\; frac\{lambda\}\{4\; pi\; R\}\; ight)^n$

where $n$ is experimentally determined, and is typically in the range of 3 to 5, and $G\_t$ and $G\_r$ 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.

Printed references

*H.T.Friis, "Proc. IRE", vol. 34, p.254. 1946.
*J.D.Kraus, "Antennas", 2nd Ed., McGraw-Hill, 1988.
*Kraus and Fleisch, "Electromagnetics", 5th Ed., McGraw-Hill, 1999.
*D.M.Pozar,"Microwave Engineering",2nd Ed., Wiley, 1998.

Online references

*Seminar Notes by Laasonen [http://www.cs.helsinki.fi/u/floreen/adhoc/laasonen.pdf]