- 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 byHarald 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 thewavelength , 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 (seeimpedance 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 andpolarized .
*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 inanechoic chamber s 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 thereflection coefficient s of the transmit and receive antennas, respectively
*mathbf{a}_t and mathbf{a}_r are thepolarization vectors of the transmit and receive antennas, respectively, taken in the appropriate directions.
*alpha is theabsorption coefficient of the intervening medium.Empirical adjustments are also sometimes made to the basic Friis equation. For example, in urban situations where there are strongmultipath 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 theHata_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]
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