Friis transmission equation
Encyclopedia
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
.
, to output power to the transmitting antenna, 
, is given by
where
and 
are the antenna gain
s (with respect to an isotropic radiator
) of the transmitting and receiving antennas respectively,
is the wavelength
, and
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 decibel
s, and the wavelength and distance units must be the same. If the gain has units of dB, the equation is slightly modified to:
(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 chamber
s specifically designed to minimize reflections.
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:
where
is experimentally determined, and is typically in the range of 3 to 5, and 
and 
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
.
Antenna (radio)
An antenna is an electrical device which converts electric currents into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver...
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
Harald T. Friis
Harald Trap Friis , who published as H. T. Friis, was a noted Danish-American radio engineer whose work at Bell Laboratories included pioneering contributions to radio propagation, radio astronomy, and radar...
at Bell Labs
Bell Labs
Bell Laboratories is the research and development subsidiary of the French-owned Alcatel-Lucent and previously of the American Telephone & Telegraph Company , half-owned through its Western Electric manufacturing subsidiary.Bell Laboratories operates its...
.
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,, to output power to the transmitting antenna, , is given bywhere
and are the antenna gainAntenna gain
In electromagnetics, an antenna's power gain or simply gain is a key performance figure which combines the antenna's directivity and electrical efficiency. As a transmitting antenna, the figure describes how well the antenna converts input power into radio waves headed in a specified direction...
s (with respect to an isotropic radiator
Isotropic radiator
An isotropic radiator is a theoretical point source of electromagnetic or sound waves which radiates the same intensity of radiation in all directions. It has no preferred direction of radiation. It radiates uniformly in all directions over a sphere centred on the source...
) of the transmitting and receiving antennas respectively,
is the wavelengthWavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
, and
is the distance between the antennas. The inverse of the factor in parentheses is the so-called free-space path lossFree-space path loss
In telecommunication, free-space path loss is the loss in signal strength of an electromagnetic wave that would result from a line-of-sight path through free space , with no obstacles nearby to cause reflection or diffraction...
. To use the equation as written, the antenna gain may not be in units of decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
s, and the wavelength and distance units must be the same. If the gain has units of dB, the equation is slightly modified to:
(Gain has units of dBDecibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
, and power has units of dBm
DBm
dBm is an abbreviation for the power ratio in decibels of the measured power referenced to one milliwatt . It is used in radio, microwave and fiber optic networks as a convenient measure of absolute power because of its capability to express both very large and very small values in a short form...
or dBW)
This simple form applies only under the following ideal conditions:
(reads as 
much greater than 
). If 
, then the equation would give the physically impossible result that the receive power is greater than the transmit power, a violation of the law of conservation of energyConservation of energyThe nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
.- The antennas are in unobstructed free space, with no multipath.
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).
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 chamber
Anechoic chamber
An anechoic chamber is a room designed to stop reflections of either sound or electromagnetic waves.They are also insulated from exterior sources of noise...
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:where
is the gain of the transmit antenna in the direction 
in which it "sees" the receive antenna.
is the gain of the receive antenna in the direction 
in which it "sees" the transmit antenna.
and 
are the reflection coefficientReflection coefficientThe reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...
s of the transmit and receive antennas, respectively
and 
are the polarization vectors of the transmit and receive antennas, respectively, taken in the appropriate directions.
is the absorption coefficient of the intervening medium.
Empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
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:
where
is experimentally determined, and is typically in the range of 3 to 5, and and 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 AreasHata Model for Urban Areas
In wireless communication, the Hata Model for Urban Areas, also known as the Okumura-Hata model for being a developed version of the Okumura Model, is the most widely used radio frequency propagation model for predicting the behaviour of cellular transmissions in built up 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