# Friis transmission equation

Not to be confused with Friis formulas for noise.

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, $P_r$, to output power to the transmitting antenna, $P_t$, is given by

$\frac{P_r}{P_t} = G_t G_r \left( \frac{\lambda}{4 \pi R} \right)^2$

where $G_t$ and $G_r$ are the antenna gains (with respect to an isotropic radiator) of the transmitting and receiving antennas respectively, $\lambda$ is the wavelength, and $R$ is the distance between the antennas. The inverse of the third factor 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:

$P_r = P_t + G_t + G_r + 20\log_{10}\left( \frac{\lambda}{4 \pi R} \right)$ (Gain has units of dB, and power has units of dBm or dBW)

In addition to the usual derivation from antenna theory, the basic equation also can be derived from principles of radiometry and scalar diffraction in a manner that emphasizes physical understanding.[1]

The simple form applies only under the following ideal conditions:

• $R\gg\lambda$ (reads as $R$ much greater than $\lambda$). If $R<\lambda$, 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 energy.
• 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 have the same polarization.
• 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:

$\frac{P_r}{P_t} = G_t(\theta_t,\phi_t) G_r(\theta_r,\phi_r) \left( \frac{\lambda}{4 \pi R} \right)^2 (1-|\Gamma_t|^2) (1-|\Gamma_r|^2) |\mathbf{a}_t \cdot \mathbf{a}_r^*|^2 e^{-\alpha R}$

where

• $G_t(\theta_t,\phi_t)$ is the gain of the transmit antenna in the direction $(\theta_t,\phi_t)$ in which it "sees" the receive antenna.
• $G_r(\theta_r,\phi_r)$ is the gain of the receive antenna in the direction $(\theta_r,\phi_r)$ 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:

$\frac{P_r}{P_t} \propto G_t G_r \left( \frac{\lambda}{ R} \right)^n$

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.

## Sources of further information

1. H.T.Friis, "Proc. IRE," vol. 34, p. 254. 1946.

2. J.D.Kraus, "Antennas," 2nd Ed., McGraw-Hill, 1988.

3. Kraus and Fleisch, "Electromagnetics," 5th Ed., McGraw-Hill, 1999.

4. D.M.Pozar, "Microwave Engineering." 2nd Ed., Wiley, 1998.

5. J.A. Shaw, "Radiometry and the Friis transmission equation," Am. J. Physics, Vol. 81, No. 33, pp. 33–37, 2013.

## Online references

• Seminar Notes by Laasonen [1]