Friis transfer equation

The Friis transmission equation (after Harald Friis , who formulated it for the first time in 1946 at Bell Laboratories ) expresses the power received by an antenna in communications technology as a function of a second transmitting antenna set up at a certain distance. The equation applies in empty space ( vacuum ) and, in addition to the free space attenuation, also takes into account the antenna gain of the transmitting and receiving antenna used.

The Friis transfer equation should not be confused with the Friis formula for calculating the noise figure, which was also developed by Harald Friis .

Mathematical formulation

In the simplest form, two antennas are installed at a distance in the otherwise empty space . The transmitting antenna radiates power with the wavelength and has an antenna gain of . The receiving antenna receives the power with an antenna gain . The Friis transfer equation can then be expressed as: ${\displaystyle R}$ ${\displaystyle P_{\text{t}}}$ ${\displaystyle \lambda }$ ${\displaystyle G_{\text{t}}}$ ${\displaystyle G_{\text{r}}}$ ${\displaystyle P_{\text{r}}}$

${\displaystyle {\begin{aligned}{\frac {P_{\text{r}}}{P_{\text{t}}}}&=G_{\text{t}}\cdot G_{\text{r}}\cdot \left({\frac {\lambda }{4\pi R}}\right)^{2}\\&={\frac {G_{\text{t}}\cdot G_{\text{r}}}{D_{\text{f}}}}\end{aligned}}}$

The reciprocal of the expression in brackets is also known as free space attenuation : ${\displaystyle D_{\text{f}}}$

${\displaystyle D_{\text{f}}=\left({\frac {4\pi R}{\lambda }}\right)^{2}}$

In practical applications, the quantities used, such as the antenna gain, are logarithmized and expressed in decibels  (dB). The powers are used dimensionless in  dBm . The equation then takes the form of a sum and represents part of a power transfer budget:

${\displaystyle {\begin{aligned}P_{\text{r dB}}&=P_{\text{t dB}}+G_{\text{t dB}}+G_{\text{r dB}}+20\cdot \log _{10}\left({\frac {\lambda }{4\pi R}}\right)\\&=P_{\text{t dB}}+G_{\text{t dB}}+G_{\text{r dB}}-D_{\text{f dB}}\\&=P_{\text{t dB}}+G_{\text{t dB}}+G_{\text{r dB}}-{\text{FSPL}}\end{aligned}}}$

with the free space damping factor :

${\displaystyle {\begin{aligned}{\text{FSPL}}&=D_{\text{f dB}}\\&=10\cdot \log _{10}(D_{\text{f}})\\&=20\cdot \log _{10}\left({\frac {4\pi R}{\lambda }}\right)\end{aligned}}}$

Boundary conditions and application

The Friis transmission equation is valid only under ideal conditions in the area of the far field with . Furthermore, there must be no multipath propagation , and the space must be free of obstacles that dampen the wave. Losses that are practically always present in the antenna feed lines and connectors are regarded as non-existent. ${\displaystyle R>2\cdot \lambda }$

Since these ideal model conditions cannot be achieved exactly, but only in approximation, the Friis transmission equation is only used as an approximation and for rough calculations in practical applications when dimensioning radio links .

literature

• Constantine A. Balanis: Antenna Theory: Analysis and Design . 3. Edition. John Wiley & Sons, 2012, ISBN 978-0-471-71461-3 .

Individual evidence

1. ↑ Harald Friis: A note on a simple transmission formula . In: Proceedings of the IRE . tape 34 , no. 5 , 1946, pp. 254-256 , doi : 10.1109 / JRPROC.1946.234568 ( online [PDF]). Online ( Memento of the original dated August 12, 2014 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.