- Free-space path loss
In

telecommunication ,**free-space path loss**(**FSPL**) is the loss insignal strength of anelectromagnetic wave that would result from a line-of-sight path throughfree space , with no obstacles nearby to causereflection ordiffraction . It does not include factors such as the gain of the antennas used at thetransmitter and receiver, nor any loss associated with hardware imperfections. A discussion of these losses may be found in the article onlink budget .**Free-space path loss formula**Free-space path loss is proportional to the square of the distance between the transmitter and receiver, and also proportional to the square of the

frequency of the radio signal.The equation for FSPL is

:$egin\{align\}\; mbox\{FSPL\}\; =\; left\; (\; frac\{4pi\; d\}\{lambda\}\; ight\; )^2\; \backslash \; =\; left\; (\; frac\{4pi\; d\; f\}\{c\}\; ight\; )^2end\{align\}$

where:

*$lambda$ is the signal wavelength (in metres),

*$f$ is the signal frequency (inhertz ),

*$d$ is the distance from the transmitter (in metres),

*$c$ is the speed of light in a vacuum, 2.99792458 × 10^{8}metres per second .This equation is only accurate in the

far field ; it does not hold close to the transmitter.**Free-space path loss in Wireless Communication**This formula applies to Wireless Communications as shown below.In highways and in rural areas where there are at most 2 multipaths ( the LOS-Line of Sight and/or the ground reflected ), this is part of the

2-ray path loss model As depicted in the figure above, it assumes that the distance between transmitting and receiving antennas 'd' is much greater than the wavelength of the EM ( Electromagnetic ) waves and the wavefronts impinging the receiving antenna are planar and not spherical and this distance is large enough to neglect the curvature of earth.The relation between the transmitted and the received power ( P

_{u}and P_{r}) in Watts is given as ::$egin\{align\}\; P\_\{r\}\; =\; P\_\{u\}left\; (\; frac\{G\_\{l\}lambda\}\{4pi\; d\}\; ight\; )^2end\{align\}$

where:

* G_{l}is the product of transmit and receive antenna radiation patternsDerivation of the formula can be found in [

*C.A. Balanis "Advanced Engineering Electromagnetics", 1989, John Wiley and Sons, Inc.*]**Free-space path loss in decibels**A convenient way to express FSPL is in terms of dB:

:$egin\{align\}mbox\{FSPL(dB)\}\; =\; 10log\_\{10\}left(left(frac\{4pi\}\{c\}df\; ight)^2\; ight)\; \backslash \; =\; 20log\_\{10\}left(frac\{4pi\}\{c\}df\; ight)\; \backslash \; =\; 20log\_\{10\}(d)\; +\; 20log\_\{10\}(f)\; +\; 20log\_\{10\}left(frac\{4pi\}\{c\}\; ight)\; \backslash \; =\; 20log\_\{10\}(d)\; +\; 20log\_\{10\}(f)\; -\; 147.56end\{align\}$

where the units are as before.

For typical radio applications, it is common to find $f$ measured in units of

MHz and $d$ inkm , in which case the FSPL equation becomes:$mbox\{FSPL(dB)\}\; =\; 20log\_\{10\}(d)\; +\; 20log\_\{10\}(f)\; +\; 32.44$

For $d$ in

statute miles , the constant becomes $36.6$ .**Physical explanation**The FSPL expression above often leads to the erroneous belief that free space attenuates an electromagnetic wave according to its frequency. This is not the case, as there is no physical mechanism that could cause this.

The expression for FSPL actually encapsulates two effects. Firstly, the spreading out of electromagnetic energy in free space is determined by the

inverse square law , i.e.:$S\; =\; P\_t\; frac\{1\}\{4\; pi\; d^2\}$

where:

* $S$ is the power per unit area (inwatt s per metre-squared) at distance $d$,

* $P\_t$ is the total power transmitted (in watts).Note that this is not a frequency-dependent effect.The second effect is that of the receiving antenna's aperture, which describes how well an antenna can pick up power from an incoming electromagnetic wave. For an isotropic antenna, this is given by

:$P\_r\; =\; S\; frac\{lambda^2\}\{4\; pi\}$

where $P\_r$ is the received power. Note that this is entirely dependent on wavelength, which is how the frequency-dependent behaviour arises.

The total loss is given by the ratio

:$mathrm\{FSPL\}\; =\; frac\{P\_t\}\{P\_r\}$

which can be found by combining the previous two expressions.

**ee also***

Friis transmission equation

*Link budget **Further Reading***C.A. Balanis, "Antenna Theory", 2003, John Wiley and Sons Inc.

* [*http://wsl.stanford.edu/Publications/Andrea/thesis.pdf Andrea J. Goldsmith, PhD Thesis*]**External references*** [

*http://www.ece.uvic.ca/~peter/35001/ass1a/node1.html Derivation of the dB version of the Path Loss Equation*]

* [*http://people.deas.harvard.edu/~jones/es151/prop_models/propagation.html#fsl Propagation Models*] Chapter on Free Space Propagation

* [*http://www.ictp.trieste.it/~radionet/ghana1998/LINKLOSS/INDEX.HTM Radio propagation primer*] A more in-depth explanation of the relationship between path loss, the inverse square law, and antenna aperture

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