Free-space loss

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In telecommunication, free-space loss is the loss in signal strength (see discussion) that would result if all absorbing, diffracting, obstructing, refracting, scattering, and reflecting influences were sufficiently removed having no effect on its propagation.

As the name implies, free space loss assumes the transmitter and receiver are both located in free space and does not consider other sources of loss such as reflections, cable, connectors etc. A discussion of these losses is included in the article on Link budget. Similarly it does not take account of gains from particular antennas.

Free space power 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 free-space loss is below, where $λ$ is the signal wavelength, f is the signal frequency, d is the distance or radius of the signal from the transmitter, and c is the speed of light (299792.458 km/s) in the signal transmission medium (note that the units used should be consistent, e.g., $\lambda\,\!$ and R in meters, and c in meters per second):

$\mbox{free space loss}=\mbox{FSL}= \left ( \frac{4\pi d}{\lambda} \right )^2 = \left ( \frac{4\pi d f}{c} \right )^2;\,\!$

Source: from Federal Standard 1037C and from MIL-STD-188

A particularly convenient way to express free space loss is in terms of dB. The loss can be expressed as:

$\begin{align}\mbox{FSL(dB)} &=10\log_{10}\left(\left(\frac{4\pi}{c}df\right)^2\right)=20\log_{10}\left(\frac{4\pi}{c}df\right),\\ &=20\log_{10}(d)+20\log_{10}(f)+20\log_{10}\left(\frac{4\pi}{c}\right)+\mbox{K};\end{align}\,\!$

where K is a constant that depends on the units used and details of the radio link.

If d is measured in meters, f in Hz, and the link uses isotropic antennas, the expression becomes:

$\begin{align}\mbox{FSL(dB)} &=20\log_{10}(d)+20\log_{10}(f)+20\log_{10}\left(\frac{4\pi}{c}\right)-120,\\ &\approx 20\log_{10}(d)+20\log_{10}(f)-147.55;\end{align}\,\!$

As an example, the FSL(dB) of a 1000 meter link operating at 1GHz using isotropic antennas is 92.5 dB.

Very useful for fast calculation is expression where d is measures in km and f in MHz (link uses isotropic antennas):

$\begin{align}\mbox{FSL(dB)} &=20\log_{10}(d)+20\log_{10}(f)+20\log_{10}\left(\frac{4\pi}{c}\right)+60,\\ &\approx 20\log_{10}(d)+20\log_{10}(f)+32.45;\end{align}\,\!$

Note: Free-space loss is primarily caused by beam divergence, i.e., signal energy spreading over larger areas at increased distances from the source, and by the inverse square law of electromagnetic radiation.