Free-space_path_loss - Wikiparency

# Free-space path loss **Free-space path loss** (FSPL) is the reduction in a radio signal's power as it travels in a straight line between two antennas through free space, arising purely from the geometric spreading of the wave rather than from any absorption, reflection or obstruction along the way. Between antennas that are ideal isotropic radiators, with no [[Directivity|directivity]] of their own, the loss grows with the square of the distance separating them and, less intuitively, with the square of the frequency as well, so a link at twice the frequency suffers four times the free-space loss at the same range. FSPL is rarely quoted on its own: it is the geometric term inside the [[Friis_transmission_equation|Friis transmission equation]], which restores the antennas' own gains, and from there it is usually the single largest entry in a [[Link_budget|link budget]], the running account of every gain and loss between a transmitter and a receiver that a [[Radio_propagation|propagation]] link must close. It describes a genuine [[Line-of-sight_propagation|line-of-sight]] path through a medium close enough to vacuum that nothing else absorbs or scatters the wave, an assumption closest to exact at [[Microwave|microwave]] frequencies and above, where most terrestrial and every satellite link is engineered around it. One step of that longer accounting, transmitter power descending through free-space path loss to a receiver's sensitivity threshold, is rendered as a waterfall chart by a three.js companion sketch elsewhere on this page. ## Free-space path loss formula Between two ideal isotropic antennas separated by a distance `d` and operating at wavelength `lambda` (equivalently frequency `f = c/lambda`, with `c` the speed of light), free-space path loss is the ratio of transmitted to received power that geometric spreading alone accounts for: `FSPL = (4*pi*d/lambda)^2`, or equivalently `FSPL = (4*pi*d*f/c)^2`. The definition deliberately excludes any gain or inefficiency of the antennas actually used, any absorption by the atmosphere, and any interaction with the ground or with obstacles, isolating the one loss mechanism, the wave simply spreading out, present on every radio path regardless of what else happens to it. Because real antennas are never perfectly isotropic, FSPL is rarely used by itself: it is the geometric factor inside the Friis transmission equation, which multiplies it by the transmitting and receiving antennas' actual gains to give the power a real link delivers, and it enters a link budget, typically summed in decibels as received power equal to transmitted power plus antenna gains minus every loss the path imposes, as the one term that depends only on distance and frequency rather than on any piece of equipment chosen for the link.[^linkbudget] In practice free-space loss is one of several standard loss models a designer chooses between: it applies over a genuine line-of-sight path with nothing in the way, while a two-ray or a knife-edge-diffraction model is substituted instead wherever a strong ground reflection or an obstruction makes the free-space assumption a poor one.[^lossmodels] ## Influence of distance and frequency Both dependences trace back to the same idea seen from opposite ends of the link. On transmit, power spreads outward over the surface of an ever-larger sphere as distance grows, so the power crossing any fixed receiving area falls as `1/d^2` regardless of frequency, ordinary inverse-square spreading. On receive, an isotropic antenna's effective aperture, the area from which it can be said to collect power, is fixed at `lambda^2/(4*pi)` rather than at some physical size a designer chooses, so the very same antenna collects less power from an identical wave as frequency rises and wavelength shrinks: the frequency-squared term is a property of the idealised receiving antenna, not of the medium the wave travelled through. A geostationary satellite link makes the distance dependence concrete without making the loss itself dramatic: at 2 gigahertz, solidly within the [[Microwave|microwave]] band, a ground station at the horizon sees a slant range of about 41,700 kilometres and a free-space loss near 190.9 dB, while the same station looking straight up at the satellite sees a range about 14 percent shorter, around 35,800 kilometres, and a loss lower by only about 1.3 dB.[^fspl-geo] The two numbers together show how forgiving the squared relationship becomes once distance is already very large: only a logarithm of the range actually matters to the loss expressed in decibels, a point the next two sections make precise. ## Derivation The formula follows directly from the two facts used above. An isotropic transmitting antenna radiating power `P_t` spreads it uniformly over the surface of a sphere of radius `d` and area `4*pi*d^2`, so the power density arriving at the receiving antenna's location is `P_t / (4*pi*d^2)`. An isotropic receiving antenna gathers power over its effective aperture `A_e = lambda^2/(4*pi)`, so the power it delivers to a matched load is `P_r = P_t * A_e / (4*pi*d^2) = P_t * lambda^2 / (4*pi*d)^2`. The ratio of transmitted to received power is then `FSPL = P_t/P_r = (4*pi*d/lambda)^2`, matching the definition given above once `lambda` is written as `c/f`. This derivation, built from an isotropic radiator's spreading and an isotropic antenna's effective aperture, is standard antenna theory and needs no further citation; the resulting formula is essentially the geometric core of the transmission formula an American radio engineer published in 1946 to relate a receiving antenna's captured power to a transmitting antenna's radiated power in one simple expression, later generalised into the Friis transmission equation used throughout link-budget work today.[^friis1946] Because the derivation never assumes anything about the medium beyond letting the wave travel in a straight line at speed `c`, the same formula applies unchanged from a few megahertz to well into the optical range, so long as the line-of-sight assumption itself continues to hold. ## Free-space path loss in decibels Because it spans many orders of magnitude, free-space path loss is almost always quoted in decibels: `FSPL_dB = 20*log10(4*pi*d*f/c)`, which for distance in kilometres and frequency in megahertz simplifies to the engineering form `FSPL_dB = 32.44 + 20*log10(d_km) + 20*log10(f_MHz)`.[^fspl-db] The logarithms make the formula's behaviour easy to read off directly: doubling either the distance or the frequency, with the other held fixed, adds `20*log10(2)`, about 6.02 decibels, of loss, and it makes no difference which of the two is doubled, since both enter the formula the same way. The two geostationary-satellite figures given above check against this form exactly: at 2,000 megahertz and the 35,800-kilometre and 41,700-kilometre ranges quoted, the formula returns losses of about 189.5 and 190.9 dB, matching the values already given.[^fspl-db] The same formula scales down comfortably to terrestrial ranges: a 2,400-megahertz Wi-Fi link ten metres, or 0.01 kilometres, from its access point works out to `32.44 + 20*log10(0.01) + 20*log10(2400)`, about 60 dB, small enough that a link budget's other terms, antenna gains, cable losses and the receiver's own sensitivity, usually decide whether the link closes rather than free-space loss alone. ## Microsims This article carries no p5.js sketch of its own. A three.js companion instead renders a complete link budget as a waterfall chart, stepping down from transmitter power through free-space path loss to a receiver's sensitivity threshold, so free-space path loss appears as it does in practice: one term in a longer chain rather than a number considered alone. *Try:* in the [[Radar]] sketch, drag a target outward and watch its echo weaken far faster than the free-space law above predicts for a one-way link: a radar signal crosses the same spreading loss twice, once out and once back, so its echo falls with the fourth power of range rather than the square. *Try:* in the [[Doppler_effect]] sketch, raise the source speed and note that the observed frequency changes while nothing there represents received power falling at all: Doppler shift and free-space path loss are independent consequences of the same relative geometry, one shifting frequency and the other only ever reducing amplitude.  **Microsim — three.js (Wikitube framework), pending deploy:** *Free-space path loss: the link budget as a waterfall* will play here once `https://wikitube-3d-microsims.netlify.app/radio/Free-space_path_loss.html` is live.  *Built from `MICROSIM_GUIDE/specs/sims/Free-space_path_loss.json`; part of the [[PORTAL_Radio|Radio]] set.*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Free-space_path_loss) : [Wikitube](https://en.wikitube.io/wiki/Free-space_path_loss) Skeleton mirrored at revision 1365112863. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Link_budget]] - [[Friis_transmission_equation]] - [[Microwave]] - [[Radio_propagation]] - [[Line-of-sight_propagation]] - [[Directivity]] - [[Multipath_propagation]] - [[Radar]] - [[Doppler_effect]] - [[Telecommunications]] ## References Standard antenna theory used above, including an isotropic radiator's spherical spreading, an isotropic antenna's effective aperture, and the resulting inverse-square and inverse-wavelength-squared dependences, is textbook material and is not separately footnoted here, per the Wikitube style guide's §6.1. Page numbers below are PDF pages of the open edition linked in Further reading. [^fspl-geo]: Nichols, R.; Mumm, H.; Lonstein, W.; Ryan, J.; Carter, C. *Cyber-Human Systems, Space Technologies, and Threats*. 2023, p. 389 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/cyber-human-systems-space-technologies-and-threats . CC BY-NC-SA. [^fspl-db]: Nichols, R.; Mumm, H.; Lonstein, W.; Ryan, J.; Carter, C. *Cyber-Human Systems, Space Technologies, and Threats*. 2023, pp. 396-398 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/cyber-human-systems-space-technologies-and-threats . CC BY-NC-SA. [^linkbudget]: Nichols, R.; Mumm, H.; Lonstein, W.; Ryan, J.; Carter, C. *Cyber-Human Systems, Space Technologies, and Threats*. 2023, p. 396 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/cyber-human-systems-space-technologies-and-threats . CC BY-NC-SA. [^lossmodels]: Nichols, R.; Mumm, H.; Lonstein, W.; Ryan, J.; Carter, C. *Cyber-Human Systems, Space Technologies, and Threats*. 2023, pp. 394-395 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/cyber-human-systems-space-technologies-and-threats . CC BY-NC-SA. [^friis1946]: Friis, H. T. "A Note on a Simple Transmission Formula." Proceedings of the IRE, vol. 34, no. 5, 1946. ## External links This article carries no p5.js or three.js sketch of its own with a published external URL; the interactive sketches referenced above are external-linked from their own articles. ## Further reading - Randall Nichols; Hans Mumm; Wayne Lonstein; Julie Ryan; Candice Carter. *Cyber-Human Systems, Space Technologies, and Threats*. 2023. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/cyber-human-systems-space-technologies-and-threats . CC BY-NC-SA. - Steven Ellingson. *Radio Systems Engineering, Revised First Edition*. 2023. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. - Christian Tiberius; Max Mulder. *Engineering Signal Analysis: From Fourier to filtering: Theory*. 2026. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. - Don Johnson. *Fundamentals of Electrical Engineering I*. 2014. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 . CC BY. - Michael Stiber; Bilin Stiber; Eric Larson. *Signal Computing: Digital Signals in the Software Domain*. 2020. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. - Allen Downey. *Think DSP: Digital Signal Processing in Python*. 2012. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/think-dsp-digital-signal-processing-in-python . CC BY-NC. - John Dyer; Chad Davis. *Measurement and Instrumentation: An Introduction to Concepts and Methods, 1st Edition*. 2020. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA.