Friis_transmission_equation

# Friis transmission equation The **Friis transmission equation** relates the power delivered to a receiving antenna's terminals to the power a transmitting antenna radiates, given the two antennas' gains, their separation and the signal's wavelength, under idealised free-space conditions. It underlies the free-space term in almost every [[Link_budget|link budget]] and is, in effect, the antenna-gain form of [[Free-space_path_loss|free-space path loss]]. A three.js companion sketch renders the same free-space geometry, folding each antenna's gain into the received-power reading it builds toward. The equation was first stated in 1946 by the radio engineer Harald T. Friis, in a form built from antenna effective area rather than gain; the two are equivalent, connected by a fixed relationship involving wavelength, and this article gives both the original and the now more common form before deriving the result from first principles and noting where it stops applying. The distinction matters historically as much as mathematically: Friis's own route into the result, by way of effective area, is closer to how a receiving antenna is naturally described, while the gain form that later became standard is closer to how a transmitting antenna's data sheet actually reports its performance. ## Friis' original formula Harald T. Friis, a Danish-American radio engineer at [[Bell_Labs|Bell Telephone Laboratories]], published this result in 1946 in a paper titled "A Note on a Simple Transmission Formula."[^friis1946] He expressed it not in terms of antenna gain but in terms of effective area: the power delivered to a receiving antenna equals the product of the transmitting and receiving antennas' effective areas, divided by the square of both the path length and the wavelength. Effective area is a property of the antenna alone, the equivalent geometric area from which it appears to draw power out of a passing wave, and it applies equally whether the antenna transmits or receives. Stated this way, the formula makes no explicit mention of gain: gain enters only once effective area is related back to it, a step that later became the more familiar route into the result. Friis wrote the formula for two antennas oriented for maximum response to one another, an idealisation the contemporary form keeps. His paper ran to little more than a page, and its brevity is part of why the result travelled so widely: it reduced a calculation that antenna engineers had previously worked out case by case into a single, general statement that needed only the two antennas' properties and their separation. That the same effective area describes an antenna whether it sends or receives is itself a consequence of reciprocity, the general rule that a passive antenna's behaviour does not depend on which of its two roles it is playing at the time, and it is this rule that lets a single measured or calculated property stand in for both directions of the link. ## Contemporary formula Today the equation is almost always written in terms of gain rather than effective area. The ratio of received power to transmitted power equals the product of the transmitting and receiving [[Antenna_(radio)|antennas]]' gains, multiplied by the square of the wavelength and divided by the square of four times pi times the distance between them. This is the same result as the original, connected to it by the fixed relationship between an antenna's gain and its effective area, and the gain form is the one used throughout modern link-budget practice.[^ellingson-ch7] Written for two antennas of unity gain, the equation reduces exactly to the free-space path-loss formula: at 2.4 GHz, a wavelength of about 12.5 centimetres, and a separation of 1 kilometre, it predicts a received-to-transmitted power ratio of roughly one part in ten billion, or about −100 dB. Raising either antenna's gain by a given number of decibels raises the received power by exactly that many decibels, which is why gain is normally the first quantity a link designer adjusts once the geometry of a path is fixed. Because every term in the gain form multiplies or divides the others, the whole equation converts cleanly into a sum of decibel quantities, which is the form a [[Radio_propagation|propagation]] engineer actually carries into a link budget rather than the multiplicative form given above. ## Derivation The result follows from two ideas, each simple alone. An antenna radiating a given power with no directional preference at all, an idealisation called an isotropic radiator, spreads that power evenly over the surface of an expanding sphere, so the power crossing each unit of area at a given distance is the transmitted power divided by the area of a sphere of that radius. A real transmitting antenna concentrates the same total power into a favoured direction instead of spreading it evenly, and its gain is defined precisely as the factor by which the power crossing a unit area in that direction exceeds the isotropic value; multiplying the isotropic figure by the transmitting antenna's gain gives the power density actually arriving at the receiving antenna's location. A receiving antenna, in turn, does not capture every bit of the power density incident on it: it captures only the fraction that its effective area intercepts, so the power delivered to its terminals is that density multiplied by its own effective area. The one remaining step is to express effective area in terms of gain rather than area outright, using the fixed relationship between the two that the reciprocity of every antenna's transmitting and receiving behaviour forces. Substituting it turns the receiving antenna's effective area into an expression built from its own gain and the wavelength, and combining every step above gives the contemporary formula. Nothing in the chain assumes anything about either antenna beyond its gain in the other's direction; the rest follows from the geometry of a wave spreading through empty space. ## Limitation of Applicability The equation assumes conditions a real link only approximates. Both antennas must be far enough apart to be in each other's far field, beyond roughly twice the square of the larger antenna's largest dimension divided by the wavelength; closer than that, gain is not yet a single well-defined number and the formula does not apply. The path between them must be free space in the literal sense: no ground reflection, no atmospheric absorption, no obstruction, and no other wave arriving by a second route to interfere with the direct one, since any such addition turns a clean two-antenna calculation into the more complicated business of multipath propagation. The antennas must also be polarisation-matched and each correctly matched to its own feedline, since the equation accounts for neither a polarisation mismatch nor a reflection at an impedance discontinuity; both appear as separate loss terms once the ideal result is put to work in a real [[Link_budget|link budget]]. Harald Friis's name also attaches to a second, unrelated result: a formula for the noise figure of a chain of amplifying stages, which he published two years earlier, in 1944. The two are occasionally confused because both belong to the vocabulary of link engineering, but the earlier formula concerns receiver noise, not free-space propagation, and has no bearing on the transmission equation at all.[^friis1944][^ellingson-cascade] In practice, none of these limitations makes the equation useless outside a laboratory; they simply mark where a designer must add a correction of its own, which is exactly the role the equation's result plays as the free-space term inside a larger link budget. ## Microsims A three.js companion sketch builds on the same free-space geometry as the [[Free-space_path_loss]] visualisation, adding each antenna's gain to its received-power reading in line with the contemporary formula above; it is described on its own rather than here, since this article's own text carries no sketch of its own. The same free-space geometry, minus the antenna gains this article adds to it, already lives in two related sketches carried by neighbouring articles, and each is worth a visit for what it shows about a signal spreading through space. *Try:* in the [[Radar]] sketch, compare its echo, which falls with the fourth power of range because the signal crosses the same free-space geometry twice, out and back, with the inverse-square falloff this equation predicts for a single one-way hop between two separate antennas. *Try:* in the [[Doppler_effect]] sketch, set the source moving and watch the received frequency shift as the geometry between source and observer changes from one instant to the next, a complication this equation assumes away entirely by holding the transmitter, the receiver and the path between them fixed for the whole calculation. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Friis_transmission_equation) : [Wikitube](https://en.wikitube.io/wiki/Friis_transmission_equation) Skeleton mirrored at revision 1285892231. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Free-space_path_loss]] - [[Link_budget]] - [[Directivity]] - [[Antenna_(radio)]] - [[Radiation_pattern]] - [[Radio_propagation]] ## References The spreading, gain and effective-area relationships used in the derivation above are standard antenna-theory results and are not separately footnoted, per Wikitube style guide §6.1. Page numbers below are PDF pages of the open editions. [^friis1946]: Friis, H. T. "A Note on a Simple Transmission Formula." Proceedings of the I.R.E., vol. 34, no. 5, May 1946, pp. 254-256. [^friis1944]: Friis, H. T. "Noise Figures of Radio Receivers." Proceedings of the I.R.E., vol. 32, no. 7, July 1944, pp. 419-422. [^ellingson-cascade]: Ellingson, S. *Radio Systems Engineering - Revised First Edition*. 2023, pp. 102-103 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ellingson-ch7]: Ellingson, S. *Radio Systems Engineering - Revised First Edition*. 2023, Ch. 7 (link budgets). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. ## Further reading - Steven Ellingson. *Radio Systems Engineering - Revised First Edition*. 2023. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. ## External links No independent external links accompany this article; its three.js companion sketch is embedded in the Microsims section above.