# Directivity **Directivity** is a measure of how strongly an antenna or another radiating system concentrates the power it radiates into one direction, defined as the ratio between the radiation intensity in a chosen direction and the radiation intensity averaged over all directions. A hypothetical isotropic radiator, one that spreads its power equally over the whole sphere around it, has a directivity of exactly 1 in every direction by definition; every real antenna does better than this reference in some directions and worse in others, and the peak value in its strongest direction is the single number usually meant when directivity is quoted without qualification. Directivity depends only on the shape of an antenna's [[Radiation_pattern|radiation pattern]], the same directional function described in the neighbouring article on the general-purpose [[Antenna_(radio)|radio antenna]]; it says nothing about how much of the power fed into the antenna is actually radiated rather than lost to heat, which is the additional factor, radiation efficiency, that turns directivity into the closely related quantity gain. Elsewhere on this page, a three.js companion sketch narrows an antenna's beam and reads out its directivity and gain together as the change happens; the neighbouring [[Radar]] and [[Sonar]] articles carry their own sketches showing what a narrow, high-directivity beam buys a working pulsed system in practice. ## Definition Formally, directivity in a given direction is the ratio `D(theta, phi) = U(theta, phi) / U_avg`, where `U(theta, phi)` is the radiation intensity, power per unit solid angle, sent in that direction and `U_avg` is the same quantity averaged over the entire sphere, equal to the total radiated power divided by 4*pi steradians. Because averaging spreads a fixed total power over the whole sphere, no direction can have a radiation intensity below what an isotropic source would produce without some other direction making up the difference; a pattern that is stronger than average somewhere is necessarily weaker than average somewhere else, so directivity always describes a redistribution of a fixed total, never a way to radiate more power overall. Directivity is a dimensionless ratio and can be quoted either as a plain number or, once converted, in decibels relative to an isotropic radiator, the unit dBi covered in its own section below. A [[Dipole_antenna|short dipole]], whose current is close to uniform along its length, has a directivity of only about 1.76 dBi, barely better than isotropic; a half-wave dipole reaches about 2.15 dBi; and a large, well-designed dish or array antenna, whose aperture spans many wavelengths, can exceed 45-50 dBi, a factor of tens of thousands over an isotropic source concentrated into a beam only a fraction of a degree wide. Unless a direction is specified, directivity by itself normally means this peak value, the ratio in the single direction the main lobe points. ## In antenna arrays An array of elements, each contributing its own pattern, can reach a far higher directivity than any single element alone by combining the elements' fields so they add constructively in one direction and destructively in most others, the same [[Beamforming|beamforming]] idea covered in its own article. For a large array of identical, uniformly excited elements spread over an aperture area `A`, directivity grows roughly in proportion to that area measured in square wavelengths, `D ~ 4*pi*A/lambda^2` for a well-illuminated aperture, which is why a large dish or a long phased-array face reaches a far higher directivity than a small one at the same frequency: doubling the linear size in each of two dimensions roughly quadruples the directivity. A parasitic array such as the [[Yagi–Uda_antenna|Yagi–Uda antenna]] reaches a comparable, if more modest, directivity increase without feeding every element directly, by letting a reflector and directors reradiate the driven element's field with a phase set by their own geometry rather than by an active feed network; the trade-off, as that article describes, is a design fixed to a single frequency and a narrower usable bandwidth than a fed [[Antenna_array|array]] of the same size can offer. Tapering the amplitude fed across an array's elements, rather than driving them all equally, trades some of this peak directivity for lower sidelobes, a compromise array designers accept routinely because a slightly lower main-lobe directivity with much weaker sidelobes usually outperforms the untapered maximum in a real, interference-filled environment. ## Relation to beam width Because directivity measures how much a [[Radiation_pattern|pattern]]'s peak exceeds its average, and a narrower main lobe concentrates the same total radiated power into a smaller solid angle, directivity and beam width move in opposite directions: halving the half-power beamwidth in both principal planes roughly quadruples the directivity, for a pattern with modest sidelobes and a single dominant lobe. A widely used approximate formula relates the two directly from the two half-power beamwidths `theta_E` and `theta_H`, measured in degrees in the two orthogonal planes: `D ~ 41253 / (theta_E * theta_H)`, accurate to within a decibel or two for a fairly clean, single-lobed pattern and progressively less accurate as sidelobes carry away a larger share of the total power. The approximation fails outright for a pattern with several comparable lobes, such as a poorly tapered array's pattern with strong grating lobes, since it implicitly assumes nearly all of the radiated power sits inside the single main lobe whose width is being measured; a designer checking a real pattern's directivity against this shortcut is really checking how much of the total power that assumption is missing to sidelobes and a back lobe. ## Expression in decibels Directivity is usually converted to a logarithmic scale for the same reason antenna gain and most other large-dynamic-range radio quantities are: `D_dB = 10 * log10(D)`, turning a range that spans many orders of magnitude, from just above 1 for a nearly isotropic source to tens of thousands for a large dish, into a range of ordinary, easily compared numbers. Expressed this way relative to an isotropic reference, the unit is written dBi, so a short dipole's 1.76-fold directivity becomes 1.76 dBi and a 50,000-fold dish becomes about 47 dBi. A second, older convention instead references a half-wave dipole rather than an isotropic source, written dBd, since a real dipole was historically easier to build and measure against than a hypothetical isotropic radiator;[^cite-dbd-history] because a half-wave dipole itself has a directivity of 2.15 dBi, converting between the two conventions is a fixed offset, `dBi = dBd + 2.15`, and a figure quoted without stating which reference is meant is a common, and sometimes costly, source of confusion in antenna specifications and in comparing one manufacturer's numbers with another's. Because the [[Decibel|decibel]] is a ratio unit, adding decibel figures for directivity, feedline loss and receiver noise directly is exactly what a [[Link_budget|link-budget]] calculation does, including the [[Friis_transmission_equation|Friis transmission equation]] that turns a transmitting and a receiving antenna's gains directly into a predicted received power, one reason directivity is normally carried in decibels rather than as a plain ratio throughout a real design. ## Accounting for polarization A real antenna radiates a specific polarization, and the field arriving from any direction can always be split into two independent, orthogonal polarization components, such as vertical and horizontal, or left- and right-hand circular. Ordinary directivity, as defined above, uses the total radiation intensity summed over both components; the IEEE's standard antenna terminology splits that single number into three related figures describing how the antenna's directive concentration is shared between an intended polarization and an unwanted cross-polarized remainder.[^ieee145-partial] ### Partial directive gain Partial directive gain in a given polarization is the ratio of the radiation intensity in that one polarization component alone, in a chosen direction, to the total radiation intensity, summed over both components, averaged over the whole sphere. Because the denominator still uses the combined total rather than only the one polarization's own share of it, the two partial directive gains in any direction, one per polarization component, add up to exactly the ordinary directivity defined above. ### Partial directivity Partial directivity instead compares a given polarization's own radiation intensity in a direction to that same polarization's own power averaged over the sphere, ignoring the other component's contribution to the total entirely, so a partial directivity describes how concentrated one polarization's radiation is on its own terms, independent of how much total power the antenna happens to radiate in the other, unwanted polarization. ### Partial gain Partial gain follows the same distinction gain always draws from directivity: multiplying a partial directive gain by the antenna's overall radiation efficiency, the fraction of input power actually radiated rather than lost to resistive and dielectric losses, gives the partial gain in that polarization, the figure that most directly predicts how strongly a receiver tuned to only one polarization will actually pick up the transmitted signal. Two orthogonally polarized antennas at the same location can therefore carry two largely independent signals with little coupling between them, a polarization-diversity trick used in some [[MIMO|MIMO]] systems to add capacity without adding physical separation between antennas. ## In other areas The same concentration-ratio idea applies wherever a source radiates or a receiver senses non-uniformly with direction, well beyond radio engineering. In [[Acoustics|acoustics]], a loudspeaker or a microphone has its own directivity, describing how much more sound pressure it produces, or how much more sensitively it picks up sound, along its principal axis compared with an idealized source radiating sound equally in every direction; a large speaker cabinet or a shotgun microphone trades a wide, even coverage pattern for a narrower, more directive one, in exactly the geometric sense this article develops for radio antennas. In optics, a laser's high directivity, concentrating light into an extremely narrow beam rather than spreading it like an ordinary lamp, is the same ratio again, computed from the same radiation-intensity-versus-average definition, only at a wavelength many orders of magnitude shorter than a radio antenna's. In radio astronomy, and in any receiver that must account for noise collected from the sky itself, an antenna's directivity plays a second, quite different role: it is the weighting function that determines how much of the sky's brightness a receiving antenna actually picks up as noise, `T_A = (1/(4*pi)) * integral T_B * D dOmega`, where `T_B` is the sky's brightness temperature in each direction and `D` is exactly the directivity defined above.[^ell-ta-weight] A narrow, highly directive beam pointed away from strong sources such as the Sun collects mostly the faint, nearly uniform cosmic microwave background at about 2.7 kelvin, while the same narrow beam pointed at the Sun collects a brightness temperature that can reach a million kelvin, so the antenna's directivity, chosen originally to maximize a communication link's [[Signal-to-noise_ratio|signal-to-noise ratio]], also sets how much unwanted sky noise the same antenna admits.[^ell-brightness] ## Microsims This article carries no p5.js sketch of its own. A three.js companion elsewhere on this page narrows an antenna's beam step by step and reads out its directivity and gain together as the change happens, so the inverse relationship between beam width and directivity described above can be watched rather than only computed from the approximate formula. *Try:* in the [[Sonar]] sketch, find `DI`, the directivity index, inside the printed sonar equation `SE = SL - 2*TL + TS - (NL - DI) - DT`: raising a receiver's directivity subtracts directly from the effective noise term `(NL - DI)`, the same trade this article expresses as a ratio rather than as a difference of decibels. *Try:* in the [[Radar]] sketch, press space to freeze the rotating beam and compare its narrow wedge with the full circle the same total transmitted power would fill if spread isotropically — the concentration this article turns into the single number `D`. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Directivity) : [Wikitube](https://en.wikitube.io/wiki/Directivity) Skeleton mirrored at revision 1307766447. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Radiation_pattern]] - [[Antenna_(radio)]] - [[Yagi–Uda_antenna]] - [[Friis_transmission_equation]] - [[Beamforming]] - [[Antenna_array]] ## References Standard antenna theory — the radiation-intensity ratio definition of directivity, the aperture–directivity relation for large arrays, and the approximate beamwidth formula — is covered in essentially every antenna-engineering textbook and is not separately footnoted here, per the Wikitube style guide §6.1. Page numbers below are PDF pages of the open edition linked in Further reading. [^cite-dbd-history]: Citation needed: a primary source pinning when and by whom the dBd (gain relative to a half-wave dipole) convention was first formally adopted has not been identified in this pass. [^ieee145-partial]: IEEE Std 145-1993, *IEEE Standard Definitions of Terms for Antennas*. Institute of Electrical and Electronics Engineers, 1993. https://ieeexplore.ieee.org/document/286098 . [^ell-ta-weight]: Ellingson, S. *Radio Systems Engineering*, Revised First Edition. 2023, p. 105 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell-brightness]: Ellingson, S. *Radio Systems Engineering*, Revised First Edition. 2023, pp. 104-105 (PDF pages). 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 <!-- Hubs: Signal_processing. Portals: PORTAL_Radio. Radio portal wave 1 · 2026-09-17 · drafted. -->