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In electromagnetics, 'directivity' is a figure of merit for an antenna. It measures the power density an actual antenna radiates in the direction of its strongest emission, relative to the power density radiated by an ideal isotropic radiator antenna radiating the same amount of total power. Mathematically, the directivity is defined as the maximum of the directive gain:
:
ight)}{mbox{Total radiated power}/left(4pi
ight)}
ight)
where

★ $heta$ and $phi$ are the standard spherical coordinates angles

★ Radiated power density is the power per unit solid angle such that $mbox\{Total\; radiated\; power\}=int\_\{phi=0\}^\{phi=2pi\}left(int\_\{\; heta=0\}^\{\; heta=pi\}mbox\{Radiated\; power\; density\}left(\; heta,phi$
ight)sin heta,d heta
ight)dphi

★ $4pi$ is the total solid angle for a sphere (also the surface area of a unit sphere, similar to $2pi$ being the total angle for a circle and a the perimeter of a unit circle).

★ The denominator, $mbox\{Total\; radiated\; power\}/left(4pi$
ight), represents the average radiated power density
The directivity is rarely expressed as a unitless number. Usually, the directivity is expressed in dBi, so that
:$left.D$
ight|_mbox{dBi} = 10log_{10}left[maxleft(rac{mbox{Radiated power density}left( heta,phi
ight)}{mbox{Total radiated power}/left(4pi
ight)}
ight)
ight]
The reason the units are dBi (decibel relative to an isotropic radiator) is that for an isotropic radiator, the radiated power density}left( heta,phi
ight) is a constant, and therefore equals the average radiated power density (the denominator). This isotropic radiator is not directive at all but has nevertheless a directivity stricto senso equal to 1. This can be misleading and is much better described in dBi.
:$displaystyle\; D\_mbox\{isotropic\; radiator\}=1mbox\{\; unitless\; \}=0mbox\{\; dB\}$

Contents

Directivity and gain

In other fields

References

Directivity and gain

An antenna's directivity is closely related to its gain. The difference between the two quantities is that for gain, the denominator equals $mbox\{Total\; power\; delivered\; to\; antenna\}/left(4pi$
ight), rather than $mbox\{Total\; radiated\; power\}/left(4pi$
ight).
If an antenna is 100% efficient, the two quantities are the same, as all the power delivered to the antenna would get radiated. Therefore, the ratio (difference in dB) between the gain and the directivity represents the antenna's efficiency.

In other fields

The term directivity is also used in acoustics, as is a measure of the radiation pattern from a source indicating how much of the total energy from the source is radiating in a particular direction. In electro-acoustics, these patterns commonly include omni-directional, cardioid and hyper-cardioid microphone polar patterns. A loudspeaker with a high degree of directivity (narrow dispersion pattern) can be said to have a high ''Q''.^[1]

References

★ An Introduction to Radio Frequency Engineering, , Christopher, Coleman, Cambridge University Press, 2004, ISBN 0-521-83481-3