# Directivity

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:
$D = maxleft\left(frac\left\{mbox\left\{Radiated power density\right\}left\left(theta,phiright\right)\right\}\left\{mbox\left\{Total radiated power\right\}/left\left(4piright\right)\right\}right\right)$
where

• $theta$ and $phi$ are the standard spherical coordinates angles
• Radiated power density is the power per unit solid angle such that $mbox\left\{Total radiated power\right\}=int_\left\{phi=0\right\}^\left\{phi=2pi\right\}left\left(int_\left\{theta=0\right\}^\left\{theta=pi\right\}mbox\left\{Radiated power density\right\}left\left(theta,phiright\right)sintheta,dthetaright\right)dphi$
• $4pi$ 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 the perimeter of a unit circle).
• The denominator, $mbox\left\{Total radiated power\right\}/left\left(4piright\right)$, represents the average radiated power density

The directivity is rarely expressed as a unitless number. Usually, the directivity is expressed in dBi, so that

$left.Dright|_mbox\left\{dBi\right\} = 10log_\left\{10\right\}left\left[maxleft\left(frac\left\{mbox\left\{Radiated power density\right\}left\left(theta,phiright\right)\right\}\left\{mbox\left\{Total radiated power\right\}/left\left(4piright\right)\right\}right\right)right\right]$

The reason the units are dBi (decibel relative to an isotropic radiator) is that for an isotropic radiator, the radiated power density 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.

$displaystyle D_mbox\left\{isotropic radiator\right\}=1mbox\left\{ unitless \right\}=0mbox\left\{ dB\right\}$

The word directivity is also sometimes used as a synonym for directive gain. This usage is readily understood, as the direction will be specified, or directional dependence implied. Later editions of the IEEE Dictionary specifically endorse this usage; nevertheless it has yet to be universally adopted.

The peak directivity of an actual antenna can vary from 1.76 dB for a short dipole, to as much as 50 dB for a large dish antenna.

## 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 $mbox\left\{Total power delivered to antenna\right\}/left\left(4piright\right)$, rather than $mbox\left\{Total radiated power\right\}/left\left(4piright\right)$.

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.

## Partial directivity and gain

The partial directive gain is the power density in a particular direction and for a particular component of the polarization, divided by the average power density for all directions and all polarizations. For any pair of orthogonal polarizations (such as left-hand-circular and right-hand-circular), the individual power densities simply add to give the total power density. Thus, if expressed as dimensionless ratios rather than in dB, the total directive gain is equal to the sum of the two partial directive gains.

The partial directivity and partial gain are similarly defined, and are similarly additive for orthogonal polarizations.

## 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.

## References

• Coleman, Christopher (2004). An Introduction to Radio Frequency Engineering. Cambridge University Press. ISBN 0-521-83481-3.

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