# Antenna gain

In electromagnetics, an antenna's power gain or simply gain is a key performance number which combines the antenna's directivity and electrical efficiency. As a transmitting antenna, the gain describes how well the antenna converts input power into radio waves headed in a specified direction. As a receiving antenna, the gain describes how well the antenna converts radio waves arriving from a specified direction into electrical power. When no direction is specified, "gain" is understood to refer to the peak value of the gain. A plot of the gain as a function of direction is called the radiation pattern.

Antenna gain is usually defined as the ratio of the power produced by the antenna from a far-field source on the antenna's beam axis to the power produced by a hypothetical lossless isotropic antenna, which is equally sensitive to signals from all directions.[1] Usually this ratio is expressed in decibels, and these units are referred to as "decibels-isotropic" (dBi). An alternative definition compares the antenna to the power received by a lossless half-wave dipole antenna, in which case the units are written as dBd. Since a lossless dipole antenna has a gain of 2.15 dBi, the relation between these units is: gain in dBd = gain in dBi − 2.15 d. For a given frequency, the antenna's effective area is proportional to the power gain. An antenna's effective length is proportional to the square root of the antenna's gain for a particular frequency and radiation resistance. Due to reciprocity, the gain of any antenna when receiving is equal to its gain when transmitting.

Directive gain or directivity is a different measure which does not take an antenna's electrical efficiency into account. This term is sometimes more relevant in the case of a receiving antenna where one is concerned mainly with the ability of an antenna to receive signals from one direction while rejecting interfering signals coming from a different direction.

## Power gain

Power gain (or simply gain) is a unitless measure that combines an antenna's efficiency Eantenna and directivity D:

$G = E_{antenna} \cdot D.$

The notions of efficiency and directivity depend on the following.

### Efficiency

A transmitting antenna accepts input power $P_{in}$ at some point along the feedline. The point is typically taken to be at the antenna (the feedpoint), thereby not counting power lost due to joule heating in the feedline and reflections back down the feedline. An antenna with efficiency $E_{antenna}$ emits a total radiated power

$P_o = E_{antenna}\cdot P_{in}$

to its environment. The environment may range from free space (the default when not specified) to an object such as a hand surrounding the antenna. Reciprocity justifies taking the properties of a receiving antenna, such as efficiency, directivity, and gain, to be those of that antenna when used for transmission.

### Directivity

Antennas are invariably directional to a greater or less extent, according to how the output power is distributed in any given direction in three dimensions. We shall specify direction here in spherical coordinates $(\theta,\phi)$, where $\theta$ is the altitude or angle above a specified reference plane (such as the ground), while $\phi$ is the azimuth as the angle between the projection of the given direction onto the reference plane and a specified reference direction (such as North or East) in that plane with specified sign (either clockwise or counterclockwise).

The distribution of output power as a function of the possible directions $(\theta,\phi)$ is given by its radiation intensity $U(\theta,\phi)$ (in SI units: watts per steradian, W⋅sr−1). The output power is obtained from the radiation intensity by integrating the latter over all directions:

$P_o =\int_{-\pi}^\pi\int_{-\pi/2}^{\pi/2}U(\theta,\phi)d\theta d\phi.$

The mean radiation intensity $\overline U$ is therefore given by

$\overline U =\frac{P_o}{4\pi}~~$   since there are 4π steradians in a sphere
$=\frac{E_{antenna}\cdot P_{in}}{4\pi}$   using the first formula for $P_o$.

The directive gain or directivity $D(\theta,\phi)$ of an antenna in a given direction is the ratio of its radiation intensity $U(\theta,\phi)$ in that direction to its mean radiation intensity $\overline U$. That is,

$D(\theta,\phi)=\frac{U(\theta,\phi)}{\overline U}.$

An isotropic antenna, meaning one with the same radiation intensity in all directions, therefore has directivity 1 in all directions independently of its efficiency. More generally the maximum, minimum, and mean directivities of any antenna are always at least 1, at most 1, and exactly 1. For the half-wave dipole the respective values are 1.64 (2.15 dB), 0, and 1.

When the directivity $D$ of an antenna is given independently of direction it refers to its maximum directivity in any direction, namely

$D=\max_{\theta,\phi}D(\theta,\phi).$

### Gain

The power gain or simply gain $G(\theta,\phi)$ of an antenna in a given direction takes efficiency into account by being defined as the ratio of its radiation intensity $U(\theta,\phi)$ in that direction to the mean radiation intensity of a perfectly efficient antenna. Since the latter equals $P_{in}/4\pi$, it is therefore given by

$G(\theta,\phi)=\frac{U(\theta,\phi)}{P_{in}/4\pi}$
$=E_{antenna}\cdot\frac{U(\theta,\phi)}{\overline U}$   using the second equation for $\overline U$
$=E_{antenna}\cdot D(\theta,\phi)$   using the equation for $D(\theta,\phi).$

As with directivity, when the gain $G$ of an antenna is given independently of direction it refers to its maximum gain in any direction. Since the only difference between gain and directivity in any direction is a constant factor of $E_{antenna}$ independent of $\theta$ and $\phi$, we obtain the fundamental formula of this section:

$G=E_{antenna}\cdot D.$

### Summary

If only a certain portion of the electrical power received from the transmitter is actually radiated by the antenna (i.e. less than 100% efficiency), then the directive gain compares the power radiated in a given direction to that reduced power (instead of the total power received), ignoring the inefficiency. The directivity is therefore the maximum directive gain when taken over all directions, and is always at least 1. On the other hand, the power gain takes into account the poorer efficiency by comparing the radiated power in a given direction to the actual power that the antenna receives from the transmitter, which makes it a more useful figure of merit for the antenna's contribution to the ability of a transmitter in sending a radio wave toward a receiver. In every direction, the power gain of an isotropic antenna is equal to the efficiency, and hence is always at most 1, though it can and ideally should exceed 1 for a directional antenna.

Note that in the case of an impedance mismatch, Pin would be computed as the transmission line's incident power minus reflected power. Or equivalently, in terms of the rms voltage V at the antenna terminals:

$P_{in} = V^2 \cdot Re \left\lbrace \frac{1}{Z_{in}} \right\rbrace$

where Zin is the feedpoint impedance.

## Numbers used for antenna gain

Published numbers for antenna gain are almost always expressed in decibels (dB), a logarithmic scale. From the gain factor G, one finds the gain in decibels as:

$G_{dBi} = 10 \cdot \log_{10}\left(G\right)$

Therefore an antenna with a peak power gain of 5 would be said to have a gain of 7 dBi. "dBi" is used rather than just "dB" to emphasize that this is the gain according to the basic definition, in which the antenna is compared to an isotropic radiator.

When actual measurements of an antenna's gain are made by a laboratory, the field strength of the test antenna is measured when supplied with, say, 1 watt of transmitter power, at a certain distance. That field strength is compared to the field strength found using a so-called reference antenna at the same distance receiving the same power in order to determine the gain of the antenna under test. That ratio would be equal to G if the reference antenna were an isotropic radiator.

However a true isotropic radiator cannot be built, so in practice a different antenna is used. This will often be a half-wave dipole, a very well understood and repeatable antenna that can be easily built for any frequency. The directive gain of a half-wave dipole is known to be 1.64 and it can be made nearly 100% efficient. Since the gain has been measured with respect to this reference antenna, the difference in the gain of the test antenna is often compared to that of the dipole. The "gain relative to a dipole" is thus often quoted and is denoted using "dBd" instead of "dBi" to avoid confusion. Therefore in terms of the true gain (relative to an isotropic radiator) G, this figure for the gain is given by:

$G_{dBd} = 10 \cdot \log_{10}\left(\frac{G}{1.64}\right)$

For instance, the above antenna with a gain G=5 would have a gain with respect to a dipole of 5/1.64 = 3.05, or in decibels one would call this 10 log(3.05) = 4.84 dBd. In general:

$G_{dBd} = G_{dBi} - 2.15dB$

Both dBi and dBd are in common use. When an antenna's maximum gain is specified in decibels (for instance, by a manufacturer) one must be certain as to whether this means the gain relative to an isotropic radiator or with respect to a dipole. If it specifies "dBi" or "dBd" then there is no ambiguity, but if only "dB" is specified then the fine print must be consulted. Either figure can be easily converted into the other using the above relationship.

Note that when considering an antenna's directional pattern, "gain with respect to a dipole" does not imply a comparison of that antenna's gain in each direction to a dipole's gain in that direction. Rather, it is a comparison between the antenna's gain in each direction to the peak gain of the dipole (1.64). In any direction, therefore, such numbers are 2.15 dB smaller than the gain expressed in dBi.

## Partial gain

Partial gain is calculated as power gain, but for a particular polarization. It is defined as the part of the radiation intensity $U$ corresponding to a given polarization, divided by the total radiation intensity of an isotropic antenna.

$G_{\theta} = 4\pi\left(\frac{U_\theta}{P_{\mathrm{in}}}\right)$
$G_{\phi} = 4\pi\left(\frac{U_\phi}{P_{\mathrm{in}}}\right)$

where $U_{\theta}$ and $U_{\phi}$ represent the radiation intensity in a given direction contained in their respective E field component.

As a result of this definition, we can conclude that the total gain of an antenna is the sum of partial gains for any two orthogonal polarizations.

$G = G_{\theta} + G_{\phi}$

## Example calculation

Suppose a lossless antenna has a radiation pattern given by:

$U = B_0\,\sin^3(\theta)$

Let us find the gain of such an antenna.

Solution:

First we find the peak radiation intensity of this antenna:

$U_{\mathrm{max}} = B_0$

The total radiated power can be found by integrating over all directions:

$P_{\mathrm{rad}} = \int_0^{2\pi}\int_0^{\pi}U(\theta,\phi)\sin(\theta) \, d\theta \,d\phi = 2 \pi B_0 \int_0^{\pi}\sin^4(\theta) \, d\theta = B_0\left(\frac{3\pi^2}{4} \right)$
$D = 4\pi\left(\frac{U_{\mathrm{max}}}{P_{\mathrm{rad}}}\right) = 4\pi\left[\frac{B_0}{B_0\left(\frac{3\pi^2}{4}\right)}\right] = \frac{16}{3\pi} = 1.698$

Since the antenna is specified as being lossless the radiation efficiency is 1. The maximum gain is then equal to:

$G = E_{antenna} \, D = (1)(1.698) = 1.698$ .
$G_{dBi} = 10 \, \log_{10}(1.698) = 2.30 \, \mathrm{dBi}$

Expressed relative to the gain of a half-wave dipole we would find:

$G_{dBd} = 10 \, \log_{10}(1.698/1.64) = 0.15 \, \mathrm{dBd}$.

## Realized Gain

According to IEEE Standard 145-1993,[1] Realized Gain differs from the above definitions of gain in that it is "reduced by the losses due to the mismatch of the antenna input impedance to a specified impedance." This mismatch induces losses above the dissipative losses described above; therefore, Realized Gain will always be less than Gain.

Gain may be expressed as absolute gain if further clarification is required to differentiate it from Realized Gain.[1]

Total radiated power is the sum of all RF power radiated by the antenna when the source power is included in the measurement. TRP is expressed in Watts, or equivalent logarithmic expressions, often dBm or dBW.[2]

TRP can be measured while in the close proximity of power-absorbing losses such as the body and hand of the Mobile Device Under Test User.[3]

The TRP can be used to determine Body Loss (BoL). The Body Loss is considered as the ratio of TRP measured in the presence of losses and TRP measured while in free space.