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# Antenna gain

Antenna Gain is defined as the ratio of the radiation intensity of an antenna in a given direction, to the intensity of the same antenna as it radiates in all directions (isotropically). Since the radiation intensity of an isotropically radiated power is equal to the power into the antenna divided by 4п (360 degrees) we can express the following equation:

$Gain = 4pileft\left(frac\left\{mbox\left\{Radiation Intensity\right\}\right\}\left\{mbox\left\{Antenna Input Power\right\}\right\}right\right)$

$Gain = 4pileft\left(frac\left\{mbox\left\{U\right\}left \left(theta,phiright\right)\right\}\left\{mbox\left\{Pin\right\}\right\}right\right) qquadqquad mbox\left\{Dimensionless Units\right\}.$

Although the gain of an antenna is directly related to its directivity, it is important to note that the antenna gain is a measure that takes into account the efficiency of the antenna as well as its directional capabilities. In contrast, directivity is defined as a measure that takes into account only the directional properties of the antenna and therefore it is only influenced by the antenna pattern. However, if we assumed an ideal antenna without losses then Antenna Gain will equal directivity as the antenna efficiency factor equals 1 (100% efficiency). In practice, the gain of an antenna is always less than its directivity.

$D\left(theta,phi\right) = 4pileft\left(frac\left\{mbox\left\{U\right\}left \left(theta,phiright\right)\right\}\left\{mbox\left\{Prad\right\}\right\}right\right)$

$D\left(theta,phi\right) = epsilon_\left\{cd\right\}left \left(4pifrac\left\{mbox\left\{U\right\}left \left(theta,phiright\right)\right\}\left\{mbox\left\{Prad\right\}\right\}right\right)$

$D\left(theta,phi\right) = epsilon_\left\{cd\right\}left \left(D\left(theta,phi\right)right\right)$

The formulas above show the relationship between antenna gain and directivity, where $epsilon_\left\{cd\right\}$ is the antenna efficiency factor, D the directivity of the antenna and G the antenna gain. In the antenna world, we usually deal with a “relative gain” which is defined as the power gain ratio in a specific direction of the antenna, to the power gain ratio of a reference antenna in the same direction. The input power must be the same for both antennas while performing this type of measurement. The reference antenna is usually a dipole, horn or any other type of antenna whose power gain is already calculated or known.

$Gain = mbox\left\{G\left(ref ant\right)\right\}left\left(frac\left\{mbox\left\{Pmax\left(AUT\right)\right\}\right\}\left\{mbox\left\{Pmax\left(ref ant\right)\right\}\right\}right\right)$

In the case that the direction of radiation is not stated, the power gain is always calculated in the direction of maximum radiation. The maximum 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. The maximum gain of a real antenna has no lower bound, and is often -10 dB or less for electrically small antennas.

Taking into consideration the radiation efficiency of an antenna, we can express a relationship between the antenna’s total radiated power and the total power input as:

$Power Radiated = mbox\left\{\left(Antenna Radiation Efficiency\right)\right\}left\left(mbox\left\{Power Input\right\}right\right)$

It is important to note that in the above formula, antenna radiation efficiency only includes conduction efficiency and dielectric efficiency and does not include reflection efficiency as part of the total efficiency factor. Moreover, the IEEE standards state that “gain does not include losses arising from impedance mismatches and polarization mismatches”.

Antenna Absolute Gain is another definition for antenna gain. However, Absolute Gain does include the reflection or mismatch losses.

$G_\left\{abs\right\}\left(theta,phi\right)= epsilon_\left\{refl\right\}G\left(theta,phi\right) = \left(1-Gamma^2\right)left\left(G\left(theta,phi\right)right\right)$

$= epsilon_\left\{refl\right\}epsilon_\left\{cd\right\}D\left(theta,phi\right) = epsilon_\left\{eff\right\}left\left(D\left(theta,phi\right)right\right)$

In this equation, $epsilon_\left\{refl\right\}$ is the reflection efficiency, and $epsilon_\left\{cd\right\}$ includes the dielectric and conduction efficiency. The term $epsilon_\left\{eff\right\}$ is the total antenna efficiency factor.

Taking into account polarization effects in the antenna, we can also define the partial gain of an antenna for a given polarization as that part of the radiation intensity corresponding to a given polarization divided by the total radiation intensity of an isotropic antenna. As a result of this definition for the partial gain in a given direction, we can conclude that the total gain of an antenna is the sum of partial gains for any two orthogonal polarizations.

$G_\left\{total\right\} = G_\left\{theta\right\} + G_\left\{phi\right\}$

$G_\left\{theta\right\} = 4pileft\left(frac\left\{U_theta\right\}\left\{mbox\left\{Pin\right\}\right\}right\right)$

$G_\left\{phi\right\} = 4pileft\left(frac\left\{U_phi\right\}\left\{mbox\left\{Pin\right\}\right\}right\right)$

The terms $U_\left\{theta\right\}$ and $U_\left\{phi\right\}$ represent the radiation intensity in a given direction contained in their respective E field component. Commonly, the gain of an antenna is expressed in terms of decibels instead of dimensionless quantities. The formula to convert dimensionless units to dB is given below:

$G_\left\{dB\right\} = 10log_\left\{10\right\}\left(epsilon_\left\{cd\right\}D_\left\{dimmensionless\right\}\right)$

$G_\left\{dB\right\} = 10log_\left\{10\right\}\left(G_\left\{dimmensionless\right\}\right)$

Example calculating antenna gain:

A lossless resonant half-wavelength dipole antenna, with input impedance of 80 ohms, is connected to a transmission line whose characteristic impedance is 50 ohms. Assuming that the pattern of the antenna is given approximately by:

$U = B_0sin^3\left(theta\right)$

Find the maximum absolute gain of this antenna.

Solution: First computing maximum directivity of antenna:

$B_0 = U_\left\{max\right\}$

$Prad = int_0^\left\{2pi\right\}int_0^\left\{pi\right\}U\left(theta,phi\right)sin\left(theta\right)d\left(theta\right)d\left(phi\right) = 2\left(pi\right)B_0int_0^\left\{pi\right\}sin^4\left(theta\right)d\left(theta\right) = B_0\left(frac\left\{3pi^2\right\}\left\{4\right\}\right)$

$D = 4pi\left(frac\left\{Umax\right\}\left\{Prad\right\}\right) = 4pi\left(frac\left\{B_0\right\}\left\{B_0\left(frac\left\{3pi^2\right\}\left\{4\right\}\right)\right\}\right) = frac\left\{16\right\}\left\{3pi\right\} = 1.698$

Since antenna is mentioned to be lossless the radiation efficiency is 1. Then maximum gain is equal to:

$G = epsilon_\left\{cd\right\}D = \left(1\right)\left(1.698\right) = 1.698$

$G_\left\{dB\right\} = 10log_\left\{10\right\}\left(1.698\right) = 2.299$

Taking into account reflection efficiency due to mismatch loss:

$epsilon_r = \left(1-|Gamma|^2\right) = left\left(1-|frac\left\{80-50\right\}\left\{80+50\right\}|^2right\right)= 0.947$

$epsilon_\left\{r\left(dB\right)\right\} = 10log_\left\{10\right\}\left(0.947\right) = -0.237$

Then the overall efficiency becomes:

$epsilon_\left\{total\right\} = epsilon_repsilon_\left\{cd\right\} = 0.947$

$epsilon_\left\{total\left(dB\right)\right\} = -0.237$

Absolute gain is calculated as:

$G_\left\{absolute\right\} = epsilon_\left\{total\right\}D = \left(0.947\right)\left(1.698\right) = 1.608$

$G_\left\{absolute\left(dB\right)\right\} = 10log_\left\{10\right\}\left(1.608\right) = 2.063$

Antenna Efficiency:

The total antenna efficiency takes into account all loses in the antenna such as reflections due to mismatch between transmission lines and the antenna, conduction and dielectric loses.

$epsilon_\left\{total\right\} = epsilon_repsilon_cepsilon_d$

Where $epsilon_\left\{total\right\}$ is the total efficiency of the antenna, $epsilon_\left\{r\right\}$ is the efficiency due to mismatch losses, $epsilon_\left\{c\right\}$ is the efficiency due to conduction losses, $epsilon_\left\{d\right\}$ is the efficiency due to dielectric losses.

Usually conduction and dielectric efficiency are determined experimentally since they are very difficult to calculate. In fact, they cannot be separated when measured and therefore, it is more helpful to rewrite the equation as:

$epsilon_\left\{total\right\} = epsilon_repsilon_\left\{cd\right\} = \left(1-|Gamma|^2\right)epsilon_\left\{cd\right\}$

Where $Gamma$ is the voltage reflection coefficient and , $epsilon_\left\{cd\right\}$ or $\left(epsilon_cepsilon_d\right)$ is the antenna radiation efficiency which is commonly used to relate the gain and directivity in the antenna.

Antenna Directivity:

Directivity is defined as the ratio of the radiation intensity of an antenna in a given direction to the radiation intensity averaged over all directions.

$D = 4pifrac\left\{U\right\}\left\{Prad\right\}$

A more general expression of directivity includes sources with radiation patterns as functions of spherical coordinate angles $theta$ and $phi$.

$D = frac\left\{4pi\right\}\left\{frac\left\{int_0^\left\{2pi\right\}int_0^\left\{pi\right\}F\left(theta,phi\right)sin\left(theta\right)d\left(theta\right)d\left(phi\right)\right\}\left\{F\left(theta,phi\right)|_\left\{max\right\}\right\}\right\} = frac\left\{4pi\right\}\left\{Omega_A\right\}$

Where $Omega_A$ is the beam solid angle and is defined as the solid angle in which if the antenna radiation intensity is constant (and maximum value), all power would flow through it. In the case of antennas with one narrow major lobe and very negligible minor lobes, the beam solid angle can be approximated as the product of the half-power beamwidths in 2 perpendicular planes.

$Omega_A = Theta_\left\{1r\right\}Theta_\left\{2r\right\}$

Where, $Omega_\left\{1r\right\}$ is the half-power beamwidth in one plane (radians) and $Omega_\left\{2r\right\}$ is the half-power beamwidth in a plane at a right angle to the other (radians). The same approximation can used for angles given in degrees as follows:

$D approx 4pifrac\left\{\left(frac\left\{180\right\}\left\{pi\right\}\right)^2\right\}\left\{Theta_\left\{1d\right\}Theta_\left\{2d\right\}\right\} = frac\left\{41253\right\}\left\{Theta_\left\{1d\right\}Theta_\left\{2d\right\}\right\}$

Where, $Omega_\left\{1d\right\}$ is the half-power beamwidth in one plane (degrees) and $Omega_\left\{2d\right\}$ is the half-power beamwidth in a plane at a right angle to the other (degrees).In planar arrays, a better approximation is:

$D approx frac\left\{32400\right\}\left\{Theta_\left\{1d\right\}Theta_\left\{2d\right\}\right\}$

Most of the time, it is desirable to express directivity in decibels instead of dimensionless quantities. Therefore:

$D_\left\{dB\right\} = 10log_\left\{10\right\}\left(D_\left\{dimenssionless\right\}\right)$

## References

• Antenna Theory (3rd edition), by C. Balanis, Wiley, 2005, ISBN 0-471-66782-X
• Antenna for all applications (3rd edition), by John de Kraus, Ronald J. Marhefka, 2002, ISBN 0-07-232103-2

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