Gain (antenna)

thumb|upright=1.5|Diagram illustrating how isotropic gain is defined. The axes represent power density in watts per square meter. is the radiation pattern of a directive antenna, which radiates a maximum power density of watts per square meter at some given distance from the antenna. The green ball is the radiation pattern of an isotropic antenna which radiates the same total power, and is the power density it radiates. The gain of the first antenna is . Since the directive antenna radiates the same total power within a small angle along the z axis, it can have a higher signal strength in that direction than the isotropic antenna, and so a gain greater than one.

In electromagnetics, an antenna's gain is a key performance parameter which combines the antenna's directivity and radiation efficiency. The term power gain has been deprecated by IEEE. In a transmitting antenna, the gain describes how well the antenna converts input power into radio waves headed in a specified direction. In 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, the gain in the direction of the antenna's main lobe. A plot of the gain as a function of direction is called the antenna pattern or radiation pattern. It is not to be confused with directivity, which does take an antenna's radiation efficiency into account.

Gain or 'absolute gain' is defined as "The ratio of the radiation intensity in a given direction to the radiation intensity that would be produced if the power accepted by the antenna were isotropically radiated".

Radiation efficiency

The radiation efficiency <math>\eta</math> of an antenna is "The ratio of the total power radiated by an antenna to the net power accepted by the antenna from the connected transmitter."

The partial gains in the <math>\theta</math> and <math>\phi</math> components are expressed as

<math display="block">G_\theta = 4\pi\left(\frac{U_\theta}{P_\text{in\right)</math>

and

<math display="block">G_\phi = 4\pi\left(\frac{U_\phi}{P_\text{in\right),</math>

where <math>U_\theta</math> and <math>U_\phi</math> represent the radiation intensity in a given direction contained in their respective <math>E</math> 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.

<math display="block">G = G_\theta + G_\phi</math>

Examples

First example

Suppose a lossless antenna has a radiation pattern given by:

<math display="block">U = B_0\,\sin^3(\theta).</math>

Let us find the gain of such an antenna. First we find the peak radiation intensity of this antenna:

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

<math display="block">\begin{align}

P_\text{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}{4}\pi^2\right) \\

D &= 4\pi\left(\frac{U_\text{max{P_\text{rad\right) = 4\pi\left[\frac{B_0}{B_0\left(\frac{3}{4}\pi^2\right)}\right] = \frac{16}{3\pi} \approx 1.698

\end{align}</math>

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

<math display="block">\begin{align}

G &= \eta D \approx (1)(1.698) = 1.698 \\

G_\text{dBi} &\approx 10\, \log_{10}(1.698) \approx 2.30\,\text{dBi}

\end{align}</math>

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

<math display="block">G_\text{dBd} = 10\, \log_{10}\left(\frac{1.698}{1.64}\right) = 0.15\,\text{dBd}.</math>

Second example

As an example, consider an antenna that radiates an electromagnetic wave whose electrical field has an amplitude <math>E_\theta</math> at a distance . That amplitude is given by:

<math display="block"> E_\theta = {A I \over r}</math>

where:

<math>I</math> is the current (in Amps) fed to the antenna, and
<math>A</math> is a characteristic constant (in Ohms) of each antenna,

for a large distance . The radiated wave can be considered locally as a plane wave. The intensity of an electromagnetic plane wave is:

<math display="block">{P\over S} = {c_\circ \varepsilon_\circ \over 2}{E_\theta}^2 = {1 \over 2} { {E_\theta}^2 \over Z_\circ }</math>

where

<math display="inline"> Z_\circ = \sqrt{ {\mu_\circ \over \varepsilon_\circ = 376.730313461\,\mathsf{\Omega}</math> is a universal constant called vacuum impedance; and
.

If the resistive part of the series impedance of the antenna is , the power fed to the antenna is . The intensity of an isotropic antenna is the power so fed divided by the surface of the sphere of radius :

<math display="block">\left({P \over S}\right)_\mathsf{iso} =

&= 60 \operatorname{Cin}(2\pi) = 60 \left[ \ln(2\pi) + \gamma - \operatorname{Ci}(2\pi) \right]

= 120 \int_{0}^{\frac{\pi}{2 \frac{\cos\left( \frac{\pi}{2}\cos\theta \right)^2 }{ \sin\theta}d\theta\ ,\\

&= 15 \left[ 2\pi^2 - \frac{1}{3} \pi^4 + \frac{4}{135}\pi^6 - \frac{1}{630}\pi^8 + \frac{4}{70875}\pi^{10} + \ldots + (-1)^{n+1}\frac{(2\pi)^{2n{n(2n)!}\right]\ ,\\

&= 73.12960 \ldots \mathsf{\; \Omega; }

\end{align}</math>

In most cases </math> = 73.130 is adequate.

<math display="block">\begin{align} G_{\frac{\lambda}{2

&= \frac{60^2}{30R_{\frac{\lambda}{2}

= \frac{3600}{30R_{\frac{\lambda}{2}

= \frac{120}{R_{\frac{\lambda}{2}

= \frac{1} \frac{ \cos \left( \frac{\pi}{2} \cos \theta \right)^2 }{ \sin \theta} d \theta\ , \\

&\approx \frac{120}{73.1296} \approx 1.6409224 \approx 2.15088\ \mathsf{\ dBi} ;

\end{align}</math>

Likewise, </math> = 1.64, or , are usually the cited values.

Sometimes, the half-wave dipole is taken as a reference instead of the isotropic radiator. The gain is then given in (decibels over dipole):

<math display="block">0\ \mathrm{dBd} = 2.15\ \mathrm{dBi}.</math>

Realized gain

Realized gain differs from gain in that it is "reduced by its impedance mismatch factor". 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.

When testing mobile devices, TRP can be measured while in close proximity of power-absorbing losses such as the body and hand of the user.

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.

References

Bibliography

Antenna Theory (3rd edition), by C. Balanis, Wiley, 2005,
Antenna for all applications (3rd edition), by John D. Kraus, Ronald J. Marhefka, 2002,

Radiation efficiency

Examples

First example

Second example

Realized gain

See also

References

Bibliography