1. Power Gain

Power gain (or simply gain) is a unitless measure that combines an antenna's efficiency [math]\displaystyle{ \epsilon_{antenna} }[/math] and directivity D:

[math]\displaystyle{ G = \epsilon_{antenna} \cdot D. }[/math]

The notions of efficiency and directivity depend on the following.

1.1. Efficiency

The efficiency [math]\displaystyle{ \epsilon_{antenna} }[/math] of an antenna is the total radiated power [math]\displaystyle{ P_o }[/math] divided by the input power at the feedpoint

[math]\displaystyle{ \epsilon_{antenna} = {P_o \over P_{in}} }[/math]

A transmitting antenna is supplied power by a feedline, a transmission line connecting the antenna to a radio transmitter. The input power [math]\displaystyle{ P_{in} }[/math] to the antenna is typically defined to be the power supplied to the antenna's terminals (the feedpoint), so antenna power losses do not include power lost due to joule heating in the feedline and reflections back down the feedline due to antenna/line impedance mismatches.

The electromagnetic reciprocity theorem guarantees that the electrical properties of an antenna, such as efficiency, directivity, and gain, are the same when the antenna is used for receiving as when it is transmitting.

1.2. Directivity

An antenna's directivity is determined by its radiation pattern, how the radiated power is distributed with direction in three dimensions. All antennas are directional to a greater or lesser extent, meaning that they radiate more power in some directions than others. The direction is specified here in spherical coordinates [math]\displaystyle{ (\theta,\phi) }[/math], where [math]\displaystyle{ \theta }[/math] is the altitude or angle above a specified reference plane (such as the ground), while [math]\displaystyle{ \phi }[/math] 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 [math]\displaystyle{ (\theta,\phi) }[/math] is given by its radiation intensity [math]\displaystyle{ U(\theta,\phi) }[/math] (in SI units: watts per steradian, W⋅sr⁻¹). The output power is obtained from the radiation intensity by integrating the latter over all solid angles [math]\displaystyle{ d\Omega = \cos\theta \, d\theta \, d\phi }[/math]:

[math]\displaystyle{ P_o =\int_{-\pi}^\pi\int_{-\pi/2}^{\pi/2}U(\theta,\phi) \, d\Omega = \int_{-\pi}^\pi\int_{-\pi/2}^{\pi/2}U(\theta,\phi) \cos \theta \, d\theta \, d\phi. }[/math]

The mean radiation intensity [math]\displaystyle{ \overline U }[/math] is therefore given by

[math]\displaystyle{ \overline U =\frac{P_o}{4\pi}~~ }[/math]   since there are 4π steradians in a sphere

      [math]\displaystyle{ =\frac{\epsilon_{antenna}\cdot P_{in}}{4\pi} }[/math]   using the first formula for [math]\displaystyle{ P_o }[/math].

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

[math]\displaystyle{ D(\theta,\phi)=\frac{U(\theta,\phi)}{\overline U}. }[/math]

An isotropic antenna, meaning one with the same radiation intensity in all directions, therefore has directivity, D = 1, in all directions independent 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 [math]\displaystyle{ D }[/math] of an antenna is given independently of direction it refers to its maximum directivity in any direction, namely

[math]\displaystyle{ D=\max_{\theta, \, \phi}D(\theta,\phi). }[/math]

1.3. Gain

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

[math]\displaystyle{ G(\theta,\phi)=\frac{U(\theta,\phi)}{P_{in}/4\pi} }[/math]

                [math]\displaystyle{ =\epsilon_{antenna}\cdot\frac{U(\theta,\phi)}{\overline U} }[/math]   using the second equation for [math]\displaystyle{ \overline U }[/math]

                [math]\displaystyle{ =\epsilon_{antenna}\cdot D(\theta,\phi) }[/math]   using the equation for [math]\displaystyle{ D(\theta,\phi). }[/math]

As with directivity, when the gain [math]\displaystyle{ G }[/math] 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 [math]\displaystyle{ \epsilon_{antenna} }[/math] independent of [math]\displaystyle{ \theta }[/math] and [math]\displaystyle{ \phi }[/math], we obtain the fundamental formula of this section:

[math]\displaystyle{ G=\epsilon_{antenna}\cdot D. }[/math]

1.4. 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, P_in 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:

[math]\displaystyle{ P_{in} = V^2 \cdot \text{Re} \left\lbrace \frac{1}{Z_{in}} \right\rbrace }[/math]

where Z_in is the feedpoint impedance.

2. Gain in Decibels

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:

[math]\displaystyle{ G_{dBi} = 10 \cdot \log_{10}\left(G\right). }[/math]

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(irad).

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:

[math]\displaystyle{ G_{dBd} = 10 \cdot \log_{10}\left(\frac{G}{1.64}\right). }[/math]

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:

[math]\displaystyle{ G_{dBd} = G_{dBi} - 2.15dB }[/math]

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.

3. Partial Gain

Partial gain is calculated as power gain, but for a particular polarization. It is defined as the part of the radiation intensity [math]\displaystyle{ U }[/math] corresponding to a given polarization, divided by the total radiation intensity of an isotropic antenna.^[1]

The partial gains in the [math]\displaystyle{ \theta }[/math] and [math]\displaystyle{ \phi }[/math] components are expressed as

[math]\displaystyle{ G_{\theta} = 4\pi\left(\frac{U_\theta}{P_{\mathrm{in}}}\right) }[/math]

and

[math]\displaystyle{ G_{\phi} = 4\pi\left(\frac{U_\phi}{P_{\mathrm{in}}}\right) }[/math],

where [math]\displaystyle{ U_{\theta} }[/math] and [math]\displaystyle{ U_{\phi} }[/math] represent the radiation intensity in a given direction contained in their respective [math]\displaystyle{ 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]\displaystyle{ G = G_{\theta} + G_{\phi} }[/math]

4. Example Calculation

Suppose a lossless antenna has a radiation pattern given by:

[math]\displaystyle{ U = B_0\,\sin^3(\theta). }[/math]

Let us find the gain of such an antenna.

Solution:

First we find the peak radiation intensity of this antenna:

[math]\displaystyle{ U_{\mathrm{max}} = B_0 }[/math]

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

[math]\displaystyle{ 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) }[/math]

[math]\displaystyle{ 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 }[/math]

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

[math]\displaystyle{ G = \epsilon_{antenna} \, D = (1)(1.698) = 1.698 }[/math] .

[math]\displaystyle{ G_{dBi} = 10 \, \log_{10}(1.698) = 2.30 \, \mathrm{dBi} }[/math]

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

[math]\displaystyle{ G_{dBd} = 10 \, \log_{10}(1.698/1.64) = 0.15 \, \mathrm{dBd} }[/math].

5. Realized Gain

According to IEEE Standard 145–1993,^[2] 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.^[2]

6. Total Radiated Power

Total radiated power (TRP) 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 the corresponding logarithmic expressions, often dBm or dBW.^[3]

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.^[4]

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.