Antenna gain is calculated by multiplying an antenna’s directivity by its radiation efficiency. In decibels, gain equals 10 × log₁₀ of the linear power ratio. The specific formula you need depends on what information you have: the antenna’s radiation pattern, its physical dimensions, or measured power readings from a test setup.
The Core Formula: Directivity Times Efficiency
Every antenna gain calculation starts from one relationship: gain equals directivity multiplied by radiation efficiency. Directivity describes how well an antenna focuses energy in a particular direction compared to spreading it equally in all directions. Radiation efficiency accounts for power lost as heat inside the antenna itself, through resistive losses in conductors and dielectric materials. A perfectly efficient antenna would have gain equal to its directivity, but real antennas always lose some power internally.
Written out, the formula is:
G(θ, φ) = e_rad × D(θ, φ)
Here, G is the gain in a given direction (defined by angles θ and φ), e_rad is the radiation efficiency as a decimal between 0 and 1, and D is the directivity in that same direction. If someone quotes a single gain number without specifying a direction, they mean the peak gain, which is the maximum value across all directions.
Converting Between Linear and Decibel Values
Gain is often expressed as a simple ratio (like 6.4 or 100), but it’s more practical to work in decibels because the numbers are easier to compare and add together in link budget calculations. To convert a linear gain ratio to decibels:
G(dBi) = 10 × log₁₀(G_linear)
So an antenna with a linear gain of 100 has a gain of 10 × log₁₀(100) = 20 dBi. Going the other direction, a gain of 6 dBi corresponds to a linear ratio of 10^(6/10) = 3.98, meaning the antenna concentrates about four times more power in its favored direction than an isotropic radiator would.
dBi vs. dBd: Two Reference Points
You’ll see antenna gain quoted in two different units, and mixing them up will throw your calculations off by a fixed amount. dBi measures gain relative to an isotropic radiator, a theoretical point source that radiates equally in every direction. dBd measures gain relative to a half-wave dipole antenna, which itself has a gain of 2.15 dBi (a linear ratio of 1.64).
The conversion is straightforward:
- dBi to dBd: subtract 2.15
- dBd to dBi: add 2.15
A half-wave dipole has a gain of 2.15 dBi, which is exactly 0 dBd, since it’s the reference for that scale. Manufacturer spec sheets sometimes leave off the “i” or “d” suffix, so if the number seems ambiguous, check which reference they’re using. An antenna listed at “6 dB gain” could mean 6 dBi or 6 dBd, and those are meaningfully different values.
Calculating Gain From Physical Size
For aperture-type antennas like parabolic dishes, horns, and flat panel arrays, you can estimate gain directly from the antenna’s physical area and operating frequency. The relationship linking gain to effective aperture is:
A_e = (λ² / 4π) × G
Rearranging to solve for gain:
G = (4π × A_e) / λ²
Here, A_e is the effective aperture in square meters and λ is the wavelength in meters. The effective aperture isn’t the full physical area of the antenna. It’s the physical area multiplied by an aperture efficiency factor that accounts for imperfect illumination, spillover, surface errors, and blockage. For a parabolic dish, the formula becomes:
G(dBi) = 10 × log₁₀[η × (π × d / λ)²]
In this version, d is the dish diameter in meters, λ is the wavelength (speed of light divided by frequency), and η is the aperture efficiency as a decimal. Well-designed parabolic dishes typically achieve aperture efficiencies between 0.55 and 0.70. A common rule-of-thumb value for quick estimates is 0.55 to 0.60.
As a worked example: a 1.2-meter dish operating at 12 GHz (λ = 0.025 m) with 60% efficiency gives G = 10 × log₁₀[0.60 × (π × 1.2 / 0.025)²] = 10 × log₁₀[0.60 × 22,619] = 10 × log₁₀(13,572) ≈ 41.3 dBi. This illustrates why satellite TV dishes can achieve such high gain from a relatively compact reflector.
Deriving Gain From the Friis Transmission Equation
If you can measure transmitted power, received power, and the distance between two antennas, the Friis transmission equation lets you extract gain values from real-world measurements. The equation describes how power transfers between a transmit and receive antenna in free space:
P_R = P_T × G_T × G_R × (λ / 4πR)²
P_T is the transmitted power, P_R is the received power, G_T and G_R are the gains of the transmit and receive antennas, λ is the wavelength, and R is the distance between them. The (λ / 4πR)² term is the free-space path loss, representing how much power spreads out over distance.
If you know the gain of one antenna and can measure everything else, you solve for the unknown gain. In practice, this is exactly how antenna engineers measure gain using the comparison method.
Measuring Gain by Comparison
The gain-comparison method (sometimes called the gain-substitution method) is the most common way to measure a real antenna’s gain without needing to map its entire radiation pattern. You need a reference antenna with precisely known gain, a transmit antenna, and a power meter or calibrated receiver.
The procedure works in two steps. First, you set up the reference antenna as the receiver at a fixed distance from the transmitter and record the received power. Then, without changing the transmitter power or the distance, you swap in the antenna you want to test and record its received power. The gain of your test antenna in decibels is:
G_test(dB) = G_reference(dB) + 10 × log₁₀(P_test / P_reference)
If your test antenna received twice the power that the reference did, it has 3 dB more gain. If it received half the power, it has 3 dB less. The beauty of this method is that the transmitter power, distance, and path loss all cancel out, since they’re identical in both measurements. The only thing that changes is the receiving antenna, so the power difference maps directly to a gain difference.
For antennas that use circular or elliptical polarization, you need to run the comparison twice, using a vertically polarized reference and then a horizontally polarized one. The total gain combines both partial measurements: G_total = 10 × log₁₀(G_vertical + G_horizontal), where G_vertical and G_horizontal are the linear (not dB) gain values from each measurement.
Quick Reference for Common Antenna Types
When you’re checking whether your calculation is in the right ballpark, these typical gain values help:
- Isotropic radiator: 0 dBi (theoretical only, used as a reference)
- Half-wave dipole: 2.15 dBi
- Quarter-wave monopole over ground plane: around 5 dBi
- Yagi-Uda (TV-style directional): 6 to 20 dBi depending on element count
- Parabolic dish: 20 to 45+ dBi depending on size and frequency
If your calculated gain falls outside the expected range for the antenna type, double-check your efficiency estimate or verify that your frequency and dimensions use consistent units. A common mistake in dish calculations is mixing centimeters and meters, which shifts the result by 20 or 40 dB.