4.4 Antenna gain and directivity

Antenna gain and directivity

Antenna gain and directivity describe how an antenna focuses electromagnetic energy into particular directions rather than radiating it uniformly. These parameters sit at the heart of link budget calculations, antenna selection, and wireless system design. This guide covers gain, directivity, efficiency, radiation patterns, effective aperture, the Friis transmission equation, antenna arrays, and measurement techniques.

Antenna gain

Gain quantifies how effectively an antenna concentrates radiated power in a given direction. It combines the antenna's directional focusing ability (directivity) with its ohmic and material losses (efficiency), making it the single most practical figure of merit for real antennas.

Definition of antenna gain

Antenna gain is the ratio of the radiation intensity in a specified direction to the radiation intensity that would be produced by a lossless isotropic radiator fed with the same total input power. An isotropic radiator is a theoretical point source that radiates equally in all directions; no physical antenna achieves this, but it serves as a universal reference.

Gain is usually expressed in dBi (decibels relative to an isotropic radiator). A gain of 10 dBi means the antenna produces 10 times the power density in its peak direction compared to an isotropic source with the same input power.

Relationship between gain and directivity

Directivity describes how tightly an antenna focuses energy, ignoring any internal losses. Gain folds in those losses through the antenna's radiation efficiency $\eta$:

$G = \eta \, D$

where $D$ is directivity and $0 < \eta \leq 1$. For a perfectly lossless antenna, $\eta = 1$ and gain equals directivity. In practice, $\eta < 1$ due to conductor resistance, dielectric loss, and other dissipative mechanisms, so gain is always less than or equal to directivity.

Units of antenna gain

• dBi uses the isotropic radiator as the reference. This is the most common convention.
• dBd uses a half-wave dipole as the reference. Because a half-wave dipole itself has a gain of 2.15 dBi, the conversion is:

$G_{\text{dBi}} = G_{\text{dBd}} + 2.15$

Always check which reference is being used when comparing datasheets. Mixing up dBi and dBd introduces a 2.15 dB error, which matters in tight link budgets.

Typical gain values for common antennas

Higher-gain antennas have narrower beams. A parabolic dish at 40 dBi produces a pencil-thin beam, while a dipole at 2.15 dBi covers a broad region around its broadside.

Directivity

Definition of directivity

Directivity is the ratio of the radiation intensity in a given direction to the radiation intensity averaged over all directions:

$D(\theta, \phi) = \frac{U(\theta, \phi)}{P_{\text{rad}} / 4\pi}$

where $U(\theta, \phi)$ is the radiation intensity (W/sr) in direction $(\theta, \phi)$ and $P_{\text{rad}}$ is the total radiated power. When stated without a direction, "directivity" usually refers to the peak value $D_{\max}$.

Directivity is dimensionless but is often quoted in dB. An isotropic radiator has $D = 1$ (0 dB).

Directivity vs. gain

These two quantities answer slightly different questions:

• Directivity asks: How well does the antenna focus energy compared to an isotropic source, assuming no losses?
• Gain asks: How well does the antenna focus energy compared to an isotropic source, including real losses?

The gap between them is entirely due to efficiency. If you measure a directivity of 15 dB and a gain of 14 dB, the antenna has about 1 dB of internal loss ($\eta \approx 0.79$).

Isotropic radiator as reference

The isotropic radiator is useful precisely because it's the simplest possible reference: uniform radiation in every direction, with $D = 1$ and a radiation pattern that's a perfect sphere. No physical antenna is isotropic, but every antenna's directivity and gain can be compared against it, giving a universal baseline.

Directivity of common antennas

For a lossless half-wave dipole, directivity and gain are both 2.15 dBi. For high-frequency dishes, the directivity can be very large because the electrical aperture is many wavelengths across.

Antenna efficiency

Efficiency captures every mechanism that converts input power into heat (or reflects it back) rather than radiating it. The overall relationship is:

$G = \eta_{\text{total}} \, D$

Radiation efficiency

Radiation efficiency ($\eta_{\text{rad}}$) is the ratio of power actually radiated to the power delivered to the antenna terminals. Losses here come from ohmic resistance in conductors and dissipation in dielectric materials. For well-designed antennas at microwave frequencies, $\eta_{\text{rad}}$ is often above 90%. Electrically small antennas or those using lossy substrates can have much lower radiation efficiency.

Conduction and dielectric losses

Conduction losses arise from finite conductivity in the antenna structure and feed network. At higher frequencies, skin effect concentrates current on conductor surfaces, increasing effective resistance. Dielectric losses come from the loss tangent of any substrate or radome material. Both reduce radiation efficiency.

Reflection efficiency

Reflection efficiency ($\eta_{\text{ref}}$) accounts for impedance mismatch between the antenna and its feed line. If the antenna input impedance doesn't match the characteristic impedance of the feed, some power reflects back toward the source. In terms of the voltage reflection coefficient $\Gamma$:

$\eta_{\text{ref}} = 1 - |\Gamma|^2$

A well-matched antenna ($|\Gamma| \approx 0$) has reflection efficiency near 100%.

Total antenna efficiency

Total efficiency combines all loss mechanisms:

$\eta_{\text{total}} = \eta_{\text{ref}} \times \eta_{\text{rad}}$

(Some texts break $\eta_{\text{rad}}$ further into separate conduction and dielectric terms, but the product structure is the same.) A typical well-matched microwave horn might have $\eta_{\text{total}} \approx 0.85$–$0.95$, while a small mobile-phone antenna on a lossy PCB might drop to 0.3–0.5.

Antenna radiation patterns

Radiation patterns are graphical maps of how an antenna distributes radiated power as a function of direction. They reveal directivity, beamwidth, sidelobe levels, and null locations.

Radiation pattern representation

Patterns are usually plotted in two standard planes:

• E-plane: the plane containing the electric field vector and the direction of maximum radiation.
• H-plane: the plane containing the magnetic field vector and the direction of maximum radiation.

Polar plots place the antenna at the origin and show field strength vs. angle around a circle. They give an intuitive sense of beam shape. Rectangular (Cartesian) plots put angle on the horizontal axis and field strength (often in dB) on the vertical axis. These make it easier to read sidelobe levels and null depths precisely.

Main lobe and side lobes

The main lobe (or main beam) is the angular region containing the direction of maximum radiation. Side lobes are secondary maxima at angles away from the main beam. Side lobes are generally undesirable because they waste power in unwanted directions and can pick up interference on receive.

The sidelobe level (SLL) is the ratio of the peak side-lobe intensity to the main-lobe peak, usually quoted in dB below the main beam. A typical design target might be SLL $\leq -20$ dB.

Half-power beamwidth (HPBW)

The half-power beamwidth is the angular width of the main lobe between the two directions where radiated power drops to half its peak value ($-3$ dB points). A narrower HPBW means higher directivity. For a uniformly illuminated circular aperture of diameter $d$:

$\text{HPBW} \approx \frac{1.02 \,\lambda}{d} \text{ (radians)}$

This inverse relationship between aperture size and beamwidth is a recurring theme: larger antennas produce narrower beams.

Front-to-back ratio

The front-to-back ratio (FBR) compares the peak radiation in the forward direction to the radiation directly behind the antenna (180° away). It's expressed in dB. A Yagi-Uda antenna might achieve an FBR of 15–25 dB, meaning signals arriving from behind are attenuated by that amount relative to the forward direction.

Effective aperture

Effective aperture connects an antenna's gain to its ability to capture power from an incoming wave. It's especially useful for analyzing receive performance.

Relationship between gain and effective aperture

The effective aperture $A_e$ is related to gain by:

$A_e = \frac{\lambda^2}{4\pi} G$

This equation holds for any antenna. It tells you that even a wire dipole, which has no obvious "area," still has a well-defined effective aperture. At longer wavelengths (lower frequencies), $\lambda^2$ grows, so a given gain corresponds to a larger effective capture area.

Effective aperture vs. physical aperture

For aperture-type antennas (dishes, horns), the physical collecting area $A_{\text{phys}}$ sets an upper bound, but the effective aperture is typically smaller because of non-uniform illumination, spillover, and ohmic losses. A wire antenna like a dipole has no meaningful physical aperture, yet it still has a calculable $A_e$.

Aperture efficiency

Aperture efficiency $\eta_{\text{ap}}$ is the ratio of effective aperture to physical aperture:

$\eta_{\text{ap}} = \frac{A_e}{A_{\text{phys}}}$

Typical values for parabolic reflectors range from 0.5 to 0.7. Achieving higher aperture efficiency requires careful feed design to illuminate the reflector uniformly without excessive spillover past the edges.

Friis transmission equation

The Friis equation predicts the received power in a free-space radio link. It ties together transmit power, antenna gains, frequency, and distance into a single expression that forms the backbone of link budget analysis.

Free-space path loss

As a wave propagates outward from a source, its power density drops with the square of distance. Free-space path loss (FSPL) captures this geometric spreading:

$\text{FSPL} = \left(\frac{4\pi d}{\lambda}\right)^2$

In dB:

$\text{FSPL}_{\text{dB}} = 20\log_{10}\!\left(\frac{4\pi d}{\lambda}\right)$

Note that FSPL increases with frequency (smaller $\lambda$). This is not because the atmosphere absorbs more at higher frequencies; it's because a fixed-gain receive antenna has a smaller effective aperture at shorter wavelengths.

The Friis equation

The received power is:

$P_r = P_t \, G_t \, G_r \left(\frac{\lambda}{4\pi d}\right)^2$

where:

• $P_t$ = transmitted power
• $G_t$ = transmit antenna gain
• $G_r$ = receive antenna gain
• $\lambda$ = wavelength
• $d$ = distance between antennas

In dB form:

$P_{r,\text{dB}} = P_{t,\text{dB}} + G_{t,\text{dB}} + G_{r,\text{dB}} - \text{FSPL}_{\text{dB}}$

This equation assumes free-space propagation (no reflections, diffraction, or atmospheric absorption), matched polarization, and antennas pointed at each other.

Link budget analysis using the Friis equation

A link budget tallies every gain and loss from transmitter to receiver:

1. Start with the transmit power $P_t$ (in dBm or dBW).
2. Add the transmit antenna gain $G_t$ (dBi).
3. Subtract the free-space path loss (dB).
4. Add the receive antenna gain $G_r$ (dBi).
5. Subtract any additional losses (cable loss, atmospheric absorption, polarization mismatch, pointing error).
6. Compare the resulting received power to the receiver sensitivity to find the link margin.

A positive link margin means the link closes; a negative margin means communication fails. Designers typically aim for several dB of margin to account for fading and other real-world impairments.

Antenna arrays

Antenna arrays combine multiple radiating elements to achieve radiation characteristics that a single element cannot provide, including higher gain, narrower beams, and electronic beam steering.

Array factor

The array factor (AF) describes the contribution of the array geometry and excitation to the overall radiation pattern. The total far-field pattern of an array is:

$E_{\text{total}}(\theta, \phi) = E_{\text{element}}(\theta, \phi) \times AF(\theta, \phi)$

This is the pattern multiplication principle: the array pattern equals the single-element pattern multiplied by the array factor. The AF depends on the number of elements, their spacing, and the amplitude and phase of each element's excitation.

Uniform linear arrays

A uniform linear array (ULA) places $N$ identical elements along a line with equal spacing $d$. For equal-amplitude excitation with progressive phase shift $\beta$ between adjacent elements, the array factor is:

$AF = \frac{\sin\!\left(\frac{N\psi}{2}\right)}{N\sin\!\left(\frac{\psi}{2}\right)}, \quad \psi = kd\cos\theta + \beta$

where $k = 2\pi/\lambda$. The main beam points in the direction where $\psi = 0$, and the beamwidth narrows as $N$ or $d/\lambda$ increases.

Phased arrays and beam steering

In a phased array, the progressive phase shift $\beta$ is adjusted electronically, steering the main beam to a desired angle $\theta_0$ without physically rotating the antenna:

$\beta = -kd\cos\theta_0$

This enables rapid, inertia-free beam steering. Phased arrays are central to modern radar, satellite communications, and 5G millimeter-wave base stations, where the beam must be redirected on millisecond timescales.

Gain enhancement using arrays

For an $N$-element array with equal excitation and optimal spacing, the maximum directivity scales roughly as $N$ times the single-element directivity. In dB:

$G_{\text{array, dB}} \approx G_{\text{element, dB}} + 10\log_{10}(N)$

Doubling the number of elements adds about 3 dB of gain. In practice, mutual coupling between elements, non-uniform excitation, and feed network losses reduce the realized gain below this ideal limit.

Measurement techniques

Accurate measurement of gain, directivity, and radiation patterns is necessary to verify designs and ensure antennas meet specifications.

Gain measurement methods

• Two-antenna method: Two identical antennas face each other at a known separation. You measure the received power and apply the Friis equation. Since both antennas are identical, the single unknown gain can be solved directly.
• Three-antenna method: Three different antennas are measured in pairwise combinations (A-B, A-C, B-C). The three Friis equations yield three unknowns, giving the gain of each antenna without needing a pre-calibrated reference.
• Gain-comparison method: The antenna under test is compared against a standard gain horn with a known, calibrated gain. The difference in received power directly gives the gain difference.

Directivity measurement methods

• Pattern integration: Measure the full 3D radiation pattern, then numerically integrate the power density over all solid angles to find total radiated power. Directivity follows from the ratio of peak intensity to the average.
• Directivity comparison: Compare the antenna under test to a reference antenna with known directivity, similar to the gain-comparison approach but focused on the pattern shape rather than absolute power.

Anechoic chambers and test ranges

An anechoic chamber is a shielded room lined with RF-absorbing material (typically pyramidal foam absorbers or ferrite tiles) on all interior surfaces. This suppresses reflections and external interference, approximating free-space conditions indoors.

For antennas that are too large for a chamber, or for lower frequencies where absorber performance degrades, outdoor test ranges are used. Elevated ranges place antennas on towers to minimize ground reflections. Compact ranges use a large parabolic reflector to create a planar wave front in a shorter distance, allowing far-field measurements in a smaller indoor space.