Antennas, Gain & Directivity

An antenna converts a guided electromagnetic mode on a feed line into radiation in space—and, by reciprocity, converts an arriving field back into a guided signal. It does not create power. It redistributes power by direction and polarization, while conductor, dielectric, matching, feeder, and radome losses decide how much input power survives the conversion.

That distinction explains most antenna-number mistakes. Directivity describes only the shape of radiation. Gain also includes radiation efficiency. Realized gain additionally charges impedance mismatch. Link calculations fail when a peak simulated directivity is substituted for installed realized gain in the needed direction.

Pattern first, scalar second #

In the far field, radiation intensity $U(\theta,\phi)$ is power per unit solid angle. Total radiated power is

$$P_{rad}=\int_{4\pi}U(\theta,\phi)\,d\Omega.$$

Directivity in a direction is

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

and peak directivity is $D_0=\max D$. An isotropic radiator has $D=1$ in every direction; it is a reference, not a physically realizable finite antenna.

Main lobe

The angular region containing the desired peak; its width sets pointing tolerance and spatial selectivity.

Half-power beamwidth

The angle between points 3 dB below the power-pattern peak. It is not the null-to-null width.

Sidelobes

Unwanted maxima that transmit interference and admit off-axis noise or interferers.

Front-to-back ratio

Forward response divided by response in the opposite direction under the stated angular definition.

Null

A deep directional minimum. Small mechanical, frequency, coupling, or environment changes can move it, so an ideal null is a fragile interference strategy.

Beamwidth and directivity trade against each other because the pattern's average over $4\pi$ is fixed at one after normalization. Concentrating more energy into a narrow main lobe necessarily reduces it elsewhere, apart from losses.

Directivity, gain, and realized gain #

Let radiation efficiency be

$$\eta_{rad}=\frac{P_{rad}}{P_{accepted}},$$

where accepted power excludes what the input mismatch reflects. Gain is

$$G(\theta,\phi)=\eta_{rad}D(\theta,\phi).$$

For feed impedance $Z_A$ connected to reference impedance $Z_0$, the reflection coefficient is

$$\Gamma=\frac{Z_A-Z_0}{Z_A+Z_0},$$

and mismatch efficiency is $\eta_m=1-|\Gamma|^2$. Realized gain is therefore

$$G_r=\eta_m\eta_{rad}D.$$

A 10 dB return loss means $|\Gamma|=10^{-10/20}=0.316$, so $\eta_m=0.90$: 90% of incident feed power is accepted, a 0.46 dB mismatch loss. A good return loss still says nothing about where the accepted power went; it can be radiated, dissipated, coupled into a chassis, or launched in an unwanted mode.

dBi, dBd, and the half-wave dipole #

A gain in dBi is referenced to an isotropic radiator:

$$G_{dBi}=10\log_{10}G.$$

An ideal thin half-wave dipole has peak directivity

$$D_0=\frac{4}{\mathop{\mathrm{Cin}}(2\pi)}=1.640922,$$

or

$$10\log_{10}(1.640922)=2.151\ \text{dBi}.$$

Therefore a dBd value, referenced to that dipole peak, converts as

$$G_{dBd}=G_{dBi}-2.15.$$

This conversion assumes the same polarization, direction, and gain definition. “6 dB antenna” is incomplete unless the reference and whether the value is directivity, gain, or realized gain are stated.

Receive aperture is the same physics #

An antenna receiving a plane wave of power flux density $S$ delivers available power

$$P_{av}=SA_e,$$

where effective aperture is related to gain by

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

This is not the physical outline of the antenna. A small resonant antenna can have an effective aperture larger than its silhouette; an aperture antenna's effective area is its physical area multiplied by aperture efficiency.

Combine isotropic spreading from a transmitting antenna with receive aperture. At distance $R$, power flux in the transmitter's direction is

$$S=\frac{P_tG_t}{4\pi R^2}.$$

Multiplying by $A_e$ gives the Friis equation

$$P_r=P_tG_tG_r\left(\frac{\lambda}{4\pi R}\right)^2.$$

Its assumptions matter: unobstructed far-field propagation, conjugate matching, aligned polarization, and gains evaluated in the actual departure/arrival directions. Free-space attenuation is not vacuum absorption; the $R^{-2}$ term is spherical spreading, while the wavelength term enters through the receiving aperture for fixed gain.

Polarization is a vector match #

Polarization describes the time trajectory of the far-field electric-field vector in a specified direction. Linear, circular, and elliptical are states of the same two-component field. For normalized transmit and receive polarization vectors $\hat e_t$ and $\hat e_r$, polarization coupling is

$$\eta_p=|\hat e_r^*\cdot\hat e_t|^2.$$

Two linear antennas misaligned by angle $\psi$ have $\eta_p=\cos^2\psi$: 45° costs 3.01 dB and 90° is an ideal null. Ideal opposite-hand circular polarizations are orthogonal, but real antennas have finite axial ratio, pattern variation, multipath, and installation errors, so cross-polar isolation is finite.

Polarization is direction-dependent. A linearly polarized element viewed off its principal direction can develop cross-polar components; a reflected path can rotate or reverse handedness. A link budget should use polarization mismatch for the actual propagation geometry, not only connector labels.

Field zones decide what can be measured #

Near an antenna, reactive electric and magnetic energy is stored and returned each cycle; the field is not a simple outgoing plane wave. Far enough away, angular pattern is essentially independent of distance and fields scale as $1/R$.

For a maximum antenna dimension $D$ with $D$ large relative to wavelength, common engineering boundaries are

$$R_{reactive}\lesssim0.62\sqrt{\frac{D^3}{\lambda}}, \qquad R_{far}\gtrsim\frac{2D^2}{\lambda}.$$

Between them lies the radiating near field (Fresnel region). These are test-range criteria, not laws that erase all errors at one radius. Electrically small antennas, scanning arrays, high-accuracy phase measurements, and large quiet-zone requirements need a stricter electromagnetic analysis.

Far-field measurements can demand an impractically long range. Near-field scanners instead sample amplitude and phase over a plane, cylinder, or sphere and mathematically transform the data to the far field. NIST gain calibration also uses three-antenna and extrapolation methods so gain and polarization can be determined without assuming a perfect reference antenna.

Arrays shape the pattern by phase #

For elements at positions $\mathbf r_n$ with complex weights $w_n$, the array factor in direction $\hat u$ is

$$AF(\hat u)=\sum_nw_ne^{jk\mathbf r_n\cdot\hat u},\qquad k=\frac{2\pi}{\lambda}.$$

Progressive phase shifts steer constructive interference toward one direction and destructive interference elsewhere. Amplitude taper reduces sidelobes but widens the main beam and usually reduces peak directivity. There is no taper that improves every pattern metric simultaneously.

Element spacing creates spatial aliasing. Spacings above roughly $\lambda/2$ can produce grating lobes when scanning broadly: multiple visible directions acquire the same inter-element phase progression. Half-wavelength spacing is a conservative wide-scan rule, not a universal optimum; mutual coupling, element size, scan range, bandwidth, and desired sidelobe level also constrain it.

Wideband arrays have another failure: phase shifters provide a fixed phase, equivalent to the desired time delay at only one frequency. Across a wide band the beam direction changes with frequency (beam squint). True-time-delay networks or subband processing address that at additional cost.

Installation turns an antenna into a system #

The installed antenna pattern includes mounting structure, feed network, enclosure, radome, and nearby platform. A catalog pattern measured on an isolated fixture is a component property, not necessarily the deployed system property.

Measurement and verification #

1. State frequency, polarization, coordinate system, reference impedance, and whether the reported quantity is directivity, gain, or realized gain.
2. Calibrate or de-embed cables, adapters, baluns, and chamber path; record uncertainty rather than only a smooth trace.
3. Measure input reflection over the full operating band and environment, but do not infer radiation efficiency from it.
4. Measure co- and cross-polar patterns over enough solid angle to integrate efficiency and find unintended lobes.
5. Satisfy far-field/quiet-zone criteria or use a validated near-field transform; suppress or time-gate range reflections where justified.
6. For arrays, test every steering angle and frequency edge; inspect element failures, quantized weights, scan loss, grating lobes, and beam squint.
7. Insert the measured directional realized gains and polarization loss into the link budget, then compare predicted and received power at multiple ranges.

An antenna is verified only when match, efficiency, full directional/polarization pattern, and installed geometry are all tied to the same frequency and configuration. Peak dBi by itself proves none of the other three.