Antenna (radio)

An antenna couples electrical circuits to propagating electromagnetic radiation for transmission or reception.

The antenna directivity compares angular radiated power $U=d{P}_{\mathrm{rad}}/d\Omega$ with its average over directions. Its integral is $4\pi$. Unlike antenna gain, it uses radiated power rather than accepted input power, so it omits dissipative losses. A narrow beam can have high directivity without a larger total radiated power.

A phased array combines radiation from spatially separated antennas with controlled relative phases. Changing those phases steers the direction of constructive interference without moving the elements. For an equally spaced linear array with spacing $d$, free-space wavenumber $k$ and phase increment $\psi$, a principal beam at angle ${\theta }_0$ from the array normal satisfies $\psi =kd\sin {\theta }_0$ when the time dependence is $e^{-i\omega t}$ and the element drives are $e^{ij\psi }$.

A grating lobe is an additional principal maximum caused by spatial sampling in a phased array. A regularly spaced linear array steered towards ${\theta }_0$ also has maxima wherever $\sin \theta =\sin {\theta }_0+m\lambda /d$ lies in $[-1,1]$. Taking $d<\lambda /2$ excludes such maxima for all steering angles; $d=\lambda /2$ is the usual limiting choice, with an endfire degeneracy at the extreme scan directions.

The array factor describes the interference between the elements of a phased array, separately from the radiation pattern of each element. For $N$ identical equally spaced elements,At $\chi =2\pi m$ the limiting value has magnitude $N$. The total intensity is proportional to $|F{|}^2$ times the single-element antenna beam pattern.

Radiation resistance represents emitted electromagnetic power as a real input resistance of an antenna. It does not describe conversion to heat inside the conductor. If peak current amplitude is used, $P_{\mathrm{rad}}=|I_0{|}^2{R}_{\mathrm{rad}}/2$. Real ohmic loss adds a separate resistance and lowers radiation efficiency. For an ideal short uniform-current element of length $d$, the Hertzian dipole gives $R_{\mathrm{rad}}=80{\pi }^2(d/\lambda {)}^2$ in ohms, using the usual vacuum impedance approximation $120\pi \Omega$.

A fixed electrically small loop of area A with peak sinusoidal current $I_0\cos \omega t$ has magnetic moment $m=AI$. The magnetic dipole radiation law gives mean power ${\mu }_0{A}^2{I}_0^2{\omega }^4/(12\pi c^3)$. Defining it as $I_0^2{R}_{\mathrm{rad}}/2$ yields $R_{\mathrm{rad}}={\mu }_0{A}^2{\omega }^4/(6\pi c^3)$. This equivalent resistance accounts for radiation, not heating in the conductor. Rotating the loop changes its vector moment and cannot in general be modelled by the same scalar current alone.

The Friis transmission equation describes a far-field free-space antenna link with aligned directions, matched polarizations and negligible additional loss. It follows by multiplying the transmitter's axial energy flux $P_t{G}_t/(4\pi d^2)$ by the receiver's antenna effective area ${\lambda }^2{G}_r/(4\pi )$. A receiving impedance mismatch multiplies the available received power by $1-|{\Gamma }_L{|}^2$. Wavelength scaling must state which gains, apertures and impedances are held fixed.

Two aligned lossless Hertzian dipoles in one another's broadside radiation zone have gain $G=3/2$ and one-way received-power fraction $\eta =G^2[\lambda /(4\pi r){]}^2$. If the second remains conjugately matched, its matched-load scattering of a Hertzian dipole reradiates the same power as its load absorbs. Applying the reciprocal far-field link on the return gives $P_{\mathrm{echo}}=P_t{\eta }^2$, proportional to $r^{-4}$. The statement concerns physical dipole reradiation; a pure antenna load-mismatch echo model omits that scattering and is not interchangeable with it.

In an ideal matched-port two-port network for an antenna pair, the incident amplitude traverses the link, reflects from the receiver's load with amplitude coefficient ${\Gamma }_L$, then traverses the reciprocal link again. If each one-way power transfer factor is $\mathcal{F}$, the returned power fraction is $|{\Gamma }_L{|}^2{\mathcal{F}}^2$. With fixed gains, $\mathcal{F}\propto {\lambda }^2$ and this echo scales as ${\lambda }^4$. Actual antenna structural scattering or an existing input mismatch can add a coherent return not specified by the Friis transmission equation alone.

An antenna aperture describes the collecting area associated with a receiving antenna. The antenna effective area need not equal its geometric projected area.

The antenna effective area converts incident energy flux into available received power for a specified direction and matched polarization of an electromagnetic wave. In a lossless free-space link, a transmitter of antenna gain $G$ at range $r$ and a receiver of effective area $A_{\mathrm{eff}}$ have $P_r/P_t=G{A}_{\mathrm{eff}}/(4\pi r^2)$. Impedance or polarization mismatch reduces the delivered power.

For a reciprocal antenna with matched polarization and load, the available antenna effective area is proportional to the directional antenna gain. For a lossless antenna, a thermal equilibrium argument fixes the constant: one polarization of Rayleigh-Jeans law radiance gives incident power $(k_{B}T/{\lambda }^2)\int A_{\mathrm{eff}}d\Omega d\nu$, and matched thermal electrical noise gives $k_{B}Td\nu$. Thus $\int A_{\mathrm{eff}}d\Omega ={\lambda }^2$. Since $\int Gd\Omega =4\pi$, the displayed relation follows. Reciprocal losses multiply both area and gain by the same efficiency.

Antennas with effective areas $A_{e,1}$ and $A_{e,2}$ receiving the same polarization-matched plane wave have available powers in that ratio. If their matched-load resistances are equal, terminal voltage magnitudes have ratio $\sqrt{A_{e,1}/A_{e,2}}$. With unequal matched resistances the ratio becomes $\sqrt{A_{e,1}{R}_1/(A_{e,2}{R}_2)}$. Open-circuit voltages additionally depend on each antenna's source impedance. Geometry alone therefore fixes available-power estimates more directly than unspecified terminal voltages.

Antenna gain compares the far-field energy flux in a specified direction with the flux that an isotropic radiator would produce using the same input power. Thus $S_r=G{P}_{\mathrm{in}}/(4\pi r^2)$. Gain includes radiation efficiency; it must be distinguished from a voltage or current amplifier gain and from directivity when losses matter.

The fraction of accepted input power radiated by an antenna. With consistent feed definitions, antenna gain is radiation efficiency times antenna directivity. Ohmic losses reduce the efficiency, while feed mismatch is usually tracked separately.

The antenna temperature expresses the noise power delivered by a receiving antenna as an equivalent input noise temperature. It depends on the sky and surroundings weighted by the antenna beam pattern, as well as losses in the antenna; it need not be the physical temperature of either the antenna or a single astronomical source.

For a uniformly illuminated circular aperture of diameter D, normalized received power is $[2J_1(x)/x{]}^2$ with $x=\pi D\sin \theta /\lambda$. The first null has $\sin \theta =1.21967\lambda /D$ and the half-power point has $\sin \theta \simeq 0.5145\lambda /D$. A pointing tolerance must name the tolerated power loss; the first-null angle is not a good low-loss tolerance. These estimates require an ideal aperture pattern and are only approximate for a dish a few wavelengths across.

For a narrow angular power pattern $\exp [-(u^2+w^2)/(2{\alpha }^2)]$ about an equatorial beam centre, the solid angle measure is locally $dudw$. Extending the small lobe to the plane gives its beam solid angle ${\Omega }_A\simeq 2\pi {\alpha }^2$. Both sides of the azimuthal lobe must be included. A lossless antenna then has peak antenna gain $G_0\simeq 2/{\alpha }^2$ and peak antenna effective area $A_0\simeq {\lambda }^2/(2\pi {\alpha }^2)$.

A Hertzian dipole is an ideal electrically short oscillating electric dipole used to model radiation from a localized current element. In vacuum its far-field electric dipole radiation has ${\sin }^2\theta$ angular power distribution and total power $P={\mu }_0|\ddot{\mathbf{p}}{|}^2/(6\pi c)$, evaluated at the retarded time. The electric-dipole approximation for radiation requires a source much smaller than its radiation wavelength.

A uniform electrically short current element of length $\ell$ with sinusoidal peak current $I_0$ has electric dipole moment amplitude $p_0=I_0\ell /\omega$. The mean electric dipole radiation power is ${\omega }^4{p}_0^2/(12\pi {ϵ}_0{c}^3)$, so fixing current makes it proportional to ${\ell }^2{\omega }^2$, rather than ${\ell }^2{\omega }^4$. Current, charge excursion and dipole moment are different quantities to hold fixed when comparing antennas. The approximation requires $\ell \ll \lambda$ and does not impose the nonuniform standing-current profile of a long antenna.

For lossless parallel Hertzian transmitting and receiving dipoles separated by distance $r$ at angle $\theta$ to their common axis, directional gains are each $G=3{\sin }^2\theta /2$. The emitted polarization matches the receiving axis's transverse projection. Matched-load power is therefore $P_L=P[3{\sin }^2\theta /2{]}^2[\lambda /(4\pi r){]}^2$. One angular factor describes transmission and the other reception. This assumes reciprocal ideal dipoles, conjugate matching, far-field separation and negligible dissipative loss.

For a lossless ideal Hertzian dipole driven by an incident wave, use its induced open-circuit voltage and its feed impedance $R_{\mathrm{rad}}+iX$. A conjugately matched load has impedance $R_{\mathrm{rad}}-iX$. The same induced current therefore delivers equal power to the load and to radiation resistance: $P_{\mathrm{sca}}=P_{\mathrm{load}}$. The scattered wave has the usual dipole angular pattern. In direction of gain $G$, its scattering cross-section is $G{A}_{\mathrm{eff}}$, since the load receives $S_{\mathrm{inc}}{A}_{\mathrm{eff}}$.

This ideal single-mode scattering is not the same as a signal that travels through an already matched antenna port, reflects from a mismatched terminating load, and returns. A zero port reflection coefficient removes that latter echo but does not make the physical dipole invisible to the incident field. Additional structural scattering, ohmic loss or a different load changes the result.