Antennas formulas

$D$ = largest dimension of the antenna (a length here, not the directivity). At $r=2D^2/\lambda$ the path-length difference between the centre and the edge of the antenna is $\lambda /16$ (phase error $\pi /8$). For an antenna large compared with the wavelength, the reactive near field extends to about $r=0.62\sqrt{D^3/\lambda }$ and the radiating near field from there to $2D^2/\lambda$. For an electrically small antenna ($D\ll \lambda$) the binding condition is $r\gg \lambda$; the radiation and induction terms of a Hertzian dipole are equal in magnitude at $r=\lambda /(2\pi )$.

The middle form holds in the far field only, where $E$ and $H$ are in phase and $|E|=\eta |H|$. $E$ is a peak phasor. In a lossless medium $P_{\text{rad}}$ is the same through every closed surface that encloses the antenna.

Valid for $\beta r\gg 1$. In free space $|E_{\theta }|=\frac{60\pi I_{0}dl}{\lambda r}\sin \theta$ V/m. The pattern is $\propto \sin \theta$: maximum broadside ($\theta =90^{\circ }$), nulls along the axis ($\theta =0^{\circ },180^{\circ }$), and the same for every $\varphi$ (omnidirectional in the plane $\theta =90^{\circ }$).

The numerical forms assume free space ($\eta =120\pi$); $I_0$ is the peak current. $R_{\text{rad}}$ grows as $(dl/\lambda {)}^2$, which is why electrically short antennas are poor radiators: a small $R_{\text{rad}}$ competes with the ohmic loss resistance.

Centre-fed wire of total length $l\ll \lambda$ (roughly $l\le \lambda /10$) with a triangular current: $I_0$ at the feed, zero at the ends. Its average current is $I_0/2$, so its far field is half that of a Hertzian dipole of the same length with current $I_0$: power and $R_{\text{rad}}$ are one quarter. The pattern ($\sin \theta$), directivity (1.5) and beamwidth ($90^{\circ }$) are unchanged.

Units W/sr. The element of solid angle is $d\Omega =\sin \theta d\theta d\varphi$; dropping the $\sin \theta$ is an easy error to make.

A pure ratio set by the shape of the pattern alone; losses do not enter. 'Directivity' with no direction stated means the maximum, $D_0$. dBi means decibels relative to an isotropic radiator; values relative to a half-wave dipole (dBd) are 2.15 dB lower.

$F_n$ is the normalized power pattern ($=|E{|}^2/|E{|}_{\max }^2$). ${\Omega }_A$ is the solid angle through which all the power would flow if $U$ stayed at $U_{\max }$ inside it and were zero outside. The approximation uses the half-power beamwidths ${\Theta }_1$ and ${\Theta }_2$ in the two principal planes, in radians, and holds only for a single narrow main lobe with small side lobes.

$41253=4\pi (180/\pi {)}^2$. It ignores the power in side lobes and so tends to overestimate $D_0$. Do not use it for broad patterns such as a dipole's; integrate instead.

$P_{\text{in}}$ = power accepted at the antenna terminals; $R_{\text{loss}}$ is referred to the same current as $R_{\text{rad}}$. $G\le D$, with equality only for a lossless antenna. Gain as defined here excludes impedance mismatch. If the feed line of impedance $Z_0$ is mismatched to the antenna impedance $Z_A$, the realized gain is $(1-|\Gamma {|}^2)G$ with $\Gamma =(Z_A-Z_0)/(Z_A+Z_0)$.

For a receiving antenna matched to its load and polarization-matched to an incident wave of power density $S_{\text{inc}}$ arriving from the direction in which the gain is $G$. For a lossless antenna $A_e={\lambda }^{2}D/(4\pi )$, the maximum effective aperture $A_{em}$. Hertzian or short dipole: $3{\lambda }^2/(8\pi )\approx 0.119{\lambda }^2$; half-wave dipole: $\approx 0.13{\lambda }^2$. For a large aperture antenna (horn, reflector) of physical area $A_p$, $A_e\le A_p$, with equality only for uniform, in-phase illumination of a lossless aperture; the ratio $A_e/A_p$ is the aperture efficiency.

Thin wire with the sinusoidal current above, far field. $l=\lambda /2$ gives the half-wave result below. For $l=\lambda$ the pattern is $[1+\cos (\pi \cos \theta )]/\sin \theta$: $D_0\approx 2.41$ (3.82 dBi), HPBW $\approx 48^{\circ }$, and $R_{\text{rad}}\approx 199\Omega$ referred to $I_m$. The broadside directivity peaks near $l=1.25\lambda$ ($D_0\approx 3.28$); longer dipoles break into more lobes and the broadside lobe shrinks.

Thin wire, free space for the second form, $I_0$ = peak current at the feed. Nulls along the wire axis, maximum at $\theta =90^{\circ }$, omnidirectional in $\varphi$; the beam is slightly narrower than the Hertzian dipole's $\sin \theta$.

In free space $R_{\text{rad}}=\frac{\eta }{4\pi }\mathrm{Cin}(2\pi )=30\times 2.4377\approx 73.1\Omega$, where $\mathrm{Cin}(x)={\int }_0^x\frac{1-\cos t}{t}dt$, and $D_0=\frac{\eta }{\pi R_{\text{rad}}}=\frac{120}{R_{\text{rad}}}\approx 1.64$. $P_{\text{rad}}=\frac{1}{2}{I}_0^2{R}_{\text{rad}}\approx 36.5I_0^2$ W. The feed sits at a current maximum, so $R_{\text{rad}}$ is also the input resistance. The half-power points are at $\theta \approx 51^{\circ }$ and $129^{\circ }$. The $+j42.5\Omega$ reactance is tuned out by making the dipole slightly shorter than $\lambda /2$ (typically about $0.47\lambda$ to $0.49\lambda$, shorter for thicker wire); the resistance at resonance is then slightly below 73 Ω.

Monopole of height $h$ over an infinite, perfectly conducting ground plane, compared with a dipole of total length $2h$ with the same current distribution. Same $U_{\max }$, half the power, hence twice the directivity. A finite or lossy ground changes these values.

$h=\lambda /4$ over an infinite perfect ground plane: half the half-wave dipole's impedance, twice its directivity. The pattern above the ground is the half-wave dipole's, with its maximum along the ground plane ($\theta =90^{\circ }$) and half power at $\theta \approx 51^{\circ }$, i.e. about $39^{\circ }$ above the ground plane.

Conditions: each antenna in the other's far field, free space with no obstacles or multipath, both antennas matched to their lines and to each other in polarization, and $G_t$, $G_r$ taken in the direction of the other antenna. Use gains, not directivities, when the antennas have loss. $(4\pi R/\lambda {)}^2$ is the free-space path loss.

$P(\text{dBm})=10{\log }_{10}(P/1\text{mW})$. The constant 32.44 dB uses $c=3\times 10^8$ m/s (the exact speed of light gives 32.45 dB). With fixed gains, $L_{\text{FS}}$ rises by 6 dB for every doubling of distance or of frequency. With fixed effective apertures instead, the received power rises with frequency, because ${\lambda }^2$ then sits in the denominator.

$N$ identical, identically oriented elements on the $z$-axis at spacing $d$, equal amplitudes, element $n$ fed with $I_0{e}^{jn\alpha }$ ($\alpha$ = progressive phase, positive when each element leads the one before it). $\theta$ is measured from the array axis; for an array along another axis use the cosine of the angle from that axis (for the $x$-axis, $\sin \theta \cos \varphi$). $|AF|$ has its maximum value $N$ at $\psi =0$, and again at $\psi =\pm 2\pi ,\pm 4\pi ,\dots$ if those values are reached.

Two equal-amplitude elements, the one at $z=d$ leading the one at $z=0$ by $\alpha$. Nulls where $\psi =\pm \pi$, provided that value lies in the visible range $\alpha -\beta d\le \psi \le \alpha +\beta d$ (for example, $d=\lambda /4$ with $\alpha =0$ has no null at all). With unequal amplitudes, $I_2/I_1=k$, $|AF|=|1+k{e}^{j\psi }|$, whose minimum (reached where $\psi =\pm \pi$ is visible) is $|1-k|$: there is no complete null unless $k=1$.

Set $\psi =0$ in the wanted direction; a real beam direction exists only if $|\alpha |\le \beta d$. With the $e^{j\omega t}$ convention the beam leans toward the end whose elements lag: for end-fire along $+z$, each element lags the one before it by $\beta d$, exactly the phase a wave travelling along $+z$ picks up between elements. Varying $\alpha$ electronically steers the beam (a phased array).

Solve $\beta d\cos \theta +\alpha ={\psi }_{\text{null}}$ for $\theta$ and keep only solutions with $|\cos \theta |\le 1$. Small arrays have higher side lobes (about $-11.3$ dB for $N=4$). With uniform amplitudes, changing $d$ or $\alpha$ only moves the side lobes in angle (or out of the visible range); lowering their level needs a tapered amplitude distribution.

Uniform array with $\alpha =0$; $m=1,2,\dots$ with $m\lambda /(Nd)\le 1$ and $m$ not a multiple of $N$. The approximate forms (in radians unless marked) need $Nd\gg \lambda$. The beamwidth depends on the array length $Nd$, not on $N$ alone.

Uniform array with $\alpha =-\beta d$, beam toward $\theta =0^{\circ }$; $m=1,2,\dots$ with $m\lambda /(Nd)\le 2$ and $m$ not a multiple of $N$. The square-root approximations (radians) need $Nd\gg \lambda$. For the same length, the end-fire beam is much wider than the broadside beam in the plane containing the axis, but it is a pencil beam around the axis instead of a fan all round it.

A grating lobe is a second full maximum ($|AF|=N$), where $\psi =\pm 2\pi$. As $\theta$ runs from $0^{\circ }$ to $180^{\circ }$, $\psi$ covers the visible range $\alpha -\beta d\le \psi \le \alpha +\beta d$; there is no grating lobe if this range contains no multiple of $2\pi$ other than the main beam's $\psi =0$. At exactly $d=\lambda$ (broadside) grating lobes appear along the axis; at exactly $d=\lambda /2$ (end-fire) a second beam appears at $\theta =180^{\circ }$.

Valid for $Nd\gg \lambda$ and no grating lobes. For a broadside array with $d=\lambda /2$, $D_0=N$ exactly, for any $N$ (see the exact formula below).

For identical, identically oriented elements; the array factor carries the feed amplitudes and phases, and mutual coupling is ignored. In dB the two patterns add. A null of either factor is a null of the total, and the total maximum lies where the product peaks, which need not be at the maximum of either factor.