Antenna Fundamentals and Radiation

A constant current produces a static magnetic field that wraps around the wire and stays put — no energy leaves. A current that oscillates is different: it accelerates charge, and accelerating charge radiates. Part of the field near the wire still stores and returns energy each cycle (the reactive near field), but a portion detaches, forms closed loops that pinch off, and propagates away as a self-sustaining electromagnetic wave that never comes back. That escaping part is radiation, and an antenna is simply a structure shaped to maximize it — a transducer between the guided waves of a line and the free-space waves of Part 5.

To compute it we need a tool that respects the finite speed of light: the field at a distant point cannot reflect what the current is doing now, only what it did a travel-time ago.

The magnetic vector potential of a current distribution, generalized to time-varying sources, is the retarded potential: each contribution is evaluated not at the present instant but at the earlier time \(t-R/u\), where \(R\) is the distance from the source element and \(u\) the wave speed. For a time-harmonic current this retardation becomes a phase factor \(e^{-j\beta R}\):

Once \(\vec{A}\) is known, the radiated fields follow from \(\vec{H}=\tfrac{1}{\mu}\nabla\times\vec{A}\) and Maxwell's curl equation for \(\vec{E}\). The single phase factor \(e^{-j\beta R}\) is the whole origin of radiation: without retardation the potential would be the static one and nothing would propagate. Apply this to the simplest possible source and the entire theory of antennas unfolds.

The Hertzian dipole is an infinitesimal current element: a short length \(dl\ll\lambda\) carrying a uniform oscillating current \(I_0\). It is to antennas what the point charge is to electrostatics — the elementary building block from which any real antenna can be assembled by superposition. Evaluating the retarded potential for it and taking the curls produces fields with terms in \(1/r\), \(1/r^2\), and \(1/r^3\). Far from the antenna (\(r\gg\lambda\)) only the \(1/r\) terms survive, and they are the radiation:

In the far field the radiated wave is locally a plane wave: \(E_\theta\) and \(H_\phi\) are perpendicular, in phase, and in the ratio \(\eta_0\), travelling radially outward. The reactive near field (\(r\ll\lambda\), the \(1/r^2\) and \(1/r^3\) terms) stores energy without radiating; the boundary between the two regions sits roughly at \(r=2D^2/\lambda\) for an antenna of size \(D\).

The far field carries a single angular factor, \(\sin\theta\): the dipole radiates most strongly broadside (\(\theta=90^\circ\), perpendicular to its axis) and not at all along its own axis (\(\theta=0,180^\circ\)). Plotted in three dimensions this is a doughnut wrapped around the wire; in any plane containing the axis it is the familiar figure-of-eight:

The squared factor \(\sin^2\theta\) is the power pattern. Its half-power beamwidth is a broad \(90^\circ\), so the short dipole spreads its energy widely — useful for coverage, poor for reaching far. Sharpening that pattern, by making the antenna larger or arraying many of them, is the route to directivity, taken up shortly and in the next chapter.

Integrating the time-average Poynting vector \(S_{\text{avg}}=|E_\theta|^2/2\eta_0\) over a large sphere gives the total radiated power. The \(\sin^2\theta\) pattern integrates to a clean constant, leaving:

To the feeding circuit, this lost power looks exactly like power dissipated in a resistor. Defining \(P_{\text{rad}}=\tfrac12 I_0^2 R_{\text{rad}}\) gives the radiation resistance — a fictitious resistance that accounts for radiated (not dissipated) power, and the single most important number for matching an antenna to its feed:

The quadratic dependence on \(dl/\lambda\) is decisive: a very short element has a tiny radiation resistance, so most of the drive current is wasted in ohmic loss rather than radiated — which is exactly why useful antennas are a sizeable fraction of a wavelength, not arbitrarily small.

How sharply an antenna concentrates its radiation is captured by a few standard parameters. The radiation intensity \(U=r^2 S_{\text{avg}}\) is power per unit solid angle (independent of \(r\)). The directivity compares the peak intensity to the average — the factor by which the antenna beats an isotropic radiator in its best direction:

Directivity is a property of the pattern alone. The gain folds in real losses through the radiation efficiency \(e\) (= \(R_{\text{rad}}/(R_{\text{rad}}+R_{\text{loss}})\)), and the effective aperture \(A_e\) — the equivalent collecting area when receiving — is tied to gain by a universal relation:

Directivities are quoted in dBi (decibels over isotropic). The Hertzian dipole has \(D=1.5\) (1.76 dBi) — barely directional. Larger and arrayed antennas reach tens of thousands (dish antennas exceed 40 dBi). Note that an antenna's transmit and receive properties are identical (reciprocity), so the same gain describes it in both roles.

The practical workhorse is the half-wave dipole, a straight wire of total length \(\lambda/2\) fed at its centre. Its current is a standing-wave cosine, peaking at the feed and vanishing at the tips, and integrating its far field gives a slightly sharper pattern than the short dipole. Two numbers define it:

Finally, the link between two antennas a distance \(R\) apart in free space is the Friis transmission equation — the foundation of every radio link budget. The received power depends on both antenna gains and an inverse-square spreading term that grows with frequency:

The bracketed factor is the free-space path loss: signals fade as \(1/R^2\) simply because the radiated power spreads over an ever-larger sphere. Rearranged in decibels it becomes the everyday link equation \(P_r=P_t+G_t+G_r-20\log_{10}(4\pi R/\lambda)\), which sizes everything from a phone link to a deep-space probe.

Track whether a current is the peak or rms value, and keep lengths in wavelengths. Convert to and from decibels confidently. Difficulty rises down the list.