Antennas and Radiation — From the Hertzian Dipole to Phased Arrays

1. The Hertzian Dipole: Radiation from an Accelerating Charge

A stationary charge produces only a static electric field. A charge moving at constant velocity produces a static field in its own rest frame plus a magnetic field — but still no radiation. Radiation only occurs when a charge accelerates. This single fact is the foundation of every antenna ever built: from Heinrich Hertz's spark-gap transmitter in 1887 to a modern 5G base station, radiation is produced by driving alternating current through a conductor, which continuously accelerates and decelerates the free electrons within it.

The simplest radiating structure is the Hertzian dipole (also called an infinitesimal or elementary dipole): a short straight wire of length dl, much shorter than the wavelength, carrying a uniform time-harmonic current I(t) = I₀cos(ωt). Solving Maxwell's equations for this current distribution gives the retarded vector potential and, from it, the radiated electric field in the far zone:

Two features of this expression define all antenna behaviour. First, the field falls off as 1/r, not 1/r² as electrostatic or magnetostatic fields do — this slower decay is what allows radiated power to propagate to infinity rather than remaining bound near the source. Second, the field is proportional to sinθ: radiation is strongest broadside to the dipole (θ = 90°) and vanishes exactly along the dipole's own axis (θ = 0° or 180°). A dipole radiates no energy off the ends of the wire.

The time-averaged Poynting vector (radiated power per unit area) follows from E and the associated magnetic field H = E/η₀:

Integrating S_avg over a sphere of radius r gives the total radiated power, which is independent of r (as it must be — energy is conserved as the wavefront expands), confirming that the Hertzian dipole genuinely radiates propagating energy rather than merely storing it in a reactive near field.

2. Near Field, Far Field, and the Radiation Zone

The exact field of any antenna contains terms that scale as 1/r, 1/r², and 1/r³. Close to the antenna, all three terms matter and the field structure is complicated; far away, the 1/r² and 1/r³ terms become negligible and only the radiating 1/r term survives. This separates space around an antenna into three regions:

• Reactive near field (r < λ/2π): energy sloshes back and forth between the antenna and the surrounding space each cycle rather than propagating away — it behaves like the field of a capacitor or inductor. This stored (reactive) energy determines the antenna's input reactance.
• Radiating near field / Fresnel region: the field pattern still depends on distance from the antenna, not just angle; this region matters for antenna measurement setups and for large aperture antennas such as radar dishes.
• Far field / Fraunhofer region (r > 2D²/λ, where D is the largest antenna dimension): the angular shape of the radiation pattern becomes independent of distance, and the field decays as a clean 1/r with E and H in phase and perpendicular, exactly like a plane wave locally.

This boundary is why antenna ranges (facilities for measuring radiation patterns) must place the receiving probe far enough away, and why compact antenna test ranges use large parabolic reflectors to synthesize a plane wave at short physical distances.

The Poynting vector in the far field points radially outward everywhere — energy streams away from the antenna and never returns, in contrast to the reactive near field where energy genuinely oscillates back into the source twice per cycle. An antenna's input impedance Z_in = R_rad + jX combines a real part (power actually radiated away, discussed in Section 4) with an imaginary reactance part (energy stored in the near field, exactly analogous to a lumped inductor or capacitor).

3. Radiation Pattern and Directivity

The radiation pattern is a plot of radiated field strength (or power density) as a function of direction (θ, φ) at a fixed distance in the far field. For the Hertzian dipole this is the familiar figure-eight (torus in 3D) proportional to sinθ, with a null along the wire axis and a broad maximum in the plane perpendicular to it.

Real antennas concentrate power into a preferred direction rather than radiating uniformly, and directivity D quantifies exactly how much. It is the ratio of the maximum radiation intensity of the actual antenna to the radiation intensity of a hypothetical isotropic radiator carrying the same total power:

For the Hertzian dipole, D = 1.5 (about 1.76 dBi — decibels relative to an isotropic radiator). Larger, more directional antennas achieve much higher directivity: a typical satellite-TV dish reaches 35–40 dBi, meaning it concentrates power roughly 3,000–10,000× more effectively than an isotropic source in its main beam direction.

The pattern is described by several standard parameters:

• Main lobe: the direction of maximum radiation.
• Half-power beamwidth (HPBW): the angular width of the main lobe between the two points where power drops to half (−3 dB) of its peak value.
• Side lobes: secondary maxima away from the main lobe; lower side-lobe levels reduce interference and improve radar/communications performance.
• Front-to-back ratio: the ratio of power radiated forward versus directly behind the antenna, important for minimizing unwanted pickup or transmission.

There is a fundamental trade-off, captured approximately by the relation D ≈ 4π/(θ_HPBW · φ_HPBW) (beamwidths in radians): narrower beams require larger antenna apertures (measured in wavelengths) and yield higher directivity. This is why high-gain microwave links use large dishes, while a Wi-Fi router uses a nearly omnidirectional low-gain antenna to cover a room in all directions.

4. Radiation Resistance and Antenna Efficiency

From a circuit perspective, an antenna looks like a load with an input impedance Z_in = R_rad + R_loss + jX. The real part of this impedance has two components: radiation resistance R_rad, an equivalent resistance that would dissipate the same power as is actually radiated away as electromagnetic waves, and loss resistance R_loss, representing genuine ohmic heating in the conductor and any surrounding dielectric.

For the Hertzian dipole, radiation resistance can be derived directly from the total radiated power P_rad = ½I₀²R_rad:

Because dl/λ << 1 for an electrically short dipole, R_rad is tiny — often a fraction of an ohm — which is why very short antennas (like the stub antenna on an early car radio) are inefficient: nearly all of the input power is dissipated as heat in the finite conductor and matching-network losses rather than radiated. Radiation efficiency is defined as:

This is why practical antennas are generally sized to be a significant fraction of a wavelength (a quarter-wave or half-wave, see Section 5) — larger R_rad relative to R_loss pushes η_rad close to 1, meaning nearly all the power delivered to the antenna terminals is actually converted to radiated electromagnetic waves.

Matching R_rad (plus any reactance X) to the impedance of the feed line or transmitter (commonly 50 Ω in RF systems, 75 Ω for video/cable) is essential: an impedance mismatch reflects power back down the feedline instead of radiating it, quantified by the reflection coefficient Γ = (Z_in − Z₀)/(Z_in + Z₀) and the resulting voltage standing wave ratio (VSWR).

5. The Half-Wave Dipole

The most common practical antenna is the half-wave dipole: a straight conductor of total length L = λ/2, fed at its centre. Unlike the infinitesimal Hertzian dipole, current is not uniform along a half-wave dipole — it follows an approximately sinusoidal distribution that is maximum at the centre feed point and falls to zero at the two open ends (where charge cannot flow off into open space):

Integrating the contributions of this current distribution over the length of the wire (treating it as a continuum of Hertzian dipole elements and summing their far-field contributions) yields the exact far-field pattern:

This pattern is very close to the sinθ pattern of the Hertzian dipole but slightly narrower — the half-power beamwidth is 78° compared with 90° for the infinitesimal dipole. The directivity works out to D = 1.64 (2.15 dBi), and this reference value is so standard in RF engineering that antenna gains are routinely quoted in dBd (decibels relative to a half-wave dipole) as an alternative to dBi, related by G(dBi) = G(dBd) + 2.15.

The radiation resistance of a thin half-wave dipole, computed at the centre feed point, is:

This value is remarkably close to common transmission line impedances, which is precisely why the half-wave dipole became the workhorse reference antenna of radio engineering — it needs little or no impedance-matching network to connect efficiently to standard 50–75 Ω feedlines. A closely related and even more common variant is the quarter-wave monopole (L = λ/4) mounted vertically over a conducting ground plane, which uses image theory to reproduce half of the dipole's radiation pattern above the ground with R_rad ≈ 36.5 Ω — the basis for most car-radio, Wi-Fi, and cellular base-station whip antennas.

6. Antenna Gain and the Friis Transmission Equation

Gain G is directivity adjusted for the antenna's radiation efficiency: G = η_rad · D. It describes how much more intensely an antenna radiates in its peak direction compared with a lossless isotropic radiator fed the same input power — a real, measurable quantity that accounts for both directional concentration and internal losses.

Antenna gain governs how much power a receiver picks up over a wireless link. The Friis transmission equation, one of the most widely used formulas in all of radio engineering, relates received power P_r to transmitted power P_t across a free-space link of distance R:

The (λ/4πR)² term is the free-space path loss, arising purely from geometric spreading of the wavefront over an ever-larger sphere of area 4πR² as it propagates — the same 1/r² law that governs the inverse-square dilution of light intensity from a lamp. In logarithmic (decibel) form, commonly used in link budgets:

Notice that path loss increases with frequency for a fixed physical antenna aperture — this is why lower-frequency signals (like AM radio at ~1 MHz) travel farther for a given transmit power than millimetre-wave 5G signals at 28 GHz, and why higher frequency systems must compensate with more directional, higher-gain antenna arrays to maintain the same link range.

The Friis equation is a design cornerstone: satellite communication links, deep-space probes such as Voyager, Wi-Fi range calculations, and radar link budgets all begin from this formula, adding system-specific factors for atmospheric absorption, rain attenuation, polarization mismatch, and implementation losses.

7. Phased Arrays and Electronic Beam Steering

A single antenna element has a fixed radiation pattern. A phased array combines N identical elements, each fed with the same signal but a controllable relative phase shift, to synthesize a much narrower, electronically steerable beam without moving any physical part. This principle underlies modern radar, 5G massive-MIMO base stations, and the Event Horizon Telescope.

Consider a linear array of N elements spaced a distance d apart along the x-axis, each driven with an incremental phase shift β relative to its neighbour. The total far-field is the single-element pattern multiplied by an array factor — this multiplicative relationship is called pattern multiplication:

The array factor has its principal maximum wherever ψ = 0, i.e. when kd cosθ₀ = −β. Crucially, θ₀ — the beam pointing direction — is set purely by the phase shift β applied electronically between elements. Sweeping β continuously steers the main beam through space in nanoseconds, with no mechanical rotation whatsoever:

Increasing N narrows the main beam and adds directivity roughly proportional to N (for fixed element spacing and no grating lobes), while also adding side lobes between the main beam and any grating lobes. To avoid grating lobes — spurious full-strength repeats of the main beam appearing at other angles — element spacing is normally kept at or below d = λ/2.

8. Practical Antenna Types

Beyond the dipole and monopole, several other antenna families dominate real-world RF systems, each trading off gain, bandwidth, size, and pattern shape for a particular application:

Yagi-Uda Antenna

A driven half-wave dipole flanked by one or more passive parasitic elements — a slightly longer reflector behind it and one or more slightly shorter directors in front. Currents are induced in the parasitic elements by mutual coupling, and their phase (set by their length) reinforces radiation forward and cancels it backward, giving strong directivity (10–20 dBi with several directors) from a lightweight, low-cost structure — the classic rooftop TV antenna.

Parabolic Reflector (Dish)

A small feed antenna illuminates a parabolic reflecting surface; because every ray from the focus travels an equal total path length to a plane in front of the dish, the reflector converts a spherical wave from the feed into a nearly plane wave, producing extremely high directivity (often 40+ dBi) for a given aperture diameter D, following the aperture-antenna relation D_max ≈ 4πA_eff/λ², where A_eff is the effective aperture area.

Patch (Microstrip) Antenna

A thin rectangular or circular conductor on a grounded dielectric substrate, resonant when its length is approximately λ/2 in the dielectric. Low profile, lightweight, and easily fabricated with printed-circuit techniques, patch antennas are the standard choice for GPS receivers, phased-array elements in 5G handsets, and any application where a flat, conformal antenna is required.

Horn Antenna

A flared waveguide that gradually expands the cross-section of a guided wave to match the impedance of free space, minimizing reflections while producing a moderately directive, very broadband, low-side-lobe pattern. Horns are the standard feed element for parabolic dishes and the reference antenna used for calibrating antenna gain measurements in EMC and RF laboratories.

Choosing among these designs is a matter of matching the radiation pattern, gain, bandwidth, polarization, size, and cost to the application — a Wi-Fi router wants broad, nearly omnidirectional coverage; a satellite ground station wants an extremely narrow, high-gain pencil beam pointed precisely at a geostationary satellite 36,000 km away.