Antenna Radiation

Acceleration, not motion, makes radiation

A stationary electron has a Coulomb field that points radially outward and falls off as 1/r². An electron moving at constant velocity carries that same field with it — the field gets boosted slightly but stays bound to the charge. Neither configuration radiates.

Accelerate the electron and something new happens. The field lines, which had been smoothly anchored on the charge, cannot keep up with the sudden change in velocity. A kink develops in the lines at the radius corresponding to where the change started — and that kink travels outward at the speed of light. The field "snaps off" and propagates as an electromagnetic wave. The Larmor formula quantifies the radiated power:

P = q² a² / (6π ε₀ c³)

Note what's there: charge squared times acceleration squared. No factor of velocity. A charge at rest doesn't radiate; a charge moving uniformly doesn't either; only when it accelerates does it pour energy into the radiation field. An antenna is fundamentally a device for forcing charges to accelerate periodically. Drive AC current up and down a wire and you accelerate electrons back and forth twice per cycle — each half-cycle launches a wave.

The Hertzian dipole and sin²θ

The simplest analytically tractable antenna is the Hertzian (infinitesimal) dipole: a short stub of length L ≪ λ with uniform current I₀ cos(ωt). Its far-field electric field at distance r and angle θ from the dipole axis is

E_θ = (jη₀ I₀ L / 2λr) sin θ · e^{−jkr}

The magnitude depends on θ as sin θ; the power per unit area as sin²θ. The pattern is doughnut-shaped: maximum perpendicular to the wire (θ = 90°), zero along the wire's axis (θ = 0°, 180°). There is no preferred azimuthal direction (the dipole is φ-symmetric), so the pattern is a torus.

For a half-wave dipole (length L = λ/2) the current distribution is not uniform — it tapers as cos(πz/L) — and the pattern sharpens slightly:

|E|² ∝ [cos((π/2) cos θ) / sin θ]²

Still zero at θ = 0°, 180° (the wire's ends); still maximum at θ = 90°. The half-power beamwidth in elevation is 78° — wider than people often guess. The half-wave dipole's directivity over isotropic is 1.64, or 2.15 dBi.

Antenna parameters at a glance

Antenna	R_rad (Ω)	Directivity	Pattern	Where used
Isotropic (theoretical)	—	1 (0 dBi)	Uniform sphere	Reference only
Hertzian dipole (L ≪ λ)	80π²(L/λ)²	1.5 (1.76 dBi)	sin²θ doughnut	Theoretical baseline
Half-wave dipole	73.1	1.64 (2.15 dBi)	Sharpened sin²θ	FM, TV, ham radio, WiFi router
Quarter-wave monopole + ground	~36.5	3.28 (5.16 dBi)	Half-pattern (above ground)	AM broadcast, car whip, walkie-talkie
Yagi-Uda (10 element)	~50	~13 dBi	Directional forward lobe	TV reception, satellite uplink
Parabolic 1 m at 10 GHz	~50 (feed)	~30 dBi	Pencil beam, ~2° wide	Satellite, radar, deep-space
Phased array (1000 elements)	—	up to 40 dBi	Steerable beam	5G NR, AEGIS radar, Starlink

The 50 Ω convention for transmission lines is a compromise: it minimizes attenuation in air-filled coax (optimal at 77 Ω for low loss) and maximizes power handling (optimal at 30 Ω for breakdown). Between 50 Ω feed and 73 Ω dipole, the mismatch is only 1.46:1 VSWR — about 4 % reflected power — small enough to tolerate in most applications without a matching network.

Worked example: half-wave FM dipole

Build an FM broadcast receive antenna for 100 MHz: λ = c/f = 3 m, so a half-wave dipole has length L = λ/2 = 1.5 m, fed at the center. The radiation resistance is 73.1 Ω. With a 50 Ω feedline the VSWR is 73.1/50 = 1.46, reflected power |Γ|² = ((1.46−1)/(1.46+1))² = 3.5 %.

Suppose the broadcaster transmits 10 kW EIRP from a vertical dipole array with 6 dBi gain. At 30 km the free-space path loss is

L_FS = 20 log₁₀(4π · 30000 / 3) = 102 dB

The received signal power at our matched 73 Ω receive antenna (gain 2.15 dBi) is

P_rx = P_tx − L_FS + G_tx + G_rx − loss = 40 dBm − 102 dB + 6 dBi + 2.15 dBi = −54 dBm

(P_tx in dBm: 10 kW = 70 dBm). −54 dBm is roughly 4 µV across 73 Ω — a strong FM signal. A consumer FM receiver needs only ~−110 dBm for noise-floor reception, so 30 km coverage from a 10 kW transmitter is generous, consistent with US FCC coverage estimates.

Near field, radiating near field, far field

Antennas have three distinct regions:

• Reactive near field (r < λ/(2π)). Strong evanescent fields, energy oscillates between antenna and surrounding space. Not propagating; not delivering power outward. This region is where capacitive and inductive coupling dominate.
• Radiating near field (λ/(2π) < r < 2D²/λ). Fields are propagating but the pattern depends on distance. The Fresnel region. Used by RFID and inductive coupling.
• Far field (r > 2D²/λ). The Fraunhofer region. E and B are perpendicular and in phase; Poynting flux is purely radial; intensity falls as 1/r²; the pattern is independent of distance. All antenna performance specifications (gain, pattern, beamwidth) refer to far-field behavior.

For a 1 m dish at 10 GHz, λ = 3 cm and the far field starts at 2 × 1²/0.03 ≈ 67 m. Outdoor antenna measurements must be conducted beyond this distance or in an anechoic chamber that compensates for it. Phased arrays with effective apertures of 10 m have far-field distances of hundreds to thousands of meters — which is why their characterization requires either an open test range or near-field-to-far-field transformation techniques.

Where antenna radiation matters

• FM broadcast and cellular base stations. A typical FM tower at 100 MHz uses stacked dipoles fed from a power divider; a 6-bay array gives ~8 dBi gain in the horizontal plane. 5G NR base stations at 3.5 GHz use 64-element phased arrays generating dozens of simultaneous narrow beams.
• WiFi and Bluetooth. The ubiquitous "rubber ducky" antenna is a half-wave dipole loaded for the 2.4 GHz band (length ~31 mm). MIMO routers use 2–8 such dipoles spaced λ/2 apart to exploit spatial diversity — the 7000-mile-per-second flight of 2.4 GHz signals is what carries your video calls.
• GPS receivers and satellites. A GPS L1 antenna at 1.575 GHz is a small ceramic patch (~25 mm square) with right-hand-circular polarization. Each GPS satellite radiates 25 W EIRP from a 12-helix array; received signal at Earth is −130 dBm, hundreds of times weaker than thermal noise but recoverable via despreading of the 1023-chip PRN code.
• Radio astronomy with the SKA. The Square Kilometre Array (SKA-Low) will use 131,072 dipole antennas in Western Australia at 50–350 MHz, summing coherently to give an effective aperture of 1 km². Sensitivity floor below 1 mJy at 100 MHz.
• Voyager 1 deep-space link. Voyager's 23 W transmitter on a 3.7 m dish radiates EIRP ~10⁹ W at 8.4 GHz. After 22 billion km of free-space path loss (~270 dB), the signal arrives at Earth's 70 m DSN dish at roughly −213 dBm — below kT noise but coherently demodulated using forward error correction. The Larmor formula running for 47 years powers the longest communication link in human history.

Variants and extensions

• Yagi-Uda array. Driven element plus parasitic reflector (behind) and one or more directors (in front). Each parasitic re-radiates with a phase set by its mutual coupling, building constructive interference forward and destructive interference rearward. 10-element Yagis typically deliver 13 dBi gain — the standard TV antenna design.
• Patch / microstrip antenna. A rectangular conductor on a grounded dielectric substrate, fed from the edge or with a coaxial probe. Bandwidth ~2 %, gain ~6 dBi. Used in phones, GPS receivers, satellite reception. Easy to mass-produce on PCB.
• Phased array. Many antennas fed with electronically controlled phase shifts. Beam direction is steered by adjusting phase — no mechanical motion. AEGIS missile-defense radar (SPY-1) uses 4 fixed phased arrays; commercial 5G mMIMO uses 64- or 192-element arrays at 28 GHz.
• Horn antenna. A flared waveguide that radiates from its open mouth. Wideband, low side-lobes, ~15 dBi for compact designs. Used as primary feeds for parabolic dishes, satellite ground stations, and standard-gain calibration.
• Fractal and metamaterial antennas. Recursive fractal geometries (Sierpiński, Koch curve) achieve multi-band resonance from a single element. Metasurface antennas using engineered impedance gratings can produce beams with arbitrary polarization and pattern from a flat substrate, important for compact satellite phased arrays.

Common pitfalls

• Treating gain as power amplification. Antenna gain is a directional concentration of the same total radiated power, not a power gain. A 20 dBi antenna doesn't deliver 100× more power; it delivers the same power into 1/100 of the sky.
• Forgetting the −90° pattern null on the dipole axis. A vertical dipole radiates nothing straight up or straight down. Aircraft and satellite installations need an antenna that points the null safely away from the link partner — a common cause of "dead zones" right under a satellite or right next to a vertical whip.
• Calculating efficiency from radiation resistance alone. A short antenna has low R_rad (e.g. 0.08 Ω at 1 MHz for a 1 m dipole) but conductor losses set by skin-effect surface resistance might be 5 Ω; only 1.6 % of input power radiates. AM transmit antennas use ground-plane systems with 120 buried radials to keep ground losses below R_rad.
• Ignoring reactive impedance. A half-wave dipole at exactly λ/2 is +42.5 Ω inductive. Shortening to 0.485 λ tunes out the reactance, leaving pure resistive 73 Ω. Off-resonance, the antenna presents a complex impedance that won't deliver power to a 50 Ω line without matching.
• Confusing dBi with dBd. dBi is gain over isotropic; dBd is gain over a half-wave dipole. The conversion: dBi = dBd + 2.15. A 5-element Yagi quoted as "8 dBd" is 10.15 dBi — and confusion between the two is a frequent source of antenna-spec disputes.