Gain, efficiency and beamwidth

Antenna gain

Gain is the ratio of the radiation intensity in a given direction to the radiation intensity that would be obtained if the input power were radiated isotropically:

G(θ,φ) = 4π U(θ,φ) / P_in
P_rad = η P_in   →   G = η D
G(dBi) = 10 log10 G

It includes losses, so G ≤ D. Example: λ/2 dipole, D = 1.64; if η = 0.97, G = 1.59 (2.0 dBi). For aperture antennas G = 4πAe/λ².

Antenna efficiency

Radiation efficiency is the ratio of power radiated to power accepted at the terminals:

η = P_rad / P_in = Rr / (Rr + RL)

Rr = radiation resistance, RL = ohmic/dielectric loss resistance. Total efficiency also includes mismatch: e0 = (1 − |Γ|²) · η.

Example: Rr = 73 Ω, RL = 2 Ω → η = 73/75 = 97.3 %.

Beamwidth

HPBW is the angle between the two directions in the main lobe where power falls to half (field to 0.707) of maximum. FNBW is the angle between the first nulls.

Derivation for a short dipole (E ∝ sinθ, power ∝ sin²θ):

sin²θ = 1/2  →  sinθ = 1/√2
θ1 = 45°, θ2 = 135°
HPBW = 135° − 45° = 90°

For a λ/2 dipole, E = cos[(π/2)cosθ]/sinθ = 0.707 gives θ ≈ 51° and 129° → HPBW ≈ 78°.

        max
       .---.
 0.707/  |  \0.707
     /<HPBW>\
    / <-FNBW->\

A narrower beam means higher directivity: D ≈ 41253/(θE° θH°).