Gain, efficiency and directivity

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 %.

Directivity

Directivity is the ratio of maximum radiation intensity to average radiation intensity:

U_avg = P_rad / 4π
D = U_max / U_avg = 4π U_max / P_rad
P_rad = ∮ U dΩ = ∫₀²π ∫₀π U(θ,φ) sinθ dθ dφ

Derivation for a short dipole (U = U0 sin²θ):

P_rad = U0 ∫₀²π ∫₀π sin³θ dθ dφ = U0 · 2π · 4/3 = 8πU0/3
D = 4π U0 / (8πU0/3) = 1.5  (1.76 dBi)

Approximation for narrow beams: D ≈ 41253/(θ1° θ2°).