Radiation pattern, gain and power flux density

Radiation pattern

The radiation pattern is a graph of the far-field radiation (field or power) of an antenna as a function of direction (θ, φ). The field pattern plots |E(θ,φ)|; the power pattern plots U(θ,φ) ∝ |E|². Normally it is normalized and drawn in the E-plane and H-plane.

For a short dipole, from the far field Eθ = jηkI0 l sinθ e^(−jkr)/(4πr):

E_n(θ) = sinθ,   P_n(θ) = sin²θ

          main lobe
           .---.
          /     \   HPBW
  side   |       |   side
  lobe .  \  o  /  . lobe
           '-+-'
          back lobe

Parts: main lobe, side lobes, back lobe, nulls, HPBW, FNBW, side lobe level.

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.

Derivation of D for the short dipole: U = U0 sin²θ, P_rad = U0 ∫∫ sin³θ dθ dφ = 8πU0/3, D = 4πU0/P_rad = 1.5; with η = 1, G = 1.5 (1.76 dBi).

Power flux density (PFD)

PFD is the power crossing unit area normal to the direction of propagation (W/m²), the magnitude of the average Poynting vector.

Power Pt radiated isotropically spreads over a sphere of area 4πd²:

PFD_iso = Pt / (4π d²)
With gain Gt: PFD = Pt Gt / (4π d²) = EIRP / (4πd²)
Also: PFD = E²rms / 120π

Example: EIRP = 1 kW, d = 10 km → PFD = 1000/(4π × 10⁸) = 7.96 × 10⁻⁷ W/m², E_rms = 17.3 mV/m.

Received power: Pr = PFD × Ae.