Infinitesimal dipole fields and basic parameters

Infinitesimal dipole

An infinitesimal (Hertzian) dipole is a very short conductor, l ≪ λ (≤ λ/50), placed along the z-axis with a uniform current I0 along its length.

Retarded vector potential:

Az = μ I0 l e^(−jkr) / (4π r)

Magnetic field (H = ∇×A/μ):

Hφ = j k I0 l sinθ e^(−jkr)/(4πr) · [1 + 1/(jkr)]
Hr = Hθ = 0

Electric field (from ∇×H = jωεE):

Er = η I0 l cosθ e^(−jkr)/(2πr²) · [1 + 1/(jkr)]
Eθ = jηkI0 l sinθ e^(−jkr)/(4πr)
     · [1 + 1/(jkr) − 1/(kr)²]
Eφ = 0

Far field (kr ≫ 1):

Eθ ≈ jη k I0 l sinθ e^(−jkr)/(4πr)
Hφ ≈ Eθ / η,  η = 120π Ω,  Er ≈ 0

The pattern is ∝ sinθ (doughnut), D = 1.5, Rr = 80π²(l/λ)² Ω.

Basic antenna parameters

Radiation pattern, directivity, gain, efficiency, beamwidth, bandwidth, input impedance, radiation resistance, polarization, effective aperture.

1. Radiation pattern: graph of the radiated field or power versus direction (θ, φ) in the far field. It shows main lobe, side lobes, back lobe and nulls.
2. Directivity (D): ratio of the maximum radiation intensity to the average radiation intensity:

D = U_max / U_avg = 4π U_max / P_rad

Isotropic: D = 1; λ/2 dipole: D = 1.64. 3. Gain (G): directivity including losses: G = η_e · D, where η_e = radiation efficiency. Expressed in dBi. 4. Radiation efficiency:

η_e = P_rad / P_in = Rr / (Rr + RL)

RL = ohmic loss resistance.

Other parameters: HPBW – angle between half-power points of the main lobe; bandwidth – frequency range where parameters stay within limits; input impedance Z_in = R_in + jX_in at the terminals; polarization – orientation of the E-field vector with time.