What is the difference between directivity and gain?

Directivity is a purely geometric property of the radiation pattern; it depends only on the shape of the pattern, not on any losses. Gain includes all losses in the antenna: ohmic losses in conductors and dielectrics, surface wave losses, feed network losses, and impedance mismatch loss. G = eta times D, where eta is the total antenna efficiency (0 to 1). For a low-loss antenna like a parabolic dish with a well-matched feed, eta is typically 0.55 to 0.7. For a small patch antenna on a lossy substrate, eta might be 0.3 to 0.5. A critical consequence: a small antenna can have high directivity but low gain if losses are high. The directivity tells you the theoretical maximum performance; the gain tells you the actual performance. In link budgets, always use gain (not directivity) because it accounts for the real power radiated.

How is directivity estimated from beamwidth?

For antennas with a single main beam and no significant sidelobes, Kraus's approximation gives: D approximately equals 41253 / (theta_E times theta_H), where theta_E and theta_H are the half-power beamwidths in degrees in the E-plane and H-plane respectively. This is equivalent to D = 4 pi / omega_A, where omega_A is the beam solid angle in steradians. For pencil beams: D approximately equals (pi d / lambda) squared for a uniformly illuminated circular aperture, yielding 52525 / theta_3dB_squared. Example: a reflector antenna with 2-degree beamwidth in both planes has D approximately equals 41253/4 = 10313 = 40.1 dBi. These approximations are within 1 to 2 dB for well-behaved patterns with sidelobe levels below minus 10 dB. For more accurate results, the full pattern must be integrated numerically.

What limits directivity?

For a given antenna size, the maximum achievable directivity is limited by the aperture area: D_max = 4 pi A / lambda squared, where A is the physical aperture area. This is the directivity of a uniformly illuminated aperture with no amplitude or phase errors. In practice, three factors reduce directivity below this limit: amplitude taper (tapering the illumination to reduce sidelobes narrows the effective aperture, reducing directivity by 1 to 3 dB, called taper efficiency), phase errors (surface roughness or distortion in reflectors causes random phase errors across the aperture, each lambda/16 RMS error reduces directivity by approximately 1 dB, per the Ruze equation), and spillover (feed radiation past the reflector edge is wasted, typically 1 to 2 dB loss). For electrically small antennas (size much less than lambda), directivity is limited to 1.5 (1.76 dBi) for a short dipole and approaches 1 (0 dBi) as the antenna shrinks, though superdirective designs can exceed this at the cost of narrow bandwidth and high sensitivity to tolerances.

Antenna Parameter

Directivity

/dih-rek-tiv-ih-tee/ — D

D = 4πU_max/P_rad. Ratio of max radiation intensity to average. Isotropic: D=1 (0 dBi). Dipole: 1.64 (2.15 dBi). Dish: D = (πd/λ)²η. Gain = ηD (efficiency × directivity). Kraus approx: D ≈ 41253/(θ_Eθ_H). Aperture limit: D_max = 4πA/λ². Independent of losses, only pattern shape.

Dipole: 2.15 dBi

G=ηD: η=0.55-0.7

Limit: 4πA/λ²

Understanding Directivity

Directivity is what makes antennas useful. An isotropic antenna (D=1) spreads power equally in all directions, wasting most of it. A high-directivity antenna concentrates power into a narrow beam, delivering far more signal to a distant receiver. A parabolic dish with 40 dBi directivity delivers 10,000 times more power in its beam direction than an isotropic antenna with the same input power.

The critical distinction from gain is that directivity is purely geometric: it depends only on the shape of the radiation pattern. An antenna with 40 dBi directivity but 50% efficiency has 37 dBi gain, with 3 dB of power lost to heat in the antenna structure. Link budgets must use gain, not directivity.

Directivity Equations

Antenna directivity:
D = 4πU_max/P_rad
= 4π/Ω_A

From beamwidths:
D ≈ 41253/(θ₁·θ₂)
θ₁,θ₂ in degrees (half-power)

Gain vs directivity:
G = η_rad·D

Directivity by Antenna Type

Antenna	Directivity	Beamwidth	Pattern	Type
Isotropic	0 dBi	360°×360°	Omnidirectional	Reference
Dipole	2.15 dBi	78°×360°	Donut	Wire
Patch	6–9 dBi	60–90°	Hemispherical	Printed
Horn	15–25 dBi	10–40°	Pencil beam	Aperture
Parabolic 1m	35–42 dBi	1–5°	Pencil beam	Reflector

Common Questions

Frequently Asked Questions

vs Gain?

Directivity: pattern shape only, loss-free. Gain: includes all losses (ohmic, dielectric, mismatch, feed). G=ηD. Dish η=0.55-0.7. Patch on lossy substrate η=0.3-0.5. Link budget: always use Gain. High-D + low-η = low-G. Directivity = theoretical max.

Beamwidth estimate?

Kraus: D ≈ 41253/(θ_E × θ_H) degrees. 2°×2° = 40 dBi. 10°×10° = 26 dBi. Accurate ±1-2 dB for SLL < −10 dB. Circular aperture: D ≈ 52525/θ². Full accuracy: numerical pattern integration.

Limits?

Aperture: D_max = 4πA/λ² (uniform illumination). Taper: −1-3 dB for SLL reduction. Phase error (Ruze): −1 dB per λ/16 RMS. Spillover: −1-2 dB. Small antennas: D ≤ 1.5 (short dipole). Superdirective: exceeds limit but narrow BW, tolerance-sensitive.

Related Terms

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