Questions: Radiation Directivity and Antenna Patterns

Question 1

Antenna X has directivity D = 20 and radiation efficiency η = 0.5. Antenna Y has directivity D = 10 and radiation efficiency η = 1.0. Which has greater gain?

• A. Antenna X, because its directivity is twice as high
• B. They have equal gain — G_X = 0.5 × 20 = 10 and G_Y = 1.0 × 10 = 10
• C. Antenna Y, because perfect efficiency always wins
• D. Cannot be determined without knowing the operating frequency

Answer: (B)

Gain G = ηD. For Antenna X: G = 0.5 × 20 = 10. For Antenna Y: G = 1.0 × 10 = 10. They are equal. This illustrates the key distinction: directivity is purely geometric (the shape of the radiation pattern relative to isotropic), while gain incorporates ohmic losses. A highly directive but lossy antenna can deliver the same on-axis power as a less directive but efficient one. Engineers use gain (not directivity) to compute link budgets via the Friis equation.

Question 2

An antenna's peak directivity is doubled from D = 10 to D = 20. What happens to the solid angle of its main beam?

• A. It doubles — more directivity means power spreads over a larger solid angle
• B. It stays the same — directivity and beam solid angle are independent
• C. It is halved — the beam narrows as directivity increases, since Ω_beam ≈ 4π/D
• D. It is quartered — the relationship between directivity and beam angle is quadratic

Answer: (C)

The tradeoff is fundamental: Ω_beam ≈ 4π/D. Total radiated power is fixed; concentrating more of it in one direction (higher D) necessarily means a smaller solid angle receives that peak power density. Doubling D halves the beam solid angle. This is the antenna analog of optical diffraction: a wider aperture produces a narrower diffraction lobe. There is no way to have both high directivity and a wide field of view simultaneously.

Question 3

An isotropic radiator has directivity D = 1 in every direction.

• T. True
• F. False

Answer: True

Directivity is defined as D(θ,φ) = (dP/dΩ) / (P_total/4π) — the ratio of actual power per steradian in a given direction to what an isotropic radiator would produce at the same total power. By definition, an isotropic antenna distributes power uniformly, so this ratio equals 1 everywhere. D = 1 is the baseline; any real antenna with a non-uniform pattern has D > 1 in its peak direction and D < 1 elsewhere, with the average over the sphere always equal to 1.

Question 4

A highly directive antenna with D = 100 necessarily has higher gain than an antenna with D = 5, regardless of their radiation efficiencies.

• T. True
• F. False

Answer: False

Gain G = ηD, so efficiency matters. An antenna with D = 100 and η = 0.01 has gain G = 1, far below an antenna with D = 5 and η = 0.9, which has G = 4.5. A directive but lossy antenna (e.g., a long wire with significant ohmic resistance) can actually deliver less power to a receiver than a simple but efficient dipole. This is why gain — not directivity — is the relevant figure for system-level link budgets.

Question 5

Why does increasing directivity necessarily result in a narrower main beam, and what physical mechanism creates high-directivity radiation patterns?

Model answer: Higher directivity means more power concentrated into a smaller solid angle, because total radiated power is conserved — Ω_beam ≈ 4π/D shrinks as D grows. Physically, high directivity arises from coherent interference among multiple current-carrying elements separated by distances comparable to the wavelength. The phase of radiation from each element depends on direction; constructive interference in the desired direction requires phases to align, which is increasingly sensitive to angle as more elements (or larger aperture) are added, creating destructive interference in all other directions. This is the same physics as diffraction from a wide aperture producing a narrow central lobe.

The fundamental limit is Fourier-dual: large aperture ↔ narrow angular spectrum, small aperture ↔ wide angular spread. An aperture of size L at wavelength λ produces a main lobe of angular width ~ λ/L. Equivalently, high directivity requires spatial coherence across a large effective aperture. Dish antennas and phased arrays exploit this by controlling the current distribution across a physical aperture; the antenna pattern is essentially the Fourier transform of the aperture current distribution.