--- title: "Radiation pattern, gain and power flux density" subject: "Propagation and Antenna" chapter: 2 key: "semester_6/propagation_and_antenna:pattern-gain-power-flux-density" source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/pattern-gain-power-flux-density site: BCT Engineering --- # Radiation pattern, gain and power flux density Propagation and Antenna · Chapter 2: Antenna Parameters and Arrays - Asked in: 2080 Asoj, 3+3+3 marks - Video: [Antenna Radiation Density & Radiated Power in Antenna and Wave Propagation by Engineering Funda](https://www.youtube.com/watch?v=T7lxZRWiQek) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/pattern-gain-power-flux-density - Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-2 ## Question Describe the radiation pattern, antenna gain and power-flux density with mathematical derivation if necessary. ## Short answer Radiation pattern is a graph of far-field field or power versus direction, showing main lobe, side lobes, back lobe and nulls; for a short dipole field pattern is sinθ and power pattern sin²θ. Gain is G = 4πU/P_in = ηD, with short dipole D = 1.5. PFD = EIRP/4πd² = E²rms/120π; for 1 kW EIRP at 10 km, PFD = 7.96×10⁻⁷ W/m². ## Answer #### 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): ```text E_n(θ) = sinθ, P_n(θ) = sin²θ ``` ```text 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: ```text 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: ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the difference between field pattern and power pattern? A. Field pattern plots |E|; power pattern plots U ∝ |E|² B. Field pattern plots |E|²; power pattern plots |E| C. Both plot only phase D. Field pattern is only for receiving Answer: A. For a short dipole they are sinθ and sin²θ. (Key point: Field vs power pattern) 2. Gain is radiation intensity relative to an isotropic radiator fed with A. The radiated power B. The input power C. The reflected power D. Half the radiated power Answer: B. So G = ηD. (Key point: Gain) 3. What is the directivity of a short dipole? A. 1.64 B. 1 C. 1.5 D. 3 Answer: C. P_rad = 8πU0/3, so D = 1.5. (Key point: Short dipole D) 4. What is PFD for gain Gt at distance d? A. Pt Gt/(2πd) B. Pt/(Gt 4πd²) C. Pt Gt 4πd² D. Pt Gt/(4πd²) = EIRP/(4πd²) Answer: D. Power spreads over a sphere of area 4πd². 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