--- title: "Antenna arrays and two-element expression" subject: "Propagation and Antenna" chapter: 2 key: "semester_6/propagation_and_antenna:define-array-two-isotropic" source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/define-array-two-isotropic site: BCT Engineering --- # Antenna arrays and two-element expression Propagation and Antenna · Chapter 2: Antenna Parameters and Arrays - Asked in: 2069 Bhadra, 6 marks - Video: [Two Elements Antenna Array (Phase Difference, Electric Field & Array Factor) Explained](https://www.youtube.com/watch?v=7eOQ2foOxxc) — 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/define-array-two-isotropic - Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-2 ## Question Define antenna arrays and also derive a mathematical expression for the array of two element isotropic radiators. ## Short answer Antenna array: multiple similar elements placed with spacing and fed with controlled amplitude/phase to shape radiation. For two isotropic sources spacing d, phase lead α of element 2, total field E = 2E0 cos[(βd cosθ + α)/2]; normalized E_n = cos(ψ/2) with ψ = βd cosθ + α. ## Answer An **antenna array** is an arrangement of two or more similar antennas (elements), placed with definite spacing and fed with definite current amplitude and phase, so that their fields add in wanted directions and cancel in others. Arrays give high directivity, narrow beams and beam steering that a single element cannot give. #### Array of two isotropic point sources Let two isotropic sources lie on the x-axis with spacing *d*, equal amplitude *E₀*, and the current of source 2 leading source 1 by phase *α*. Point P is in the far field at angle θ from the array axis. The midpoint O is the phase reference. ```text P (far field) / / θ 1 ●-----------O-----------● 2 ---> axis <--- d/2 ---><--- d/2 ---> extra path from 1 compared to 2 = d cosθ ``` Total phase difference between the two waves at P: ```text ψ = βd cosθ + α , β = 2π/λ ``` With O as reference, source 1 contributes a phase −ψ/2 and source 2 a phase +ψ/2: ```text E = E₀ e^(-jψ/2) + E₀ e^(jψ/2) = E₀ [2 cos(ψ/2)] = 2E₀ cos(ψ/2) ``` **E = 2E₀ cos[(βd cosθ + α)/2]**, normalised **E_n = cos(ψ/2)**. #### Special cases | Case | Field | Maximum | |---|---|---| | α = 0, d = λ/2 | cos(π/2 · cosθ) | θ = 90°, 270° (broadside) | | α = π, d = λ/2 | sin(π/2 · cosθ) | θ = 0°, 180° (along axis) | | α = −π/2, d = λ/4 | cos[π/4 (cosθ − 1)] | θ = 0° only (cardioid, end fire) | Main points: - The pattern depends only on d/λ and α, not on E₀. - Maximum occurs where ψ = 0, i.e. cosθₘ = −α/(βd). - If the sources are not isotropic, the total pattern = element pattern × this array factor (pattern multiplication). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. An antenna array is A. Two or more similar elements with definite spacing, amplitudes and phases whose fields add in wanted directions B. A single element with a reflector C. Elements fed at random phases D. A feed line only Answer: A. Fields add in wanted directions and cancel in others. (Key point: Definition) 2. The total field of two equal isotropic sources is A. E = 2E0 sin(βd cosθ) B. E = 2E0 cos[(βd cosθ + α)/2] C. E = E0 cos(α) D. E = 2E0 cosθ Answer: B. Adding e^(-jψ/2) and e^(jψ/2) gives 2cos(ψ/2). (Key point: Total field) 3. The direction of maximum is given by A. cosθm = α/(βd) B. cosθm = βd/α C. cosθm = -α/(βd) D. θm = α Answer: C. Maximum when ψ = 0. (Key point: Direction of maximum) 4. For α = π and d = λ/2, the field is A. cos[(π/2)cosθ] with maxima at 90° B. sin[π cosθ] with a null at 0° C. cos(θ) D. sin[(π/2)cosθ] with maxima at 0° and 180° Answer: D. cos[(π cosθ + π)/2] = |sin(π cosθ/2)|. 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