--- title: "Pattern multiplication, quadrature-phase sources" subject: "Propagation and Antenna" chapter: 2 key: "semester_6/propagation_and_antenna:pattern-multiplication-quadrature" source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/pattern-multiplication-quadrature site: BCT Engineering --- # Pattern multiplication, quadrature-phase sources Propagation and Antenna · Chapter 2: Antenna Parameters and Arrays - Asked in: 2080 Asoj, 2+6 marks - Video: [Array of two isotropic point sources with opposite phase and phase quadrature | Array | Lecture 15 |](https://www.youtube.com/watch?v=ytEcjF03yPo) — Dopamine - 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-multiplication-quadrature - Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-2 ## Question Define pattern multiplication. Derive the expression for far field pattern of an array of two isotropic point sources having equal amplitude and quadrature phase with d = λ/4. ## Short answer Pattern multiplication: array field pattern equals element pattern times array factor of isotropic sources. For two isotropic sources with equal amplitude, quadrature phase (α=-π/2), d=λ/4: array factor E_n = cos[(π/4)(cosθ-1)]. Maximum at θ=0° (toward lagging source), null at θ=180°, forming a cardioid unidirectional end-fire pattern. ## Answer #### Pattern multiplication **Pattern multiplication** states that the field pattern of an array of identical, similarly oriented non-isotropic elements is the product of the pattern of one element and the pattern of the array made of isotropic point sources (array factor). Total phase = element phase + array phase. ```text E(θ) = E_element(θ) × E_array(θ) ``` #### Two isotropic sources, equal amplitude, quadrature phase, d = λ/4 Let source 1 and source 2 lie on the x-axis, spacing *d*, with the current of source 2 **lagging** source 1 by 90° (α = −π/2). θ is measured from the axis, midpoint O as reference. ```text P (far field) / / θ 1 ●-----O-----● 2 -> axis (θ = 0°) <- d/2 -><- d/2-> path difference = d cosθ ``` Total phase difference at P: ```text ψ = βd cosθ + α βd = (2π/λ)(λ/4) = π/2 , α = -π/2 ψ = (π/2) cosθ - π/2 = (π/2)(cosθ - 1) ``` Adding the two fields: ```text E = E₀ e^(-jψ/2) + E₀ e^(jψ/2) = 2E₀ cos(ψ/2) E = 2E₀ cos[(π/4)(cosθ - 1)] Normalised: E_n = cos[(π/4)(cosθ - 1)] ``` #### Pattern values | θ | cosθ − 1 | E_n | |---|---|---| | 0° | 0 | 1.000 | | 60° | −0.5 | 0.924 | | 90° | −1 | 0.707 | | 120° | −1.5 | 0.383 | | 180° | −2 | 0 | Maximum at θ = 0° (towards the lagging source, along the axis); null at θ = 180°. The pattern is a **cardioid** (unidirectional end fire). ```text 90° (0.707) .--''--. .' '. 180° 0 1 2 > 0° (max = 1) '. .' '--..--' 270° ``` Physical reason: towards θ = 0°, the wave from source 1 travels λ/4 (90°) to reach source 2, which exactly cancels source 2's 90° lag, so the fields add. Towards θ = 180°, the travel delay and current lag add to 180°, so the fields cancel. If source 2 leads instead (α = +π/2), the cardioid points to θ = 180°. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Pattern multiplication is A. E = element pattern × array pattern, with phases added B. Element pattern + array pattern C. Array pattern squared D. Element pattern ÷ array pattern Answer: A. It applies to identical similarly oriented elements. (Key point: Definition) 2. For d = λ/4 and α = -π/2, ψ equals A. (π/2)(cosθ + 1) B. (π/2)(cosθ - 1) C. π cosθ D. (π/4) cosθ Answer: B. βd = π/2 and α = -π/2. (Key point: Phase difference) 3. Where is the null of E_n = cos[(π/4)(cosθ - 1)]? A. θ = 0° B. θ = 90° C. θ = 180° D. θ = 60° Answer: C. At 180° the argument is -π/2. (Key point: Null direction) 4. Why is the field maximum toward θ = 0°? A. The λ/4 travel delay cancels source 2's 90° lag, so fields add B. Because the sources are isotropic C. Because the delay doubles the lag D. Because the amplitudes differ Answer: A. At 180° the delay and lag add to 180° and cancel. 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