Antenna Arrays

A single antenna element has limited ability to focus radiated power in a desired direction. Antenna arrays solve this by combining multiple radiating elements arranged in a specific geometric pattern, so that their individual fields add constructively in desired directions and cancel in unwanted directions. This spatial interference of electromagnetic waves gives arrays the ability to achieve very high directivity and steerable radiation patterns without changing physical antenna shape.

Core Concept Explanation

In a linear array, antenna elements are arranged along a straight line at regular spacing d. Each element is fed with a signal of the same frequency, but the phase of excitation is progressively shifted by an amount δ (delta) from one element to the next. The total far-field radiation pattern of the array results from the superposition of fields from all elements. This is governed by the principle of pattern multiplication.

Pattern multiplication states that the total radiation pattern of an array of identical elements equals the product of the element pattern and the array factor (AF). The element pattern depends on the shape of each individual antenna (dipole, patch, etc.), while the array factor depends entirely on the geometry of the array — number of elements, spacing, and phase excitation. This separation is extremely powerful: it allows the engineer to optimize the array geometry independently of the element design.

The two most important special cases are the broadside array and the end-fire array. In a broadside array, maximum radiation occurs perpendicular to the array axis. This requires all elements to be excited in phase (δ = 0). In an end-fire array, maximum radiation is along the array axis (parallel to the line of elements), which requires δ = -kd, where k = 2π/λ is the wavenumber.

By continuously varying the phase shift δ electronically (rather than mechanically rotating the antenna), the beam can be steered to any angle in space. This is the basis of phased array technology used in radar, 5G base stations, and satellite communication systems.

Mathematical Expression

For a uniform linear array of N isotropic elements with spacing d and progressive phase shift δ, the array factor is given as AF = sin(Nψ/2) / (N × sin(ψ/2)), where ψ = kd cosθ + δ, k = 2π/λ, and θ is the angle measured from the array axis. The normalized array factor divides by N so the maximum value is always 1. The main lobe maximum occurs when ψ = 0, giving the condition for the beam direction.

The half-power beamwidth (HPBW) of the main lobe decreases as N increases or as d increases. For a broadside array, the approximate HPBW = 0.886 λ/(Nd) radians. The number of grating lobes (unwanted secondary main lobes) can be controlled by choosing d less than or equal to λ/2. When d = λ/2, no grating lobes appear in the visible region for any scan angle.

Practical Understanding

In practice, antenna arrays are widely used when a single element cannot produce enough gain or directivity. Increasing the number of elements N increases array gain by approximately N (or 10 log N in dB). A 16-element array, for example, can theoretically provide 12 dB more gain than a single element, which is very significant for long-range communication links.

The spacing d = λ/2 is the most commonly chosen value in practice. It provides adequate spatial sampling of the wave, prevents grating lobes for all scan angles, and keeps the array physically compact. Smaller spacing causes mutual coupling between elements to increase significantly, which can distort patterns and affect impedance matching.

Modern 5G massive MIMO antennas use two-dimensional (planar) arrays with hundreds of elements, enabling simultaneous steering of multiple beams toward different users. The mathematics extends directly from the 1D linear array to 2D planar arrays using the product of two array factors (one per axis).

Given:
N = 4 elements, spacing d = λ/2, broadside array (δ = 0)
Find: array factor magnitude at θ = 90° (broadside direction)

Why this formula applies:
At broadside, ψ = kd cosθ + δ. With θ = 90°, cos90° = 0, so ψ = 0.
The AF formula reduces to its maximum value.

Formula:
AF = sin(Nψ/2) / (N × sin(ψ/2))
As ψ → 0: AF_max = 1 (normalized)

Substitution:
ψ = (2π/λ)(λ/2)(cos90°) + 0 = 0
AF = lim(ψ→0) sin(4×0/2) / (4 × sin(0/2)) = 4/4 = 1

Calculation:
Normalized |AF| = 1.0 (maximum, main beam)

Final Answer:
At broadside (θ = 90°), |AF| = 1 (maximum radiation)
Array gain over single element = 10 log(4) ≈ 6 dB

Exam Tip: For GATE, remember the two key conditions — broadside requires δ = 0 (all in phase), end-fire requires δ = -kd = -2πd/λ. For d = λ/2 end-fire, δ = -π. These are frequently tested as direct substitution questions.

• Total array pattern = Element pattern × Array factor (AF). The two can be designed and optimized independently.
• Array factor depends on: number of elements N, spacing d, progressive phase δ, and observation angle θ.
• Broadside condition: δ = 0 (all elements in phase), maximum at θ = 90° to array axis.
• End-fire condition: δ = -kd, maximum along array axis (θ = 0° or 180°).
• Beam steering in phased arrays is achieved by electronically varying δ without mechanical movement.
• Spacing d = λ/2 is the standard choice to prevent grating lobes across all scan angles.

Quick Revision

• Pattern multiplication: Total pattern = Element pattern × AF. AF is independent of element type.
• AF for N-element uniform linear array: AF = sin(Nψ/2) / [N sin(ψ/2)], where ψ = kd cosθ + δ.
• Broadside: δ = 0, beam at θ = 90°. End-fire: δ = -kd, beam at θ = 0°.
• Array gain over single element ≈ N (linear) or 10 log N dB.
• Grating lobes prevented when d ≤ λ/2.
• Exam trap: Pattern multiplication applies only when all elements are identical and similarly oriented.
• HPBW ≈ 0.886 λ/(Nd) radians for broadside, decreases as N or d increases.