Undergraduate Electrical Engineering

Introduction to Array Antennas

A comprehensive guide to the principles of antenna arrays, radiation patterns, and beam steering. Master the mathematics behind the Array Factor and explore practical configurations used in modern radar and communication systems.

1. Overview & Motivation

An Antenna Array is a set of multiple connected antennas (elements) which work together as a single antenna, to transmit or receive radio waves. The purpose is to achieve directionality, gain, and beam steering that is impossible with a single element.

Why use Arrays?

High Gain

Concentrating energy in a specific direction increases the effective radiated power without increasing input power.

Beam Steering

By adjusting the phase of the signal to each element, the main lobe of the radiation pattern can be steered electronically.

Pattern Shaping

Placing nulls in the direction of interferers or jammers to improve signal-to-noise ratio.

Reliability

If one element fails, the array continues to operate, albeit with slightly degraded performance.

2. The Array Factor (AF)

The total field of an array is determined by the Pattern Multiplication Theorem:

Etotal = Eelement (Single Element Pattern) × AF (Array Factor)

The Array Factor (AF) is the factor by which the field of a single element must be multiplied to get the total field. For a linear array of N isotropic elements spaced d apart:

Uniform Linear Array (ULA)

Assuming identical amplitudes I0 and a progressive phase shift β between elements:

AF = ∑n=0N-1 ejn(kd cosθ + β) = ej(N-1)ψ/2 [ sin(Nψ/2) sin(ψ/2) ]

Where ψ = kd cosθ + β

The magnitude of the Array Factor is often what we plot:

|AF| = | sin(Nψ/2)sin(ψ/2) |

3. Key Parameters

HPBW

Half-Power Beamwidth

The angular width of the main lobe where the power drops to half (-3dB) of the maximum value.

HPBW ≈ 50.8° ( λ / L )

L is the total length of the array.

SLL

Side Lobe Level

The ratio (in dB) of the amplitude of the largest side lobe to the amplitude of the main lobe.

SLL ≈ -13.5 dB (for uniform array)

Can be reduced using amplitude tapering (windowing).

BWFN

First Null Beamwidth

The angular separation between the first nulls on either side of the main lobe.

BWFN = 2 × sin-1( λ / Nd )
D

Directivity

The ratio of the radiation intensity in a given direction to the average radiation intensity.

D ≈ 2N ( d / λ )

Increases with number of elements N.

Interactive Array Lab

Visualize the Array Factor in real-time

Live Render
5
0.50
0°

Controls beam steering direction

Current Max Direction: 90°
Polar Plot: |AF| vs Angle
Tip: Increase spacing (d) beyond 0.5λ to observe Grating Lobes. Adjust phase (β) to steer the beam.

4. Common Array Types

Broadside Array

Maximum radiation occurs perpendicular to the axis of the array (at θ = 90°).

Condition: β = 0°

End-Fire Array

Maximum radiation occurs along the axis of the array (at θ = 0° or 180°).

Condition: β = -kd (for θ=0°)

Phased (Scanning) Array

The phase β is varied electronically to steer the main beam without physically moving the antenna.

Scan Angle: cosθ0 = -β / kd

5. Pattern Synthesis

Pattern Synthesis is the process of finding the current distribution (amplitude and phase) that produces a desired radiation pattern. It is essentially the inverse of pattern analysis.

Fourier Transform Method

The array factor is the Fourier transform of the current distribution. For a continuous source, this is exact. For discrete arrays, it is an approximation.

Note: A uniform current distribution yields the narrowest beamwidth but highest side lobes (-13.2 dB). To reduce side lobes, we must taper the current distribution towards the edges (e.g., Binomial or Dolph-Chebyshev distributions).

Summary

  • Arrays combine simple elements to create complex, directive radiation patterns.
  • The Array Factor depends on geometry (N, d) and excitation (β).
  • Beam steering is achieved by varying the phase shift β between elements.
  • Trade-offs exist between beamwidth, side lobe level, and directivity.