Antenna Arrays

This page provides a comprehensive look at the theory of uniform linear antenna arrays (ULA). Use the interactive simulator below to alter the number of elements, spacing, and progressive phase shift to monitor how radiation patterns and beamforming behavior update in real-time[cite: 1].

Introduction — Basic Theory

An antenna array is a set of individual radiating elements (e.g., dipoles) arranged in space. By controlling the relative amplitude and phase of each element, the combined radiation pattern can be shaped and electronically steered.

A simple and widely-studied configuration is the uniform linear array (ULA), where elements are placed along a line with constant spacing \(d\). If the \(n\)-th element is at position \(x_n\) (measured along the array axis), then:

\[ x_n = n\,d, \qquad n = 0,1,\dots,N-1. \]

A progressive phase excitation introduces a phase term between adjacent elements. For an observation angle \(\theta\) (measured from the array axis), the progressive phase between adjacent elements is:

\[ \Psi(\theta) \;=\; k d \cos\theta \;+\; \beta, \qquad \text{where } k=\frac{2\pi}{\lambda}. \]

The array factor (AF) for a uniform linear array with equal amplitudes and progressive phase \(\beta\) is expressed by the geometric sum:

\[ AF(\theta)\;=\;\left| \sum_{n=0}^{N-1} e^{\,i n \Psi(\theta)} \right| \;=\;\left|\frac{\sin\left(\tfrac{N\Psi}{2}\right)}{\sin\left(\tfrac{\Psi}{2}\right)}\right|. \]

Here \(N\) represents the number of elements, \(d\) the spacing in wavelengths, and \(\beta\) the progressive phase. Setting \(\beta=0\) generates broadside radiation, whereas adjusting \(\beta\) steers the main beam electronically—the foundation of beamforming[cite: 1].

Array elements & controls

Value: 8
Value: 0.25 λ
Value: 0.00 rad
Tip: set \(d=0.25\), \(\beta=0\) and increase \(N\) to watch directivity increase. For end-fire steer try \(\beta \approx kd\)[cite: 1].
The element diagram maps element positions along the array axis and labels spacing \(d\)[cite: 1].

Radiation pattern (polar)

\[ AF(\theta)=\left|\frac{\sin\left(\tfrac{N\Psi}{2}\right)}{\sin\left(\tfrac{\Psi}{2}\right)}\right|, \qquad \Psi = 2\pi d \cos\theta + \beta \]
Red dot marks main-lobe direction. Pattern is normalized for display[cite: 1].

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