Antenna Arrays

Theory and an interactive HTML5 simulation of a uniform linear antenna array (ULA). Use the sliders to change number of elements, spacing and progressive phase; results update in real-time.

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 plane wave (or a progressive phase excitation) introduces a phase term between adjacent elements. For example, for observation angle \(\theta\) (measured from the array normal or as appropriate), 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 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 \(N\) is the number of elements, \(d\) the spacing in wavelengths, and \(\beta\) the progressive phase between elements. Setting \(\beta=0\) typically gives broadside radiation. Changing \(\beta\) electronically steers the main beam — this is the fundamental concept of beamforming in array antennas.

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 \(\beta \approx kd\).
The element diagram shows element positions along the array axis and labels the spacing \(d\) between elements.

Radiation pattern (polar)

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