Phased array

Contents

1 Description 1.1 Usage 2 Mathematical Treatment 3 References

Description

In telecommunication, a phased array is a group of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and suppressed in undesired directions. This design is used in radar, and is generalized in interferometric radio antennas (usually in radio telescopes).

Usage

The relative amplitudes of – and constructive and destructive interference effects among – the signals radiated by the individual antennas determine the effective radiation pattern of the array. A phased array may be used to point a fixed radiation pattern, or to scan rapidly in azimuth or elevation.

The phased array is used for instance in optical telecommunication as a wavelength selective splitter.

Phased arrays are required to be used by many AM broadcast stations to enhance signal coverage in the city of license, while minimizing interference to other areas. Due to the differences between daytime and nightime ionospheric propagation at AM broadcast frequencies, it is common for AM broadcast stations to change between day and night radiation patterns by switching the phase and power levels supplied to the individual antenna elements daily at sunrise and sunset.

Mathematical Treatment

A phased array is an example of N-slit diffraction. Since each individual antenna acts as a slit, emitting radio waves, their diffraction pattern can be calculated by adding the phase shift Φ to the fringing term.

We will begin from the N-slit diffraction pattern derived on the diffraction page.

<math> \psi ={{\psi }_0}\left[\frac{\sin \left(\frac{{\pi a}}{\lambda }\theta \right)}{\frac{{\pi a}}{\lambda }\sin\theta}\right]\left[\frac{\sin \left(\frac{N}{2}{kd}\sin\theta\right)}{\sin \left(\frac{{kd}}{2}\sin\theta \right)}\right] <math>

Now, adding a Φ term to the <math>\begin{matrix}kd\sin\theta\,\end{matrix}<math> fringe effect in the second term yields:

<math>\psi ={{\psi }_0}\left[\frac{\sin \left(\frac{{\pi a}}{\lambda }\sin \theta\right)}{\frac{{\pi a}}{\lambda }\sin\theta}\right]\left[\frac{\sin \left(\frac{N}{2}\big(\frac{{2\pi d}}{\lambda }\sin\theta + \phi \big)\right)}{\sin \left(\frac{{\pi d}}{\lambda }\sin\theta +\phi \right)}\right] <math>

Taking the square of the wave function gives us the intensity of the wave. <math>I = I_0{{\left[\frac{\sin \left(\frac{\pi a}{\lambda }\sin\theta\right)}{\frac{{\pi a}}{\lambda } \sin [\theta ]}\right]}^2}{{\left[\frac{\sin \left(\frac{N}{2}(\frac{2\pi d}{\lambda} \sin\theta+\phi )\right)}{\sin \left(\frac{{\pi d}}{\lambda } \sin\theta+\phi \right)}\right]}^2} <math>

<math> I =I_0{{\left[\frac{\sin \left(\frac{{\pi a}}{\lambda } \sin\theta\right)}{\frac{{\pi a}}{\lambda } \sin\theta}\right]}^2}{{\left[\frac{\sin \left(\frac{\pi }{\lambda } N d \sin\theta+\frac{N}{2} \phi \right)}{\sin \left(\frac{{\pi d}}{\lambda } \sin\theta+\phi \right)}\right]}^2} <math>

Now space the emitters a distance <math> d=\begin{matrix}\frac{\lambda}{4}\end{matrix}<math> apart. This distance is chosen for simplicity of calculation but can be adjusted as any scalar fraction of the wavelength.

<math>I =I_0{{\left[\frac{\sin \left(\frac{\pi }{\lambda } a \theta \right)}{\frac{\pi }{\lambda } a \theta }\right]}^2}{{\left[\frac{\sin \left(\frac{\pi }{4} N \sin\theta+\frac{N}{2} \phi \right)}{\sin \left(\frac{\pi }{4} \sin\theta+ \phi \right)}\right]}^2}<math>

Sin achieves its maximum at <math>\begin{matrix}\frac{\pi}{2}\end{matrix}<math> so we set the numerator of the second term = 1.

<math> \frac{\pi }{4} N \sin\theta+\frac{N}{2} \phi = \frac{\pi }{2} <math>

<math> \sin\theta=\Big(\frac{\pi }{2} - \frac{N}{2} \phi \Big)\frac{4}{N \pi } <math>

<math> \sin\theta=\frac{2}{N}-\frac{2\phi }{\pi } <math>

Thus as N gets large, the term will be dominated by the <math>\begin{matrix}\frac{2\phi}{\pi}\end{matrix}<math>. As sin can oscillate between -1 and 1, we can see that setting <math>\phi=-\begin{matrix}\frac{\pi}{2}\end{matrix}<math> will send the maximum energy on a angle given by <math>\theta = \sin^{-1}(1) = \begin{matrix}\frac{\pi}{2}\end{matrix} = 90<math>degrees. Additionally, we can see that if we wish to adjust the angle at which the maximum energy is emitted, we need only to adjust the phase shift φ between successive antennae. Indeed the phase shift corresponds to the negative angle of maximum signal.

A similar calculation will show that the denominator is minimized by the same factor.

References

Source: from Federal Standard 1037C[1] (http://glossary.its.bldrdoc.gov/fs-1037/dir-027/_3979.htm) and from MIL-STD-188