Phased array antennas are amongst cheapest and most used antennas in the market. In this blog we will describe how they work and how to model them.

# Modelling of Phased Array Antenna

Previously, we have covered how to model the reflection of an electromagnetic wave on a surface (Radar Modelling – Reflection of an EM wave – Claytex). Following up on the modelling of radars, we will now address the antenna beam pattern.

One of the most common (and cheapest) types of antenna are arrays, i.e. a set of radiating elements connected to a transmitter (receiver). In this blog we will explain how they work and how to simulate them.

### Interference

Huygens principle: The waves emitting from the aperture can be thought of as being produced by many individual radiating elements separated by a wavelength or less.

Phased array antennas are constructed as an array of radiating elements that interfere each other to create a desired beam pattern.

Note: We usually use the term “phased array antenna” and “antenna arrays” interchangeably, but a phased array antenna is an antenna array where the elements are connected to a phase shifter that allows for dynamic control of the beam pattern.

Each element generates a circular wave, much like the drop of a finger in water. On every point in space, the wave will be the superposition of each wave from each receiver. If the phase difference between the two waves is 0 modulo π (usually, the distance from receiver A to receiver B is the same modulo λ), then the waves will add up (“constructive interference”). If they are out of phase (usually the same distance modulo λ/2), then they will cancel out (“destructive interference”).

Figure 1 : Interference of two radiating elements

The circular wave from each element combines into a plane wave. In the plane of the wave you will have an alternate of constructive and destructive interference. The overall result is a lobe shaped beam pattern with a main and side lobes.

### Modelling

The important elements in the modelling that influence the beam pattern are:

N : number of radiating elements in the array

d : spacing between those elements

θ0 Azimuth : the orientation of the antenna in azimuth to the referential.

θ0 Elevation : the orientation of the antenna in elevation to the referential.

λ : wavelength of the cast wave.

It is also good to note that an antenna beam will be the same in emission and reception due to the reciprocity principle.

We then need to emulate the electrical field resulting from the interference mentioned above :

This allows us to display a beam pattern in one dimension (Azimuth in this case), and to illustrate the influence of each parameter.

Figure 2 : Beam pattern for various number of radiating elements

Figure 3 : Beam pattern for various antenna orientation

Figure 3 is important as it illustrates that, with a phase shifter, the orientation of the antenna can be modified to point in a specific direction if, for instance, we are looking at a target of interest.

In order to obtain a full EM field, we need to combine the phenomenon in azimuth and elevation (usually the beam in azimuth will be much larger).

Figure 4 : Complete beam pattern (azimuth and elevation)

Figure 5 : Complete beam pattern (azimuth and elevation)

Usually, we want to work with a main beam as narrow as possible to increase directivity and sidelobes as low as possible to reduce residual radiation in unwanted directions. In order to achieve that as per fig. 2, we need to increase the width and therefore the number of radiating elements of the antenna.

### Verification

Array antennas are well documented, which makes the validation of their model simpler. Here are a few resources used to validate this model :

Phased Array Antenna Patters – Part 1: Linear Array Beam Characteristics and Array Factor, which (with different definitions and therefore formulas) achieve the same result.

Phased.Array.Antenna.Handbook.Artech.House.Publishers.Second.Edition.eBook-kB.pdf (free.fr), which approximates the half power beam to be 0.886 * λ/ L  (page 17), that is 12.6910°. Figure 2 shows the beam pattern obtained in the same conditions (6.34 x 2 degrees). Figure 3 also shows the power obtained for the second lobes (around -13 dBs)

Antenna Array Analysis with Custom Radiation Pattern – MATLAB & Simulink – MathWorks United Kingdom, Matlab library “phased”, which matches this 3d pattern with this set up.

Written by Geoffroy Heurtier – Project Engineer at Claytex

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