Fractal Mathematics and its Application in
Electronics Engineering (Antenna Design)

Geometry is an important branch of mathematics that deals in mathematical computations and analysis of shape, size and relative position of the objects (2D - Triangle, Square, Circle etc and 3D - sphere, cylinder, pyramid etc). We are very much familiar with this and we know geometry finds applications in various fields like art, architecture, physics, engineering etc.

There is another fascinating branch of mathematics (or geometry) that deals in never ending patterns or 'self similar' patterns. It is called fractal mathematics. It is relatively new area and is finding lots of applications in many fields like earth sciences, chemistry, physics, engineering etc.

Fractal Image is a never ending image, means if you zoom a fractal image you will see a same pattern, no matter how much you zoom it. To clarify the distinction, here is an example - if you zoom any image (any standard image or geometry), with every zoom you will see the finer details of the image coming out and becoming visible (try zooming any picture). But if you zoom a fractal image no matter how many times you zoom it, same pattern is repeated again and again and again.

There are two simple examples of fractal images - Koch Curve and Hilbert Curve. We will try to explain Koch Curve as it is the simplest of the all fractal images.

Take an equilateral triangle and divide each of its edge in three equal segments. Use a middle segment as base and draw 3 more equilateral triangles pointing outwards (one on each middle segment on each edge). Repeat this process again and again and you obtain your fractal!

Application of Fractal in Antenna Design

Fractal Antenna is an antenna that uses fractal geometry or self repeating geometry to maximize the size and perimeter of the conductor (cooper etc.) for transmitting and receiving electromagnetic radiations. Fractal Antennas are very compact in size, are wide band (or multiband) and are used in applications like cellular mobile communication and microwave communication.

A good example of a fractal antenna as a spacefilling curve is in the form of a shrunken fractal helix (see image below). Here, each line of copper is just a small fraction of a wavelength.

Originally antennas are designed with the help of coils, some of the examples are microstrip patch antennas, meander line antenna, stubby coil antenna etc. They are effective in simpler applications of single band communication. But as the applications today are becoming more complex and demanding, fractal offers a promising solution in antenna design. Some benefits of using fractals are:

• Multiband and Broadband characteristics which were derived from the inherent properties of the fractal geometry of the antenna
• Along with maintaining excellent efficiencies and gains, fractal antennas are compact and provide miniaturization
• Generally fractal antennas are mechanically simple and robust

Some of the areas where fractal antennas are being deployed these days are - Surveillance and information gathering, satellite communication, medical applications (detection of cancer and tumors) etc

By Sagar Juneja - Research Associate, Chitkara University,
Jasmine Kaur Boparai - Assistant Professor, Chitkara University H.P.

References:

https://en.wikipedia.org/wiki/Fractal_antenna

https://en.wikipedia.org/wiki/Koch_snowflake