# Experimenting with inductors blog Part 1: Introduction to Fourier Series

Posted by rsjawale24 in Experimenting with Inductors on Jan 29, 2020 11:04:44 AMHi there!

This is the first part of my blog in Experimenting with Inductors. Here's a short introduction of me and what I do.

I'm a MS by Research Scholar from India working on Radio frequency antennas. I have been working on electronics since last 7-8 years when I picked up interest in Radio Frequency circuits. I decided to pursue my masters in RF.

Apart from Antennas, I have also worked on Embedded systems, Analog and RF circuits, SDRs and some general electronics projects.

I have entered the Experimenting with Inductors challenge to showcase a very beautiful and fundamental topic studied by every electrical engineer i.e. Fourier Series.

When I say Fourier series the first thing that comes in mind is a series of trigonometric functions!

Yes, it is math but do you know that you can prove or verify the Fourier series experimentally using inductors?(Well, not just inductors but some other passive components)

I have always believe that engineering education should be a bit fun by practically verifying the theories learnt in class.

## Introduction to Fourier Series

We all have studied Fourier series at some point of time in our engineering studies. It says that any periodic signal can be represented as a sum of harmonics of sines and cosines.

In other words, every periodic signal consists of harmonics of sines/cosines. **Does this mean we can extract sine from a square wave or a ramp?**

**Yes, we can! **

Let's take a look at the formula for Fourier series

Where a0, an and bn are the fourier coefficients which needs to be solved using integrals. I won't go into much of math here. I will directly show the results.

Now let's take an example of Fourier series representation of a periodic square wave

Assuming a even symmetric square wave, for even signals, bn= 0 as bn is the coefficient of sin(nx) which is an odd signal.

The Fourier series expansion is

which shows that a even square wave can be represented as the sum of infinite harmonics of cosines. In other words, a square wave can be made by adding infinite cosines.

## Experiment to prove Fourier series

Above, we saw that a square wave is made up of infinite cosines. **Is it possible to extract a cosine wave from a square wave?**

**Yes, it is. But how?**

Firstly, as we can see each cosine wave in the square wave has a different frequency (harmonics) which means we will need a frequency selective circuit that can separate these harmonics.

A filter is one such frequency selective circuit which can allow or stop a certain frequency or a band or frequencies depending upon type of filter.

So, in my experiment I plan to use a LC filter using Inductors from the Kemet Kit which I received for the Experimenting with inductors challenge.

A single LC filter provides two poles which means it provides a roll off of -40dB/dec at the cutoff frequency. I will be using a LC low pass filter with a cutoff frequency equal to the fundamental frequency of the square wave which we want to separate into cosines.

The low pass filter will filter out the fundamental frequency from the square wave.

Similarly, for the harmonics, a LC band pass filter will be used to separate the various harmonics of cosines from the square wave.

I will cover the pole-zero diagram, filter design details, calculations and circuits in the next part of the blog.

Till then, if you have any queries please post them below.

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