Electronic – the function of a Fourier Series


What is a Fourier Series? What it is used for?

Best Answer

The Fourier series:

\$ V_t = \dfrac{a_0}{2} + \displaystyle \sum_{i=1}^{\infty}[a_i sin(i \omega_0 t) + b_i cos(i \omega_0 t) ] \$

The term \$\dfrac{a_0}{2}\$ is a constant, that's the DC level. It could also have been written without dividing by two, but this is the convention. The terms of the infinite sum are the sum of a weighted sine and a weighted cosine with the same frequency. If you would draw these as phasors in the complex Argand plane you'd see that the result is again a sine, but with a different amplitude, and phase shifted. Therefore the equation can also be written as

\$ V_t = \dfrac{a_0}{2} + \displaystyle \sum_{i=1}^{\infty}[a_i sin(i \omega_0 t + \phi_i) ] \$

So we have the sum of sines, all multiple frequencies of a fundamental frequency \$\omega_0\$, each of them with its own amplitude and phase.

Fourier proved that you can describe every repetitive function this way. Sometimes the series is infinite, sometimes it has a finite number of terms. Sometimes terms are missing, which means their amplitude is zero.

One of the best known Fourier series is that of a square wave:

\$ V_t = \displaystyle \sum_{i=1}^{\infty}\left[\dfrac{sin((2i - 1) \omega_0 t)}{2i - 1} \right] \$

or, expanded:

\$ V_t = sin(\omega_0 t) + \dfrac{1}{3} sin(3 \omega_0 t) + \dfrac{1}{5} sin(5 \omega_0 t) + \dfrac{1}{7} sin(7 \omega_0 t) + ...\$

So this is such a series with missing terms: a square wave has no even harmonics. The following image shows what it looks like in the time domain:

enter image description here

The top drawing shows the sum of the first two terms, then a third and at the bottom a fourth term is added. Each added term will bring the waveform closer to a square wave, and you'll need the limit of the series to infinity to get a perfect square wave.

Sometimes it's difficult to see the fundamental sine in it. Take for instance the sum of a 3Hz sine and a 4Hz sine. The resulting waveform will repeat once every second, that's 1Hz. The 1Hz is the fundamental, even if its amplitude is zero. The series can be written as

\$ V_t = 0 \cdot sin(\omega_0 t) + 0 \cdot sin(2 \omega_0 t) + sin(3 \omega_0 t) + sin(4 \omega_0 t)\$

All the following terms also have zero amplitude.