# Fourier Series of an Inverter Circuit

One of the simplest inverters switches between polarities of a DC input to create a square wave. This wave can be filtered to give an approximate sinusoidal output. Let’s find the Fourier Series of a square wave of radian frequency $\omega$. We will also determine which components are wanted and the amplitude of the wanted components. These specifications will have a large impact on later stages of the design as we start looking at physical hardware to build our designed circuit.

An example of a general square wave.

A square wave of amplitude $V_0$ and frequency $\omega$ will exhibit period $T = 2\pi/\omega$. This is plotted in the figure to the right. The general square wave function with amplitude 1 is sometimes given the symbol $sq(\omega t)$, and is defined as follows:

$sq(\omega t) = sgn[cos(\omega t)]$

where

The average of $sq(\omega t)$ is zero. As in the full wave rectifier case, the symmetry about the y-axis is such that $b_n = 0$ for all $n$. With the change of variables $\theta = \omega t$, and the choice $\tau = -T/4$, the remaining coefficients $a_n$ of $V_) sq(\theta)$ are

$a_n = \frac{1}{\pi} \int_{-\pi/2}^{3pi/2} V_0 sq(\theta) cos(n\theta) d\theta$

$= \frac{1}{\pi} \int_{-\pi/2}^{3pi/2} V_0 cos(n\theta) d\theta - \frac{1}{\pi} \int_{\pi/2}^{3pi/2} V_0 cos(n\theta) d\theta$

$= \frac{4V_0}{n\pi} sin(\frac{n\pi}{2})$

Since the component amplitudes decrease as $1/n$, the $n= 1$ term is the largest. Presumably, this is the one to use in the inverter application. Therefore, the fundamental is also the wanted component. Its amplitude is $4V_0/\pi$.

In a typical power converter application, the designer identifies the wanted component, then tries to generate a waveform from a switch matrix such that the wanted component is large and the unwanted components are small. This is not a trivial task, but Fourier analysis helps to point in the right direction.