## 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 . 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 will exhibit period . This is plotted in the figure to the right. The general square wave function with amplitude 1 is sometimes given the symbol , and is defined as follows:

where

The average of is zero. As in the full wave rectifier case, the symmetry about the y-axis is such that for all $n$ . With the change of variables , and the choice , the remaining coefficients $a_n$ of are

Since the component amplitudes decrease as , the 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 .

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.