Fourier Series of an Inverter Circuit

Power Inverter Circuit
Fourier Series Representations of Switching Functions
Power and Average Power in Fourier Series

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.

Square wave example.

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 Sign Function

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.

Fourier Series Representations of Switching Functions
Power and Average Power in Fourier Series

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