Fourier Series

Phasors generally give any budding electrical engineer an effective tool to analyze circuits with sources that have sinusoidal waveforms. However, as the discipline advances, phasors will become obsolete. This is due to the presence of sources that do not have sinusoidal waveforms. This limitation can be overcame with another method, that is, using Fourier series.

Fourier series, developed by Jean-Baptiste Joseph Fourier, is simply an expanded representation of a particular waveform. During his experiment on heat conduction, he managed to explore a new mathematical principle, that is an infinite number of sinusoids. Basically, the Fourier series tells us that a nonsinusoid waveform is simply a collection of infinite number of sinusoids that vary in amplitude and frequency. This fact is illuminating since the individual sinusoids can be converted to phasors and the nonsinusoid waveform can be represented as a collection of these phasors.

However, when dealing with complex circuits, utilizing the idea of Fourier series alone is impractical. Thus, the Fourier transform was introduced. The Fourier transform is another excellent tool, especially for nonperiodic waveforms, that can be used to analyze AC circuits with more than two storage elements. It is given by the equation that relates the function in the frequency domain and the integration of function in the time domain multiplied to a damped exponential function of time evaluated from negative infinity to positive infinity.

Fourier transform can be characterized by different properties. One is the equation stating that the Fourier transform of a function is a complex number, that is having imaginary quantities. Furthermore, the real part of the Fourier transform is the integration of the function in time domain and a cosine function at the fundamental frequency evaluated from negative infinity to positive infinity. The same goes for the imaginary part, though, with the addition of a negative factor and a sine function instead of the cosine.

Posted 2010-12-14 and updated on Dec 14, 2010 6:19am by crisd

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