12. FOURIER ANALYSIS CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press.

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12. FOURIER ANALYSIS CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Analysis Techniques single-sided: defined over [0,∞] double-sided: defined over [ − ∞,∞] All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

1. Periodic Excitation: Solution Method: Fourier series + Phasor Analysis 2. Nonperiodic Excitation: Solution Method: Fourier Transform Fourier Analysis All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Fourier Series Analysis Technique (details later) Example Cont. All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Fourier Series Analysis Technique (cont.) Cont. All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Fourier Series Analysis Technique (cont.) All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Fourier Series: Cosine/Sine Representation The Fourier theorem states that a periodic function f(t) of period T can be cast in the form All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example Fourier series: All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example 12-1: Sawtooth Waveform All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Fourier Series: Amplitude/Phase Representation All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example 12-2: Line Spectra (cont.) All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Symmetry Considerations dc All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Even & Odd Symmetry All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

This oscillatory behavior of the Fourier series in the neighborhood of discontinuous points is called the Gibbs phenomenon. Example 12-3: M-Waveform All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Circuit Applications All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Cont. All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example 12-5: RC Circuit cont. Cont. All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example 12-5: RC Circuit cont. Cont. All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Average Power All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Fourier Series: Exponential Representation All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Fourier Transform Fourier Series Analysis Technique Fourier Transform Analysis Technique All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example 12-8: Pulse Train Note that: All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Line Spectrum of Pulse Train Spacing between adjacent harmonics is : spectrum becomes continuous All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Derivation Of Fourier Transform Fourier Transform Pair All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example 12-9: Rectangular Pulse The wider the pulse, the narrower is its spectrum, and vice versa All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Circuit Analysis with Fourier Transform vs(t) = cos 4t Example Cont. All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Circuit Analysis with Fourier Transform Applying Inverse Fourier Transform: All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Summary All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

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