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

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

Analysis Techniques Circuit Excitation Method of Solution Chapter All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Analysis Techniques Circuit Excitation Method of Solution Chapter 1. dc (w/ switches) Transient analysis 5 & 6 2. ac Phasor-domain analysis 7 - 9 ( steady state only) 3. Any waveform LaplaceTransform 6 (single-sided only) (transient + steady state) 4. Any waveform Fourier Transform 12 (double-sided) (transient + steady state) This chapter single-sided: defined over [0,∞] double-sided: defined over [−∞,∞]

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

Fourier Series Analysis Technique Example (details later) 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 Cont.

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 Example 12-2: Line Spectra (cont.)

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

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

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Example 12-3: M-Waveform This oscillatory behavior of the Fourier 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.

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 Cont.

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

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

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

© 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 Series Analysis Technique Fourier Transform Analysis Technique

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 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

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

Example 12-9: Rectangular Pulse 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

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Circuit Analysis with Fourier Transform Example 12-11 vs(t) = 10 + 5 cos 4t All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Cont.

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