12. fourier analysis circuits by ulaby & maharbiz all rights reserved. do not copy or...

12. FOURIER ANALYSIS

CIRCUITS by Ulaby & MaharbizAll rights reserved. Do not copy or

distribute. © 2013 National Technology and

Science Press

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© 2013 National

Technology and Science

Press

Analysis Techniques

Circuit Excitation Method of SolutionChapter

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 [−∞,∞]

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1. Periodic Excitation:

Solution Method: Fourier series + Phasor Analysis

2. Nonperiodic Excitation:

Solution Method: Fourier Transform

Fourier Analysis

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Fourier Series Analysis Technique

(details later)

Example

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Fourier Series Analysis Technique (cont.)

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Fourier Series Analysis Technique (cont.)

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Fourier Series: Cosine/Sine Representation

The Fourier theorem states that a periodic function f(t) of period T can be cast in the form

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Example

Fourier series:

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Example 12-1: Sawtooth WaveformAll rights reserved. Do

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Fourier Series: Amplitude/Phase Representation

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Example 12-2: Line Spectra (cont.)

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Symmetry Considerations

dc

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Even & Odd Symmetry

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This oscillatory behavior of the Fourierseries in the neighborhood of discontinuous points is called the Gibbs phenomenon.

Example 12-3: M-WaveformAll rights

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Circuit Applications

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

Example 12-5: RC Circuit cont.

Cont.

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Example 12-5: RC Circuit cont.

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Average Power

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Fourier Series: Exponential Representation

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Fourier Transform

Fourier Series Analysis Technique

Fourier Series Analysis Technique

Fourier Transform Analysis Technique

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Example 12-8: Pulse Train

Note that:

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Line Spectrum of Pulse Train

Spacing between adjacent harmonics is :

spectrum becomes continuous

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Derivation Of Fourier Transform

Fourier Transform Pair

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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

vs(t) = 10 + 5 cos 4t

Example 12-11

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Circuit Analysis with Fourier Transform

Applying Inverse Fourier Transform:

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Summary

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