Post on 12-Jan-2016
12. FOURIER ANALYSIS
CIRCUITS by Ulaby & MaharbizAll rights reserved. Do not copy or
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Science Press
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Technology and Science
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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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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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