Engineering Circuit Analysis

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Engineering Circuit Analysis
CH8 Fourier Circuit Analysis
8.1 Fourier Series 8.2 Use of Symmetry
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series

• Most of the functions of a circuit are periodic
  functions
• They can be decomposed into infinite number of
  sine and cosine functions that are harmonically
  related.
• A complete responds of a forcing function
  
• Partial response to each
  harmonics.

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Ch8 Fourier Circuit Analysis
8.1 Fourier Series

• Harmonies Give a cosine function
• fundamental frequency ( is
  the fundamental wave form)
• Harmonics have frequencies

Freq. of the 1st harmonics (fund. freq)
Freq. of the 3rd harmonics
Freq. of the 4th harmonics
Freq. of the 2nd harmonics
Amplitude of the nth harmonics (amplitude of the
fundamental wave form)
Freq. of the nth harmonics
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series
Example
Fundamental v1 2cosw0t
v3a cos3w0t
v3b 1.5cos3w0t
v3c sin3w0t
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series
- Fourier series of a periodic function Given
a periodic function can be
represented by the infinite series as
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series
Example 12.1
Given a periodic function
It is knowing
It can be seen , we can evaluate
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series

• Review of some trigonometry integral observations
• (a)
• (c)
• (d)

(b)
(e)
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series

• Evaluations of

Based on (a) (b)
( is also called the DC component of
)
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series
Based on (b)
Based on (c)
Based on (e)
When kn
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series
Based on (a)
Based on (c)
Based on (d)
When kn
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series
Harmonic amplitude
Phase spectrum
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Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry
- Depending on the symmetry (odd or even), the
Fourier series can be further simplified. Even
Symmetry Observation rotate the function curve
along axis, the curve will overlap with
the curve on the other half of . Example
Odd Symmetry Observation rotate the function
curve along the axis, then along the
axis, the curve will overlap with the curve on
the other half . Example
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Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry

• Symmetry Algebra
• odd func. odd func. even func.
• Example
• (b) even func. odd func. odd func.
• Example
• (c) even func. even func. even func.
• Example

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Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry
(d) even func. const. ? even func. (No odd
func.) Example (e) odd func. ?odd func.
Example
odd func.
odd func.
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Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry
Apply the symmetry algebra to analyze the Fourier
series. If is an even function If
is an odd function
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Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry
Half-wave symmetry
f(t) -f(t - ) or f(t) -f(t )
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Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry
Fourier series