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Chapter 18
Fourier Circuit
Analysis
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A general periodic function of period T=2/0 can be represented by an infinite sum of
harmonic sines and cosines.
The harmonics of
v1(t) = cos(0t)
have frequencies n0, where 0 is the
fundamental frequency and n = 1, 2, 3, . . . .
The sum (green) of a
fundamental (blue) and a
third harmonic (red) can look
very different, depending on
the amplitude and phase of
the harmonic.Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for
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Any normal periodic function f(t) can be
expressed as a Fourier series:
The period T and fundamental frequency 0 satisfy
T=2/0
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Find the Fourier Series of the half-wave rectified
sine wave shown.
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The discrete-line spectrum with Vm=1
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Even: f(t)=f(-t)
FS: bn=0
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Odd:
f(t)=-f(-t)
FS: an=0
Find i(t).
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A more compact and simpler method of
expressing the Fourier series is to use complex
exponentials instead of sine and cosine:
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Determine the cn values for v(t).
Answer: 2/(n) sin(n/2) for n odd, 0 otherwise
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The Fourier Series concept can be extended to
include non-periodic waveforms using the
Fourier Transform:
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Parsevals Theorem
allows us to think of |F(j)|2 as the energy
density of f(t) at .
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The Unit-Impulse:
Cosine:
Other transform pairs are derived in Section
18.7 and summarized in Table 18.2
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(t t0)e jt0
cos(0t) (0 ) (0 )
The Fourier Transform also exists for periodic
functions, although we must resort to using the
impulse function to represent it:
With this knowledge, Fourier Series can be
ignored in favor of the Fourier Transform.Copyright 2013 The McGraw-Hill Companies, Inc. Permission required for
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The system function H(j), defined as the
Fourier transform of the impulse response
allows the calculation of the output of a system
given the Fourier Transform of its input:
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h(t)H( j)
the system function and the transfer function
are identical: H( j) = G()[The fact that one argument is while the other is indicated by j is
immaterial and arbitrary; the j merely makes possible a more direct
comparison between the Fourier and Laplace transforms.]
Our previous work on steady-state sinusoidal
analysis using phasors was but a special case
of the more general techniques of Fourier
transform analysis.
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Find v0(t) using Fourier techniques.
Method: find
H(j) by assuming
Vo and Vi are
sinusoids.
So: H(j)=j2/(4 + j2)
and using FT tables and partial fractions:
vo(t)=5(3e3t 2e2t )u(t)
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