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EEB 311 (2013/2014)Electrical Network Theory
The Fourier Series 1.Introduction2.Trigonometric Fourier Series3.Symmetry Considerations4.Circuit Applications
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• Previously, have considered analysis of circuits with sinusoidal sources.
• The Fourier series provides a means of analyzing circuits with periodic non-sinusoidal excitations.
• Fourier is a technique for expressing any practical periodic function as a sum of sinusoids.
• Fourier representation + superposition theorem, allows to find response of circuits to arbitrary periodic inputs using phasor techniques.
IntroductionIntroduction
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• The Fourier series of a periodic functionperiodic function f(t) is a representation that resolves f(t) into a dc componentdc component and an ac componentac component comprising an infinite series of harmonic sinusoids.
• Given a periodic function f(t)=f(t)=f(t+nT) ) where nn is an integer and TT is the period of the function.
where w0=2π/T is called the fundamental frequency in radians per second.
Trigonometric Fourier Series
ac
nnn
dc
tnbtnaatf
1
000 )sincos()(
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• and Fourier coefficients, an and bn , are:
Trigonometric Fourier SeriesTrigonometric Fourier Series
T
on dttntfT
a0
)cos()(2
)(tan , 1n
22
n
nnnn a
bbaA
T
on dttntfT
b0
)sin()(2
• in alternative form of f(t)
where
ac
nnn
dc
tnAatf
1
00 )cos(()(
T
dttfT
a00 )(
1
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ConditionsConditions ((Dirichlet conditions conditions)) on on f(t)f(t) to yield a to yield a convergent Fourier seriesconvergent Fourier series:
1. f(t) is single-valued everywhere.
2. f(t) has a finite number of finite discontinuities in any one period.
3. f(t) has a finite number of maxima and minima in any one period.
4. The integral
Trigonometric Fourier SeriesTrigonometric Fourier Series
.any for )( 0
0
0
tdttfTt
t
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Example
Determine the Fourier series of the waveform shown below. Obtain the amplitude and phase spectra
Trigonometric Fourier SeriesTrigonometric Fourier Series
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Solution:
Trigonometric Fourier Series
)2()( and 21 ,0
10 ,1)(
tftft
ttf
evenn ,0
oddn,/2)sin()(
2
and 0)cos()(2
0 0
0 0
ndttntf
Tb
dttntfT
a
T
n
T
n
1
12 ),sin(12
2
1)(
k
kntnn
tf
evenn ,0
oddn,90
evenn ,0
oddn,/2
n
n
nA
Truncating the series at N=11
a) Amplitude andb) Phase spectrum
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Three types of symmetry
1. Even Symmetry : a function f(t) if its plot is symmetrical about the vertical axis.
In this case,
Symmetry Considerations
)()( tftf
0
)cos()(4
)(2
2/
0 0
2/
00
n
T
n
T
b
dttntfT
a
dttfT
a
Typical examples of even periodic function
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2. Odd Symmetry : a function f(t) if its plot is anti-symmetrical about the vertical axis.
In this case,
Symmetry Considerations
)()( tftf
2/
0 0
0
)sin()(4
0
T
n dttntfT
b
a
Typical examples of odd periodic function
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3. Half-wave Symmetry : a function f(t) if
Symmetry Considerations
)()2
( tfT
tf
even n for ,
odd n for ,
even n for ,
odd n for ,
0
)sin()(4
0
)cos()(4
0
2/
0 0
2/
0 0
0
T
n
T
n
dttntfTb
dttntfTa
a
Typical examples of half-wave odd periodic functions
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Example
Find the Fourier series expansion of f(t) given below.
Symmetry Considerations
1 2sin
2cos1
12)(
n
tnn
ntf
Ans:
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Example
Determine the Fourier series for the half-wave cosine function as shown below.
Symmetry Considerations
1
2212 ,cos
14
2
1)(
k
knntn
tf
Ans:
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Circuit ApplicationsCircuit Applications
Steps for Applying Fourier Series
1. Express the excitation as a Fourier series.Example, for periodic voltage source:
2. Transform the circuit from the time domain to the frequency domain.
3. Find the response of the dc and ac components in the Fourier series.
4. Add the individual dc and ac response using the superposition principle.
ac
1nn0n
dc
0 )tncos(V(V)t(v
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Example
Find the response v0(t) of the circuit below when the voltage source vs(t) is given by
Circuit Applications
12 ,sin12
2
1)(
1
kntnn
tvn
s
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Solution
Phasor of the circuit
For dc component, (n=0 or n=0), Vs = ½ => Vo = 0
For nth harmonic,
In time domain,
Circuit Applications
s0 V25
2V
nj
nj
)5
2tan(c
425
4)(
1
1
220
k
ntnos
ntv
s22
1
0 V425
5/2tan4V ,90
2V
n
n
nS
Amplitude spectrum of the output voltage
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Given:
Useful Formula