EE2003
Circuit Theory
Chapter 17
The Fourier Series
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The Fourier Series - Chapter 17
17.1
17.2
17.3
17.4
17.5
17.5
Trigometric Fourier Series
Symmetry Considerations
Circuit Applications
Average Power and RMS Values
Exponential Fourier Series
Applications
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17.1 Trigometric Fourier Series (1)
• The Fourier series of a periodic function f(t)
is a representation that resolves f(t) into a dc
component and an ac component comprising
an infinite series of harmonic sinusoids.
• Given a periodic function f(t)=f(t+nT) where
n is an integer and T is the period of the
function.

f (t )  a0   (a0 cos nw0t  bn sin nw0t )
 n1
dc



ac
where w0=2∏/T is called the fundamental
frequency in radians per second.
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17.1 Trigometric Fourier Series (1)
• and an and bn are as follow
2 T
an   f (t ) cos( nwot )dt
T 0
2 T
bn   f (t ) sin( nwot )dt
T 0
• in alternative form of f(t)

f (t )  a0   ( An cos( nw0t  n )
 n 1
dc

ac
where
An  an2  bn2 , n   tan 1 (
bn
)
an
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17.1 Trigometric Fourier Series (2)
Conditions (Dirichlet conditions) on f(t) to
yield a convergent 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

t 0 T
t0
f (t ) dt   for any t0 .
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17.1 Trigometric Fourier Series (3)
Example 1
Determine the Fourier series of the waveform
shown below. Obtain the amplitude and phase
spectra
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17.1 Trigometric Fourier Series (4)
Solution:
1, 0  t  1
f (t )  
and f (t )  f (t  2)
0, 1  t  2
2 T
an   f (t ) cos( nw0t )dt  0 and
T 0
 2 / n , n  odd
2 T
bn   f (t ) sin( nw0t )dt  
n  even
T 0
0,
 2 / n , n  odd
An  
n  even
 0,
  90, n  odd
n  
n  even
 0,
a) Amplitude and
b) Phase spectrum
1 2  1
f (t )    sin( nt ), n  2k  1
2  k 1 n
Truncating the series at N=11
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17.2 Symmetry Considerations (1)
Three types of symmetry
1. Even Symmetry : a function f(t) if its plot
is symmetrical about the vertical axis.
f (t )  f (t )
In this case,
2 T /2
a0   f (t )dt
T 0
4 T /2
an   f (t ) cos( nw0t )dt
T 0
bn  0
Typical examples of even periodic function
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17.2 Symmetry Considerations (2)
2. Odd Symmetry : a function f(t) if its plot is
anti-symmetrical about the vertical axis.
f (t )   f (t )
In this case,
a0  0
4 T /2
bn   f (t ) sin( nw0t )dt
T 0
Typical examples of odd periodic function
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17.2 Symmetry Considerations (3)
3. Half-wave Symmetry : a function f(t) if
a0  0
T
f (t  )   f (t )
2
 4 T /2

f (t ) cos( nw0t )dt , for n odd
an   T 0

0
, for an even
 4 T /2

f (t ) sin( nw0t )dt , for n odd
bn   T 0

0
, for an even
Typical examples of half-wave odd periodic functions
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17.2 Symmetry Considerations (4)
Example 2
Find the Fourier series expansion of f(t)
given below.
Ans:
f (t ) 
1
n   n 
1

cos
t

 sin 

 n 1 n 
2   2 
2

*Refer to in-class illustration, textbook
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17.2 Symmetry Considerations (5)
Example 3
Determine the Fourier series for the halfwave cosine function as shown below.
Ans:
1 4
f (t )   2
2 

1
cos nt , n  2k  1

2
k 1 n
*Refer to in-class illustration, textbook
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17.3 Circuit Applications (1)
Steps for Applying Fourier Series
1. Express the excitation as a Fourier series.
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.
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17.3 Circuit Applications (2)
Example 4
Find the response v0(t) of the circuit below
when the voltage source vs(t) is given by
1 2  1
vs (t )    sin nwt , n  2k  1
2  n 1 n
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17.3 Circuit Applications (3)
Solution
Phasor of the circuit
V0 
j 2n
Vs
5  j 2n
For dc component, (wn=0 or n=0), Vs = ½
=> Vo = 0
For nth harmonic,
2
4  tan 1 2n / 5
VS 
  90, V0 
Vs
2 2
n
25  4n 
In time domain,

v0 (t )  
k 1
4
25  4n 2 2
cos(nt  tan 1
2n
)
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Amplitude spectrum of
the output voltage
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17.4 Average Power and RMS Values (1)
Given:


n 1
n 1
v(t )  Vdc   Vn cos( nw0t   n ) and i (t )  I dc   I m cos( mw0t  m )
The average power is
1 
P  Vdc I dc   Vn I n cos( n  n )
2 n 1
The rms value is

Frms  a   (an2  bn2 )
2
0
n 1
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17.4 Average Power and RMS Values (2)
Example 5:
Determine the average power supplied to
the circuit shown below if
i(t)=2+10cos(t+10°)+6cos(3t+35°) A
Ans: 41.5W
*Refer to in-class illustration, textbook
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17.5 Exponential Fourier Series (1)
• The exponential Fourier series of a periodic function f(t)
describes the spectrum of f(t) in terms of the amplitude
and phase angle of ac components at positive and
negative harmonic.
f (t ) 

jnwo t
c
e
n
n  
1
cn 
T

T
0
f (t )e  jnw0t dt , where w0  2 / T
• The plots of magnitude and phase of cn versus nw0 are
called the complex amplitude spectrum and complex
phase spectrum of f(t) respectively.
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17.5 Exponential Fourier Series (1)
• The complex frequency spectrum of the function
f(t)=et, 0<t<2 with f(t+2)=f(t)
(a) Amplitude spectrum;
(b) phase spectrum
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17.6 Application – filter (1)
•Filter are an important component of electronics and
communications system.
•This filtering process cannot be accomplished without
the Fourier series expansion of the input signal.
•For example,
(a) Input and output spectra of a lowpass filter, (b) the
lowpass filter passes only the dc component when wc << w0
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17.6 Application – filter (2)
(a) Input and output spectra of a bandpass filter, (b) the
bandpass filter passes only the dc component when B << w0
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