Transcript Ch.17

Fundamentals of
Electric Circuits
Chapter 17
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Overview
• This chapter introduces the Fourier series.
• The definition and properties of the series
will be introduced.
• Symmetry considerations for different
waveforms will be covered.
• The general concept of applying the Fourier
series to circuit analysis is discussed.
2
Trigonometric Fourier Series
• While studying heat flow, Fourier discovered
that a nonsinusoidal periodic function can be
expressed as an infinite sum of sinusoidal
functions.
• Recall that a periodic function satisfies:
f t   f t  nT 
• Where n is an integer and T is the period of
the function.
3
Trigonometric Fourier Series
• According to the Fourier theorem, any
practical periodic function of frequency ω0
can be expressed as an infinite sum of sine
or cosine functions.

f  t   a0    an cos n0t  bn sin n0t 
dc
n 1
ac
• Where ω0=2/T is called the fundamental
frequency in radians per second.
4
Harmonics
• The sinusoid sin(nω0t) or cos(nω0t) is called
the n’th harmonic of f(t).
• If n is odd, the function is called the odd
harmonic.
• If n is even, the function is called the even
harmonic.
• The equation on the last slide is called the
trigonometric Fourier series of f(t).
• The constants an and bn are called the
Fourier coefficients.
5
Fourier Series
• The Fourier series of a function is a
representation that resolves the function into
a dc component and an ac component.
• For a function to be expressed as a Fourier
series it must meet certain requirements:
1. f(t) must be single valued everywhere.
2. It must have a finite number of finite
discontinuities per period.
3. It must have a finite number of maximum and
minima per period.
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Trigonometric Fourier Series
• The last requirement is that
t 0 T
 f t dt  
t0
• For any t0
• These conditions are called the Dirichlet
conditions.
• A major task in Fourier series is the
determination of the Fourier coefficients.
• The process of finding these is called Fourier
analysis.
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Trigonometric Fourier Series
• To find a0:
1
a0 
T
T
 f  t  dt
o
• To find an:
T
2
an   f  t  cos n0tdt
To
• To find bn:
T
2
bn   f  t  sin n0tdt
T o
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Amplitude-Phase Form
• An alternative is called the amplitude phase
form:

f  t   a0   An cos  n0t  n 
n 1
• Where:
An  a  b
2
n
2
n
bn
n   tan
an
1
• The frequency spectrum of a signal consists
of the plots of amplitude and phases of the
harmonics versus frequency
9
Example 1
Determine the Fourier series of the waveform shown
below. Obtain the amplitude and phase spectra
10
Solution:
1, 0  t  1
f (t )  
and f (t )  f (t  2)
0, 1  t  2
2 T
an   f (t ) cos(n0t )dt  0 and
T 0
 2 / n , n  odd
2 T
bn   f (t ) sin(n0t )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
11
Symmetry Considerations
• If one looks at a typical Fourier
series, for the square wave, for
example.
• The series consists of only sine
terms.
• If the series contains only sine or
cosine, it is considered to have a
certain symmetry.
• There is a technique for identifying
the three symmetries that exist,
even, odd, and half-wave.
12
Even Symmetry
• In the case of even
symmetry, the function is
symmetrical about the
vertical axis:
f t   f  t 
• A main property of an even
function is that:
T /2

T /2
T /2
f e  t  dt  2  f e  t  dt
0
13
Even Symmetry
• The Fourier coefficients for an even function
become:
2
T /2
a0 
4
an 
T
T
 f  t  dt
0
T /2
 f  t  cos n tdt
0
0
bn  0
• Note that this become a Fourier cosine
series.
• Since Cosine is an even function, one can
see how this series is called even.
14
Odd Symmetry
• A function is said to be odd if its plot is
antisymmetrical about the vertical axis.
f  t    f t 
• Examples of odd functions are t, t3, and sin t
• An add function has this major
characteristic:
T /2

f 0  t  dt  0
T /2
15
Odd Symmetry
• This comes about because
the integration from –T/2 to 0
is the negative of the
integration from 0 to T/2
• The coefficients are:
a0  0 an  0
4
bn 
T
T /2
 f  t  sin n tdt
0
0
• This gives the Fourier sine
series,
16
Properties of Odd and Even
1. The product of two even functions is also
an even function.
2. The product of two odd functions is an even
function.
3. The product of an even function and an odd
function is an odd function.
4. The sum (or difference) of two even
functions is also an even function.
17
Properties of Odd and Even II
5. The sum (or difference) of two odd
functions is an odd function.
6. The sum (or difference) of an even function
and an odd function is neither even nor odd.
18
Half Wave Symmetry
• Half wave symmetry compares one half of a
period to the other half.
• In this context, a half wave (odd) symmetric
function has the following property:
 T
f  t     f t 
 2
• This means that each half-cycle is the mirror
image of the next half-cycle.
19
Half Wave Symmetry
• The Fourier coefficients for the half wave
symmetric function are:
a0  0
 4 T /2
f  t  cos n0tdt for n odd

an   T 0
for n even

0

 4 T /2
f  t  sin n0tdt for n odd

bn   T 0
for n even

0

• Note that the half wave symmetric functions
only contain the odd harmonics .
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Common Functions
21
Common 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
23
24
Circuit Applications
• Fourier analysis can be helpful in analyzing
circuits driven by non-sinusoidal waves.
• The procedure involves four steps:
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 responses
using the superposition principle.
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Circuit Applications
• The first step is to determine the Fourier
series expansion of the excitation.
• For example a periodic voltage can be
expressed as a Fourier series as follows:

v  t   V0  Vn cos  n0t  n 
n 1
• On inspection, this can be represented by a
dc source and a set of sinusoidal sources
connected in series.
• Each source would have its own amplitude
and frequency.
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Circuit Applications II
• Example of a Fourier series expanded
periodic voltage source.
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Circuit Applications III
• Each source can be analyzed on its own by
turning off the others.
• For each source, the circuit can be
transformed to frequency domain and solved
for the voltage and currents.
• The results will have to be transformed back
to the time domain before being added back
together by way of the superposition
principle.
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Circuit Applications IV
29
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t , n  2k  1
2  n1 n
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Solution
Phasor of the circuit
V0 
j 2n
Vs
5  j 2n
For dc component, (n=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
)
5
Amplitude spectrum of the
output voltage
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Average Power and RMS
• Fourier analysis can be applied to find
average power and RMS values.
• To find the average power absorbed by a
circuit due to a periodic excitation, we write
the voltage and current in amplitude-phase
form:

v  t   Vdc  Vn cos  n0t  n 
n 1

i  t   I dc   I m cos  m0t  m 
m 1
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Average Power and RMS
• For periodic voltages and currents, the total
average power is the sum of the average
powers in each harmonically related voltage
and current:
1 
P  Vdc I dc  Vn I n cos  n  n 
2 n 1
• A RMS value is:
Frms
1  2 2
 a   an  bn
2 n1
2
0


• Parseval’s theorem defines the power
dissipated in a hypothetical 1Ω resistor
2
p1  Frms

1
 a02   an2  bn2
2 n1


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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
*
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35
Exponential Fourier Series
• A compact way of expressing the Fourier
series is to put it in exponential form.
• This is done by representing the sine and
cosine functions in exponential form using
Euler’s law.
1 jn0t
cos n0t  e
 e  jn0t 
2
1 jn0t  jn0t
e

sin n0t 
e
2j
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Exponential Fourier Series
• This can be rewritten as:
f t  

jn0t
c
e
 n
n 
• This is the complex or exponential Fourier
series representation.
• The values of cn are:
T
1
cn   f  t  e  jn0t dt
T 0
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Exponential Fourier Series
• The exponential Fourier series of a periodic
function describes the spectrum in terms of
the amplitude and phase angle of ac
components at positive and negative
harmonic frequencies.
• The coeffcients of the three forms of Fourier
series (sine-cosine, amplitude-phase, and
exponential form) are related by:
Ann  an  jbn  2cn
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