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EEEB123EEEB123Circuit Analysis 2Circuit Analysis 2

Chapter 17Chapter 17

The Fourier SeriesThe Fourier Series

Materials from Fundamentals of Electric Circuits (4th Edition), Alexander & Sadiku, McGraw-Hill Companies, Inc.

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The Fourier Series - Chapter 17The Fourier Series - Chapter 17

17.1 Introduction 17.2 Trigonometric Fourier Series 17.3 Symmetry Considerations 17.4 Circuit Applications

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• Previously, have considered analysis of circuits with sinusoidal sources.

•The Fourier series provides a means of a means of analyzing circuits with periodic non-analyzing circuits with periodic non-sinusoidal excitationssinusoidal 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.

17.1 Introduction(1)17.1 Introduction(1)

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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+nT) f(t)=f(t+nT) where nn is an integer and TT is the period of the function.

where 0=2/T is called the fundamental fundamental frequency frequency in radians per second.

17.2 Trigonometric Fourier Series (1)17.2 Trigonometric Fourier Series (1)

ac

nn

dc

tnbtnaatf

1

0000 )sincos()(

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• and Fourier coefficients, an and bn , are:

17.2 Trigonometric Fourier Series (2)17.2 Trigonometric Fourier Series (2)

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(()(

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Conditions (Conditions (Dirichlet conditionsDirichlet conditions) on ) on f(t)f(t) to yield a convergent Fourier seriesto 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

17.1 Trigometric Fourier Series (3)17.1 Trigometric Fourier Series (3)

.any for )( 0

0

0

tdttfTt

t

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Example 17.1Example 17.1

Determine the Fourier series of the waveform shown below. Obtain the amplitude and phase spectra

17.2 Trigometric Fourier Series (4)17.2 Trigometric Fourier Series (4)

*Refer to textbook, pg. 760

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SolutionSolution:

17.2 Trigonometric Fourier Series (5)17.2 Trigonometric Fourier Series (5)

)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 symmetryThree types of symmetry

1. Even Symmetry Even Symmetry : a function f(t) if its plot is symmetrical about the vertical axis.

In this case,

17.3 Symmetry Considerations (1)17.3 Symmetry Considerations (1)

)()( 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 SymmetryOdd Symmetry : a function f(t) if its plot is anti-symmetrical about the vertical axis.

In this case,

17.3 Symmetry Considerations (2)17.3 Symmetry Considerations (2)

)()( 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 SymmetryHalf-wave Symmetry : a function f(t) if

17.3 Symmetry Considerations (3)17.3 Symmetry Considerations (3)

)()2

( tfT

tf

evenan for , 0

oddn for , )sin()(4

evenan for , 0

oddn for , )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 17.3Example 17.3

Find the Fourier series expansion of f(t) given below.

17.3 Symmetry Considerations (4)17.3 Symmetry Considerations (4)

1 2sin

2cos1

12)(

n

tnn

ntf

Ans:

*Refer to textbook, pg. 771

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Example 17.4Example 17.4

Determine the Fourier series for the half-wave cosine function as shown below.

17.3 Symmetry Considerations (5)17.3 Symmetry Considerations (5)

1

2212 ,cos

14

2

1)(

k

knntn

tf

Ans:

*Refer to textbook, pg. 772

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17.4 Circuit Applications (1)17.4 Circuit Applications (1)

Steps for Applying Fourier SeriesSteps 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 17.6Example 17.6

Find the response v0(t) of the circuit below when the voltage source vs(t) is given by

17.4 17.4 Circuit Applications (2)Circuit Applications (2)

12 ,sin12

2

1)(

1

kntnn

tvn

s

*Refer to textbook, pg. 775

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SolutionSolution

Phasor of the circuit

For dc component, (n=0 or n=0), Vs = ½ => Vo = 0

For nth harmonic,

In time domain,

17.4 17.4 Circuit Applications (3)Circuit Applications (3)

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:

Ch 17 Useful FormulaCh 17 Useful Formula