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Alexander-SadikuAlexander-Sadiku

FuFundamentals of Electric Circuitsndamentals of Electric Circuits

Chapter 17Chapter 17

The Fourier SeriesThe Fourier Series

Copyright © The McGraw-ill Co!panies" #nc$ %er!ission re&uired 'or reproduction or display$


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

17$1 Trigo!etric Fourier Series

17$( Sy!!etry Considerations 17$) Circuit Applications

17$* A+erage %ower and ,MS alues

17$. /xponential Fourier Series 17$. Applications


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)

0 The Fourier series o' a periodic 'unction f(t) is a representation that resol+es f(t) into a dcco!ponent and an ac co!ponent co!prisingan in'inite series o' har!onic sinusoids$

0 Gi+en a periodic 'unction f(t)=f(t+nT) wheren is an integer and T is the period o' the'unction$

where ω2(34T is called the 'unda!ental

're&uency in radians per second$

17$1 Trigo!etric Fourier Series 51617$1 Trigo!etric Fourier Series 516

ac

n

n

dc

t nbt naat f ∑∞

=++=

1

0000 )sincos()( ω ω


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*

0 and an and n are as 'ollow

17$1 Trigo!etric Fourier Series 51617$1 Trigo!etric Fourier Series 516

∫ = T

on dt t nt f T

a0

)cos()(2

ω

)(tan , 1n22

n

nnnn

a

bba A −−=+= φ

∫ =T

on dt t nt f T

b0

)sin()(2

ω

0 in alternati+e 'or! o' f(t)

where

ac

n

nn

dc

t n Aat f ∑∞

=

++=1

00 )cos(()( φ ω


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Conditions 58irichlet conditions6 on f(t) toyield a con+ergent Fourier series9

1.f(t) is single-+alued e+erywhere$

2.f(t) has a 'inite nu!er o' 'initediscontinuities in any one period$

3.f(t) has a 'inite nu!er o' !axi!a and!ini!a in any one period$

*$The integral

17$1 Trigo!etric Fourier Series 5(617$1 Trigo!etric Fourier Series 5(6

.anyfor)( 00

0

t dt t f T t

t ∞


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:

/xa!ple 1

8eter!ine the Fourier series o' the wa+e'or!shown elow$ ;tain the a!plitude and phase

spectra

17$1 Trigo!etric Fourier Series 5)617$1 Trigo!etric Fourier Series 5)6


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7

Solution9

17$1 Trigo!etric Fourier Series 5*617$1 Trigo!etric Fourier Series 5*6

)2()(and 21 ,0

10 ,1)( +=


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Three types o' sy!!etry

1$/+en Sy!!etry 9 a 'unction f(t) i' its plotis sy!!etrical aout the +ertical axis$

#n this case"

17$( Sy!!etry Considerations 51617$( Sy!!etry Considerations 516

)()( t f t f −=

0

)cos()(4

)(2

2/

00

2/

00

=

=

=

∫ ∫

n

T

n

T

b

dt t nt f T

a

dt t f

T

a

ω

Typical examples of even periodic function


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=

($;dd Sy!!etry 9 a 'unction f(t) i' its plot isanti-sy!!etrical aout the +ertical axis$

#n this case"

17$( Sy!!etry Considerations 5(617$( Sy!!etry Considerations 5(6

)()( t f t f −=−

∫ =

=

2/

00

0

)sin()(4

0

T

n dt t nt f T

b

a

ω

Typical examples of odd periodic function


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)$al'-wa+e Sy!!etry 9 a 'unction f(t) i'

17$( Sy!!etry Considerations 5)617$( Sy!!etry Considerations 5)6

)()2( t f

T

t f −=−

=

=

=

∫

∫

evenanfor, 0

oddnfor, )sin()(4

evenanfor, 0

oddnfor, )cos()(4

0

2/

00

2/

00

0

T

n

T

n

dt t nt f T b

dt t nt f T a

a

ω

ω

Typical examples of half-wave odd periodic functions


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/xa!ple (

Find the Fourier series expansion o' '5t6gi+en elow$

17$( Sy!!etry Considerations 5*617$( Sy!!etry Considerations 5*6

∑∞

=

−=

1 2sin2

cos112

)(n

t nn

nt f

π π

π Ans:

*Refer to in-class illustration, textbook


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/xa!ple )

8eter!ine the Fourier series 'or the hal'-wa+e cosine 'unction as shown elow$

17$( Sy!!etry Considerations 5.617$( Sy!!etry Considerations 5.6

∑∞

=

−=−=122

12,cos14

2

1)(

k

k nnt n

t f π

Ans:



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17$) Circuit Applications 51617$) Circuit Applications 516

Steps 'or Applying Fourier Series

1$/xpress the excitation as a Fourier series$

($Trans'or! the circuit 'ro! the ti!e do!ain tothe 're&uency do!ain$

)$Find the response o' the dc and ac co!ponentsin the Fourier series$

*$Add the indi+idual dc and ac response usingthe superposition principle$


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1*

/xa!ple *

Find the response v 0(t) o' the circuit elow

when the +oltage source v s(t) is gi+en y

17$)17$) Circuit Applications 5(6Circuit Applications 5(6

( ) 12 ,sin122

1)(

1

−=+= ∑∞

=

k nt nn

t vn

s πω π


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1.

Solution

%hasor o' the circuit

For dc co!ponent" 5ωn2 or n26" s 2 >2? o 2

For nth har!onic"

#n ti!e do!ain"

17$)17$) Circuit Applications 5)6Circuit Applications 5)6

s0 V25

2V

π

π

n j

n j

+=

)5

2tan(c

425

4)(1

1

220 ∑

∞

=

−−+

=k

nt nos

nt v

π π

π

s22

1

0 V425

5/2tan4V ,90

2V

π

π

π n

n

nS

+

−∠=°−∠=

−

Amplitude spectrum of

the output voltage


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Gi+en9

The a+erage power is

The r!s +alue is

17$*17$* A+erage %ower and ,MS alues 516A+erage %ower and ,MS alues 516

∑∑∞

=

∞

=

−+=−+=1

0mdc

1

0ndc )cos(II)( and)cos(VV)(n

m

n

n t mt it nt v φ ω θ ω

)(1

222

0 ∑∞

=

++=n

nnrms baa F

∑∞

=

−+=1

nndcdc )cos(IV2

1IVP

n

nn φ θ


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17

/xa!ple .9

8eter!ine the a+erage power supplied to thecircuit shown elow i' i(t)=2+10cos(t+10°)

+6cos(3t+35°) A

17$*17$* A+erage %ower and ,MS alues 5(6A+erage %ower and ,MS alues 5(6

Ans: 41!"



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0 The exponential Fourier series o' a periodic 'unction f(t) descries the spectru! o' f(t) in ter!s o' the a!plitudeand phase angle o' ac co!ponents at positi+e andnegati+e har!onic$

0 The plots o' !agnitude and phase o' c n +ersus nω 0 are

called the co!plex a!plitude spectru! and co!plexphase spectru! o' f(t) respecti+ely$

17$.17$. /xponential Fourier Series 516/xponential Fourier Series 516

∫ == −T

t jn

n T dt et f T

c0

0 /2 where,)(1

0 π ω ω ∑

∞

−∞=

=n

t jn

noect f

ω )(


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17$.17$. /xponential Fourier Series 516/xponential Fourier Series 516

0 The co!plex 're&uency spectru! o' the 'unction

f(t)=et " 0


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0Filter are an i!portant co!ponent o' electronics and

co!!unications syste!$0This 'iltering process cannot e acco!plished withoutthe Fourier series expansion o' the input signal$

0For exa!ple"

17$:17$: Application @ 'ilter 516Application @ 'ilter 516

#a$ 'nput and output spectra of a lowpass filter( #&$ thelowpass filter passes only the dc component when c )) *


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17$:17$: Application @ 'ilter 5(6Application @ 'ilter 5(6

#a$ 'nput and output spectra of a &andpass filter( #&$ the&andpass filter passes only the dc component when ))

*

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