bab 17 sistem linier
TRANSCRIPT
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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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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:
*Refer to in-class illustration, textbook
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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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/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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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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/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!"
*Refer to in-class illustration, textbook
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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 ))
*