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8/9/2019 Engineering Circuit Analysis-CH8
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Engineering Circuit AnalysisEngineering Circuit Analysis
CH8 Fourier Circuit AnalysisCH8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series8.2 Use of Symmetry8.2 Use of Symmetry
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Ch8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series
- Most of the functions of a circuit are periodic functions
- They can be decomposed into infinite number of sine and
cosine functions that are harmonically related.
- A complete responds of a forcing function =
Partial response to each harmonics.∑ erpositonsup
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Harmonies: Give a cosine function
- : fundamental freuency ! is the fundamental "ave form#
- Harmonics have freuencies
( )
$
$
$%
&%:
&:
cos&
w f T T
w f f
t wt v
π
π
==
==
$w ( )t v%
( ) t nwat v nn $cos=
Amplitude of the nth harmonics
!amplitude of the fundamental "ave form#
'(')'&' $$$$ wwww
*re. of the %st harmonics
!=fund. fre#
*re. of the &nd
harmonics
*re. of the nthharmonics
*re. of the )rd harmonics
*re. of the (th harmonics
Ch8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series
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8.1 Fourier Series8.1 Fourier Series Example *undamental: v
%
= &cosw$
t
v)a = cos)w$t v)b = %.+cos)w$t
v)c = sin)w$t
Ch8 Fourier Circuit Analysis
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- *ourier series of a periodic function
Given a periodic function
can be represented by the infinite series as
! # ! # ! #T t f t f t f ,=:
( )t f
( )
( )∑∞
=
++=
++++++=
%
$$$
$&$%
$&$%$
sincos
&sinsin&coscos
n
nn t nwbt nwaa
t wbt wbt wat waat f
$
=
=
=
n
n
b
a
a
Ch8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series
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/ample %&.%
( )
≤≤
≤≤−==
).$%.$'$
%.$%.$'+cos
t
t t V t V
m π
π
mV
a =$
Ch8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series
Given a periodic function
0t is 1no"ing
&%
mV
a =
( )( )%n % &
cos&& >−= n
nV
a mn
π
π $$=b
0t can be seen ' "e can evaluateπ ω +$=
π )
&&
mV a = $)=a
π %+
&(
mV a −= $+=a
π )+
&2
mV a =
( )
++
−++=
t V
t V
t V
t V V
t V
m
mmmm
π
π
π π
π π
π π
)$cos
)+
&
&$cos%+
&%$cos
)
&+cos
&
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-3evie" of some trigonometry integral observations
!a#
!c#
!d#
$sin$
$ =∫ dt t nwT
$cos$
$ =∫ dt t nwT
!b#
!e#
( ) ( )[ ] $sinsin&
%cossin
$$$
$$$ =−++= ∫ ∫ dt t wnk t wnk dt t nwt kw
T T
( ) ( )[ ]
≠
==
+−−=
∫
∫
nk if
nk if T
dt t wnk t wnk
dt t nwt kw
T
T
'$
'&
coscos&
%
sinsin
$ $$
$ $$
( ) ( )[ ]
≠
==
++−=
∫
∫
nk if
nk if T
dt t wnk t wnk
dt t nwt kw
T
T
'$
'&
coscos&
%
cossin
$ $$
$ $$
Ch8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series
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-valuations of nn baa ''$
$a ( ) ( )∫ ∑∫ ∫ ∞
=
++= T
n
nn
T T
dt t nwbt nwadt adt t f $
%
$$$
$$
sincos
4ased on !a# !b#
( ) $sincos$
%
$$ =+∫ ∑∞
=
T
n
nn dt t nwbt nwa ( )∫ = T
dt t f T
a$
$
%
$a! is also called the 56 component of #( )t f
Ch8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series
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na
4ased on !b#
( )
∫ ∑
∫ ∑∫ ∫ ∞
=
∞
=
+
+=
T
n
n
T
n
nT T
dt t kwt nwb
dt t kwt nwadt t kwatdt kwt f
$%
$$
$%
$$$
$$$
$
cossin
coscoscoscos
( )∫
∫ ∑
=
=∞
=
T
n
n
T
n
n
tdt nwt f T
a
aT
dt t kwt nwa
$ $
$%
$$
cos&
&coscos
$cossin$
%
$$ =∫ ∑∞
=
T
n
n dt t kwt nwb
$cos$
$$ =∫ dt t kwaT
4ased on !c#
4ased on !e#
Ch8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series
7hen k =n
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nb
4ased on !a#
( )
∫ ∑
∫ ∑∫ ∫ ∞
=
∞
=
+
+=
T
n
n
T
n
n
T T
dt t kwt nwb
dt t kwt nwadt t kwatdt kwt f
$%
$$
$%
$$$
$$$
$
sinsin
sincossinsin
( )∫
∫ ∑
=
=∞
=
T
n
n
T
n
n
tdt nwt f T
b
bT
dt t kwt nwb
$ $
$%
$$
sin&
&sinsin
$sincos$
%
$$ =∫ ∑∞
=
T
n
n dt t kwt nwa
$sin$
$$ =∫ dt t kwaT
4ased on !c#
4ased on !d#
Ch8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series
7hen k =n
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( )nnnnn t nwbat nwbt nwa φ ++=+ $&&
$$ cossincos
#! Hz f
&&
nnn bav +=
Harmonicamplitude
8v
$8 f $2 f $+ f $( f $) f $& f $ f
+v
2v
(v)v&v%v
Phase spectrum
$
f $
& f $
) f $
( f $
+ f $
2 f $
8 f #! Hz f
n
φ
n
nn
ab−= −%tanφ
Ch8 Fourier Circuit Analysis
8.1 Fourier Series8.1 Fourier Series
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- 5epending on the symmetry !odd or even#' the *ourier series can be further simplified.
Even Symmetry
Observation: rotate the function curve along a/is' the curve
"ill overlap "ith the curve on the other half of .
/ample :
Odd Symmetry
Observation: rotate the function curve along the a/is' then along
the a/is' the curve "ill overlap "ith the curve on the other
half .
/ample :
( ) ( )t f t f −=
( ) ( )t f t f −−=
( ) wt t f cos=
( )t f
( )t f
( ) wt t f sin=
t
( )t f
Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry8.2 Use of Symmetry
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9ymmetry Algebra
!a# odd func. =odd func. × even func.
/ample:
!b# even func. =odd func. odd func.
/ample:
!c# even func. =even func. even func./ample:
t ωt ωt ω
cossin=&
&sin
t ωt ωt ω
coscos=&
%,&cos
t ωt ωt ω
sinsin=&
&cos-%
Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry8.2 Use of Symmetry
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!d# even func. =const. ,∑ even func. !;o odd func.#
/ample:
!e# odd func. =∑odd func.
/ample:
t ωt ωt ω && sin&-%=%-cos&=&cos
! # φt ωφt ωφt ω sincos,cossin=,sin
odd func. odd func.
Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry8.2 Use of Symmetry
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Apply the symmetry algebra to analy
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Half-"ave symmetry f !t # = - f !t - # or f !t # = - f !t , #&T &
T
Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry8.2 Use of Symmetry
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( )
( )
=
=
∫
∫
evenisn
odd isntdt nwt f T b
evenisn
odd isntdt nwt f
T a
T
n
T
n
$
sin(
$
cos(
$
$
$
$
*ourier series:
Ch8 Fourier Circuit Analysis
8.2 Use of Symmetry8.2 Use of Symmetry