Engineering Circuit Analysis-CH8

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


3/17

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




4/17

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



5/17

- *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




6/17

/ample %&.%

( )

≤≤

≤≤−==

).$%.$'$

%.$%.$'+cos

t

t t V t V

m π

π

mV

a =$



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

&


7/17

-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

$ $$

$ $$




8/17

-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




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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#



7hen k =n


10/17

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#



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




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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


8.2 Use of Symmetry8.2 Use of Symmetry


13/17

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-%




14/17

!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.




15/17

Apply the symmetry algebra to analy


16/17

Half-"ave symmetry f !t # = - f !t - # or f !t # = - f !t , #&T &

T




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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:



Engineering Circuit Analysis-CH8

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