bee lecture5

8/9/2019 BEE Lecture5

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20/09/2013 Bahman R. Alyaei 1

Chapter 4

AC Network Analysis

Part 2


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4.3 Solutions of Circuits

Containing Dynamic Elements

• The analysis of the

resistive element

circuits

• KVL and KCL.

• The equatin! that

re!ult "rm a##lyin$

Kir%hh""&! la'! are

al$e(rai% equatin!.

• The analysis of

!ynamic element

circuits

• KVL and KCL.

• The equatin! that

re!ult "rm a##lyin$

Kir%hh""&! la'! are

di""erential

equatin!.



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Continue"

• Cn!ider the series RC

%ir%uit) a##lyin$ KVL*

• +(!er,in$ that i R = i C )

hen%e

• The a(,e equatin i! an

inte$ral equatin.



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Continue"

• Thi! equatin %an (e %n,erted t di""erential

equatin (y di""erentiatin$ (th !ide! " the

equatin) then

20/09/2013 Bahman R. Alyaei -

where theargument #t $

has been

dropped for

ease of

notation.


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Continue"

• hat i" 'e a##ly KCL at the

nde %nne%tin$ the re!i!tr

t the %a#a%itr) then

• ither equatin KVL r KCL i! !u""i%ient) t

determine all ,lta$e! and %urrent! in the %ir%uit.

20/09/2013 Bahman R. Alyaei


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4.3.% &orce! 'esponse of Circuits

E(cite! )y Sinusoi!al Sources

• Let v s#t $ the &orcing &unction (e

!inu!idal !i$nal

• e n' that

• Then



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Continue"

• 5in%e the "r%in$ "un%tin i! a !inu!id

• Then) the !lutin may al! (e a!!umed

t (e " the !ame "rm.• There"re) an e6#re!!in "r v C #t $ i! then

the "ll'in$*

• hi%h i! equi,alent t



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Continue"

• 5u(!titutin$ thi! equatin in the di""erential

equatin "r v C #t $ and !l,in$ "r the

%e""i%ient! A and B



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Continue"

• Rearran$in$

• The %e""i%ient! " !inωt and %!ωt must

both be zero in order for the above equatin

t hld) thu!)



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Continue"

• Then) A and B are $i,en (y

• Thu!) the !lutin "r

v C #t $ may (e 'ritten a!*



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Continue"

• 'emarks

• The!e (!er,atin! indi%ate that three #arameter!

uniquely de"ine a !inu!id*

1.Frequency,

2.Ampitude

!."hase.



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Continue"

• Then) i! it ne%e!!ary t %arry the e6%e!!

lu$$a$e): that i!) the !inu!idal "un%tin!;

•


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4.4 *hasors an! +mpe!ance

• ,e will represent sinusoi!al signals as

compe# numbers, and to eiminate the

need for sovin$ differentia e-uations.

• Read Appendi# A "r %m#lete treatment "%m#le6 num(er!.

%.A!!ition an! Su)traction.

./ultiplication an! Division.3.Con0ugate.

4.*olar an! 'ectangular forms.



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4.4.% Euler1s +!entity

• %uer&s identity "rm!

the (a!i! " #ha!r

ntatin.

• ?t !tate! that) the

Cm#le6 6#nential

"un%tin i! de"ined a!

20/09/2013 Bahman R. Alyaei 1-


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Continue"

• Cn!ider a ,e%tr " len$th A making an

angle ' with the real axis.

• The "ll'in$ equatin illu!trate! the

relatin!hi# (et'een the re%tan$ular and#lar "rm!*

• ?n e""e%t) %uer&s identity i! !im#ly atri$nmetri% relatin!hi# in the %m#le6

#lane.



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4.4. *hasors

• A methd thru$h 'hi%h %m#le6 num(er!

%an (e u!ed t re#re!ent !inu!idal

!i$nal!.

• Re'rite the e6#re!!in "r a $enerali@ed

!inu!id in li$ht " uler&! equatin*



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Continue"

• The comple( phasor correspon!ing to

the sinusoi!al signal Acos#(t ) *$ is

therefore defined t (e the %m#le6

num(er Ae +**

• ?t i! im#rtant t e6#li%itly #int ut that

thi! i! a definition.


φ ∠ A


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Continue"

• The %m#le6 #ha!r ntatin i! the !im#li"i%atin "

the %m#le6 ntatin 'e2 Ae + #(t)*$) a! "ll'*

• The rea!n "r thi! !im#li"i%atin i! !im#ly

mathemati%al %n,enien%e.

• Remem(er that the e +(t term that 'a! rem,ed "rm

the %m#le6 "rm " the !inu!id i! really !till

#re!ent.


φ φ ω je A j X =∠=)(


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• Summary



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E 4.5 Superposition of Two AC

Sources #Same &re-uency$

• >ind the equi,alent #ha!r ,lta$e

v st re!ultin$ "rm the !erie!

%nne%tin " t' !inu!idal

,lta$e !ur%e! $i,en (y*

• Solution• rite the t' ,lta$e! in #ha!r

"rm a! "ll'*



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Cn,ert the #ha!r ,lta$e! "rm #lar t re%tan$ular "rm*

Add them t $et

' 'e %an %n,ert 6S jω t it! timedmain "rm*


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Note

e 'ill (tained the !ame re!ult (y addin$ the t' !inu!id! in

the time dmain) u!in$ tri$nmetri% identitie!*

Cm(inin$ lie term!) 'e (tain


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'emarks

?n $eneral) #ha!r analy!i! $reatly !im#li"ie! %al%ulatin! related t!inu!idal ,lta$e! and %urrent!.


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4.4.3 Superposition of AC

Signals

• A mre $eneral %a!e i! t deal 'ith thesuperposition of sinusoi!s oscillating at!ifferent fre-uencies.

• The -uestion is how to a!! them inphasor notation7

• The %ir%uit !h'n de#i%t! a

lad e6%ited (y t' %urrent !ur%e! %nne%ted

in #arallel) 'here



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The lad %urrent i! equal t the !um " the t' !ur%e %urrent! that i!)

?n thi! %a!e) the #ha!r "rm i! $i,en (y

+8D % jω1 E jω2

here)

'emarks

?n rder t %m#lete the

analy!i! " any %ir%uit

'ith multi#le !inu!idal

!ur%e! at di""erent"requen%ie! u!in$

#ha!r!) it i! ne%e!!ary

t !l,e the %ir%uit

!e#arately "r ea%h

!i$nal and then add the

indi,idual an!'er!

(tained "r the di""erent

e6%itatin !ur%e!.


[ ]

[ ]

2

222

0

222

1

111

0

111

I

2, 0

Re)(

I

2, 0

Re)(

2

1

==∠=

=

==∠=

=

f A

ee A j I

f A

ee A j I

t j j

t j j

π ω

ω

π ω

ω

ω

ω


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E 4.%9 Superposition of Two AC

Sources #Different &re-uency$

• Cm#ute the ,lta$e!

v R1t and v R2 t in the

%ir%uit " >i$ure -.30.

• The !ur%e! are $i,en

(y



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Solution5in%e the t' !ur%e! are at di""erent "requen%ie!) then) 'e a##ly

superposition theory t %m#ute a !e#arate !lutin "r ea%h and

then %m(ine the re!ult.

1 Cn!ider the %urrent !ur%e*

rite the !ur%e %urrent in #ha!r

ntatin*

[ ]

s

1

0

11

I

rad/s200,A05.0

Re)( 1

=

=∠=

=

π ω

ω ω t j j s ee A j I

Then) v R1t and v R2 t due t i st in #ha!r "rm i! $i,en (y


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( )( ) rad/s200, V075.1815005.04

1

I11

1

)I(V

1

1s

21

1sR1

π ω =∠=∠

=

+= R

R R

R

( )( ) rad/s200, V075.185005.04

3

I11

1

)I(V

1

2s

21

2sR2

π ω =∠=∠

=

+= R

R R

R



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2 Cn!ider the ,lta$e !ur%e*

Then) v R1t and v R2 t due t v st in #ha!r "rm i! $i,en (y

rite the !ur%e %urrent in #ha!r ntatin*

[ ]

s

2

0

22

V

rad/s 2000, 020

Re)( 2

==∠=

=π ω

ω ω t j j

s ee A jV

( )

( ) rad/s2000, V5050204

1

)V()V(V

rad/s2000, V0150204

3

V)V(V

2

s

21

2sR2

2

s

21

1sR1

π ω π

π ω

=∠=∠−=∠−

=

−

+=

=∠=∠

=

+=

R R

R

R R

R



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' 'e %an determine the ,lta$e a%r!! ea%h re!i!tr (y addin$

the %ntri(utin! "rm ea%h !ur%e and %n,ertin$ the #ha!r "rm

t timedmain re#re!entatin*

Comments

te that it i! im#!!i(le t !im#li"y the "inal e6#re!!in any "urther)(e%au!e the t' %m#nent! " ea%h ,lta$e are at di""erent

"requen%ie!.



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4.4.4 +mpe!ance• e n' analy@e the i :v relatin!hi# " the three ideal

%ir%uit element! in li$ht " the #ha!r ntatin.• Re!i!tr!) %a#a%itr!) and indu%tr! 'ill (e de!%ri(ed

(y a #arameter %alled +mpe!ance.

• +mpe!ance may )e viewe! as comple( resistance.

• The impe!ance concept is e-uivalent to statingthat capacitors an! in!uctors act as fre-uency:!epen!ent resistors.

• That is; as resistors whose resistance is a function

of the fre-uency of the sinusoi!al e(citation.



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'esistive; +n!uctive; an! Capacitive AC Circuits

&igure 4.33 The im#edan%e element


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4.4.4.% The 'esistor ?n the %a!e " !inu!idal !ur%e!)

then) the %urrent "l'in$ thru$h

the re!i!tr i! $i,en (y

Cn,ertin$ the ,lta$e v st and the

%urrent i t t #ha!r ntatin) 'e

(tain the "ll'in$ e6#re!!in!*


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Then) the im#edan%e " the re!i!tr dented (y Z R jω i! de"ined a!

the rati " the #ha!r ,lta$e a%r!! the re!i!tr t the #ha!r

%urrent "l'in$ thru$h it)

'emarks

%. The a)ove e-uation correspon!s to


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4.4.4. The +n!uctor


e n' that

Thu!) "r the %ir%uit !h'n) v Lt

D v st and i Lt D i t ) hen%e


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'emarks

1. te h' a de#enden%e n the radian "requen%y " the

!ur%e i! %learly #re!ent in the e6#re!!in "r the

indu%tr %urrent.. &urther; the in!uctor current is shifte! in phase #)y

59=$ with respect to the voltage.

F!in$ #ha!r ntatin*

2/)(

0)(

π ω

ω

ω

−∠=

∠=

L

A j I

A jV

s

s


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Thu!) the im#edan%e " the indu%tr) Z L jω i! de"ined a! "ll'!*

'emarks

%. The in!uctor )ehaves like a comple( fre-uency:

!epen!ent 'esistor.

. The magnitu!e of this comple( resistor; (-; is

proportional to the signal fre-uency; (.

3. At low signal fre-uencies; an in!uctor actssomewhat like a short circuit.

4. At high fre-uencies it ten!s to )ehave more as an

open circuit.


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4.4.4.3 The Capacitor



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?n #ha!r "rm)

The im#edan%e " the ideal %a#a%itr) Z C jω) i! there"re de"ined

a! "ll'!*

e ha,e u!ed the "a%t that


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Continue"

• 'emarks

1. The im#edan%e " a %a#a%itr i! a"requen%yde#endent %m#le6 quantity.

2. The im#edan%e " a %a#a%itr ,arie! a!an in,er!e "un%tin " "requen%y.

3. A capacitor acts like a short circuit at

high fre-uencies.4. +t )ehaves more like an open circuit at

low fre-uencies.

20/09/2013 Bahman R. Alyaei -0


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• 5in%e #ra%ti%al %ir%uit! are made u# "mre r le!! %m#le6 inter%nne%tin!" di""erent %ir%uit element!.

• The im#edan%e " a %ir%uit element i!de"ined a! the !um " a real #art andan ima$inary #art*

R jω G i! the real #art " Z jω and %alled

the AC re!i!tan%e

H jω G i! the ima$inary #art " Z jω and

%alled the rea%tan%e.'eactance coul! )e in!uctive

which is )ve or capacitive which

is ve.


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E 4.%3 +mpe!ance of a

Comple( Circuit

• >ind the equi,alent im#edan%e "

the %ir%uit !h'n i" ( > %94 ra!?s.


Solution

e determine "ir!t the #arallel im#edan%e

" the R 2 C %ir%uit) IJJ.


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e6t) 'e determine the equi,alent im#edan%e) Ieq*

'emarks

At the "requen%y u!ed in thi! e6am#le) the %ir%uit ha! an

indu%ti,e im#edan%e) !in%e the rea%tan%e i! #!iti,e r)alternati,ely) the #ha!e an$le i! #!iti,e.



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4.4.@ A!mittance

• The Con!uctance) / ) is !efine! as the

inverse of the resistance.

• The A!mittance) ) is !efine! as the

inverse of the +mpe!ance.

/ i! %alled the AC%ndu%tan%e.

B i! %alled the

!u!%e#tan%e.20/09/2013 Bahman R. Alyaei --


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E 4.%4 A!mittance

• >ind the equi,alent

admittan%e " the t'

%ir%uit! !h'n in >i$ure

-.-1.

• The data i! a! "ll'*

ω D 2 103 rad/! R 1

D 10 M L D 14 mN R 2 D 100 M) C D 3 O>



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Circuit #a$

>ir!t) determine the equi,alent im#edan%e " the %ir%uit*

Circuit #)$

>ir!t) determine the equi,alent im#edan%e " the %ir%uit*

te that the unit! " admittan%e are !iemen!) that i!) the !ame a!

the unit! " %ndu%tan%e.


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4.@ AC Circuit Analysis /etho!s

• The AC %ir%uit analy!i! #r(lem " intere!t

in thi! !e%tin %n!i!t! " determinin$ the

unn'n ,lta$e r %urrent! in a %ir%uit

%ntainin$ linear #a!!i,e %ir%uit element!R ) L) C and e6%ited (y a !inu!idal

!ur%e.

• The #r%edure "r AC Cir%uit Analy!i! i!e6#lained in ne6t !lide.



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E 4 %@ *h A l i f AC


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E 4.%@ *hasor Analysis of AC

Circuit

• A##ly the #ha!r analy!i!

methd t determine the

!ur%e %urrent i st .

• Solution


e"ine the ,lta$e v t at the t# nde and u!e ndal analy!i! t

determine v t ) then

e t e "ll the !te#!


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e6t) 'e "ll' the !te#!

5te# 1*

5te# 2*

5te# 3*

5te# -* e6t) 'e !l,e "r the !ur%e %urrent u!in$ ndal analy!i!.

>ir!t 'e "ind V*


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>ir!t 'e "ind V*

Then 'e %m#ute ?5*

5te# * Cn,ert the #ha!r an!'er t time dmain


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4.@.% AC E-uivalent Circuits

• The %m#utatin " an equi,alent im#edan%e

i! %arried ut in the !ame 'ay a! that "

equi,alent re!i!tan%e in the %a!e " re!i!ti,e

%ir%uit!.• 5hrt%ir%uit all ,lta$e !ur%e!) and #en

%ir%uit all %urrent !ur%e!.

• Cm#ute the equi,alent im#edan%e (et'eenlad terminal!) 'ith the lad di!%nne%ted.

• Cm#ute 6T r +N a! (e"re.



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20/09/2013 Bahman R. Alyaei

E 4 %B S l ti f AC Ci it


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E 4.%B Solution of AC Circuit

)y No!al Analysis

• The ele%tri%al %hara%teri!ti%! "

ele%tri% mtr! %an (e

a##r6imately re#re!ented (y

mean! " a series ':8 %ir%uit.


• ?n thi! #r(lem 'e analy@e the %urrent! dra'n (y

t' di""erent mtr! %nne%ted t the !ame AC

,lta$e !u##ly.

• R D 0. M R 1 D 2 M R 2 D 0.2 M) L1 D 0.1N L2

D 20 mN. v t D 1 %!377t V

• >ind the mtr lad %urrent!) i 1t and i 2 t .

Solution


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Solution

>ir!t) 'e %al%ulate the im#edan%e! " the !ur%e and " ea%h mtr*

The !ur%e ,lta$e i!

e6t) 'e a##ly KCL at the t# nde) 'ith the aim " !l,in$ "r the

nde ,lta$e 6*


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>inally) 'e %an 'rite the timedmain e6#re!!in! "r the %urrent!*

Na,in$ the #ha!r nde ,lta$e) 6) the #ha!r mtr %urrent!) +%

and +*

E 4 % Thvenin E-uivalent of


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E 4.% Thvenin E-uivalent of

AC Circuit

• Cm#ute the ThQ,enin

equi,alent " the %ir%uit "

&igure 4.@9.

• Z 1 D M Z 2 D j 20 M.

v t D 110 %!377t V.


Solution


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>ir!t) 'e rem,e the lad) !hrt%ir%uit the ,lta$e !ur%e) and

%m#ute the equi,alent im#edan%e !een (y the lad

e6t) 'e %m#ute the #en%ir%uit ,lta$e) (et'een terminal! a

and ( *

bee lecture5

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