2012/11/13

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Sinusoids and Phasors•Introduction•Sinusoids•Phasors•Phasor Relationships for Circuit Elements•Impedance and Admittance•Kirchhoff’s Laws in the Frequency Domain•Impedance Combinations•Applications

Introduction•AC is more efficient and economical to transmit

power over long distance. (Transformer is the key.)•A sinusoid is a signal that has the form of the sine

or cosine function.•Circuits driven by sinusoidal current or voltage

sources are called ac circuits.•Why sinusoid is important in circuit analysis?–Nature itself is characteristically sinusoidal.–A sinusoidal signal is easy to generate and transmit.–Easy to handle mathematically.

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Sinusoids

)(2

2

seconds.everyitselfrepeatssinusoidThe

sinusoidtheofargumentthe)(radians/sfrequencyangularthe

sinusoidtheofamplitudethe

where

sin)(voltagesinusoidalheConsider t

T:periodω

TωT

T

ωt

V

tVtv

m

m

)(

)2sin(

)2

(sin

)(sin)(:Proof

)()(

tv

ntVω

ntV

nTtVnTtv

tvnTtv

m

m

m

Sinusoids (Cont’d)•A periodic function is one that satisfies

f(t) = f(t+nT), for all t and for all integers n.–The period T is the number of seconds per cycle.–The cyclic frequency f = 1/T is the number of cycles per

second.

(Hz)hertz:(rad/s)secondperradians:

where2

1

f

fT

f

Phase:Argument:)(

where

)sin()(asgivenisexpressiongeneralmoreA

t

tVtv m

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Sinusoids (Cont’d)

byby

say thatWe

21

12

vlagsvvleadsv

0if,areand0if,areand

say thatWe

21

21

seout of phavvin phasevv

Sinusoids (Cont’d)•To compare sinusoids.–Use the trigonometric identities.–Use the graphical approach.

BABABABABABA

sinsincoscos)cos(sincoscossin)sin(

:identitiesricTrigonomet

sin)90cos(cos)90sin(

cos)180cos(sin)180sin(

tttt

tttt

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The Graphical Approach

)1.53cos(5sin4cos3

ttt

tanwhere

sincos)cos(sincos

1

22

AB

BAC

θCBθ, CAtCtBtA

Phasors•Sinusoids are easily expressed by using phasors•A phasor is a complex number that represents the

amplitude and the phase of a sinusoid.•Phasors provide a simple means of analyzing linear

circuits excited by sinusoidal sources.

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Phasors (Cont’d)

j

j

rejrrjyxz

ryrxyxr

xy

yxr

ryx

zzr

rer

jyxz

z

)sin(cos

sin,cosasandobtaincanwe,andknowweIf

tan,

asandgetcanwe,andGiven

ofphase:ofmagnitude:

where,formlExponentia:

formPolar:formrRectangula:

:numbercomplexarepresenttowaysThree

122

x

y

r

z

Important Mathematical Properties

)(

)()())((

)()()()(

2121

)(2121

12212121

221121

212121

212121

2121

rrerrerer

yxyxjyyxxjyxjyxzz

yyjxxzzyyjxxzz

jjj

:tionMultiplica

:onSubstracti:Addition

2

1

222222

111111

j

j

j

errjyxz

errjyxz

rerjyxz

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Important Mathematical Properties

rjyxz

rzrz

rr

err

erer

yxyxyx

jyx

yyxxjyxjyxjyxjyx

jyxjyx

zz

jj

j

2

11

)(

))(())((

212

1)(

2

1

2

1

22

22

211222

22

2121

2222

2211

22

11

2

1

21

2

1

:ConjugateComplex

:RootSquare

:Reciprocal

:Division

Phasor Representation

.)(sinusoidtheoftionrepresentaphasortheis

)Re()(

)Re()Re(

)cos()(

)Im(sin)Re(cos

sincos

)(

tv

VeV

etv

eeVeV

tVtv

ee

je

mj

m

tj

tjjm

tjm

m

j

j

j

V

V

V

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Phasor Representation (Cont’d)

Phasor Diagram

mVV

mII

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Sinusoid-Phasor Transformation

)90()90cos(

)90()90cos(

)cos()(PhasorsSinusoids

haveFinally we

)Re()Re(

)(Re)Re(

)90cos()sin()(

where)Re()cos()(

9090

mm

mm

mm

tjtj

tjjm

jjjtjm

mm

mj

mtj

m

Vj

tV

vdt

VjtVdtdv

VtVtv

eej

eeVeeeeV

tVtVdt

tdv

VeVetVtv

V

V

V

VV

VV

Phasor Relationship for Resistor

IV

I

RRItRIiRv

ItIi

mm

mm

)cos(law,sOhm'By

)cos(isresistorrough thecurrent ththeIf

Time domain Phasor domain Phasor diagram

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Phasor Relationship for Inductor

Time domain Phasor domain Phasor diagram

IV

I

LjtLIdtdi

Lv

ItIi

m

mm

)90cos(

isinductortheacrossvoltageThe)cos(

isinductorrough thecurrent ththeIf

Phasor Relationship for Capacitor

Phasor diagramTime domain Phasor domain

VI

V

CjtCVdtdv

Ci

VtVv

m

mm

)90cos(

iscapacitorrough thecurrent thThe

)cos(iscapacitortheacrossvoltagetheIf

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Impedance and Admittance

1

1

1AdmittanceImpedanceElement

currentphasortheistagephasor voltheis

rewhe

(S)1

:Admittance,)(:Impedance

CjCj

C

LjLjL

RRR

YZ

YZ

YZ

IV

ZY

IV

Z

Impedance and Admittance (Cont’d)

Cj1

Z

LjZ

0

0

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Impedance and Admittance (Cont’d)

voltageleadscurrentsinceleadingorcapacitive:

voltagelagscurrentsincelaggingorinductive:

thenpositive,isIf

negativeiswhencapacitivepositiveiswheninductive

betosaidisimpedanceThe

reactance:resistance:

where

jXR

jXR

X

XX

XR

jXR

Z

Z

Z

sincos

and

tanwhere

1

22

ZZ

Z

ZZ

XR

RXXR

jXR

Impedance and Admittance (Cont’d)

22

22

22

11

esusceptanc:econductanc:

where

1

XRX

B

XRR

G

XRjXR

jXRjXR

jXRjXRjBG

BG

jBGZ

Y

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KVL and KCL in the Phasor Domain

teeV

eVeV

eeVeeVeeV

tVtVtV

vvv

vvv

tj

jmn

jm

jm

tjjmn

tjjm

tjjm

nmnmm

n

n

n

n

anyfor0Re

0)Re()Re()Re(

asrewrittenbecanThis

0)cos()cos()cos(form.cosine

inwrittenbemaygeeach voltastate,steadysinusoidalIn the

0

loop.closedaaroundvoltagesthebe,,...,,letKVL,For

21

21

21

21

2211

21

21

KVL and KCL (Cont’d)

0phasor.forholdsKCLmanner,similaraIn

phasor.)forholds(KVL0

)(0)2(

)()2(1

0)(cos(1)solutionsPossible

where

0)cos(Re

ReRe

then,Let

21

21

21

)(

21

n

jTn

T

TT

nj

TT

TTtj

T

tjT

tjn

jmkk

T

T

T

k

eV

V

ktt

eV

tVeV

ee

eV

III

VVV

VVVV

VVVV

V

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Series-Connected Impedance

)(

givesKVLApplying

21

21

n

n

ZZZI

VVVV

V

ZZ

VZV

I

ZZZIV

Z

eq

kk

eq

neq

,

21

Parallel-Connected Impedance

)111

(

givesKCLApplying

21

21

n

n

ZZZV

IIII

IYY

IYI

V

YYYVI

Y

eq

kk

eq

neq

,

21

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Y-Transformations

3

133221

2

133221

1

133221

ZZZZZZZ

Z

ZZZZZZZ

Z

ZZZZZZZ

Z

c

b

a

ion:Δ ConversY

cba

ba

cba

ac

cba

cb

ion:Y ConversΔ

ZZZZZ

Z

ZZZZZ

Z

ZZZZZ

Z

3

2

1

Example 1

07.1122.3811

1082310

108||2310

2.08||101

32

1

|| H2.08F103F2in

jj

jjj

jjj

jmjmj

mm

ZZZZ

rad/s.50forFind in

Z

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Example 2(t).Find ov

)96.154cos(15.17)(96.1515.17

152096.308575.0

152010060

10025||2060

25||204,1520

)154cos(20

ttv

jj

jjjj

tvSol:

o

so

s

s

VV

V

Example 3.Find I

-Y transformation

204.4666.3204.464.13

050204.464.1316.13

86||312

2.36.110

)42(8

2.310

)8(48.06.1

8424)42(4

ZV

I

ZZZZ

Z

Z

Z

j

jj

jj

jj

jjjjj

Sol:

cnbn

an

cn

bn

an

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Applications: Phase Shifters

i

ii

iio

RCCR

RC

CRjRCRC

CRRCjRCj

RCjRCj

CjR

R

V

VV

VVV

1tan

1

1)1(

1)1(

11

1

222

222222

Output leads input.

Phase Shifters (Cont’d)

i

iiio

RCCR

CRRCj

RCjCj

R

Cj

V

VVVV

1

222

222

tan1

1

11

11

1

1

Output lags input.

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Example

9031

4532

4522

4522

202020

4532

2412412

20

4122040

)2020(20)2020(||20

11

1

io

iii

j

jj

j

jj

jj

Sol:

VVVV

VVVZ

ZV

Z

leading.90ofphaseaprovideto

circuitanDesign

RC

Applications: AC Bridges

21

3132

321

2

32

21

21

21:conditionBalanced

ZZZ

ZZZZZZZ

ZZZ

Z

VZZ

ZVV

ZZZ

V

VV

xxx

x

sx

xs

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AC Bridges (Cont’d)

sx

sx

LRR

L

LjRLjR

1

2

21

Bridge for measuring L Bridge for measuring C

sx

sx

CRR

C

CjRCjR

2

1

21

Summary•Transformation between sinusoid and phasor is

given as

•Impedance Z for R, L, and C are given as

•Basic circuit laws apply to ac circuits in the samemanner as they do for dc circuits.

CjLjR CLR

1

,, ZZZ

mm VtVtv V)cos()(