Intro

ductio

n:

Stea

dy-S

tate

Analysis

Linear

Circuit

vo- +

t = 0

Asin

(ωt)

•Con

sider

agen

erallin

earcircu

itw

ithsom

ecu

rrent

orvoltage

ofin

teresttreated

asth

eou

tput

•If

the

input

isa

sinusoid

alsou

rce(eith

ervoltage

orcu

rrent)

applied

att=

0,th

enth

eou

tput

respon

secan

be

divid

edin

totw

ocom

pon

ents:

–T

he

steady-sta

teresp

onse

isth

epart

ofth

eresp

onse

that

remain

sas

t→∞

–T

he

transien

tresp

onse

isth

epart

that

approach

eszero

ast→

∞

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

73

Overview

ofSin

uso

ids

and

Phaso

rs

•Sin

usoid

alstead

y-statean

alysis

•Phasors

•Circu

itelem

ent

defi

nin

geq

uation

srevisited

•K

irchhoff

’slaw

srevisited

•Im

ped

ance

combin

ations

•Phasor

analysis

•M

any

examples

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

71

Exa

mple

1:

Stea

dy-S

tate

Analysis

t = 0

vo- +

1kΩ

sin(1

000t)

1µF

Given

vo (0)

=0,

solvefor

vo (t)

fort≥

0.

vo (t)

=12e −

t/0.0

01+

1√2sin(1000t−

45 )

=vtr (t)

+vss (t)

vtr (t)

=12e −

t/0.0

01

vss (t)

=1√2

sin(1000t−45 )

Portla

nd

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teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

74

Intro

ductio

n:

Sin

uso

ids

0

−A 0 A

Tim

e(s)

Voltage (V)

π2π

3π

4π

5π

Con

sider

avoltage,

v(t)=

Asin(ω

t)w

here

ω=

πrad

s/s.

Portla

nd

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teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

72

Phaso

rs

Acos(ω

t+θ)⇔

A∠

θ

•For

historical

reasons,

ωw

illalw

ayshave

units

ofrad

s/san

dθ

will

always

have

units

ofdegrees

•Phaso

r:a

complex

num

ber

that

represents

the

amplitu

de

and

phase

ofa

sinusoid

•You

will

need

tolearn

how

tom

anip

ulate

complex

num

bers

efficien

tly.T

he

advan

cedscien

tific

calculators

shou

ldm

aketh

ism

uch

easier.

•W

ew

illuse

j=

√−1

•W

hy

not

use

ilike

math

ematician

s?

•Exam

ple:

z=

x+

jy

•Real

operator

example:

Rez

=x

•Im

aginary

operator

example:

Imz

=y

Portla

nd

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teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

77

Exa

mple

1:

Contin

ued

01

23

45

67

89

10−

0.8

−0.6

−0.4

−0.2 0

0.2

0.4

0.6

0.8 1T

otalT

ransientSteady State

Tim

e(m

s)

vo(t) (V)

vo (t)

=12e −

t/0.0

01+

1√2sin(1000t−

45 )

=vtr (t)

+vss (t)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

75

Com

plex

Num

bers

Review

Three

representation

sof

complex

num

bers:

z=

x+

jyz

=r∠

φz

=re

jφ

wherer

=√

x2

+y2

φ=

angle(x,y)

Euler’s

iden

tity:

e ±jφ

=cos

φ±jsin

φ

Re

ejφ

=cos

φ

Imejφ

=sin

φ

Com

plex

Con

jugate:

z ∗=

x−jy

z ∗=

r∠−

φz ∗

=re −

jφ

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

78

Stea

dy-S

tate

Analysis

Com

men

ts

•T

his

chap

terwe

will

discu

sscircu

itsdriven

bysin

usoid

alsou

rcesexclu

sively

•W

ew

illon

lysolve

forth

estead

y-statecom

pon

ent

ofth

etotal

respon

se

•In

ECE

222we

will

learnhow

tosolve

forth

etotal

respon

sefor

any

type

ofdrivin

gsign

al(n

on-sin

usoid

al)

•T

he

limitation

sof

phasors

do

not

make

them

useless

–Sin

usoid

alstead

y-statean

alysisis

usefu

lfor

pow

ersystem

s&

comm

unication

scircu

its

–Easier

toap

ply

than

the

techniq

ues

we

will

discu

ssnext

term

•You

shou

ldalread

ybe

familiar

with

sinusoid

san

dtrigon

ometry

(Section

9.2)

Portla

nd

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teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

76

Exa

mple

2:

Unit

Circle

on

the

Com

plex

Pla

ne

Imaginary(z)

Real(z)

1

j-j

-1

Fin

deq

uivalen

texpression

sfor

the

followin

g.

1=

e −j90

=1∠

135 =

1∠45

=

ej270

=1∠

180 =

1∠−

180 =

1∠90

=

Portla

nd

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teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

711

Euler’s

Iden

tityExercised

e ±jφ

=cos

φ±jsin

φ

ejφ

+e −

jφ

=(cos

φ+

jsin

φ)+

(cosφ−

jsin

φ)=

2cos

φ

cosφ

=12 (e

jφ

+e −

jφ)

=R

e ejφ

ejφ−

e −jφ

=(cos

φ+

jsin

φ)−(cos

φ−jsin

φ)=

2jsin

φ

sinφ

=12j (e

jφ−

e −jφ)

=Im

ejφ

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

79

Phaso

rTra

nsfo

rm

The

phasor

transform

ofa

sinusoid

,

v(t)=

Acos(ω

t+φ)

isgiven

by

V=

Pv(t)=

PAcos(ω

t+φ)

=A

ejφ

The

inverse

phasor

transform

ofa

phasor

isgiven

by

v(t)=

P−

1V=

P−

1Aejφ

=R

e A

ejφejω

t

=R

e A

ej(ω

t+φ)

=A

cos(ωt+

φ)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

712

The

Com

plex

Pla

ne

y

x

z

r

Imaginary(z)

Real(z)

φ

z=

x+

jyz

=r∠

φz

=re

jφ

•Com

plex

angle:

φ=

angle(x,y)

•Com

plex

Am

plitu

de:

r=

√x

2+

y2

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

710

Phaso

rs&

Circu

itA

nalysis

•To

use

phasors

forcircu

itan

alysis,we

need

tokn

owhow

the

laws

ofcircu

itan

alysisap

ply

inth

ephasor

dom

ain

•W

hat

areth

edefi

nin

geq

uation

sfor

circuit

elemen

ts?

•H

owdo

Kirch

hoff

’slaw

sap

ply?

Portla

nd

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teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

715

Phaso

rTra

nsfo

rmCom

men

ts

Acos(ω

t+φ)

=R

e A

ej(ω

t+φ)

PAcos(ω

t+φ)

=A

∠φ

=A

ejφ

•A

phasor

transform

represents

the

amplitu

de

and

phase

ofsin

usoid

sby

asin

glecom

plex

num

ber

•Sin

ceall

curren

tsan

dvoltages

operate

atth

esam

efreq

uen

cy,ω

isom

ittedfrom

the

phasor

representation

•W

euse

phasors

becau

seth

eym

akestead

y-statesin

usoid

alcircu

itan

alysis“easy”

Portla

nd

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teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

713

Phaso

rTra

nsfo

rm:

Inducto

rs

i(t)

v(t)

-

+

LI

V

-

+

?

Ifi(t)

=A

cos(ωt+

φ),w

hat

isv(t)?

v(t)=

Ldi(t)dt

=L

(−ωA

sin(ωt+

φ))=

−ωL

Acos(ω

t+φ−

90 )

What

isth

erelation

ship

ofth

ephasors

Ian

dV

?

I=

Aejφ

V=

−ωL

Aej(φ−

90

)=

−ωL

Aejφe −

jπ2

V=

−ωL

Aejφ(−

j)=

jωL

(Aejφ)

V=

jωL

I

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

716

Phaso

rTra

nsfo

rmExa

mples

Acos(ω

t+φ)

⇔A

∠φ

Asin(ω

t+φ)

=A

cos(ωt+

φ−90 )

⇔A

∠(φ−

90 )

=A

ej(φ−

90

)

=A

ejφe −

j90

=A

ejφ (cos

π2 −jsin

π2 )

=−

jAejφ

•You

rcalcu

latorssh

ould

be

able

tocon

vertbetw

eenrectan

gular

and

phasor

representation

seasily

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

714

Phaso

rTra

nsfo

rm:

Sum

mary

Elem

ent

Equation

Phasor

Equation

Phasor

Circu

it

Resistor

v(t)=

Ri(t)

V=

RI

I

V

-

+

R

Inductor

v(t)=

Ldi(t)dt

V=

jωL

II

V

-

+

jω

L

Cap

acitori(t)

=C

dv(t)

dt

V=

1jω

CI

I

V

-

+

1jω

C

All

ofth

eseare

inth

eform

V=

ZI.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

719

Phaso

rTra

nsfo

rm:

Resisto

rs

i(t)

v(t)

-

+

RI

V

-

+

?

Ifi(t)

=A

cos(ωt+

φ),w

hat

isv(t)?

v(t)=

Ri(t)

=R

Acos(ω

t+φ)

What

isth

erelation

ship

ofth

ephasors

Ian

dV

?

I=

Aejφ

V=

R(A

ejφ)

V=

RI

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

717

Phaso

rA

nalysis:

Imped

ance

and

Adm

ittance

•In

the

phasor

dom

ain,th

ereis

alin

earrelation

ship

betw

eenI

and

Vfor

allth

reecircu

itelem

ents:

V=

ZI

orI

=Y

V

•T

his

isa

generalization

ofO

hm

’slaw

•In

the

phasor

dom

ain,th

econ

stant

coeffi

cients

Zan

dY

may

be

complex

•Z

iscalled

imped

ance

(ohm

s-Ω

)

•Y

iscalled

adm

ittance

(siemen

s-

S)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

720

Phaso

rTra

nsfo

rm:

Capacito

rs

i(t)

v(t)

-

+

CI

V

-

+

?

Ifv(t)

=A

cos(ωt+

φ),w

hat

isi(t)?

i(t)=

Cdv(t)

dt

=C

(−ωA

sin(ωt+

φ))=

−ωC

Acos(ω

t+φ−

90 )

What

isth

erelation

ship

ofth

ephasors

Van

dI?

V=

Aejφ

I=

−ωC

Aej(φ−

90

)=

−ωC

Aejφe −

jπ2

I=

−ωC

Aejφ(−

j)=

jωC

(Aejφ)

I=

jωC

VV

=1

jω

CI

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

718

Phaso

rTra

nsfo

rm:

KCL

The

same

argum

ents

canbe

applied

toK

CL.Recall

that

KCL

statesth

esu

mof

curren

tsleavin

g(or

enterin

g)a

node

iseq

ual

tozero.

0=

i1 (t)+

i2 (t)+

···+iN

(t)0

=A

1cos(ω

t+φ

1 )+

A2cos(ω

t+φ

2 )+···+

A4cos(ω

t+φ

N)

0=

Re

A1 e

j(ω

t+φ

1)+

A2 e

j(ω

t+φ

2)+

···+A

Nej(ω

t+φ

N)

0=

Re (A

1 ejφ

1+

A2 e

jφ

2+···+

AN

ejφ

N )ejω

t

0=

Re (I

1+

I2

+···+

IN

)ejω

t

ejω

t=0,

so

0=

I1

+I2

+···+

IN

Thus,

KCL

applies

inth

ephasor

dom

ainas

well

asth

etim

edom

ain.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

723

Phaso

rA

nalysis:

Imped

ance

and

Adm

ittance

Contin

ued

•In

general,

imped

ance

and

adm

ittance

will

be

complex

num

bers

•Z

=R

+jX

and

Y=

G+

jB

•R

calledresista

nce

(Ω)

•X

calledrea

ctance

(Ω)

•G

calledco

nducta

nce

(S)

•B

calledsu

scepta

nce

(S)

•W

hat

abou

tK

irchhoff

’slaw

s?

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

721

Phaso

rCircu

itA

nalysis

Overview

Phasor

analysis

(akasin

usoid

alstead

y-statean

alysis)con

sistsbasically

offou

rstep

s.

1.Tran

sformall

indep

enden

tsou

rcesto

phasors

2.Calcu

lateth

eim

ped

ance

ofall

passive

circuit

elemen

ts

3.A

pply

analysis

meth

ods

that

we

learned

earlierth

isterm

4.A

pply

inverse

phasor

transform

toob

taintim

e-dom

ainexpression

forcu

rrents

and

voltagesof

interest

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

724

Phaso

rTra

nsfo

rm:

KV

L

What

isth

eeq

uivalen

tof

KV

Lin

the

phasor

dom

ain?

Recall

KV

Lstates

the

sum

ofvoltages

around

aclosed

path

iseq

ual

tozero.

0=

v1 (t)

+v2 (t)

+···+

vN

(t)0

=A

1cos(ω

t+φ

1 )+

A2cos(ω

t+φ

2 )+···+

AN

cos(ωt+

φN

)

0=

Re

A1 e

j(ω

t+φ

1)+

A2 e

j(ω

t+φ

2)+

···+A

Nej(ω

t+φ

N)

0=

Re (A

1 ejφ

1+

A2 e

jφ

2+···+

AN

ejφ

N )ejω

t

0=

Re (V

1+

V2

+···+

VN

)ejω

t

ejω

t=0

ingen

eral,so

0=

V1

+V

2+···+

VN

Thus,

KV

Lap

plies

inth

ephasor

dom

ainas

well

asth

etim

edom

ain.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

722

Exa

mple

3:

Work

space

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

727

Phaso

rCircu

itA

nalysis

Overview

Contin

ued

•Phasor

circuit

analysis

isvery

similar

tow

hat

we

have

already

discu

ssed

•K

eydiff

erences

–Circu

itelem

ents

now

have

complex

values

–W

ehave

afew

extrastep

s

•Everyth

ing

that

we

learned

earlierth

isterm

stillap

plies

•T

he

only

idea

that

isa

littletricky

ism

aximum

pow

ertran

sfer

•T

he

next

fewlectu

resw

illcon

sistof

examples

ofhow

toap

ply

phasors

forsin

usoid

alstead

y-statecircu

itan

alysis

•T

his

will

alsoserve

asa

review

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

725

Exa

mple

4:

Equiva

lent

Imped

ance

b a

5Ω

10

Ω

10

Ω

20

Ω

-j40

Ω

-j10

Ωj30

Ω

j20

Ω

Fin

dth

eeq

uivalen

tin

put

imped

ance.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

728

Exa

mple

3:

Equiva

lent

Imped

ance

b a

4Ω

5Ω

-j12.8

Ω

6Ω

j12

Ω

j10

Ω

-j2

Ω

13.6

Ω

Fin

dth

eeq

uivalen

tin

put

adm

ittance.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

726

Exa

mple

5:

Work

space

(1)

Portla

nd

Sta

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niv

ersity

EC

E221

Phaso

rsVer.

1.4

731

Exa

mple

4:

Work

space

Portla

nd

Sta

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niv

ersity

EC

E221

Phaso

rsVer.

1.4

729

Exa

mple

5:

Work

space

(2)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

732

Exa

mple

5:

Equiva

lent

Imped

ance

25 nF

15 i1

i1

b a

25

Ω

25

µH

Fin

dth

eeq

uivalen

tin

put

imped

ance

when

the

circuit

isop

erating

ata

frequen

cyof

1.6M

rad/s.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

730

Phaso

rCircu

itA

nalysis

Step

s

Phasor

(sinusoid

alstead

y-state)an

alysisgen

erallycon

sistsof

four

steps.

1.Tran

sformall

indep

enden

tsou

rcesto

their

phasor

equivalen

t

2.Calcu

lateth

eim

ped

ance

(Z)

ofall

passive

circuit

elemen

ts

3.A

pply

analysis

meth

ods

that

we

learned

earlierth

isterm

4.A

pply

inverse

phasor

transform

toob

taintim

e-dom

ainexpression

forcu

rrents

and

voltagesof

interest

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

735

Exa

mple

6:

Equiva

lent

Imped

ance

Fin

dth

eeq

uivalen

tim

ped

ance

ofa

10µF

capacitor

isin

seriesw

itha

100m

Hin

ductor

when

excitedw

itha

sinusoid

alsou

rceop

erating

at1000

rads/sec.

Fin

dth

eeq

uivalen

tw

hen

the

capacitor

isin

parallel

with

the

inductor?

What

areeach

ofth

eseeq

uivalen

tto?

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

733

Exa

mple

7:

Volta

ge

Divid

er

31.25 nF

500 mH

vg

vo- +

2kΩ

Fin

dth

estead

y-stateexpression

forv

o (t)if

vg (t)

=64

cos(8000t).

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

736

Exa

mple

6:

Work

space

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

734

Exa

mple

8:

Work

space

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

739

Exa

mple

7:

Work

space

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

737

Exa

mple

9:

Source

Tra

nsfo

rmatio

n

15 mH

v1

v2

vo- +

20

Ω

30

Ω25/6

µF

Use

source

transform

ations

tosolve

forth

estead

y-statepart

ofv

o (t).T

he

sinusoid

alvoltage

sources

are:

v1 (t)

=240

cos(4000t+

53.13 )V

v2 (t)

=96

sin(4000t)V

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

740

Exa

mple

8:

Curren

tD

ivider

1 H

ig

io

50

Ω250

Ω

20

µF

Fin

dth

estead

y-stateexpression

forio (t)

ifig (t)

=125

cos(500t)m

A.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

738

Exa

mple

10:

Kirch

hoff’s

Law

s

Vg

IaIb

Ic

5Ω

15

Ω25

Ω

j25Ω

-j15Ω

2∠45

A

The

phasor

curren

tIb

is5∠

45 A

.

1.Fin

dIa ,

Ic ,

and

Vg .

2.If

ω=

800rad

s/s,w

riteth

eexpression

sfor

ia (t),ic (t),

and

vg (t).

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

743

Exa

mple

9:

Work

space

(1)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

741

Exa

mple

10:

Work

space

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

744

Exa

mple

9:

Work

space

(2)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

742

Exa

mple

12:

Mesh

-Curren

tM

ethod

Ib Ia

Ic

Id

5Ω

5Ω

j5

Ω-j

5Ω

2∠0

A

50∠

0 V

100∠

0 V

Use

the

mesh

-curren

tm

ethod

tofind

the

branch

curren

tsIa ,

Ib ,

Ic ,

and

Id .

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

747

Exa

mple

11:

Node-V

olta

ge

Meth

od

Vo

-+

5Ω

j2

Ω

j3

Ω

-j3

Ω

5∠0

A5∠

-90

V

Use

the

node-voltage

meth

od

tofind

the

phasor

voltageV

o .

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

745

Exa

mple

12:

Work

space

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

748

Exa

mple

11:

Work

space

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

746

Exa

mple

14:

Theven

in&

Norto

nEquiva

lents

ab

1Ω

12

Ω

12

Ω

12

Ω12

Ω

j12

Ω

-j12

Ω

87∠

0 V

Fin

dth

eT

heven

inan

dN

ortoneq

uivalen

tsof

the

circuit

inth

ephasor

dom

ain.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

751

Exa

mple

13:

Node-V

olta

ge

Meth

od

Vo- +

2.5 I1I1

8Ω

j5

Ω

-j10

Ω15∠

0 A

Use

the

node-voltage

meth

od

tofind

the

phasor

voltageV

o .

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

749

Exa

mple

14:

Work

space

(1)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

752

Exa

mple

13:

Work

space

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

750

Exa

mple

15:

Work

space

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

755

Exa

mple

14:

Work

space

(2)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

753

Exa

mple

16:

Superp

ositio

n

15 mH

v1

v2

vo- +

20

Ω

30

Ω25/6

µF

Use

superp

ositionto

solvefor

the

steady-state

part

ofv

o (t).T

he

sinusoid

alvoltage

sources

are:

v1 (t)

=240

cos(2000t+

53.13 )V

v2 (t)

=96

sin(8000t)V

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

756

Exa

mple

15:

Theven

in&

Norto

nEquiva

lents

b a

0.02 Vo

Vo- +

40

Ω

600

Ωj150

Ω-j

150

Ω

75∠

0 V

Fin

dth

eT

heven

inan

dN

ortoneq

uivalen

tsof

the

circuit

inth

ephasor

dom

ain.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

754

Exa

mple

17:

Opera

tionalAm

plifi

ers

100 pF

50 pFv

gv

o- +

10

kΩ

20

kΩ

25

kΩ

40

kΩ

Fin

dth

estead

y-stateexpression

forv

o (t)given

that

vg (t)

=2

cos(105t)

V.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

759

Exa

mple

16:

Work

space

(1)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

757

Exa

mple

17:

Work

space

(1)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

760

Exa

mple

16:

Work

space

(2)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

758

Exa

mple

18:

Work

space

(1)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

763

Exa

mple

17:

Work

space

(2)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

761

Exa

mple

18:

Work

space

(2)

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

764

Exa

mple

18:

Opera

tionalAm

plifi

ers

0.1 nF

vo- +

vg

20

kΩ

80

kΩ

160

kΩ

200

kΩ

Fin

dth

estead

y-stateexpression

forv

o (t)w

hen

vg (t)

=20

cos(106t)

V.

Portla

nd

Sta

teU

niv

ersity

EC

E221

Phaso

rsVer.

1.4

762