phasorsweb.cecs.pdx.edu/~ece2xx/ece221/lectures/phasorsx4.pdfportland state university ece 221...
TRANSCRIPT
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
Sta
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
Sta
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
Sta
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
Sta
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
Sta
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
Sta
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
Sta
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
teU
niv
ersity
EC
E221
Phaso
rsVer.
1.4
731
Exa
mple
4:
Work
space
Portla
nd
Sta
teU
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