§ 5-4 The driven RL circuit
)()()1( ta l lf o rtueRVi tLR???
r e s po n s ec om pl e t e
eRV tL
R
???
? ? )1(
RV/? r e s p o n s en a t u r a l
eRV tL
R
???
? ?
i
t
0
RV/
r es p o n s ef o r cedRV ???/
0)(0 ?? titat
ktRiVRLdtRiV L d iVdtdiLRitf o r ????????? )l n (0
VRLkii ln0)0(0)0( ???????
tLRe
V
RiVtVRiV
R
L ??????? ]ln)[ l n (
)0()1( ????? ?? teRVeRVRVior tLRtLR
The complete response=the natural response+forced response
fn iii ??
)/()1()1( RLeRVeRVi ttLR ????? ?? ??
)0( ??? ? dtdiLRiAei tLRn )( ??? tRVi f
R
VAeiii tLR
fn ????
?
i
t
?0
RV/
RV /632.0
RVi
RLt
/6 3 2.0)(
,/
??
??
?
?
R
VA
R
VAi ?????? 00)0(
Example,Find i(t).
)()()1(2525 5.0 ta l lf o rAtueior t????
:So lu tio n
fn iii ??
sR LAei
eq
t
n 25.1/3 ????
? ??
505.1/75 ??fi
)0(502 ???? ? tAei t
)0(25 ?? tAi
)0(25502550 5.02 ????? ?? tAeei tt
255025
252/50)0()0(
??????
??? ??
AA
ii
Time
0s 2s 4s 6s 8s 10s
I(L1)
20A
30A
40A
50A
V1
I
2
6 3H
1
2
50 V
0
)0()1(5025
)0(2550
5.05.0
5.0
????
???
??
?
teeior
tei
tt
t
r es p o n s e
i n p u tz er o ?
r es p o n s e
s t a t ez er o ?
Zero input response--term due to the excitation of
the network by a nonzero initial condition.
Zero state response--terms due to the excitation of
the network by sources.
Aeti
Aii
t
225)(
25)0()0(
?
??
?
??
Aeti
RVii
t
eqeq
)1(50)(
5.1750)0()0(
2
?
??
??
????? ?
Aeeti
tt
)1(5025)( 22 ?? ???
The complete response=zero input response+zero state response.
1,With all independent sources killed,simplify
the circuit to determine Req,Leq,and the time
constant τ = Leq / Req.
eqL
)0( ?Li
2,Viewing as a short circuit,use dc-analysis
methods to find,the inductor current just prior
to the discontinuity.
?
eqL
??
3,Again viewing as a short circuit,use dc-
analysis methods to find the forced response,This is
the value approached by f(t) as t ; we represent it
by f ( ).
)()]()0([)(),()0( / ???????? ??? fefftfa n dfAf t ?6,Then,
).()( / ??? ? fAetf t ?
4,Write the total response as the sum of the natural
and forced responses
)0( ?f )0( ?Li
)0( ?Li
)0( ?Li
)0( ?Li
eqL
5,Find by using the condition that =,
If desired,may be replaced by a current source
[an open circuit if = 0] for this calculation,With
the exception of inductor currents (and capacitor
voltages),other currents and voltages in the circuit may
change abruptly.
v a lu ein itia l ? v al u ef or c e d ? tc o n st i m e t a n?
)()]()0([)( / ????? ?? fefftf t ?
DP 1,Find iL and,
L?
)0(8)(
8)()(4
???
??
? tVet
tti
or
t
L
LL
?
?
)0(22)(
24/8)(
0)0()0(
14/4
:
????
???
??
??
?
??
tAeti
Ai
ii
s
S o l u t i o n
t
L
L
LL
?
)0(8)( ??? ? tVedtdiLt tLL?
DP 2:
1
1
1
,:,2::)3(
,:,)(2::)2(
,:,)(2::)1(
iif i n dAil e t
iif i n dAtuil e t
iif i n dAtuil e t
Ls
Ls
Ls
??
??
?
A)t(u)e.(i
dt
)t(di
.i
A)t(ue)t(i
)(i)(i
A/)(i
s.:)(
t
L
t
L
LL
L
1 0 0
1
1
1 0 0
6722
602030
44
000
423060
0101
?
?
??
???
??
??
??
????
??
Atuei
dt
tdi
iAtueti
iiis
t
Lt
L
LLL
)().(
)(
.)()(
)()()(.:)(
10 0
1
1
10 0
3312
6020302
20000103
?
?
??
???
????
???????
Ae)(i
/)(i
)(i
))(i()(i
)t(i
)(i)(i
)(i
s.:)(
t
L
LL
L
2222
23060
20
602060030
0
000
0
0102
1 0 0
1
1
1
11
?????
???
?
???
?
??
??
?
?
?
??
??
?
DP 2:
1
1
1
,:,2::)3(
,:,)(2::)2(
,:,)(2::)1(
iif i n dAil e t
iif i n dAtuil e t
iif i n dAtuil e t
Ls
Ls
Ls
??
??
?