§ 5-2 The source-free RL circuit
00 000 I)(iIt)t(i,tat
c u r r e n ty i n gv a rt i m e)t(i
??
?
?
??
tLReItit
L
RIti ??????
00 )(ln)(ln
00,???? dtdiLRiorvvK V L LR
?? ??????? t)t(iI dtLRididtLRidiordtLRidi 000
)(tL?
?
?
)(tR?
?
?
R
)(ti
L
tLR
R eRIRip
2
2
0
2 ???
Let us assume a solution i(t)=A,where A and
s1 are constant to be determined.
tse1
0)()( 11 ?? dtAedLAeR tsts
0)(0 111 11 ???? tststs e
L
RsAoreL A sR A e
L
Rsa r es o l u t i o nt h e ??
1
tLRAe)t(i ???
tLReIti ???
0)(
)21(21 2020
2
0
2
00 LIwLIdteRIdtpw L
tLR
RR ?????
??? ??
00 IAI0t)t(iif ????
Time
0s 10us 20us 30us 40us 50us
I(L1)
0A
1.0A
2.0A
0
I
L1
10 m H
1
2
R1
1k
The series RL circuit,tLReIti ??
0)(
At zero time,the current is the I0,And as time increases,
the current decreases and approaches zero.
i
t
0
0I
The initial rate of decay is found by evaluating the
derivative at zero time.
L
R
t
e
L
R
tdt
I
id
t
L
R
??
?
??
?
?
00
)(
0
1
t
0
0/Ii
?
We designated the value of time it takes for i/I0 to drop
from unity to zero,assuming a constant rate of decay,by
the Greek letter, ?
The value of i/I0 at t=?
368.0)( 1
0
??? ?? eeIi ?
??
In one time constant the
response has dropped to 36.8
percent of its initial value.
If =1 or =L/R--time-constant?
L
R ?
?
t
eIti ??? 0)(
0000 8.36368.0)( IIior ???
368.0
t
0
0/ Ii
? ?2 ?3
135.0
05.0
1
If a circuit contains any number of resistors and
one inductor,we fix our attention on the two terminals
of the inductor and determine the equivalent
resistance across these terminals,The circuit is
reduced to simple series case.
Why does a larger value of the time constant L/R
produce response curves,which decays more slowly?
(L↑ ↑ wL=Li2/2↑ t↑or R ↓ ↑ p=Ri2↓ t↑),? ?
??
t
LL
eq
eq eiiR
L
RR
RR
RRR
?
????
?
??? )0(
21
21
43
??
t
L
t
L e)(iRR
Rian de)(i
RR
Ri ??
?????? 00 21
1
2
21
2
1
)/()0()( 1
2
21
eq
t
L RLeiR
RRtian d ???? ?? ??
2
11
2
)0()0(
R
iRit h e n ?? ?
)(i
R
RR
)](i)(i[
)(i)(i LL
?
??
?
?
??
???
?
0
00
00
1
2
21
21
??
tt
eiRRtieiti ???? ??? )0()()0()( 1
2
1
211
If we are given the initial value have i1 as,)0(
1 ?i
Since the inductor current decays exponentially as,then
every current and voltage in the resistive network must have
the same function behavior.
?
te?
Example 2,Find iL and i1 for t>0.
110
9060120
90)60120(
50
?
??
??
??eqR
Ae.)t(ia n dAe.)t(i ttL 50 00 0150 00 0 240360 ?? ????
AiiAi LLL 36.0)0()0(36.0)0( ???? ???
A.)(i)(ian dA./)(i L 2409018 018 0002090180 11 ??????? ???
:so lu tio n
sRL
eq
eq 53 102
110
1022 ?? ?????,? Li
Ri
36.0
240.?
20.
mHL eq 2.2132 32 ?????
I
R1
50
R2
90
L1
1m H
1 2
R4
12 0
R3
60
18 V
L2
1,2m H
1
2
U1
T OPEN = 0
1 2
0
Time
0s 20us 40us 60us 80us 100us
I(L1)
0A
200mA
400mA
If iL1(0-)=0.16A,iL2(0-)=0.2A,find,iL1 and iL2.
Aiit LL 056.0)()( 21 ???????
te 50 0003 )50000(36.010
32
32 ?? ???
?
??
dtLdidtdiL LL
1
1
1
1
?? ???
? ? ???? t LL idtLi 0 1
1
1 )0(
1 ?
? ? ???? t LL idtLi 0 2
2
2 )0(
1 ?
Aeiii tLLL 50 00 021 36.0 ????
dt
diL L
12??
Ve t500006.21 ???
0 5 6.02 1 6.0 5 0 0 0 0 ?? ? te
? ? ??? ?t t dte0 50 00 0 16.06.2121 0 0 0
0 5 6.01 4 4.0 5 0 0 0 0 ?? ? te
? ? ??? ?t t dte0 50000 2.06.2131000