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ENE 104
Electric Circuit Theory
Lecture 06:
Basic RL and RC Circuits
http://webstaff.kmutt.ac.th/~dejwoot.kha/
Week #6 : Dejwoot KHAWPARISUTH
Objectives : Ch8
Week #6
Page 2
ENE 104
• time constants for RL and RC circuits
• the natural and forced response
• the total response
• the effect of initial conditions on circuit
response
• RL and RC circuit response
Basic RL and RC Circuits
The Source-Free RL Circuit:
Week #6
Page 3
ENE 104
• A natural response
• A transient response
• A forced response
Basic RL and RC Circuits
The Source-Free RL Circuit:
Week #6
Page 4
ENE 104
A series RL circuit for which i(t) is
to be determined, subject to the
initial condition that i(0) = I0.
tti
t
tdL
R
i
id
dtL
R
i
di
iL
R
dt
di
dt
diLRivRi
0
)(
L
0
0
LRteItiso
tL
RIi
/
0
0
)(
0 lnln
Basic RL and RC Circuits
The Source-Free RL Circuit:
Week #6
Page 5
ENE 104
A series RL circuit for which i(t) is
to be determined, subject to the
initial condition that i(0) = I0.
Basic RL and RC Circuits
LRteIti /
0 )(
The power: L
Rt
R eRIRip2
2
0
2
The total energy:
0
2
2
0
0
dteRIdtpw L
Rt
RR
2
0
0
2
2
02
1
2LIe
R
LRI L
Rt
Example 8.1:
Week #6
Page 6
ENE 104 Basic RL and RC Circuits
find the current through the 5-H inductor at t = 200ms
L
Rt
L eIti
0)(
Example :
Week #6
Page 7
ENE 104 Basic RL and RC Circuits
find the current through the 5-H inductor at t = 200ms
L
Rt
L eIti
0)(
4.210
240 I
.8.3244.2.)200( 10 mAemsti t
L
Practice: 8.1 Page 8
ENE 104 Week #6
Determine the energy remaining in the inductor at t = 2 ns
if it is initially storing 7 µJ
Basic RL and RC Circuits
Practice: 8.1 Page 9
ENE 104 Week #6 Basic RL and RC Circuits
7 µJ stored in a 500 nH inductor
a current magnitude =
6
9
2 7 105.292 A
500 10
9/ 500 10
( ) where 500 ps1000
t
o
Li t I e
R
9
12
2 10(2 ) 5.292exp 96.93 mA
500 10
i ns
2 9 3 21(2 ns) 0.5(500 10 )(96.93 10 )
2w Li
Properties of the Exponential resp:
Week #6
Page 10
ENE 104 Basic RL and RC Circuits
L
Rt
L eIti
0)(
The initial rate of decay:
L
Re
L
R
I
i
dt
d
t
L
Rt
t
000
R
L
3679.0)( 1
0
eI
i 03679.0)( Ii or
Properties of the Exponential resp:
Week #6
Page 11
ENE 104
A plot of the exponential response versus time.
Basic RL and RC Circuits
Practice: 8.2 Page 12
ENE 104 Week #6
In a source-free series RL circuit, find the numerical value
of the ratio: (a) 𝑖(2𝜏)/𝑖(𝜏), (b) 𝑖(0.5𝜏)/𝑖(0), and (c) 𝑡/𝜏 if
𝑖 𝑡
𝑖 0= 0.2; (d) 𝑡/𝜏 if 𝑖 0 − 𝑖 𝑡 = 𝑖 0 𝑙𝑛2 .
Basic RL and RC Circuits
/( ) t
oi t I e (0) oi I
2
1
(2 )
( )
i e
i e
0.5(0.5 )
(0)
i
ei
/( )0.2, so 0.2
(0)
ti t te n
i
The Source-Free RC Circuit:
Week #6
Page 13
ENE 104
A parallel RC circuit for which
v(t) is to be determined,
subject to the initial condition
that v(0) = V0.
Basic RL and RC Circuits
0R
v
dt
dvC
0RC
v
dt
dv
RC
t
RC
t
eVevtv
0)0()(
The Source-Free RC Circuit:
Week #6
Page 14
ENE 104 Basic RL and RC Circuits
RC
Practice: 8.3 Page 15
ENE 104 Week #6
Find 𝑣 0 and 𝑣(2𝑚𝑠)
Basic RL and RC Circuits
Practice: 8.3 Page 16
ENE 104 Week #6 Basic RL and RC Circuits
With no current flow permitted through the capacitor
(it is assumed any transients have long since died
out), v = 50 V. After the switch is thrown, the only remaining circuit is a simple source-free RC circuit.
1.6 ms RC
3/
3
2 10( ) (0) so (2ms) 50exp
1.6 10
tv t v e v
Example 8.2:
Week #6
Page 17
ENE 104 Basic RL and RC Circuits
determine both i1 and iL in the circuit
Example :
Week #6
Page 18
ENE 104 Basic RL and RC Circuits
determine both i1 and iL in the circuit
.36050
18,0 mAit L
.20090
181 mAi
Example :
Week #6
Page 19
ENE 104 Basic RL and RC Circuits
determine both i1 and iL in the circuit
For t>0, .2.2132
32mHLeq
11050
18090
)12060(90eqR
.20110
102.2 3
sR
L
eq
eq
LRteIti /
0 )(
Example :
Week #6
Page 20
ENE 104 Basic RL and RC Circuits
determine both i1 and iL in the circuit
For t>0,
LRteIti /
0 )(
.360)0()0( mAii LL
.360360360)( 50000mAeeeti t
t
L
Rt
L
Example :
Week #6
Page 21
ENE 104 Basic RL and RC Circuits
determine both i1 and iL in the circuit
For t>0,
LRteIti /
0 )(
.2409060120
60120)0()0(1 mAii L
.240)( 50000
1 mAeti t
Example :
Week #6
Page 22
ENE 104 Basic RL and RC Circuits
determine both i1 and iL in the circuit
)0(.360
)0(.360)(
50000 tmAe
tmAti
tL
)0(.240
)0(.200)(
500001tmAe
tmAti
t
Practice: 8.4 Page 23
ENE 104 Week #6
at t = 0.15s. find the value of iL, i1, i2
Basic RL and RC Circuits
Example :
Week #6
Page 24
ENE 104 Basic RL and RC Circuits
at t = 0.15 s. find the value of iL, i1, i2
For t<0,
.4.0.228
21 AAi
0)0()0(.6.1.228
8IiiAAi LLL
Example :
Week #6
Page 25
ENE 104 Basic RL and RC Circuits
at t = 0.15s. find the value of iL, i1, i2
For t>0, LRteIti /
0 )( 2eqR
.756.06.16.1.)15.0( 4.0
2
Aeesti
t
L
Rt
L
.244.122 Aii L
Example 8.3:
Week #6
Page 26
ENE 104 Basic RL and RC Circuits
find and if )0( v )0(1
i 0)0( Vv
312 // RRRReq
0)0()0( Vvv CC
t
eiti
)0()( 11
CReq
31
3
2
01
31
31
)0(RR
R
R
Vi
RRRR
Practice: 8.5
Week #6
Page 27
ENE 104 Basic RL and RC Circuits
find vC and v0 at t = 0-, 0+, 1.3ms.
Example :
Week #6
Page 28
ENE 104 Basic RL and RC Circuits
find vC and v0 at t = 0-, 0+, 1.3ms.
For t<0,
)0()0(.100.1202501250
1250
CCC vvVVv
.4.384002100400
2120600
2100400
2)100400()0(
1
Vk
k
k
kvo
Example :
Week #6
Page 29
ENE 104 Basic RL and RC Circuits
find vC and v0 at t = 0-, 0+, 1.3ms.
For t=0+,
.100)0()0( Vvv CC
.6.254002100400
2
2//500600250
.100)0( V
k
k
k
Vvo
Example :
Week #6
Page 30
ENE 104 Basic RL and RC Circuits
find vC and v0 at t = 0-, 0+, 1.3ms.
For t=1.3ms.,
625
400850//1250
)400100//(2600250//1250 kReq
0025.0)4()625( FCReq
4001
.100)0()( 400Veevtv t
t
CC
.45.59.)3.1( VmstvC
.22.156.25)0(.)3.1( 400400
00 Veevmstv tt
The Unit-Step Function:
Week #6
Page 31
ENE 104 Basic RL and RC Circuits
0
0
01
0)(
tt
ttttu
01
00)(
t
ttu
The Unit-Step Function:
Week #6
Page 32
ENE 104
(a) A voltage-step forcing function is shown as the source driving a
general network. (b) A simple circuit which, although not the exact
equivalent of part (a), may be used as its equivalent in many cases.
(c) An exact equivalent of part (a).
Basic RL and RC Circuits
The Unit-Step Function:
Week #6
Page 33
ENE 104 Basic RL and RC Circuits
The Rectangular Pulse Function:
Week #6
Page 34
ENE 104
The rectangular pulse
function
tt
ttt
tt
ttv
1
100
0
0
0
V
0
)(
Basic RL and RC Circuits
Practice: 8.6
Week #6
Page 35
ENE 104 Basic RL and RC Circuits
Evaluate each of the following at 𝑡 = 0.8:
(a) 3𝑢 𝑡 − 2𝑢 −𝑡 + 0.8𝑢(1 − 𝑡); (b) 4𝑢 𝑡 𝑢(−𝑡);
(c) 2𝑢 𝑡 𝑠𝑖𝑛𝜋𝑡.
(a) 3 - 0 + 0.8 = 3.8
(b) [4] (0) = 0 (c) 2 sin 0.8 ¶ = 1.176
Driven RL Circuits:
Week #6
Page 36
ENE 104 Basic RL and RC Circuits
)()(
)( 0 tuVdt
tdiLtRi
0,0)( tti
for positive time; 0t
0
)()( V
dt
tdiLtRi
yielding; dttRiV
tLdi
)(
)(
0
integrated; ktRiVR
L 0ln
Driven RL Circuits:
Week #6
Page 37
ENE 104 Basic RL and RC Circuits
ktRiVR
L 0ln
setting i = 0 for t = 0, we obtain;
kVR
L 0ln
And, hence,
tVRiVR
L 00 lnln
L
Rt
eV
RiV
0
0
or )()( 00 tueR
V
R
Vti L
Rt
Practice: 8.7
Week #6
Page 38
ENE 104 Basic RL and RC Circuits
The voltage source 60 − 40𝑢(𝑡) V is in series with a 10-
resistor and a 50-mH inductor. Find the magnitudes of the
inductor current and voltage at 𝑡 equal to (a) 0−; (b) 0+;
(c) ∞; (d) 3𝑚𝑠.
Practice: 8.7
Week #6
Page 39
ENE 104 Basic RL and RC Circuits
แหล่งจ่ายแรงดันขนาด V ต่ออนุกรมอยู่กับตัว
ต้านทานขนาด 10- และตัว inductor ขนาด 50-mH ให้หา
ขนาดของกระแสในตัว inductor และค่าแรงดันท่ี t เท่ากับ (a)
; (b) ; (c) ; (d) 3 ms
)(4060 tu
00
Natural and Forced Response:
Week #6
Page 40
ENE 104 Basic RL and RC Circuits
The solution of any linear differential equation
may be expressed as the sum of two parts:
- the complementary solution (natural response) - the particular solution (forced response)
let consider; QPi
dt
di
dtQdtPidi
PtPtPt AedtQeei
;Q = a forcing function
Natural and Forced Response:
Week #6
Page 41
ENE 104 Basic RL and RC Circuits
PtPtPt AedtQeei
For a source free circuit, Q must be zero, and the solution is the natural response:
Pt
n Aei
The first term is also called the steady state
response, the particular solution, or the particular
integral. When Q is a constant,
P
Qi f
Natural and Forced Response:
Week #6
Page 42
ENE 104 Basic RL and RC Circuits
PtPtPt AedtQeei
or the complete response;
Pt
nf AeP
Qiiti )(
Determination of the complete res.:
Week #6
Page 43
ENE 104 Basic RL and RC Circuits
nf iiti )(
L
Rt
n Aeti
)(
R
Vi f
0
R
VAeititi L
Rt
fn0)()(
apply initial condition to evaluate A, i = 0 when t = 0;
R
VA 00 so
R
Ve
R
Vti L
Rt
00)(
Determination of the complete res.:
Week #6
Page 44
ENE 104 Basic RL and RC Circuits
R
Ve
R
Vti L
Rt
00)(
Example 8.4:
Week #6
Page 45
ENE 104
Circuit for which a
complete response i(t) is
desired.
The desired current
response as a function of
time.
Basic RL and RC Circuits
Example :
Week #6
Page 46
ENE 104 Basic RL and RC Circuits
0,)(
tAeAetit
L
tR
n
eq
5.1
62
626//2eqR
nf iiti )(
.502
100Ai f
0,50)( 3
5.1
tAetit
To find A, @t=0-; .252
50)0( Ai
A 5025 25; A
0,2550)( 5.0 teti t or .)(12525)( 5.0 Atueti t
Example :
Week #6
Page 47
ENE 104 Basic RL and RC Circuits
.)(12525)( 5.0 Atueti t
Practice: 8.8
Week #6
Page 48
ENE 104 Basic RL and RC Circuits
A voltage source, 𝑣𝑠 = 20𝑒−100𝑡𝑢(𝑡) V, is in series with a
200-Ω resistor and a 4-H inductor. Find the magnitude of
the inductor current at 𝑡 equal to (a) 0−; (b) 0+; (c) 8 ms;
(d) 15 ms.
(0 ) 0 so (0 ) S Lv i
(0 ) (0 ) so (0 ) L L Li i i
Practice: 8.8
Week #6
Page 49
ENE 104 Basic RL and RC Circuits
Practice: 8.8
Week #6
Page 50
ENE 104 Basic RL and RC Circuits
Example 8.5:
Week #6
Page 51
ENE 104 Basic RL and RC Circuits
find the current response
Example :
Week #6
Page 52
ENE 104 Basic RL and RC Circuits
find the current response
)()()( 21 tititi
0,1)( 01
teR
Vti L
Rt
0
)(
02 ,1)(
0
tteR
Vti L
ttR
00 0,1)( tte
R
Vti L
Rt
and 0
)(
00 ,11)(0
tteR
Ve
R
Vti L
ttR
L
Rt
Example :
Week #6
Page 53
ENE 104 Basic RL and RC Circuits
find the current response
00 0,1)( tte
R
Vti L
Rt
0
)(
00 ,11)(0
tteR
Ve
R
Vti L
ttR
L
Rt
Summary:
Week #6
Page 54
ENE 104 Basic RL and RC Circuits
Practice: 8.9
Week #6
Page 55
ENE 104 Basic RL and RC Circuits
The circuit shown has been in the form shown for a very
long time. The switch opens at 𝑡 = 0. find 𝑖𝑅 at 𝑡 equal to
(a) 0−; (b) 0+; (c) ∞; (d) 1.5 ms.
Practice: 8.9
Week #6
Page 56
ENE 104 Basic RL and RC Circuits
Driven RC Circuits: Example
Week #6
Page 57
ENE 104
Find the capacitor voltage vC(t) and the current i(t) in the 200-
resistor of the circuit shown in (a).
(a) Original circuit; (b) circuit valid for t 0; (c) circuit for t 0.
Basic RL and RC Circuits
Driven RC Circuits: Example
Week #6
Page 58
ENE 104
Find the capacitor voltage vC(t) and the current i(t) in the 200-
resistor of the circuit shown in (a).
Basic RL and RC Circuits
)0(1001201050
50)0(
CC vv
CnCfC vvtv )(
.3.192260
50)0( mAi
Driven RC Circuits: Example
Week #6
Page 59
ENE 104
Find the capacitor voltage vC(t) and the current i(t) in the 200-
resistor of the circuit shown in (a).
Basic RL and RC Circuits
50//200//60eqR
241
501
2001
601
2.1)(t
CR
t
Cn AeAetv eq
.205060 )20050(
20050
)20050(20050
VvCf
CnCfC vvtv )(
Driven RC Circuits: Example
Week #6
Page 60
ENE 104
Find the capacitor voltage vC(t) and the current i(t) in the 200-
resistor of the circuit shown in (a).
Basic RL and RC Circuits
2.120)(
t
C Aetv
To find A, @t=0;
Avv CC 20100)0()0(
80A
2.18020)(;
t
C etv
Driven RC Circuits: Example
Week #6
Page 61
ENE 104
Find the capacitor voltage vC(t) and the current i(t) in the 200-
resistor of the circuit shown in (a).
Basic RL and RC Circuits
To find A, @t=0;
.5.0200
100
200
)0()0( A
vi C
.1.020050
50
60
50)(
)20050(20050
Atii f
2.11.0)()(t
nf Aetiiti
Ai 1.05.0)0(
4.0A 0,4.01.0)( 2.1
tetit
Driven RC Circuits: Example
Week #6
Page 62
ENE 104
Find the capacitor voltage vC(t) and the current i(t) in the 200-
resistor of the circuit shown in (a).
Basic RL and RC Circuits
2.18020)(t
C etv
2.14.01.0)(t
eti
Practice: 8.10
Week #6
Page 63
ENE 104 Basic RL and RC Circuits
For the circuit, find 𝑣𝐶(𝑡) at 𝑡 equal to (a) 0−; (b) 0+; (c) ∞;
(d) 0.08 s.
Practice: 8.10
Week #6
Page 64
ENE 104 Basic RL and RC Circuits
Summary :
Week #6
Page 65
ENE 104 Basic RL and RC Circuits
Reference: Page 66
ENE 104
W.H. Hayt, Jr., J.E. Kemmerly, S.M. Durbin, Engineering Circuit Analysis, Sixth Edition.
Copyright ©2002 McGraw-Hill. All rights reserved.
Circuit Analysis and Electrical Engineering
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