07NANO107 Transient Analysis of RC-RL Circuits
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Transcript of 07NANO107 Transient Analysis of RC-RL Circuits
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Transient Analysis of RC circuits
Nano107
Chapter 7
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Why there is a transient response?
• The voltage across a capacitor cannot be changed instantaneously.
)0()0( CC VV
• The current across an inductor cannot be changed instantaneously.
)0()0( LL II
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Figure 5.9,
5.10
(a) Circuit at t = 0
(b) Same circuit a long time after the switch is closed
The capacitor acts as open circuit for the steady state condition
(a long time after the switch is closed).
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(a) Circuit for t = 0
(b) Same circuit a long time before the switch is opened
The inductor acts as short circuit for the steady state condition
(a long time after the switch is closed).
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)1(
t
C eEv
/tc eE
dt
dv
0
0
tc
tc
dt
dv
EE
dt
dv
RCTime Constant
_ US I1
I1
I1
I1
I1 I1 I1 I1
R
R1
R1 R1
5
5
5 5 +
+
_
US
IS
E
U1
+
- 1 U
C
2
K
E
Time
0s 1ms 2ms 3ms 4ms 5ms 6ms 7ms 8ms 9ms 10ms
V(2)
0V
5V
10V
SEL>>
RC
R=2k
C=0.1F
Transient Response of RC Circuits
vc
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WHAT IS TRANSIENT RESPONSE
Figure 5.1
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Solution to First Order Differential Equation
)()()(
tfKtxdt
tdxs
Consider the general Equation
Let the initial condition be x(t = 0) = x( 0 ), then we solve the
differential equation:
)()()(
tfKtxdt
tdxs
The complete solution consists of two parts:
• the homogeneous solution (natural solution)
• the particular solution (forced solution)
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The Natural Response
/)(,)(
)(
)(
)(,
)()(0)(
)(
t
N
N
N
N
NNNN
N
etxdt
tx
tdx
dt
tx
tdxtx
dt
tdxortx
dt
tdx
Consider the general Equation
Setting the excitation f (t) equal to zero,
)()()(
tfKtxdt
tdxs
It is called the natural response.
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The Forced Response
0)(
)()(
tforFKtx
FKtxdt
tdx
SF
SFF
Consider the general Equation
Setting the excitation f (t) equal to F, a constant for t 0
)()()(
tfKtxdt
tdxs
It is called the forced response.
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The Complete Response
)(
)()(
/
/
xe
FKe
txtxx
t
St
FN
Consider the general Equation
The complete response is:
• the natural response +
• the forced response
)()()(
tfKtxdt
tdxs
Solve for ,
)()0(
)()0()0(
0
xx
xxtx
tfor
The Complete solution:
)()]()0([)( / xexxtx t
/)]()0([ texx called transient response
)(x called steady state response
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KVL around the loop: EvRi Cc
EvRdt
dvC c
c
EAev RC
t
C
Initial condition 0)0()0( CC vv
)1()1(
t
RC
t
C eEeEv
dt
dvCi c
c
t
eR
E
Switch is thrown to 1
RCCalled time constant
Transient Response of RC Circuits
EA
_ US I1
I1
I1
I1
I1 I1 I1 I1
R
R1
R1 R1
5
5
5 5 +
+
_
US
IS
E
U1
+
- 1 U
C
2
K
E
cc
dvi C
dt
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)1(
t
C eEv
/tc eE
dt
dv
0
0
tc
tc
dt
dv
EE
dt
dv
RCTime Constant
_ US I1
I1
I1
I1
I1 I1 I1 I1
R
R1
R1 R1
5
5
5 5 +
+
_
US
IS
E
U1
+
- 1 U
C
2
K
E
Time
0s 1ms 2ms 3ms 4ms 5ms 6ms 7ms 8ms 9ms 10ms
V(2)
0V
5V
10V
SEL>>
RC
R=2k
C=0.1F
Transient Response of RC Circuits
vc
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Switch to 2
_ US I1
I1
I1
I1
I1 I1 I1 I1
R
R1
R1 R1
5
5
5 5 +
+
_
US
IS
E
U1
+
- 1 U
C
2
K
E RC
t
c Aev
Initial condition
Evv CC )0()0(
0 Riv cc
0dt
dvRCv c
c
// tRCt
c EeEev
/t
c eR
Ei
Transient Response of RC Circuits
cc
dvi C
dt
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RCTime Constant
_ US I1
I1
I1
I1
I1 I1 I1 I1
R
R1
R1 R1
5
5
5 5 +
+
_
US
IS
E
U1
+
- 1 U
C
2
K
E
R=2k
C=0.1F
Time
0s 1.0ms 2.0ms 3.0ms 4.0ms 5.0ms 6.0ms 7.0ms 8.0ms
V(2)
0V
5V
10V
SEL>>
t
RC
t
C EeEetv
)(
E
dt
dv
t
C 0
0
t
C
dt
dv
E
Transient Response of RC Circuits
vc
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Response to square wave input
Time
0s 0.5ms 1.0ms 1.5ms 2.0ms 2.5ms 3.0ms 3.5ms 4.0ms 4.5ms 5.0ms 5.5ms 6.0ms
V(2) V(1)
0V
2.0V
4.0V
6.0V
vR
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Transient Response of RL Circuits Switch to 1
_ US I1
I1
I1
I1
I1 I1 I1 I1
R
R1
R1 R1
5
5
5 5 +
+
_
US
IS
E
U1
+
- 1 U
L
2
K
E
dt
diLvL
KVL around the loop: EviR L
iRdt
diLE
Initial condition 0)0()0(,0 iit
Called time constant RL /
/
/
/
(1 ) (1 )
(1 )
1
Rt
tL
t
R
R Rt t
L LL
t
L
E Ei e e
R R
v iR E e
di d E E Rv L L e L e
dt dt R R L
v Ee
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Transient Response of RL Circuits
Switch to 2
tL
R
Aei
dtL
R
i
di
iRdt
diL
0
Initial condition R
EIt 0,0
/tt
L
R
eR
Ee
R
Ei
_ US I1
I1
I1
I1
I1 I1 I1 I1
R
R1
R1 R1
5
5
5 5 +
+
_
US
IS
E
U1
+
- 1 U
L
2
K
E
0
0
0
0
0
: 0
:
1
ln
i t t
I
i t t
I
t t
i I i t
Rdi dt
i L
Ri t
L
tL
R
I
ti
0
)(ln
tL
R
eIti
0)(
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_ US I1
I1
I1
I1
I1 I1 I1 I1
R
R1
R1 R1
5
5
5 5 +
+
_
US
IS
E
U1
+
- 1 U
L
2
K
E
Transient Response of RL Circuits
Time
0s 1ms 2ms 3ms 4ms 5ms 6ms 7ms 8ms 9ms 10ms
I(L1)
0A
2.0mA
4.0mA
SEL>>
Time
0s 1ms 2ms 3ms 4ms 5ms 6ms 7ms 8ms 9ms 10ms
I(L1)
0A
2.0mA
4.0mA
SEL>>
Input energy to L
Switch to 2
_ US I1
I1
I1
I1
I1 I1 I1 I1
R
R1
R1 R1
5
5
5 5 +
+
_
US
IS
E
U1
+
- 1 U
L
2
K
E
Switch to 1
iL
iL
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Basic RL and RC Circuits: Summary The Time Constant
• For an RC circuit, = RC
• For an RL circuit, = L/R
• -1/ is the initial slope of an exponential with an initial value of 1
• Also, is the amount of time necessary for an exponential to decay
to 36.7% of its initial value
• When a sudden change occurs, only two types of quantities will remain the same as before the change.
– IL(t), inductor current
– Vc(t), capacitor voltage
• Find these two types of the values before the change and use them as the initial conditions of the circuit after change.
How to determine initial conditions for a transient circuit?
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Summary
Initial Value
(t = 0)
Steady Value
(t )
time
constant
RL
Circuits
Source
(0 state)
Source-
free (0 input)
RC
Circuits
Source
(0 state)
Source-
free (0 input)
00 iR
EiL
R
Ei 0
0i
00 v Ev
Ev 0 0v
RL /
RL /
RC
RC