Post on 06-Feb-2016
description
Transient Analysis of RC circuits
Nano107
Chapter 7
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
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).
(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).
)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
WHAT IS TRANSIENT RESPONSE
Figure 5.1
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)
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.
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.
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
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
)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
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
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
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
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
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)(
_ 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
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?
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