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unit6 RL-transient.ppt
1. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Transients Analysis
Presented by
ReddyPrasad Reddivari
Assistant Professor
EEE Dept.
2. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Solution to First Order Differential Equation
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
Consider the general Equation
Let the initial condition be x(t = 0) = x( 0 ), then we solve the
differential equation:
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
The complete solution consists of two parts:
• the homogeneous solution (natural solution)
• the particular solution (forced solution)
3. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
The Natural Response
/
)
(
,
)
(
)
(
)
(
)
(
,
)
(
)
(
0
)
(
)
(
t
N
N
N
N
N
N
N
N
N
e
t
x
dt
t
x
t
dx
dt
t
x
t
dx
t
x
dt
t
dx
or
t
x
dt
t
dx
Consider the general Equation
Setting the excitation f (t) equal to zero,
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
It is called the natural response.
4. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
The Forced Response
0
)
(
)
(
)
(
t
for
F
K
t
x
F
K
t
x
dt
t
dx
S
F
S
F
F
Consider the general Equation
Setting the excitation f (t) equal to F, a constant for t 0
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
It is called the forced response.
5. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
The Complete Response
)
(
)
(
)
(
/
/
x
e
F
K
e
t
x
t
x
x
t
S
t
F
N
Consider the general Equation
The complete response is:
• the natural response +
• the forced response
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
Solve for ,
)
(
)
0
(
)
(
)
0
(
)
0
(
0
x
x
x
x
t
x
t
for
The Complete solution:
)
(
)]
(
)
0
(
[
)
( /
x
e
x
x
t
x t
/
)]
(
)
0
(
[ t
e
x
x
called transient response
)
(
x called steady state response
6. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
WHAT IS TRANSIENT RESPONSE
Figure 5.1
7. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Figu
re
5.2,
5.3
Circuit with switched DC
excitation
A general model of the
transient analysis problem
8. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
In general, any circuit containing energy storage element
Figure 5.5, 5.6
9. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
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).
10. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
(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).
11. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Why there is a transient response?
• The voltage across a capacitor cannot be
changed instantaneously.
)
0
(
)
0
(
C
C V
V
• The current across an inductor cannot be
changed instantaneously.
)
0
(
)
0
(
L
L I
I
13. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Transients Analysis
1. Solve first-order RC or RL circuits.
2. Understand the concepts of transient
response and steady-state response.
3. Relate the transient response of first-order
circuits to the time constant.
14. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Transients
The solution of the differential equation
represents are response of the circuit. It is
called natural response.
The response must eventually die out, and
therefore referred to as transient response.
(source free response)
15. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Discharge of a Capacitance through a Resistance
ic iR 0
,
0
R
C i
i
i
0
R
t
v
dt
t
dv
C C
C
Solving the above equation
with the initial condition
Vc(0) = Vi
16. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Discharge of a Capacitance through a Resistance
0
R
t
v
dt
t
dv
C C
C
0
t
v
dt
t
dv
RC C
C
st
C Ke
t
v
0
st
st
Ke
RCKse
RC
s
1
RC
t
C Ke
t
v
K
Ke
V
v
RC
i
C
/
0
)
0
(
RC
t
i
C e
V
t
v
17. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
RC
t
i
C e
V
t
v
Exponential decay waveform
RC is called the time constant.
At time constant, the voltage is 36.8%
of the initial voltage.
)
1
( RC
t
i
C e
V
t
v
Exponential rising waveform
RC is called the time constant.
At time constant, the voltage is
63.2% of the initial voltage.
18. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
RC CIRCUIT
for t = 0-, i(t) = 0
u(t) is voltage-step function
Vu(t)
R
C
+
VC
-
i(t)
t = 0
+
_
V
R
C
+
VC
-
i(t)
t = 0
+
_
Vu(t)
19. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
RC CIRCUIT
Vu(t)
0
)
(
,
,
)
(
t
for
V
t
u
v
V
v
dt
dv
RC
dt
dv
C
i
R
v
t
vu
i
i
i
C
C
C
C
C
R
C
R
Solving the differential equation
20. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Complete Response
Complete response
= natural response + forced response
• Natural response (source free response) is due
to the initial condition
• Forced response is the due to the external
excitation.
21. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Figure
5.17,
5.18
5-8
a). Complete, transient and steady
state response
b). Complete, natural, and forced
responses of the circuit
22. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Circuit Analysis for RC Circuit
Vs
+
Vc
-
+ VR -
R
C
iR
iC
s
R
C
C
C
R
s
R
C
R
v
RC
v
RC
dt
dv
dt
dv
C
i
R
v
v
i
i
i
1
1
,
Apply KCL
vs is the source applied.
23. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Solution to First Order Differential Equation
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
Consider the general Equation
Let the initial condition be x(t = 0) = x( 0 ), then we solve the
differential equation:
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
The complete solution consits of two parts:
• the homogeneous solution (natural solution)
• the particular solution (forced solution)
24. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
The Natural Response
/
)
(
)
(
)
(
0
)
(
)
(
t
N
N
N
N
N
e
t
x
t
x
dt
t
dx
or
t
x
dt
t
dx
Consider the general Equation
Setting the excitation f (t) equal to zero,
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
It is called the natural response.
25. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
The Forced Response
0
)
(
)
(
)
(
t
for
F
K
t
x
F
K
t
x
dt
t
dx
S
F
S
F
F
Consider the general Equation
Setting the excitation f (t) equal to F, a constant for t 0
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
It is called the forced response.
26. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
The Complete Response
)
(
)
(
)
(
/
/
x
e
F
K
e
t
x
t
x
x
t
S
t
F
N
Consider the general Equation
The complete response is:
• the natural response +
• the forced response
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
Solve for ,
)
(
)
0
(
)
(
)
0
(
)
0
(
0
x
x
x
x
t
x
t
for
The Complete solution:
)
(
)]
(
)
0
(
[
)
( /
x
e
x
x
t
x t
/
)]
(
)
0
(
[ t
e
x
x
called transient response
)
(
x called steady state response
27. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Example
+
Vc
-
+ VR -
100 k
ohms
0.01
microF
iR
iC
100V
Initial condition Vc(0) = 0V
s
C
C
C
C
C
s
R
C
R
v
v
dt
dv
RC
dt
dv
C
i
R
v
v
i
i
i
,
100
10
100
10
01
.
0
10
3
6
5
C
C
C
C
v
dt
dv
v
dt
dv
28. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Example
+
Vc
-
+ VR -
100 k
ohms
0.01
microF
iR
iC
100V
Initial condition Vc(0) = 0V
and
3
3
10
10
100
100
100
100
0
,
0
)
0
(
100
t
c
c
t
c
e
v
A
A
v
As
Ae
v
)
(
)
(
)
(
t
f
K
t
x
dt
t
dx
s
)
(
)
(
)
(
/
/
x
e
F
K
e
t
x
t
x
x
t
S
t
F
N
100
10 3
C
C v
dt
dv
29. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
Energy stored in capacitor
2
2
)
(
)
(
2
1
o
t
t
t
t
t
t
t
v
t
v
C
vdv
C
dt
dt
dv
Cv
pdt
dt
dv
Cv
vi
p
o
o
o
If the zero-energy reference is selected at to, implying that the
capacitor voltage is also zero at that instant, then
2
2
1
)
( Cv
t
wc
30. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
R C
Power dissipation in the resistor is:
pR = V2/R = (Vo
2 /R) e -2 t /RC
2
0
/
2
2
0
/
2
2
0
2
1
|
)
2
1
(
o
RC
t
o
RC
t
o
R
R
CV
e
RC
R
V
R
dt
e
V
dt
p
W
RC CIRCUIT
Total energy turned into heat in the resistor
31. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
RL CIRCUITS
L
R
-
VR
+
+
VL
-
i(t)
Initial condition
i(t = 0) = Io
equation
al
differenti
the
Solving
i
dt
di
R
L
dt
di
L
Ri
v
v L
R
0
0
32. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
RL CIRCUITS
L
R
-
VR
+
+
VL
-
i(t)
Initial condition
i(t = 0) = Io L
Rt
o
o
t
o
i
I
t
o
t
i
I
e
I
t
i
t
L
R
I
i
t
L
R
i
dt
L
R
i
di
dt
L
R
i
di
i
L
R
dt
di
o
o
/
)
(
)
(
ln
ln
|
|
ln
,
0
33. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
RL CIRCUIT
L
R
-
VR
+
+
VL
-
i(t)
Power dissipation in the resistor is:
pR = i2R = Io
2e-2Rt/LR
Total energy turned into heat in the resistor
2
0
/
2
2
0
/
2
2
0
2
1
|
)
2
(
o
L
Rt
o
L
Rt
o
R
R
LI
e
R
L
R
I
dt
e
R
I
dt
p
W
It is expected as the energy stored in the inductor is 2
2
1
o
LI
34. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
RL CIRCUIT
Vu(t)
R
L
+
VL
-
i(t)
+
_
Vu(t)
k
t
Ri
V
R
L
sides
both
g
Integratin
dt
Ri
V
Ldi
V
dt
di
L
Ri
)
ln(
,
0
,
]
ln
)
[ln(
ln
,
0
)
0
(
/
/
t
for
e
R
V
R
V
i
or
e
V
Ri
V
t
V
Ri
V
R
L
V
R
L
k
thus
i
L
Rt
L
Rt
where L/R is the time constant
35. Department of Electronic Engineering
BASIC ELECTRONIC ENGINEERING
DC STEADY STATE
The steps in determining the forced response for RL or
RC circuits with dc sources are:
1. Replace capacitances with open circuits.
2. Replace inductances with short circuits.
3. Solve the remaining circuit.