401062-CHAPTER 1-TRANSIENT ANALYSIS IN TIME DOMAIN.ppt

giai tich

1. CHAPTER I
TRANSIENT ANALYSIS IN TIME
DOMAIN
Ths. Nguyễn Thị Phương Thảo

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TON DUC THANG UNIVERSITY
FACULTY OF ELECTRICAL AND ELECTRONICS
ENGINEERING
ELECTRICAL ENGINEERING DEPARTMENT

2. COURSE OBJECTIVES
Remember the features of the transient circuits: RL,
RC, RLC
Understand two methods: “classical method and
Laplace method” to analyze the transient circuits
Apply analysis techniques to design the efficient
circuits, the electrical systems and calculate their
appropriate parameters.
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3. 1.1. Transient analysis in time domain using the
classical method.
1.1.1.Concept:
Steady-state and appearance of the transient state
in electrical circuit
First order circuits and second order circuits
1.1.2.Transient analysis using classical method:
The forced response of circuits
The general solution for natural responses,
characteristic equation
CHAPTER CONTENTS
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4. CHAPTER CONTENTS
Independent and dependent initial conditions.
Transient analysis for the first order circuit
Transient analysis for the second order circuit
1.2. Transient analysis using the Laplace
transform techniques:
Laplace transform of some simple time functions
Circuit analysis with the Laplace transform.
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5. 1. Resistor:
The equation U relate to I showing through Ohm’s Law :
2. Inductor:
The voltage cross the inductor:
The change of current in an inductor brings about the induced
voltage of magnitude
The current flow through the inductor:

THE RELATIONSHIP BETWEEN U
AND I ON ELEMENTS R,L,C
u = R.i
()
()
dit
ut L
dt

1
() () ( )
0
0
t
i t u t dt i t
Lt
 
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6. THE RELATIONSHIP BETWEEN U
AND I ON ELEMENTS
3. Capacitor :
The current flow through capacitor, change of voltage
brings about infinite current.
Voltage
()
()
dq dut
it C
dt dt
 
1
() () ( )
0
0
t
ut it dt ut
Ct
 
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7. TRANSIENT ANALYSIS USING THE
CLASSICAL METHOD.
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8. 1.1.1.Concepts
Steady-state
1.1.TRANSIENT ANALYSIS USING
THE CLASSICAL METHOD.
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Appearance of the transient state in electrical circuit
The transient phenomenan will occur when there is any
switching, interrupting, short-circuiting as well as any
sudden changes in the structure of an electric circuit..
 These energy redistributions cannot take place
instantaneously but it remains during some period of
time of the transient-state.

9. 1.1.1.Concepts
First order circuits and second order circuits
1.1.TRANSIENT ANALYSIS USING
THE CLASSICAL METHOD.
Any circuit with a single energy storage element,
is a circuit of order 1.
R
C
v
s
(t)
+
–
v
c(t)
+ –
+
–
Example
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10. 1.1.1.Concepts
First order circuits and second order circuits
1.1. TRANSIENT ANALYSIS USING
THE CLASSICAL METHOD.
Any circuit with a single capacitor, a single inductor,
is a circuit of order 2
Example
v
s(t)
R
C
L
+
–
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11. 1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
The general solution for forced response
Let consider the circuit R, L, C series
as the figure. Find i(t) through the circuit
when closing the switch k. Assume
that at t<0 the switch k has been opened

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1.1.2.Transient analysis using classical method:

12. 1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
The general solution for forced response
Apply Kirchhoff’s voltage Law:

(1)
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1.1.2.Transient analysis using classical method:
R L C
v v v v
t
  
1
()L R
L
C
di
v i idt vt
dt
    

13. 1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
The variable x(t) could be voltage or current.
The total solution to any differential equation consists of two
parts:
x(t) = x
f(t) + x
n(t)
Particular (forced) solution is x
f
(t)
Response particular to a given source
Complementary (natural) solution is x
n
(t)
Response common to all sources, that is, due to the
“passive” circuit elements
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1.1.2.Transient analysis using classical method:

14. 1.1 TRANSIENT ANALYSIS USING
CLASSICAL METHOD
At t=0 magnetic energy and electrical energy don’t
change instantly. Therefore we have
At t=0 voltage across capacitor and current flows inductor
don’t change
Initial conditions
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1.1.2.Transient analysis using classical method:

15. 1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Steps to analyze the transient circuits
Determine the independent initial conditions
 Find the forced response of circuit basing on steady –
state analysis method
 Determine the characteristic equation and evaluation of
its roots.
Solve the differential equation to find the natural response
The desired response is the sum of the natural response
and the forced response.
Determine the integration constants.
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1.1.2.Transient analysis using classical method:

16. R
LE
K
+When the switch K is opened i=0
+ After closing switch K
E
i
R

At t= 0 close K, find i(t)? Assume
that at t<0, k has been opened a
long time ago
i(t
)
RL circuits
Transient analysis for the first order circuit
1.1 TRANSIENT ANALYSES
USING CLASSICAL METHOD
Solution
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17. At the time close K
R L
Eu u 
R
u =iR
E
dt
di
LiR 
Differential
equation
To determine responses i(t) we need to solve the above
differential equation
Substituting
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
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;u =L
L
di
dt

18. i(t) : consists of two parts
I
f
: (Depend on the sources of the circuit) The current of
circuit after switching k a long time
I
n
: (Depend on the circuit parameters, and there isn’t
effect of power) is the root of the first order differential
equation.
i(t) i
f i
n= +
()
() 0 (1)
di t
i t R L
dt
 
1.1 TRANSIENT ANALYSES
USING CLASSICAL METHOD
Transient analysis for the first order circuit
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19. The root of the first order differential equation:
( )
(2)
pt
ptn
pt
pt
n
did Ke
Kpe
dt dt
Ke
i dt Ke dt
p

 




 


 
0
pt pt
RKe LpKe 
( ) 0
pt
Ke R pL  
( ) 0R pL  Characteristic equation
pt
n
i Ke
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
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20. Find K? Apply the initial conditions to find K
At the time t=0 close K. We have i = 0
R
p
L
 
()
R
t
L
f n
E
i t i i Ke
R

    
0
E E
K K
R R
   
( ) (1 )
R R
t t
L L
f n
E E E
i t i i e e
R R R
 
     
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
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Find K:

21. Assume:
Denote: time constant of circuit
Response i(t)
0, () 0
, ()
t i t
E
t i t
R
 
 
u
c
(t)
E/R
t
0
R
L

() (1 )
t
E
i t e
R


  
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
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22. +Before closing K: Uc(t)= 0
+After closing Ku E
At t= 0, close K. Find u
c(t)

E
R
C
K
Uc(t)
RC Circuits
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
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23. Close k
R c
E u u 
R
u iR
c
c
du
i C
dt

Eu
dt
du
RC
c
c

Differential
equation
To find response u
c(t) we need to solve the differential equation
above.
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
Substituting
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24. i(t) consist of 2 components:
u
c
(t) u
cf u
cn= +
()
() 0 (3)
C
C
du t
u t RC
dt
 
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
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25. The roots of the first order differential equation :
( )
(4)
pt
ptcn
pt
pt
cn
dud Ke
Kpe
dt dt
Ke
u dt Ke dt
p

 




 


 
0
pt pt
Ke RCpKe 
(1 ) 0
pt
Ke pRC  
(1 ) 0pRC  Differential equation
pt
cn
u Ke
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
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26. Find K? Applying the initial conditions. At t=0 close K, uc = 0
1
p
RC
 
1
()
t
RC
f n
u t u u E Ke

   
0E K K E   
() (1 )
R R
t t
L L
f n
u t u u E Ee E e
 
     
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
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Tìm k

27. Assume
This value of time is called the time constant
Response UC(t)
Etut
tut


)(,
0)(,0
u
c
(t)
E
t
0
RC
() (1 )
t
c
u t E e


  

1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the first order circuit
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28. 2
( ) ( )
( ) (5)
C C
c
du t d u t
E RC LC u t
dt dt
  
R
L
C
K
At t= 0 close K. Let determine voltage
across the capacitor
E
RLC Circuits
Then:
Apply K2:
We have:
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the second order circuit
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( ) ( ) ( )
c cf cn
u t u t u t 

29. Assumes:
Resonance frenquency
Coefficient
We have the characteristic equation:
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the second order circuit
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2
2
2
( 1) 0
( 1) 0
1
0
pt
Ke LCp RCp
LCp RCp
R
p p
L LC
  
   
   
2
R
L

2
R
L


30. After resolving the above equation, if the characteristic
equation have 2 roots then we obtain the natural response
Uc(t) as following
1.1 TRANSIENT ANALYSES USING
CLASSICAL METHOD
Transient analysis for the second order circuit
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1 2
1 2
()
pt p t
n
u t ke ke 

31. Lect12
EXERCISES
Do examples 7.1-7.4; 8.1- 8.6
Suggestion: Drill assessment Problems 7.1 – 7.5; 8.1-
8.6
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32. ASSIGNMENT
Assess your understanding of this material by doing homework
7.6 – 7.10
Students read the next lesson in the text books in advance to
more understand the next lesson.
Read the next slide of this chapter of this subject.
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33. CIRCUIT ANALYSIS USING LAPLACE
TRANSFORM
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34. This chapter will apply the Laplace transform to analyze the
transient circuit
The equivalent models for the resistor, capacitor, and
inductor will be introduced.
Setting up proper initial conditions will be covered.
Transfer functions and state variables are also discussed.
Finally circuit stability and network synthesis will be covered.
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35. 1.1 Laplace transform
Laplace transform of some simple functions
Inverse laplace transform
1.2 Transient analysis using Laplace transform
Transform all elements of the circuit into s domain
Apply some circuit analysis methods to find the
necessary variables.
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36. We will now look at how to apply Laplace
transforms to circuit.
Transform the circuit from time domain to the s-domain.
Solve the circuit using nodal analysis, mesh analysis, source
transformation, superposition, or any circuit analysis technique
with which we are familiar.
Take the inverse transform of the solution and thus obtain the
solution in the time domain.
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37. -st
0
L(f (t))= f (t)e dt


1.2 LAPLACE TRANSFORM
The Laplace transform of a function is given by the expression
Where : the symbol is read “the Laplace transform of ” f(t).
The Laplace transform of is also denoted F(s)
Definition
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38. 1.2 LAPLACE TRANSFORM
Table 3.1 Laplace Transforms for Various Time-Domain
Functions
f(t) F(s)
1 (Step) 1/s
t (Ramp) 1/s
2
t
n
(n!)/s
(n+1)
(exponential) e
-at
1/(s+a)
(damped ramp) t
n
e
-at
(n!)/(s+a)
(n+1)
sinat a/(s
2
+a
2
)
(damped sine) e
-t
. Sinat a/((s+)
2
+a
2
)
cosat s/(s
2
+a
2
)
(damped cosine) e
-t
.cosat S+/((s+)
2
+a
2
)
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39. For example,
1.2 LAPLACE TRANSFORM
The denominator has four roots. Two of these roots are distinct—
namely at s =-3. s = 0, and A multiple root of multiplicity 2 occurs
at s =-1
Inverse laplace transform
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40. The key to the partial fraction technique for finding inverse
transforms lies in recognizing the corresponding to each term
in the sum of par-tial fractions. From Table 12.1 you should be
able to verify that f(t)
1.2 LAPLACE TRANSFORM
Inverse laplace transform
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41. Find the initial conditions of the circuit in the negative steady-
state
Transform Laplace of the sources of excitation and all the
elements in the circuit in the positive steady state
Find the output U(S), I(S) in the Laplace freq domain using
the circuit analysis methods in steady – state
Obtain the time response u(t), i(t) by taking the inverse
Laplace.
1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM
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42. Sources in the s Domain.
i(t) I(s)
+
_
+
_v(t) V(s)
Transforming laplace of circuit elements in the s
domain
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1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM

43. A resistor in the s domain & ohm’s law
In the time domain:
In the s-domain:
i(t)
+ v(t)-
R
v(t)=i(t)R
I(s)
+ V(s)-
R
V(s)= I(s)R
Transforming laplace of circuit elements in the s domain
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1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM

44. In the time domain: Inductor’s voltage:
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1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM
Transforming laplace of circuit elements in the s
domain

45. In the time domain:
In the s-domain:
()
L
di
v t L
dt

( ) ( ) (0 )
L L L
V s sLI s Li

 
Transforming laplace of circuit elements in the s
domain
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1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM

46. In the time domain: In the s-domain:
Transforming laplace of circuit elements in the s
domain
Capacitor
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1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM

47. In the time domain:
In the s-domain:
dt
dv
Cti
c)(
 
1 1
0
c c c
V(s) I (s) v( )
sC s

 
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1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM
Transforming laplace of circuit elements in the s
domain

48. Example
Find v
0
(t) if the initial voltage is given u(0
-
)=5 V
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1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM
48

49. s-Domain Circuit
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1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM

50. Apply nodal analysis method
1 1
( 2) 2.5
10 1
o
V s
s
   

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0 0 0
0 0 0
10/ 1
2 0.5
10 10 10/
1
2.5
10 1 10 10
V s V V
s
V V sV
s
 
   
    

1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM

51. 1 2
25 35
( 1)( 2) 1 2
o
s K K
V
s s s s

  
   
Rewrite V
0(s) using PFE:
Solved for K
1 and K
2:
1 2
10; 15K K 
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0
10
( 2) 25
1
25 35
( 1)( 2)
o
V s
s
s
V
s s
  



 
Apply nodal analysis method
1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM

52. Obtain v
0(s) and v
0(t)
2
15
1
10
)(




ss
sV
o
Calcula te V
0(s):
Ob tain V
0(t) using loo k up table:
2
() (10 15 ) ()
t t
o
v t e e ut
 
 
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1.2 TRANSIENT ANALYSIS USING
LAPLACE TRANSFORM

53. 401062- Transient analysis in time domain 5330/07/2020
SUMMARY
Inductors and capacitors are passive elements; they can store
and release energy, but they cannot generate or dissipate
energy
The natural response is the currents and voltages that exist
when stored energy is released to a circuit that contains no
independent sources.

54. 401062- Transient analysis in time domain 5430/07/2020
SUMMARY
Inductors and capacitors are passive elements; they can store
and release energy, but they cannot generate or dissipate
energy
The natural response is the currents and voltages that exist
when stored energy is released to a circuit that contains no
independent sources.

55. 401062- Transient analysis in time domain 5530/07/2020
SUMMARY
Inductors and capacitors are passive elements; they can store
and release energy, but they cannot generate or dissipate
energy
The natural response is the currents and voltages that exist
when stored energy is released to a circuit that contains no
independent sources.

56. 401062- Transient analysis in time domain 5630/07/2020
SUMMARY
The step response is the currents and voltages that result
from abrupt changes in dc sources connected to a circuit.
Stored energy may or may not be present at the time the
abrupt changes take place.
The solution for either the natural or step response of both RL
and RC circuits involves finding the initial and final value of the
current or voltage of interest and the time constant of the
circuit.

57. 401062- Transient analysis in time domain 5730/07/2020
ASSIGNMENT
To assess your understanding of this material, students
should drill homework 12.27 – 12.31, 13.3-13.8
To make sure more understand the next lesson, students
need to read the contents of the next lesson in the text book
in advance
Read the slide chapter 2 of this subject.