Basic Elements Modeling-R,L,C Solved Examples with RLC circuit L, C Modeling with Non-Zero Initial condition

1. Modeling of Electrical System
•Basic Elements Modeling-R,L,C
•Solved Examples with RLC circuit
•L, C Modeling with Non-Zero Initial condition
1

2. Modeling of Electrical System
• The time domain expression relating voltage and current for the
resistor is given by Ohm’s law i-e
R
t
i
t
v R
R )
(
)
( 
• The Laplace transform of the above equation is
R
s
I
s
V R
R )
(
)
( 
R
s
I
s
V R
R 
)
(
/
)
(

3. Basic Elements of Electrical Systems
• The time domain expression relating voltage and current for the
Capacitor is given as:
dt
t
i
C
t
v c
c 
 )
(
)
(
1
• The Laplace transform of the above equation (assuming there is no
charge stored in the capacitor) is
)
(
)
( s
I
Cs
s
V c
c
1

Cs
s
I
s
V c
c
1
)
(
/
)
( 

4. Basic Elements of Electrical Systems
• The time domain expression relating voltage and current for the
inductor is given as:
dt
t
di
L
t
v L
L
)
(
)
( 
• The Laplace transform of the above equation (assuming there is no
energy stored in inductor) is
)
(
)
( s
LsI
s
V L
L 
Ls
s
I
s
V L
L 
)
(
/
)
(

5. V-I and I-V relations
5
Component Laplace V-I Relation I-V Relation
Resistor R
Capacitor
Inductor
dt
t
di
L
t
v L
L
)
(
)
( 
dt
t
i
C
t
v c
c 
 )
(
)
(
1
R
t
i
t
v R
R )
(
)
( 
R
t
v
t
i R
R
)
(
)
( 
dt
t
dv
C
t
i c
c
)
(
)
( 
dt
t
v
L
t
i L
L 
 )
(
)
(
1
Ls
Cs
1

6. Example-1
• The two-port network shown in the following figure has vi(t) as
the input voltage and vo(t) as the output voltage. Find the
transfer function Vo(s)/Vi(s) of the network.
6
C
i(t)
vi( t) vo(t)


 dt
t
i
C
R
t
i
t
vi )
(
)
(
)
(
1

 dt
t
i
C
t
vo )
(
)
(
1

7. Example-1
• Taking Laplace transform of both equations, considering initial
conditions to zero.
• Re-arrange both equations as:
7


 dt
t
i
C
R
t
i
t
vi )
(
)
(
)
(
1

 dt
t
i
C
t
vo )
(
)
(
1
)
(
)
(
)
( s
I
Cs
R
s
I
s
Vi
1

 )
(
)
( s
I
Cs
s
Vo
1

Cs
s
I
s
V
s
I
s
CsV
o
o
/
)
(
)
(
)
(
)
(


)
)(
(
)
(
Cs
R
s
I
s
Vi
1



8. Example-1
• .
8
Cs
s
I
s
Vo /
)
(
)
( 
)
)(
(
)
(
Cs
R
s
I
s
Vi
1


)
(
/
)
( s
V
s
V
nction
TransferFu i
o

)
1
)(
(
/
)
(
)
(
)
(
Cs
R
s
I
Cs
s
I
s
V
s
V
i
o


RCs
s
V
s
V
i
o


1
1
)
(
)
(
)
1
(
1
)
(
)
(
Cs
R
Cs
s
V
s
V
i
o



9. Mathematical modeling of electrical systems
9
Example-2

10. Mathematical modeling of electrical systems
10

11. Mathematical modeling of electrical systems
Where:-
11
Example-4

12. Mathematical modeling of electrical systems
12

13. Mathematical modeling of electrical systems
13

14. Mathematical modeling of electrical systems
14

15. Example-4
Use Laplace transforms to find v(t) for t > 0.
+
_
 

t = 0 6 k 
3 k 
100 F
+
_
v(t)
12 V
volts
v 4
)
0
( 

16. Circuit theory problem:
+
_
vc(t) i(t)
3 k 
100 F
6 k 
 
 
  0
5
)
(
0
6
^
10
*
100
*
3
^
10
*
2
)
(
0
)
(
0
)
(
)
(
)
(
)
(
)
(
)
(
)
arg
(
)
(
)
(














t
v
dt
t
dv
t
v
dt
t
dv
RC
t
v
dt
t
dv
t
v
dt
t
dv
RC
dt
t
dv
RC
t
v
R
t
i
t
v
ing
disch
dt
t
dv
C
t
i
c
c
c
c
c
c
c
c
c
c
c
c
c
c
Take the Laplace transform
of this equations including
the initial conditions on vc(t)

17. Circuit theory problem:
)
(
4
)
(
5
4
)
(
0
)
(
5
4
)
(
0
)
(
5
)
(
5
t
u
e
t
v
s
s
V
s
V
s
sV
t
v
dt
t
dv
t
c
c
c
c
c
c










18. An inductor in the s domain
 iv-relation in the time domain
v(t)  L
d
i(t).
dt
 By operational Laplace transform:
Lv(t) LLi(t) L Li(t),
V(s)  LsI(s)  I0  sL I(s)  LI0.
initial current
1

19. Equivalent circuit of an inductor
 Series equivalent:  Parallel equivalent:
Thévenin 
Norton
1

20. A capacitor in the s domain
 iv-relation in the time domain
i(t)  C
d
v(t).
dt
 By operational Laplace transform:
Li(t) LCv(t) C  Lv(t),
 I(s)  C sV(s) V0  sC V (s)  CV0.
initial voltage
2

21. Equivalent circuit of a capacitor
 Parallel equivalent:  Series equivalent:
Norton 
Thévenin
2