The document describes the mathematical modeling of electrical components like resistors, inductors and capacitors using Laplace transforms. It provides an example of modeling an RC circuit. The RC circuit is represented by two equations in the time domain, which are then transformed to the s-domain using Laplace transforms. This yields the transfer function of the circuit, relating the output voltage to the input current. The circuit and transfer function are then represented as a block diagram.

1. Model of Resistance
Laplace Transform
Mathematical model
R

2. Model of Resistance
R

3. Model of Resistance
R
i(t)

4. Model of Resistance
R
i(t)
v(t)

5. Model of Resistance
R
i(t)
v(t) v(t)= Ri(t)

6. Model of Resistance
R
i(t)
v(t)
Take Laplace :
   
( ) ( )
( ) ( )
( ) ( )
v t Ri t
v t R i t
V s RI s
=
=
=
L L

7. Model of Resistance
R
I(s)
V(s)
Impedance term:
( )
( )
V s
R
I s
=
s-domain

8. Model of Inductance
Laplace Transform
Mathematical model
L

9. Model of Inductance
L
i(t)

10. Model of Inductance
L
i(t)
v(t)

11. Model of Inductance
L
i(t)
v(t) ( ) ( )
d
v t L i t
dt
=

12. Model of Inductance
L
i(t)
v(t)
 
( ) ( )
( ) ( )
( ) ( )
d
v t L i t
dt
d
v t L i t
dt
V s LsI s
=
 
=  
 
=
L L
Take Laplace :

13. Model of Inductance
Ls
I(s)
V(s)
( )
( )
V s
Ls
I s
=
Impedance term:
s-domain

14. Model of Capacitance
Laplace Transform
Mathematical model
C

15. Model of Capacitance
C

16. Model of Capacitance
C
i(t)

17. Model of Capacitance
C
i(t)
v(t)
0
1
( ) ( )
v t i t dt
C

= 

18. Model of Capacitance
C
i(t)
v(t)  
0
0
1
( ) ( )
1
( ) ( )
1 1
( ) ( )
v t i t dt
C
v t i t dt
C
V s I s
C s


=
 
=  
 
=


L L
Take Laplace :

19. Model of Capacitance
I(s)
V(s)
1 ( )
( )
V s
Cs I s
=
1
Cs
Impedance term:
s-domain

20. Conclusion
Time-domain s-domain
Impedance term:
R
L
C
R
Ls
1
Cs
Laplace
Transform
L

21. Example : Electrical Circuit
RC Circuit :
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t

22. Example : Electrical Circuit
RC Circuit :
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t
1 2 1
0
1
( ) ( ) ( ) ( )
R i t R i t i d v t
C

 
+ + =

Mathematical Model time-domain

23. Example : Electrical Circuit
RC Circuit :
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t
1 2 1
0
2 2
0
1
( ) ( ) ( ) ( )
1
( ) ( ) ( )
R i t R i t i d v t
C
R i t i d v t
C


 
 
+ + =
+ =


Mathematical Model time-domain

24. Example : Electrical Circuit
RC Circuit :
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t
1 2 1
0
2 2
0
1
( ) ( ) ( ) ( )
1
( ) ( ) ( )
R i t R i t i d v t
C
R i t i d v t
C


 
 
+ + =
+ =


Mathematical Model time-domain
Two equations form the mathematical model of the circuit

25. Example : Electrical Circuit
RC Circuit :
 
 
1 2 1
0
2 2
0
1
( ) ( ) ( ) ( )
1
( ) ( ) ( )
R i t R i t i d v t
C
R i t i d v t
C


 
 
 
+ + =
 
 
 
+ =
 
 


L L
L L
Taking the Laplace Transform
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t

26. Example : Electrical Circuit
RC Circuit :
     
   
1 2 1
0
2 2
0
1
( ) ( ) ( ) ( )
1
( ) ( ) ( )
R i t R i t i d v t
C
R i t i d v t
C


 
 
 
=
 
 
 
+ =
 
 


L + L + L L
L L L
Taking the Laplace Transform
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t

27. Example : Electrical Circuit
RC Circuit :
     
   
1 2 1
0
2 2
0
1
( ) ( ) ( ) ( )
1
( ) ( ) ( )
R i t R i t i d v t
C
R i t i d v t
C


 
 
 
=
 
 
 
+ =
 
 


L + L + L L
L L L
Taking the Laplace Transform
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t

28. Example : Electrical Circuit
RC Circuit :
1 2 1
2 2
1 1
( ) ( ) ( ) ( )
1 1
( ) ( ) ( )
R I s R I s I s V s
C s
R I s I s V s
C s
=
+ =
+ +
Taking the Laplace Transform
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t

29. Example : Electrical Circuit
RC Circuit :
1 2 1
2 2
( )
( ) ( ) ( )
( )
( ) ( )
I s
R I s R I s V s
Cs
I s
R I s V s
Cs
=
+ =
+ +
Taking the Laplace Transform
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t

30. Example : Electrical Circuit
RC Circuit :
1 2 1
2 2
( )
( ) ( ) ( )
( )
( ) ( )
I s
R I s R I s V s
Cs
I s
R I s V s
Cs
=
+ =
+ +
Circuit Transform
Time-domain to s-domain
1( )
v t 2 ( )
v t
1
R
2
R
C
( )
i t
1( )
V s 2 ( )
V s
1
R
2
R
1
Cs
( )
I s

31. Example : Electrical Circuit
RC Circuit :
1 2 1
2 2
( )
( ) ( ) ( )
( )
( ) ( )
I s
R I s R I s V s
Cs
I s
R I s V s
Cs
=
+ =
+ +
s-domain
1( )
V s 2 ( )
V s
1
R
2
R
1
Cs
( )
I s
Next step : Transfer Function

32. Example : Electrical Circuit
RC Circuit :
1 2 1
2 2
( )
( ) ( ) ( )
( )
( ) ( )
I s
R I s R I s V s
Cs
I s
R I s V s
Cs
+ + =
+ =
s-domain
1( )
V s 2 ( )
V s
1
R
2
R
1
Cs
( )
I s
Input Output
Transfer Function = Output
Input

33. Example : Electrical Circuit
RC Circuit :
1 2 1
2 2
( )
( ) ( ) ( )
( )
( ) ( )
I s
R I s R I s V s
Cs
I s
R I s V s
Cs
+ + =
+ =
s-domain
1( )
V s 2 ( )
V s
1
R
2
R
1
Cs
( )
I s
Input Output
Transfer Function = Output
Input
2
1
( )
( )
V s
V s
=

34. Example : Electrical Circuit
RC Circuit :
s-domain
1( )
V s 2 ( )
V s
1
R
2
R
1
Cs
( )
I s
Input Output
Transfer Function = Output
Input
2
1
( ) ( )
( ) ( )
V s C s
V s E s
= =
Output = C(s)
Input = E(s)
E(s) C(s)

35. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
Systems
Transfer Function = Output
Input
( )
( )
C s
E s
=
Input Output

36. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
Input Output

37. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
1 2 1
1 2 1
1
1 2
( )
( ) ( ) ( )
1
( ) ( )
( )
( )
1
I s
R I s R I s V s
Cs
I s R R V s
Cs
V s
I s
R R
Cs
+ + =
 
+ + =
 
 
=
+ +
2 2
2 2
2
2
( )
( ) ( )
1
( ) ( )
( )
( )
1
I s
R I s V s
Cs
I s R V s
Cs
V s
I s
R
Cs
+ =
 
+ =
 
 
=
+

38. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
1
1 2
( )
( )
1
V s
I s
R R
Cs
=
+ +
2
2
( )
( )
1
V s
I s
R
Cs
=
+

39. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
1 2
1 2 2
( ) ( )
1 1
V s V s
R R R
Cs Cs
=
+ + +

40. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
2
2
1
1 2
1
( )
1
( )
R
V s Cs
V s R R
Cs
+
=
+ +

41. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
2
2
1 2
1
1
( )
1
( )
R Cs
V s Cs
R Cs R Cs
V s
Cs
+
=
+ +

42. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
2 2
1 1 2
( ) 1
( ) 1
V s R Cs
V s R Cs R Cs
+
=
+ +

43. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
( )
2 2
1 1 2
( ) 1
( ) 1
V s R Cs
V s R R Cs
+
=
+ +

44. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
( )
2 2
1 1 2
( ) 1
( )
( ) ( ) 1
V s R Cs
C s
E s V s R R Cs
+
= =
+ +

45. Example : Electrical Circuit
RC Circuit :
Block Diagram
E(s) C(s)
RC Circuit
( )
2
1 2
1
( )
( ) 1
R Cs
C s
E s R R Cs
+
=
+ +

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