Mathematical Modeling of Electrical and Electronics Control Systems

1. 1
Feedback Control Systems (FCS)
Lecture-7-8
Mathematical Modelling of Electrical & Electronic Systems

2. 2
Outline of this Lecture
•Part-I: Electrical System
•Basic Elements of Electrical Systems
•Equations for Basic Elements
•Examples
•Part-II: Electronic System
•Operational Amplifiers
•Inverting vs Non-inverting
•Examples

3. 3
ELECTRICAL SYSTEMS
Part-I

4. Basic Elements of Electrical Systems
• 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 )
(
)
( 

5. 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


6. 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 

7. 7
V-I and I-V relations
Component Symbol V-I Relation I-V Relation
Resistor
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

8. 8
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.
C
i(t)
vi( t) v2(t)


 dt
t
i
C
R
t
i
t
vi )
(
)
(
)
(
1

 dt
t
i
C
t
vo )
(
)
(
1

9. 9
Example#1
• Taking Laplace transform of both equations, considering initial
conditions to zero.
• Re-arrange both equations as:


 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

)
(
)
( s
I
s
CsVo 
)
)(
(
)
(
Cs
R
s
I
s
Vi
1



10. 10
Example#1
• Substitute I(s) in equation on left
)
(
)
( s
I
s
CsVo 
)
)(
(
)
(
Cs
R
s
I
s
Vi
1


)
)(
(
)
(
Cs
R
s
CsV
s
V o
i
1


)
(
)
(
)
(
Cs
R
Cs
s
V
s
V
i
o
1
1


RCs
s
V
s
V
i
o


1
1
)
(
)
(

11. 11
Example#1
• The system has one pole at
RCs
s
V
s
V
i
o


1
1
)
(
)
(
RC
s
RCs
1
0
1 





12. 12
Example#2
• Design an Electrical system that would place a pole at -3 if
added to another system.
• System has one pole at
• Therefore,
C
i(t)
vi( t) v2(t)
RCs
s
V
s
V
i
o


1
1
)
(
)
(
RC
s
1


3
1



RC
pF
C
and
M
R
if 333
1 



13. 13
Example#3
• Find the transfer function G(S) of the following
two port network.
i(t)
vi(t) vo(t)
L
C

14. 14
Example#3
• Simplify network by replacing multiple components with
their equivalent transform impedance.
I(s)
Vi(s) Vo(s)
L
C
Z

15. 15
Transform Impedance (Resistor)
iR(t)
vR(t)
+
-
IR(S)
VR(S)
+
-
ZR = R
Transformation

16. 16
Transform Impedance (Inductor)
iL(t)
vL(t)
+
-
IL(S)
VL(S)
+
-
LiL(0)
ZL=LS

17. 17
Transform Impedance (Capacitor)
ic(t)
vc(t)
+
-
Ic(S)
Vc(S)
+
-
ZC(S)=1/CS

18. 18
Equivalent Transform Impedance (Series)
• Consider following arrangement, find out equivalent
transform impedance.
L
C
R
C
L
R
T Z
Z
Z
Z 


Cs
Ls
R
ZT
1




19. 19
Equivalent Transform Impedance (Parallel)
C
L
R
T Z
Z
Z
Z
1
1
1
1



C
L
R
Cs
Ls
R
ZT
1
1
1
1
1




20. 20
Equivalent Transform Impedance
• Find out equivalent transform impedance of
following arrangement.
L2
L2
R2
R1

21. 21
Back to Example#3
I(s)
Vi(s) Vo(s)
L
C
Z
L
R Z
Z
Z
1
1
1


Ls
R
Z
1
1
1


RLs
RLs
Z


1

22. 22
Example#3
I(s)
Vi(s) Vo(s)
L
C
Z
RLs
RLs
Z


1
)
(
)
(
)
( s
I
Cs
Z
s
I
s
Vi
1

 )
(
)
( s
I
Cs
s
Vo
1


23. 23
Example#4
Vin
C
R
L
Vout
• Find transfer function Vout(s)/Vin(s) of the following electrical
network

24. 24
Example#5
Vin
C1
R
L
Vout
• Find transfer function Vout(s)/Vin(s) of the following electrical
network
C2
C3

25. 25
ELECTRONIC SYSTEMS
Part-II

26. 26
Operational Amplifiers
1
2
Z
Z
V
V
in
out


1
2
1
Z
Z
V
V
in
out



27. 27
Example#6
• Find out the transfer function of the following
circuit.
1
2
Z
Z
V
V
in
out



28. 28
Example#7
• Find out the transfer function of the following
circuit.
v1

29. 29
Example#8
• Find out the transfer function of the following
circuit.
v1

30. 30
Example#9
• Find out the transfer function of the following
circuit and draw the pole zero map.
10kΩ
100kΩ