This document provides an overview of mathematical modeling of electrical and electronic systems. It discusses: - The basic elements of electrical systems including resistors, capacitors, and inductors and their voltage-current relationships. - Examples of modeling simple RC and RLC circuits and calculating their transfer functions. - Operational amplifiers and examples of inverting and non-inverting configurations. - Worked examples of calculating transfer functions for various circuits containing resistors, capacitors, inductors, and operational amplifiers.

1. Feedback Control Systems (FCS)
Lecture-6
Mathematical Modelling of Electrical & Electronic Systems
Dr. Imtiaz Hussain
email: imtiaz.hussain@faculty.muet.edu.pk
URL :http://imtiazhussainkalwar.weebly.com/
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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
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3. Part-I
ELECTRICAL SYSTEMS
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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
v R (t )
iR (t )R
• The Laplace transform of the above equation is
VR ( s)
I R ( s )R

5. Basic Elements of Electrical Systems
• The time domain expression relating voltage and current for the
Capacitor is given as:
vc (t )
1
i c ( t )dt
C
• The Laplace transform of the above equation (assuming there is no
charge stored in the capacitor) is
Vc ( s )
1
Cs
Ic(s)

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

7. V-I and I-V relations
Component
Resistor
Capacitor
Inductor
Symbol
V-I Relation
v R (t )
vc (t )
v L (t )
iR (t )R
1
C
L
i c ( t )dt
di L ( t )
dt
I-V Relation
v R (t )
iR (t )
ic ( t )
iL (t )
R
C
dv c ( t )
dt
1
L
v L ( t )dt
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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.
vi( t)
vi (t )
vo (t )
i( t ) R
i(t)
1
C
vo(t)
i ( t ) dt
C
1
i ( t ) dt
C
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9. Example#1
vi (t )
i( t ) R
1
i ( t ) dt
C
1
vo (t )
i ( t ) dt
C
• Taking Laplace transform of both equations, considering initial
conditions to zero.
Vi ( s )
I ( s )R
1
I(s)
Vo ( s )
Cs
1
I(s)
Cs
• Re-arrange both equations as:
Vi ( s )
I ( s )( R
1
Cs
)
CsV
o
(s)
I(s)
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10. Example#1
Vi ( s )
I ( s )( R
1
)
CsV o ( s )
Cs
I(s)
• Substitute I(s) in equation on left
Vi ( s )
1
CsV o ( s )( R
Vo ( s )
Vi ( s )
)
Cs
1
1
Cs ( R
)
Cs
Vo ( s )
Vi ( s )
1
1
RCs
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11. Example#1
Vo ( s )
Vi ( s )
1
1
RCs
• The system has one pole at
1
RCs
0
s
1
RC
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12. Example#2
• Design an Electrical system that would place a pole at -3 if
added to another system.
Vo ( s )
Vi ( s )
1
1
RCs
vi( t)
i(t)
v2(t)
C
• System has one pole at
1
s
RC
• Therefore,
1
3
if
R
1 M
and
C
333 pF
RC
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13. Example#3
• Find the transfer function G(S) of the following
two port network.
L
vi(t)
i(t)
C
vo(t)
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14. Example#3
• Simplify network by replacing multiple components with
their equivalent transform impedance.
L
Z
Vi(s)
I(s)
C
Vo(s)
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15. Transform Impedance (Resistor)
iR(t)
IR(S)
+
+
Transformation
vR(t)
-
ZR = R
VR(S)
-
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16. Transform Impedance (Inductor)
iL(t)
IL(S)
+
vL(t)
-
ZL=LS
+
VL(S)
LiL(0)
-
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17. Transform Impedance (Capacitor)
ic(t)
Ic(S)
+
vc(t)
-
ZC(S)=1/CS
+
Vc(S)
-
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18. Equivalent Transform Impedance (Series)
• Consider following arrangement, find out equivalent
transform impedance.
ZT
ZT
ZR
R
ZL
Ls
ZC
L
C
1
R
Cs
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19. Equivalent Transform Impedance (Parallel)
1
1
1
1
ZT
ZR
ZL
ZC
1
1
1
R
Ls
C
1
ZT
L
1
Cs
R
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20. Equivalent Transform Impedance
• Find out equivalent transform impedance of
following arrangement.
L2
L2
R1
R2
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21. Back to Example#3
L
Z
Vi(s)
I(s)
C
Vo(s)
1
1
1
Z
ZR
ZL
1
1
1
Z
R
Ls
Z
RLs
1
RLs
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22. Example#3
Z
Vi(s)
Vi ( s )
I ( s )Z
RLs
1
RLs
L
Z
C
I(s)
1
Cs
I(s)
Vo(s)
Vo ( s )
1
I(s)
Cs
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23. Example#4
• Find transfer function Vout(s)/Vin(s) of the following electrical
network
Vin
C
R
L
Vout
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24. Example#5
• Find transfer function Vout(s)/Vin(s) of the following electrical
network
C1
R
Vin
C3
L
Vout
C2
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25. Part-II
ELECTRONIC SYSTEMS
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26. Operational Amplifiers
V out
Z2
V out
V in
Z1
V in
1
Z2
Z1
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27. Example#6
• Find out the transfer function of the following
circuit.
V out
Z2
V in
Z1
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28. Example#7
• Find out the transfer function of the following
circuit.
v1
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29. Example#8
• Find out the transfer function of the following
circuit.
v1
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30. Example#9
• Find out the transfer function of the following
circuit and draw the pole zero map.
100kΩ
10kΩ
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31. To download this lecture visit
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END OF LECTURES-6
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