Dr' B'K' Sridhara, Head, Department of Mechanical Engineering, NIE, Mysore

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Welcome to e-learning session onCONTROL
ENGINEERING (ME 55)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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ByDr. B.K. Sridhara HeadDepartment of
Mechanical EngineeringThe National Institute of
EngineeringMysore 570 008
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Session 6 CHAPTER II MATHEMATICAL
MODELING (Continued)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Recap of Session V

• Models for system with more than one mass
• Both Translational and Rotational
• Geared systems
• Equivalent Inertia, Equivalent Stiffness and
  Equivalent Damping
• Equivalence of Rotational / Translational systems

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematical Modeling of Electrical Systems

• Represented by Three Elements
• Resistance
• Inductance
• Capacitance

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Fundamental Law Kirchoffs Voltage
Law Algebraic Sum of Voltages around a closed
Loop is Zero
?e 0
The applied voltage is equal to the sum of the
voltages around the circuit
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Resistance
Idealized electrical element that represents the
resistance offered to the flow of current
o
e
e i R --- (a)
i
R
o
R Resistance in Ohms
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Inductance
It is the idealized electrical circuit element
that represents storage of energy in the magnetic
field associated with the field due to moving
charges (current)
e L di/dt L Inductance in Henrys
o
e
i
L
--- (b)
o
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Capacitance
It is the idealized electrical circuit element
that represents the storage of energy in the
electric field associated with electric charges
(static)
o
e
i
C
o
--- (c)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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R-L Circuit
Apply KVL ?e 0 e eR eL 0
Example 1
R
o
e
i
L
o
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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R -L -C Circuit
e iR
Example 2
R
L
e L.dt/dt
o
e
i
i
C
i
e 1/C ? i.dt
o
Apply KVL ?e 0 e eR eL eC 0
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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q charge in Coulombs
But
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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is the Governing differential equation
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Example 3
Obtain the governing differential equation
assuming a high impedance at the output. Also
establish the relation between the input voltage
ei and the output voltage e0
R
o
o
C
i1
e0
ei
i
i1
i1
o
o
Loop (1)
Loop (2)
Apply KVL for Loop 1 ?e 0 ei eR eC 0
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Also
i i1 i2
Due to high Impendence at the output
?i i1
--- (1)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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R
o
o
C
i1
e0
ei
i
i1
i1
o
o
Loop (1)
Loop (2)
Apply KVL for Loop 2 ?e 0 e0 eC 0
--- (2)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematical Models
--- (1)
--- (2)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Relation between the input voltage ei and the
output voltage e0
adopt operator notation
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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?From equation (1)
--- (1)
On simplification
ei
--- (a)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Similarly equation 2 becomes
--- (b)
e0
Now divide Equation (b) by (a) We get

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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?RCP
Governing differential equation
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Example 4
Obtain the governing differential equation for
the given circuit and establish the relation
between the input voltage ei and the output
voltage e0
L
R
o
o
i1
i2
i1
e0
ei
i1
C
i2
o
o
Loop (1)
Loop (2)
We can have i i1 i2 (Kirchoffs current
law)
Due to high impendence at the output
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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R
L
o
o
i1
i2
i1
e0
ei
i1
C
i2
o
o
Loop (1)
Loop (2)
Apply KVL for Loop 1 ?e 0 ei eR eL - eC
0 ei eR eL eC
--- (1)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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R
L
o
o
i1
i2
i1
e0
ei
i1
C
i2
o
o
Loop (1)
Loop (2)
Apply KVL for Loop 2 ?e 0 e0 eC 0 e0 eC
--- (2)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematical Models
--- (1)
--- (2)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Relation between the input voltage ei and the
output voltage e0
Using operator notation
--- (a)
--- (b)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Now divide Equation (b) by (a) We get
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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?e0
Governing differential equation
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Analogous Systems
..
.
C
K
M
F
..
.
kt
b
J
T
R
L
o
..
.
e
i
i
C
i
o
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Mathematical Models are identical Such systems
are Analogous systems Force / Torque is
Analogous to voltage Analog concepts Analysis
simple Having solution for one system solution
for others can be obtained by Analogy Models for
Electrical systems are easy to obtain
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Network approach Equivalent electrical systems
are considered to get GDE and the behaviour of
the system is obtained by a analogy
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Analogous Quantities
.
.
.
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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Summary

• Mathematical Model for Electrical Systems
• Resistance, Capacitance and Inductance
• R-L Circuit, R-L-C Circuit
• Analogous System and Network Approach

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
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THANK YOU
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore