Modeling of Electrical Elements g ```NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Modeling
g of Electrical Elements
D Bi
Dr.
Bishakh
h kh Bh
Bhattacharya
tt h
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
Module 1- Lecture 7
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
This Lecture Contains
 Modeling of electrical elements
Kirchoff’s Laws and modeling electrical circuit
A
A multi‐loop system
lil
Modeling of a DC servomotor
Joint Initiative of IITs and IISc - Funded by MHRD
Module 1- Lecture 7
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 1- Lecture 7
Modeling
g of Electrical Elements
In the last lecture we have discussed about modeling of dynamic mechanical
systems. However, today you will hardly find dynamic systems which are purely
mechanical in nature. There will be invariably electrical systems coupled with the
mechanical elements. Similar to mechanical systems electrical systems consist of
three basic electrical elements
elements. These are: Resistors (R)
(R), Capacitors (C) and
Inductors (L)
Resistors are electric elements for which voltage
g across it is proportional
p p
to the
current passing through it. The constant of proportionality is known as Resistance.
For a wire of length ‘l’, cross-sectional area –’A’ and resistivity – ‘ρ’:
R
Joint Initiative of IITs and IISc ‐ Funded by MHRD
l
A
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 1- Lecture 7
Electrical Elements – Resistors
Resistors are dissipative in nature. Energy (E) dissipated from a resistor (R) could
be modeled using ‘Joule Effect’. Accordingly:
2
V
E  i2R t 
t
R
where ‘i’ is the current p
passed through
g the resistor for a time ‘t’ and ‘V’ is the
applied Voltage. Figure below shows different types of resistors.
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 1- Lecture 7
Electrical Elements: Capacitors
p
Capacitors are electrical elements used to store the electrostatic energy. Voltage
(V) across a capacitor of capacitance (C) changes according to the following
equation:
t
1
V (t )   i dt
C0
Th energy stored
The
t
d in
i a capacitor
it could
ld be
b expressed
d as:
1
E  CV 2
2
Capacitance of commonly used configuration of a capacitor and the corresponding
capacitance is shown below:
oint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 1- Lecture 7
Electrical Elements: Inductors
Inductors refer to the coiled conductors where a variable current generates
voltage, which, for a linear inductor is proportional to the current rate.
Accordingly, the voltage (V) is given by:
V (t )  L
d i (t )
dt
Where, L refers to the inductance of the coil. For a cylindrical coil of diameter d
length
g l,, no. of turns n,, and magnetic
g
permeability
p
y µ, the inductance is given
g
by:
y
L
  n2 d 2
4l
The magnetic energy stored by an inductor is:
1
E  L i 2 (t )
2
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 1- Lecture 7
Modeling of Electrical Circuits
When the electrical elements get connected into a circuit, the dynamic behavior
of the electrical system could be modeled by using the following governing laws:
•
Kirchoff’s Current Law: The algebraic sum of currents leaving a junction or
node equals the algebraic sum of currents entering the node.
•
Kirchoff s Voltage Law: The algebraic sum of all voltages taken around a closed
Kirchoff’s
path in a circuit is zero.
Parameters
Mechanical
Electrical
Static storage of Static
storage of
Energy Spring
Capacitor
Dynamic Storage of Inertia
gy
Energy
Inductor
Energy Dissipation
Dashpot/Damper
Resistor
Excitation
Force
Voltage
R
Response
Di l
Displacement
t
Ch
Charge
Joint Initiative of IITs and IISc ‐ Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Modeling
g of a LRC Circuit
The governing equation of the electrical circuit could be obtained using Kirchoff ‘s Laws and is given by:
d q (t )
dt
d 2 q (t )
d q (t ) 1
V (t )  L

R
 q
d t2
dt
C
i (t ) 
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 1- Lecture 7
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Impedance
p
based representation
p
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 1- Lecture 7
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
A multi-loop presentation
Similar to multiple DOF mechanical systems, one can have multi‐loop electrical circuits. Figure below shows a multi‐loop electrical circuit in time and frequency domain and the corresponding transfer function for the system
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 1- Lecture 7
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Assignment: How to Model a DC Servomotor?
The example below shows an electromechanical system. Find out the
governing equations of the system (electrical and mechanical).
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 1- Lecture 7
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Hints: Development
p
of state-space
p
representation
p
•
Torque developed by the motor Tm = Kt im
•
Back emf em = Kb dΘ/dt
•
Kirchoff’s Law: ea = Lm dim/dt + Rm im + em
•
Force Balance: Tm – Tl = J d2Θ/dt2 + B dΘ/dt
•
Take states as x1 = Θ, x2 = dΘ/dt, x3=im, Obtain EOM in state-space form
Joint Initiative of IITs and IISc ‐ Funded by MHRD
Module 1- Lecture 7
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System
Module 1- Lecture 7
Special
p
References for this lecture:
 System Dynamics for Engineering Students: Nicolae Lobontiu,
Publisher
 Feedback Control of Dynamic Systems
Systems, Frankline,
Frankline Powell and Emami
Emami-Naeini,
Naeini
Pearson Publisher
 Control
C
lS
Systems Engineering,
E i
i
N
Norman
S Ni
Nise, J
John
h Wil
Wiley & Sons
S
 Systems Dynamics and Response, S. Graham Kelly, Thomson Publisher
Joint Initiative of IITs and IISc ‐ Funded by MHRD
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