mechatronics

1. Mechatronic Applications
EN 83002
Lecture 02
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2. Outline
• Basic System Models
• Mathematical Models
• Electrical System Building Blocks
• System Models
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3. Mathematical models
• A mathematical model of a system is a description of it in terms of
equations relating inputs and outputs so that outputs can be predicted
from inputs.
• In devising a mathematical model of a system it is necessary to make
assumptions and simplifications and a balance has to be chosen between
simplicity of the model and the need for it to represent the actual real-
world behavior.
• For example the rotation of a motor shaft controlled by a microcontroller.
The speed will not immediately assume the full-speed value but will only
attain that speed after some time
• Any mathematical model is provided by the fundamental physical laws that
govern the behavior of the system.
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4. System Building Blocks
• Systems can be made up from a range of building blocks.
• Each building block is considered to have a single property or
function.
• A system is called lumped parameter system if each parameter of
function is considered independently.
• Example- in an electrical system the basic building blocks can be combined in
different ways to make a variety of electric circuit systems.
• Overall input/output relationships of the electrical system can be obtained by
combining in an appropriate way, the relationships for the building blocks.
• Thus a mathematical model for the system can be obtained.
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5. Electrical System Building Blocks
• The basic building blocks of electrical systems are inductors,
capacitors and resistors.
• Assumptions – resistor building block is assumed to have purely the
properties of resistance, the capacitor purely that of capacitance and
the inductor purely that of inductance.
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6. Inductor
• For an inductor the potential difference 𝑣 across it any instant
depends on the rate of change of current (di/dt) through it.
𝑣 = 𝐿
𝑑𝑖
𝑑𝑡
L is the inductance.
The direction of the potential difference is in the opposite direction to
the potential difference used to drive the current through the inductor,
hence the term back e.m.f. is used.
• The equation can be rearranged to give :
𝑖 =
1
𝐿
‫׬‬ 𝑣 𝑑𝑡
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7. Capacitor
• For a capacitor , the potential difference across it depends on the charge 𝑞 on the capacitor plates at the
instant concerned.
𝑣 =
𝑞
𝐶
C is the capacitance.
• The current 𝑖 to or from the capacitor is the rate at which charge moves to or from the capacitor plates.
𝑖 =
𝑑𝑞
𝑑𝑡
Total charge 𝑞 on the plates is :
𝑞 = ‫׬‬ 𝑖 𝑑𝑡
So
𝑣 =
1
𝐶
‫׬‬ 𝑖 𝑑𝑡
Alternatively since 𝑣 = 𝑞/C then,
𝑑𝑣
𝑑𝑡
=
1
𝐶
𝑑𝑞
𝑑𝑡
=
1
𝐶
𝑖
And so
𝑖 = 𝐶
𝑑𝑣
𝑑𝑡
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8. Resistor
• For a resistor, the potential difference 𝑣 across it at any instant depends on the current 𝑖 through it :
𝑣 = R𝑖 R is the resistance.
• Both the inductor and capacitor store energy which can be released at a later time. A resistor just
dissipates it.
• The energy stored by an inductor when there is current 𝑖 is :
𝐸 =
1
2
𝐿𝑖2
• The energy stored by a capacitor when there is a potential difference 𝑣 across it is :
𝐸 =
1
2
𝐶𝑣2
• The power 𝑃 dissipated by a resistor when there is a potential difference 𝑣 across it is :
𝑃 = 𝑖𝑣 =
𝑣2
𝑅
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9. Summary
Summary of the equations defining the characteristics of the electrical building blocks when the input
is current and the output is potential difference.
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10. Building up a model for an electrical system
• The electrical building blocks can be combined using Kirchhoff’s Law
as outlined below:
• Law 1: the total current flowing towards a junction is equal to the total
current flowing from the junction, i.e. the algebraic sum of the currents at the
junction is zero.
• Law 2 : in a closed circuit of loop, the algebraic sum of the potential
difference across each part of the circuit is equal to the applied e.m.f.
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11. Example
• Consider a simple electrical system consisting of a resistor and capacitor in series.
Applying Kirchhoff’s second law to the circuit loop :
𝑣 = 𝑣𝑅 + 𝑣𝑐
Since this is just a single loop, the current i through all the circuit elements will
be the same.
If the output from the circuit is the potential difference across the capacitor:
Since 𝑣𝑅 = 𝑖𝑅 and 𝑖 = C (d𝑣𝑐/dt)
𝑣 = 𝑅𝐶
𝑑𝑣𝑐
𝑑𝑡
+ 𝑣𝑐
the relationship between the output 𝑣𝑐 and the input 𝑣 and is a
first-order differential equation.
Resistor–capacitor system
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12. Example
• Resistor-inductor-capacitor System.
If the output from the circuit is the potential difference across the capacitor, 𝑣𝑐 :
Applying Kirchhoff ’s second law to this circuit loop :
𝑣 = 𝑣𝑅 + 𝑣𝐿 + 𝑣𝑐
Since there is just a single loop, the current i will be the same through all circuit
elements.
𝑣 = 𝑖𝑅 + 𝐿
𝑑𝑖
𝑑𝑡
+ 𝑣𝑐
But i = C(d𝑣𝑐 /dt) and so
𝑑𝑖
𝑑𝑡
= 𝐶
𝑑( Τ
𝑑𝑣𝑐 𝑑𝑡)
𝑑𝑡
=
𝑑2𝑣𝑐
𝑑𝑡2
Hence
𝑣 = 𝑅𝐶
𝑑𝑣𝑐
𝑑𝑡
+ 𝐿𝐶
𝑑2𝑣𝑐
𝑑𝑡2 + 𝑣𝑐
This is a second-order differential equation 12

13. Example
• Resistor-inductor system
Relationship between the output, the potential difference
across the inductor of 𝑉𝐿 and input 𝑣.
Applying Kirchhoff ’s second law to the circuit loop gives,
𝑣 = 𝑣𝑅 + 𝑣𝐿
then the relationship between the input and output is
𝑣 =
𝑅
𝐿
න 𝑣𝐿𝑑𝑡 + 𝑣𝐿
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14. Example
• Resistor –capacitor-inductor system.
Consider the relationship between the output, the potential difference vC
across the capacitor and the input v .
Applying Kirchhoff ’s law 1 to node A gives
𝑖1 = 𝑖2 + 𝑖3
𝑖1 =
𝑣 − 𝑣𝐴
𝑅
𝑖2 =
1
𝐿
න 𝑣𝐴𝑑𝑡
𝑖3 = 𝐶
𝑑𝑣𝐴
𝑑𝑡
With some replacements and rearrangements ,the relationship between
the input and output is
𝑣 = 𝑅𝐶
𝑑𝑣𝐶
𝑑𝑡
+ 𝑣𝐶 +
𝑅
𝐿
න 𝑣𝐶 𝑑𝑡
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15. Non Electrical Systems
• There are other than electrical systems that can be mathematically
modeled.
• Mechanical Systems – translational, rotational
• Fluid (Hydraulic ) Systems
• Thermal Systems
• There are many elements in mechanical, electrical, fluid and thermal systems
which have similar behaviors. Thus, for example, mass in mechanical systems has
similar properties to capacitance in electrical systems, capacitance in fluid
systems and capacitance in thermal systems.
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16. 16

17. System Models
• Many systems encountered in engineering involve aspects of more
than just electrical systems.
• Building blocks of translational mechanical, rotational mechanical,
electrical, fluid and thermal systems may need to be considered.
• Example – Electric motor
• Single-discipline building blocks can be combined to give models for
multidiscipline systems.
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18. Electro-mechanical Systems
• Electromechanical devices transform electrical signals to rotational
motion or vice versa.
• Example – Potentiometers, Motors , Generators.
• Potentiometer – has input of a rotation and an output of a potential
difference.
• Electric motor – has input of a potential difference and output of
rotation of a shaft.
• Generator – has input of rotation of a shaft and an output of a
potential difference.
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19. Potentiometer
• The rotary potentiometer is a potential divider.
• The relationship can be given by:
𝑣𝑜
𝑉
=
𝜃
𝜃𝑚𝑎𝑥
• V – potential difference across the full length of the potentiometer track.
• 𝜃𝑚𝑎𝑥 - total angle swept out by the slider in being rotated from one end of the track to the other.
• 𝑣𝑜 - output for the input 𝜃.
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20. Direct Current Motor
• In a d.c. motor a current through the armature coil of the motor
results in a shaft being rotated and hence the load rotated.
Motor driving a load One wire of armature coil
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21. Direct Current Motor
Direct current motor circuits
The motor basically consists of a coil, the
armature coil, which is free to rotate. This
coil is located in the magnetic field
provided by a current through field coils or
a permanent magnet.
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22. Direct Current Motors
System model for armature-controlled d.c. motor.
Armature is a coil rotating in a magnetic field, a voltage will be induced in it as a consequence of
electromagnetic induction. This voltage will be in such a direction as to oppose the change producing it and is
called the back e.m.f.
Two equations that describe the conditions occurring for an armature-controlled motor are:
𝑣𝑎 − 𝑘3𝜔 = 𝐿𝑎
𝑑𝑖𝑎
𝑑𝑡
+ 𝑅𝑎𝑖𝑎 and 𝐼
𝑑𝜔
𝑑𝑡
= 𝑘4𝑖𝑎 − 𝑐𝜔
The equation relating to the output 𝜔 with the input 𝑣𝑎 to the system by eliminating 𝑖𝑎 using Laplace
transform.
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23. Direct Current Motors
System model for field-controlled d.c. motor.
Here armature current is held constant and the motor is controlled by varying field voltage.
The conditions occurring for a field-controlled motor are thus described by the equations :
𝑣𝑓 = 𝐿𝑓
𝑑𝑖𝑓
𝑑𝑡
+ 𝑅𝑓𝑖𝑓 and 𝐼
𝑑𝜔
𝑑𝑡
= 𝑘5𝑖𝑓 − 𝑐𝜔
The equation relating the output 𝜔 with the input 𝑣𝑓 to the system can be obtained by eliminating 𝑖𝑓
using Laplace transform.
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24. Linearity
• Often real components are not linear.
• Non-linear models are much more difficult to deal with and non-
linear systems might be approximated by a linear model.
• Example – A simple model for a spring assumes that force and extensions are
proportional regardless of how large force was. Thus the mathematical model
can be a simplification of a real spring.
• In combining blocks to create models of systems it is assumed that
the relationship for each block is linear.
• It is required to make a linear approximation for a non-linear item.
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25. Linearity
• A system needs to satisfy Principle of superposition for it to
be termed as linear system.
• Example if an ideal spring system is given by F = kx, then if force
F1 produces extension x1 and force F2 produces extension x2 ,
then force equal to (F1+F2) will produce an extension (x1+x2).
• If force F1 produces an extension x1, then an input cF1 will
produce an output cx1 ,where c is a constant multiplier
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26. Linearity
Ideal springs. Real springs.
For many system
components, linearity
can be assumed for
operations within a
range of values of the
variable about some
operating point
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27. Linearity
Non-linear relationship.
For such components the best
that can be done to obtain a
linear relationship is just to work
with the straight line which is the
slope of the graph at the
operating point.
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