control system engineering Electromechanical system and Thermal solutions

1. Topics: Mathematical Modeling of
Control System Engineering
PE-3032
Prof. CHARLTON S. INAO
Defence University
College of Engineering
Debre Zeit , Ethiopia

2. Electromechanical System
Transfer Functions
• Systems that are hybrids of electrical and
mechanical variables is the electromechanical
systems.
• Other applications for systems with
electromechanical components are robot
controls, sun and star trackers, and computer
tape and disk-drive position controls.

3. • An example of a control system that uses
electromechanical components is shown in Figure 1
NASA flight simulator robot arm with electromechanical control system components.

4. Potentiometer

5. Eq. 1

6. Motors

7. Motor Construction

9. DC Motor Modeling
From W. Bolton

11. Eq. 2
Eq. 3

12. Eq. 4

14. Eq. 5

16. Eq. 6

17. Eq. 7

19. Eq. 8

20. DC Motor
Modeling
Prof. Charlton Inao
Electromechanical Modeling

22. DC Motor

23. Where
Θ= angular displacement
Vf=voltage input at the field circuit
Km= motor constant in Nm/A
J=moment of inertia=I
b=rotary viscous coefficient at the bearing and shaft
Lf=inductance in the field circuit
Rf=resistance in the field circuit
s=laplace operator =d/dt

25. Figure 1

26. If(s)(L f s + R f)=U(s)
If(s)(=U(s)/ (L f s + R f)

27. voltage

29. 77

30. Figure 1

35. Modeling
of
Thermal Systems

36. Mode of Heat Transfer
• Conduction
• Convection
• Radiation

37. Conduction
Conduction is the transfer of energy from the more energetic
particles of a substance to the adjacent less energetic ones as a
result of interactions between the particles. Conduction can take
place in solids, liquids, or gases.
In gases and liquids, conduction is due to the collisions and
diffusion of the molecules during their random motion. In solids, it
is due to the combination of vibrations of the molecules in a lattice
and the energy transport by free electrons.
A cold canned drink in a warm room, for example, eventually
warms up to the room temperature as a result of heat transfer
from the room to the drink through the aluminum can by
conduction.

39. Thermal Conductivities

40. CONVECTION
Convection is the mode of energy transfer between a solid surface
and the adjacent liquid or gas that is in motion, and it involves the
combined effects of conduction and fluid motion. The faster the
fluid motion, the greater the convection heat transfer.

41. RADIATION
Radiation is the energy emitted by
matter in the form of
electromagnetic waves (or
photons) as a result of the changes
in the electronic configurations of
the atoms or molecules. Unlike
conduction and convection, the
transfer of energy by radiation
does not require the presence of
an intervening medium. In fact,
energy transfer by radiation is
fastest (at the speed of light) and it
suffers no attenuation in a vacuum.
This is how the energy of the sun
reaches the earth.

43. 1 Thermal resistance
The thermal resistance R is the resistance
offered to the rate of flow of heat q (Figure a)
and is defined by:
where T1- T2 is the temperature difference
through which the heat flows.
R=L/kA, oC/W

45. Application Example of Thermal
Resistance
Sample No.1

46. Application Example Thermal
ResistanceSample No.2

48. 2 Thermal capacitance
The thermal capacitance (Figure b) is a measure of the store of
internal energy in a system. If the rate of flow of heat into a
system is q1 and the rate of flow out q2 then the rate of change of
internal energy of the system is q1 - q2. An increase in internal
energy can result in a change in temperature:
change in internal energy = mc x change in temperature
where m is the mass and c the specific heat capacity. Thus the
rate of change of internal energy is equal to mc times the rate of
change of temperature. Hence:
This equation can
be written as:
where the capacitance C = mc.

49. Example
Develop a model for the
simple thermal system of a
thermometer at
temperature T being used
to measure the
temperature of a liquid
when it suddenly changes
to the higher temperature
of TL(Figure ).

52. Example
• Determine a model for
the temperature of a
room (Figure) containing
a heater which supplies
heat at the rate q1 and
the room loses heat at the
rate q2.

55. Thermal Modeling
• The figure shows a thermal system involving two compartments with one
containing a heater. The temperature of the compartment containing the
heater is T1, the temperature of the other compartment is T2 and the
temperature surrounding the compartment is T3, develop equations how
temperatures T1 and T2 will vary with time.
• All the walls of the containers have the same resistance and negligible
capacity. The two containers have the same capacity C. Also R1=R2=R
Compartment 1 :RC(dT1/dt)=
Rq-2T1 +T2 + T3
Compartment 2: RC(dT2/dt)
=T1-2T2 + T3
Answer

56. Nomenclature/Given Data
T1=temperature at chamber 1
T2=temperature at chamber 2
T3=outside temperature of the chambers
q=heat from the heater located at chamber 1
q 1=heat coming out from chamber 1
q2=heat at chamber 2
q3= heat coming out from chamber 2
R=R1=R2

57. Solution
@ the first chamber
• q-q1-q2= Cd/dtT1
But let us define first q1 and q2 in terms of
thermal resistance q=deltaT/R.
Therefore q1=T1-T3/R
And q2=T1-t2/R then substitute from the first
equation
Q-(T1-T3/R) – (T1-T2/R)= Cd/dtT1

58. • Multiply both sides by R so that R will be taken
away from denominator;
• Rq – T1 + T3-T1+T2=RCd/dtT1
RCd/dtT1=Rq-2T1 +T2+T3

59. @ the second chamber
From thermal resistance and thermal
capacitance equation;
q2- q3=Cd/dtT2 (thermal capacitance)
Substituting values of q1 and q2 in terms of
thermal resistance, we have:
Q2=T1-T2/R and Q3=T2-T3/R; therefore
Ti-T2/R-(T2-T3/R) =Cd/dt T2
T1-T2-T2+T3/R = Cd/dt T2
T1-2T2+T3=RCd/dtT2 or
RCd/dtT2=T1-2T2+T3