1) The document discusses modeling systems in the frequency domain using Laplace transforms. It reviews key concepts like the Laplace transform, transfer functions, and partial fraction expansions. 2) Methods for deriving transfer functions for various systems are presented, including electrical circuits using mesh/nodal analysis, mechanical translational/rotational systems using Newton's laws, and electromechanical systems using motor dynamics. 3) The document also covers modeling nonlinearities through linearization and using transfer functions to represent interconnected subsystems like those with gear trains.

1. LCS
MARYAM IQBAL
ASSISTANT PROFESSOR
EE DEPARTMENT
BUIC

2. MODELING IN THE
F
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E
Q
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C
Y
D
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1
Review of the Laplace transform
Learn how to find a mathematical model,
called a transfer function, for linear, time-
invariant electrical, mechanical, and
electromechanical systems
Learn how to linearize a nonlinear system in
order to find the transfer function

3. 2
INTRODUCTION TO
MODELING
• we look for a mathematical representation where the input, output, and
system are distinct and separate parts
• also, we would like to represent conveniently the interconnection of
several subsystems. For example, we would like to represent cascaded
interconnections, where a mathematical function, called a transfer
function, is inside each block

4. 3
LAPLACE TRANSFORM REVIEW
where s =  + j, is a complex variable.
• a system represented by a differential equation is difficult to model
as a block diagram
• on the other hand, a system represented by a Laplace transformed
differential equation is easier to model as a block diagram
The Laplace transform is defined as

5. 4
LAPLACE TRANSFORM REVIEW
The inverse Laplace transform, which allows us
to find f (t) given F(s), is
where

6. 5
Laplace transform table
LAPLACE TRANSFORM
REVIEW

7. 6
Problem: Find the Laplace transform of
Solution:

8. 7
LAPLACE TRANSFORM
REVIEW
Laplace transform theorems

9. 8
LAPLACE TRANSFORM
REVIEW
Laplace transform theorems

10. 9
PARTIAL-FRACTION
EXPANSION
• To find the inverse Laplace transform of a complicated function,
we can convert the function to a sum of simpler terms for which we
know the Laplace transform of each term.
Case 1: Roots of the Denominator of F(s) Are Real and Distinct
Case 2: Roots of the Denominator of F(s) Are Real and Repeated
Case 3: Roots of the Denominator of F(s) Are Complex or Imaginary
F(s) 
N(s)
D(s)

11. 10
CASE 1: ROOTS OF THE DENOMINATOR OF F(S) ARE
REALAND DISTINCT
If the order of N(s) is less than the order of D(s), then
Thus, if we want to find Km, we multiply above equation by (s  pm )

12. 11
IF WE SUBSTITUTE S   PM IN THE ABOVE EQUATION,
THEN WE CAN FIND KM

13. 12
Problem Given the following differential equation, solve for y(t) if
all initial conditions are zero. Use the Laplace transform.
Solution

14. 13

15. 14
CASE 2: ROOTS OF THE DENOMINATOR OF F(S) ARE
REALAND REPEATED
First, we multiply by and we can solve immediately for K1 if s= - p1
1
(sp)r

16. 15
• we can solve for K2 if we differentiate F1(s) with respect to s and
then let s approach –p1
• subsequent differentiation allows to find K3 throughKr
• the general expression for K1 through Kr for the multiple roots is

18. 16
Case 3: Roots of the Denominator of F(s) Are Complex or Imaginary
(K2 s  K3 )
is put in the form of by completing
(s2
 as b)
the squares on (s2
 as b) and adjusting the numerator
has complex or pure imaginary roots
• the coefficients K2, K3 are found through balancing the coefficients
of the equation after clearing fractions

20. 18
THE TRANSFER
FUNCTION (1)
Let us consider a general nth-order, linear, time-invariant differential
equation is given as:
where c(t) is the output, r(t) is the input, and ai, bi are coefficients.
Taking the Laplace transform of both sides (assuming all initial
conditions are zero) we obtain

21. 19
THE TRANSFER
FUNCTION (2)
Transfer function is the ratio G(s) of the output transform, C(s),
divided by the input transform, R(s)
m m1
C(s)
 G(s) 
(bms  bm1s ... b0)
R(s) (ansn
 an1sn1
 ... a0 )

22. 20
ELECTRIC NETWORK TRANSFER
FUNCTIONS
Voltage-current, voltage-charge, and impedance relationships for capacitors,
resistors, and inductors

23. 21
Problem Find the transfer function relating the capacitor voltage,
Vc(s), to the input voltage, V(s)
Solution:

24. 22

25. 23
TRANSFER FUNCTION-SINGLE LOOP VIA
TRANSFORM METHODS

26. 24
COMPLEX ELECTRIC CIRCUITS VIA MESH
ANALYSIS
1. Replace passive element values with their impedances.
2. Replace all sources and time variables with their Laplace
transform.
3. Assume a transform current and a current direction in each
mesh.
4. Write Kirchhoff's voltage law around each mesh.
5. Solve the simultaneous equations for the output.
6. Form the transfer function

27. 25
PROBLEM: FIND THE TRANSFER
FUNCTION, I2(S)/V(S)

28. 26

29. 27
COMPLEX ELECTRIC CIRCUITS VIA NODAL
ANALYSIS
1. Replace passive element values with their admittances
2. Replace all sources and time variables with their Laplace
transform.
3. Replace transformed voltage sources with transformed current
sources.
4. Write Kirchhoff's current law at each node.
5. Solve the simultaneous equations for the output.
6. Form the transfer function.

30. 28
OPERATIONAL
AMPLIFIERS
Inverting OPAMP

31. 30
EXAMP
LE

32. 29
NON-INVERTING OP
AMP
Example

33. 1 𝑧2 =
1
𝑅2
+ 𝑠𝐶2
𝑧2 =
1
1
𝑅2
+ 𝑠𝐶2

35. 31
TRANSLATIONAL MECHANICAL SYSTEM
TRANSFER FUNCTIONS
Newton's law: The sum of forces on a body equals zero; the sum of
moments on a body equals zero.
K… spring
constant
fv … coefficient of
viscious friction
M … mass

36. 32

37. 33
Problem Find the transfer function, X(s)/F(s), for the system
Solution:

38. 34
TRANSLATIONAL MECHANICAL SYSTEM
TRANSFER FUNCTIONS
• The required number of equations of motion is equal to the
number of linearly independent motions. Linear independence
implies that a point of motion in a system can still move if all
other points of motion are held still.
• Another name for the number of linearly independent motions
is the number of degrees of freedom.

39. 35
PROBLEM: FIND THE TRANSFER FUNCTION,
X2(S)/F(S), OF THE SYSTEM
Solution
• we draw the free-body diagram for each point of motion and then use
superposition
• for each free-body diagram we begin by holding all other points of
motion still and finding the forces acting on the body due only to its
own motion.
• then we hold the body still and activate the other points of motion one
at a time, placing on the original body the forces created by the
adjacent motion

40. C
36
a. Forces on M1 due only
to motion of M1
b. forces on M1 due only to
motion of M2
ontrol Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
c. all forces on M1

41. 37
a. Forces on M2 due only
to motion of M2;
b. forces on M2 due only
to motion
c. all forces on M2

42. 38

43. 39

44. 40
ROTATIONAL MECHANICAL SYSTEM
TRANSFER
FUNCTIONS
Torque-angular velocity, torque-angular displacement, and
impedance rotational relationships for springs, viscous
dampers, and inertia
• the mass is replaced by inertia
• the values of K, D, and J are
called spring constant, coefficient
of viscous friction, and moment of
inertia, respectively
• the concept of degrees of freedom
is the same as for translational
movement

45. 41
PROBLEM FIND THE
TRANSFER FUNCTION
Solution
a. Torques on J1
due only to the
motion of J1

46. 42
b. torques on J1
due only to the
motion of J2
c. final free-body
diagram for J1

47. 43
a.Torques on J1
due only to the
motion of J1
b.torques on J1
due only to the
motion of J2
c.final free-body
diagram for J1

48. 44
a.Torques on J2
due only to the
motion of J2
b.torques on J2
due only to the
motion of J1
c.final free-body
diagram for J2

49. 45

50. 46

51. 47
TRANSFER FUNCTIONS FOR SYSTEMS
WITH GEARS
- the distance travelled along each gear's circumference is the same
2

r1

N1
1 r2 N2
- the energy into Gear 1 equals the energy out of Gear 2
T2

1

N2
T1 2 N1

52. 48
T2

1

N2
T1 2 N1

53. 49
ROTATIONAL MECHANICAL IMPEDANCES CAN BE
REFLECTED THROUGH GEAR TRAINS BY MULTIPLYING
THE MECHANICAL IMPEDANCE BY THE RATIO

54. 50
A GEAR TRAIN IS USED TO IMPLEMENT LARGE GEAR
RATIOS BY CASCADING SMALLER GEAR RATIOS.

55. 51
Problem Find the transfer function,
Solution:

56. 52
ELECTROMECHANICAL SYSTEM TRANSFER
FUNCTIONS
Electromechanical systems: robots, hard disk drives, …
Motor is an electromechanical component that yields a displacement output
for a voltage input, that is, a mechanical output generated by an electrical
input.

57. 53
where Kb is a constant of proportionality called the back emf constant;
is the angular velocity of the motor
dm / dt  m (t)
DERIVATION OF THE TRANSFER
FUNCTION OF MOTOR
Back electromotive force (back emf):
Taking the Laplace transform, we get:
Torque:

58. 54
THE RELATIONSHIP BETWEEN THE ARMATURE CURRENT, IA(T), THE
APPLIED ARMATURE VOLTAGE,
EA(T), AND THE BACK EMF, VB(T),
t
m
a
K
T (s)
I (s) 

59. 55
We assume that the armature inductance, La, is small compared to the armature
resistance, Ra,

60. 56
The mechanical constants Jm and Dm can be found as:
Electrical constants Kt/Ra and Kb can be found through a dynamometer test of
the motor.

61. 57
Kt
Ra ea

Tstall
b
ea
noload
K 


62. 58
NONLINEARITIES
• A LINEAR SYSTEM POSSESSES TWO PROPERTIES: SUPERPOSITION AND
HOMOGENEITY.
• SUPERPOSITION - THE OUTPUT RESPONSE OF A SYSTEM TO
THE SUM OF INPUTS IS THE SUM OF THE RESPONSES TO THE
INDIVIDUAL INPUTS.
• HOMOGENEITY - DESCRIBES THE RESPONSE OF THE SYSTEM TO
A MULTIPLICATION OF THE INPUT BY A SCALAR.
MULTIPLICATION OF AN INPUT BY A SCALAR YIELDS A RESPONSE
THAT IS MULTIPLIED BY THE SAME SCALAR.
Linear system Nonlinear system

63. 59
NONLINE
ARITIES

64. 60
LINEARIZATI
ON
• if any nonlinear components are present, we must linearize the system
before we can find the transfer function
• we linearize nonlinear differential equation for small-signal inputs about
the steady-state solution when the small signal input is equal to zero
ma is the slope of the curve at pointA

65. 61
Solution:
At 2 df/dx = - 5