1. Mechatronics Engineering
MT-144
NETWORK ANALYSIS
Mechatronics Engineering
(12) 1

2. Laplace Transform

3. Definition of Laplace Transform
The Laplace Transform is an integral transformation of a function f(t) from the time domain into the complex frequency domain, giving F(s)
s: complex frequency
Called “The One-sided or unilateral Laplace Transform”.
In the two-sided or bilateral LT, the lower limit is -. We do not use this.

4. Definition of Laplace Transform
Example 1
Determine the Laplace transform of each of the following functions shown below:

5. Definition of Laplace Transform
Solution:
The Laplace Transform of unit step, u(t) is given by

6. Definition of Laplace Transform
Solution:
The Laplace Transform of exponential function, e-atu(t),a>0 is given by

7. Definition of Laplace Transform
Solution:
The Laplace Transform of impulse function,
δ(t) is given by

8. Functional Transform

9. TYPE
f(t)
F(s)
Impulse
Step
Ramp
Exponential
Sine
Cosine

10. TYPE
f(t)
F(s)
Damped ramp
Damped sine
Damped cosine

11. Properties of Laplace Transform
Step Function
The symbol for the step function is K u(t).
Mathematical definition of the step function:

12. f(t) = K u(t)

13. Properties of Laplace Transform
Step Function
A discontinuity of the step function may occur at some time other than t=0.
A step that occurs at t=a is expressed as:

14. f(t) = K u(t-a)

15. Ex:

16. Three linear functions at t=0, t=1, t=3, and t=4
Y=mx+c

17. Expression of step functions
Linear function +2t: on at t=0, off at t=1
Linear function -2t+4: on at t=1, off at t=3
Linear function +2t-8: on at t=3, off at t=4
Step function can be used to turn on and
turn off these functions

18. Step Functions
ON OFF

19. Properties of Laplace Transform
Impulse Function
The symbol for the impulse function is (t).
Mathematical definition of the impulse function:

20. Properties of Laplace Transform
Impulse Function
The area under the impulse function is constant and represents the strength of the impulse.
The impulse is zero everywhere except at t=0.
An impulse that occurs at t = a is denoted K (t-a)

21. f(t) = K (t)

22. Properties of Laplace Transform
Linearity
If F1(s) and F2(s) are, respectively, the Laplace Transforms of f1(t) and f2(t)
Example:

23. Properties of Laplace Transform
Scaling
If F (s) is the Laplace Transforms of f (t), then
Example:

24. Properties of Laplace Transform
Time Shift
If F (s) is the Laplace Transforms of f (t), then
Example:

25. Properties of Laplace Transform
Frequency Shift
If F (s) is the Laplace Transforms of f (t), then
Example:

26. Properties of Laplace Transform
Time Differentiation
If F (s) is the Laplace Transforms of f (t), then the Laplace Transform of its derivative is
Example:

27. Properties of Laplace Transform
Time Integration
If F (s) is the Laplace Transforms of f (t), then the Laplace Transform of its integral is
Example:

28. Properties of Laplace Transform
Frequency Differentiation
If F(s) is the Laplace Transforms of f (t), then the derivative with respect to s, is
Example:

29. Properties of Laplace Transform
Initial and Final Values
The initial-value and final-value properties allow us to find the initial value f(0) and f(∞) of f(t) directly from its Laplace transform F(s).
Initial-value theorem
Final-value theorem

30. The Inverse Laplace Transform
Suppose F(s) has the general form of
The finding the inverse Laplace transform of F(s) involves two steps:
Decompose F(s) into simple terms using partial fraction expansion.
Find the inverse of each term by matching entries in Laplace Transform Table.

31. The Inverse Laplace Transform
Example 1
Find the inverse Laplace transform of
Solution:

32. Partial Fraction Expansion
Distinct Real Roots of D(s)
s1= 0, s2= -8
s3= -6

33. 1) Distinct Real Roots
To find K1: multiply both sides by s and evaluates both sides at s=0
To find K2: multiply both sides by s+8 and evaluates both sides at s=-8
To find K3: multiply both sides by s+6 and evaluates both sides at s=-6

34. Find K1

35. Find K2

36. Find K3

37. Inverse Laplace of F(s)

38. 2) Distinct Complex Roots
S2 = -3+j4
S3 = -3-j4

39. Partial Fraction Expansion
Complex roots appears in conjugate pairs.

40. Find K1

41. Find K2 and K2* Coefficients associated with conjugate
pairs are themselves conjugates.

42. Inverse Laplace of F(s)

43. Inverse Laplace of F(s)

44. Useful Transform Pairs

45. The Convolution Integral
It is defined as
Given two functions, f1(t) and f2(t) with Laplace Transforms F1(s) and F2(s), respectively
Example:

46. Operational Transform

47. Operational Transforms
Indicate how mathematical operations performed on either f(t) or F(s) are converted into the opposite domain.
The operations of primary interest are:
Multiplying by a constant
Addition/subtraction
Differentiation
Integration
Translation in the time domain
Translation in the frequency domain
Scale changing

48. OPERATION
f(t)
F(s)
Multiplication by a constant
Addition/Subtraction
First derivative (time)
Second derivative (time)

49. OPERATION
f(t)
F(s)
n th derivative (time)
Time integral
Translation in time
Translation in frequency

50. OPERATION
f(t)
F(s)
Scale changing
First derivative (s)
n th derivative
s integral

51. Translation in time domain
If we start with any function:
we can represent the same function translated in time by the constant a, as:
In frequency domain:

52. Ex:

53. Translation in frequency domain
Translation in the frequency domain is defined as:

54. Ex:

55. Ex:

56. APPLICATION

57. Problem
Assumed no initial energy is stored in the circuit at the instant when the switch is opened.
Find the time domain expression for v(t) when t≥0.

58. Integrodifferential Equation
A single node voltage equation:

59. s-domain transformation=0

61. Ex Obtain the Laplace transform for the function below: 1 2 3 t h(t) 41 2 3 t
h(t) 4 5

62. Find the expression of f(t):
Expression for the ramp function with slope, m =2 and period, T=2:
For a periodic ramp function, we can write:

63. Different time occurred:
Expanding:
Different time occurred:
t=0 and t=1

64. Equal time shift:

65. Inverse Laplace using translation in time property:

66. Time periodicity property: