1. Chap2. Modeling in the Frequency Domain
Automatic Control
Chap2. Modeling in the Frequency Domain
Kim, Do Wan
HANBAT
NATIONAL UNIVERSITY

2. Outline
Laplace transform
Electrical network
Mechanical system

3. Why do we study Laplace transform

4. Why do we study Laplace transform
Differential
Equation

5. Why do we study Laplace transform
Transfer
Function
Differential
Equation
Solution

6. Why do we study Laplace transform
Transfer
Function
Differential
Equation
Algebraic
Equation
Solution

7. Why do we study Laplace transform
Transfer
Function
Time-domain
Frequency-domain
Differential
Equation
Algebraic
Equation
Solution

8. Why do we study Laplace transform
Transfer
Function
Arrangement
Differential
Equation
Algebraic
Equation
Solution

9. Partial Fraction Expansion
Laplace transform
Why do we study Laplace transform
Transfer
Function
Arrangement
Differential
Equation
Algebraic
Equation
Algebraic
manipulation
Solution
Partial Fraction Expansion

10. Partial Fraction Expansion
Laplace transform
Why do we study Laplace transform
Transfer
Function
Arrangement
Differential
Equation
Algebraic
Equation
Algebraic
manipulation
Solution
Partial Fraction Expansion
Look-up
table

11. Inverse definition (Inverse Laplace transform)

12. Laplace transform
Example 2.1:
Solution:

13. Laplace transform theorems
Laplace transform table

14. Laplace transform
Example 2.1:
Solution:

15. Example 2.2: Find the inverse Laplace transform of
Solution:

16. Partial-fraction Expansion
Laplace transform
Partial-fraction Expansion
To find the inverse Laplace transform of a complicated function
with
Case 1: Roots of the denominator of are real and distinct.
Example:
where hence

17. Laplace transform

18. Case 2: Roots of the denominator of are real and repeated.
Laplace transform
Case 2: Roots of the denominator of are real and repeated.
Example:

19. Case 3: Roots of the denominator of are complex or imaginary
Laplace transform
Case 3: Roots of the denominator of are complex or imaginary
Example:

20. Laplace transform

21. Skill- Assessment Exercise 2. 1 Skill- Assessment Exercise 2. 2
Laplace transform
Skill- Assessment Exercise 2. 1
Skill- Assessment Exercise 2. 2

22. Transfer function Definition Pole, Zero Block diagram
Laplace transform
Transfer function
Definition
Pole, Zero
Block diagram

23. Laplace transform
Evaluation
Pole, Zero

24. Skill-Assessment Exercise 2.3 Skill-Assessment Exercise 2.4
Laplace transform
Example 2.5
Skill-Assessment Exercise 2.3
Skill-Assessment Exercise 2.4
Skill-Assessment Exercise 2.5
Example 2.3 (Solution)

25. Impedance Relationships
Electrical Network
Impedance Relationships

26. Example: Transfer function relating the input voltage to the current
Electrical Network
Example:
Transfer function relating the input voltage to the current
Kirfchhoff's voltage law: LT
Transfer function

27. Electrical Network
Examples 2.6, 2.10

28. Mechanical system
Translational System

29. Mechanical system
Example 2.16

30. Governing Principle: sum of all forces are zero. Now,
Mechanical system
Governing Principle: sum of all forces are zero.
Now,
So, the differential equation is

31. Homework
Problems 1, 2, 7, 8, 16, 22
Read Chap. 3.2 carefully!