EKT 119 ELECTRIC CIRCUIT II

1. EKT 119ELECTRIC CIRCUIT II Chapter 2 Laplace Transform

2. 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.

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

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

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

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

7. Functional Transform

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

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

12. 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: Properties of Laplace Transform

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

14. Ex:

15. Three linear functions at t=0, t=1, t=3, and t=4

16. Step function can be used to turn on and turn off these functions 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

17. Step Functions

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

19. 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)

20. f(t) = K (t)

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

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

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

24. 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.

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

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

27. 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

28. Find K1

29. Find K2

30. Find K3

31. Inverse Laplace of F(s)