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EKT919 ELECTRIC CIRCUIT II

EKT919 ELECTRIC CIRCUIT II. Chapter 2 Laplace Transform. 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

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EKT919 ELECTRIC CIRCUIT II

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  1. EKT919ELECTRIC 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

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

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

  10. 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

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

  12. Ex:

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

  14. 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

  15. Step Functions

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

  17. 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)

  18. f(t) = K (t)

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

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

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

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

  23. 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:

  24. 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:

  25. 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:

  26. 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

  27. 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.

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

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

  30. 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

  31. Find K1

  32. Find K2

  33. Find K3

  34. Inverse Laplace of F(s)

  35. S1 = -6 S2 = -3+j4 S3 = -3-j4 2) Distinct Complex Roots

  36. Partial Fraction Expansion • Complex roots appears in conjugate pairs.

  37. Find K1

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

  39. Inverse Laplace of F(s)

  40. Inverse Laplace of F(s)

  41. Useful Transform Pairs

  42. 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:

  43. Operational Transform

  44. 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

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

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