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TRANSPORMASI

TRANSPORMASI. LAPLACE. Ditemukan oleh Piere Simon Maequis de Laplace tahun (1747-1827) seorang ahli astronomi dan matematika Prancis Menurut; fungsi waktu atau f(t) dapat ditranspormasi menjadi fungsi komplek atau F(s) Dimana s bilangan komplek dari s = s + j 2 p f atau s + j 

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TRANSPORMASI

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  1. TRANSPORMASI LAPLACE

  2. Ditemukan oleh Piere Simon Maequis de Laplace tahun (1747-1827) seorang ahli astronomi dan matematika Prancis • Menurut; fungsi waktu atau f(t) dapat ditranspormasi menjadi fungsi komplek atau F(s) • Dimana s bilangan komplek dari s = s + j2pf atau s + j • = frekuensi neper = neper/detik •  = frekuensi radian = radian/detik

  3. Hasil TL dari f(t) di beri nama F(s) • Tanda TL diberikan dengan £ atau L, dan fungsinya di tulis f(t): nilai komplek dari fungsi sebuah fariabel t F(s): Nilai komplek dari fungsi sebuah fariabel s

  4. Inverse Transformasi Laplace • Inverse (Bilateral) Transform • Notation F(s) = L{f(t)} variable t tersirat untuk L f(t) = L-1{F(s)} variable s tersirat untuk L-1

  5. Contoh: Transpormasi Laplace • f(t) = A • Jawab

  6. Contoh 2. f(t) = At Jawab Dibantu dengan formula integral partsiel yaitu

  7. Contoh 3 f(t) = e-at jawab

  8. Contoh 4 : f(t) = t.e-at

  9. f(t) = Sin(t) • f(t) = Cos (t) • f(t) = Sin(t+) • f(t) = e-at.Sin(t)

  10. Contoh 9; • f(0+) artinya harga nol untuk fungsi, jika didekati dari arah positif

  11. Contoh 10;

  12. f(t) F(s)=L[f(t)] Some useful Laplace transforms

  13. f(t) F(s)=L[f(t)] Some useful Laplace transforms

  14. f(t) L F(s) Laplace Transform Properties • Linear atau Nonlinear? • Linear operator

  15. contoh • Seperti gambar disamping, muatan awal kapasitor = 0. Tentukan persamaan arusnya;

  16. Transpormasi Laplace

  17. Pembalikan transpormasi laplace • Lihat tabel

  18. Contoh 2 • Gambar RL seperti gambar disamping, jika saklar s di on-kan maka tentukan persamaan arunya

  19. Persamaan rangkaian • Transpormasi Laplace

  20. Transpormasi dari cos t

  21. Laplace transform f(t) t Definition of function f(t) • f(t)=0 for t<0 • defined for t>=0 • possibly with discontinuities • f(t)<Mexp(t)[exponential order] • s: real or complex Examples Definition of Laplace transform

  22. Laplace transform f(t) Dirac t f(t) t Examples

  23. Laplace transform Heaviside f(t) t f(t) t Examples

  24. Laplace transform f(t) Ramp t Examples

  25. Laplace transform properties •Linearity

  26. Example Laplace transform properties • Translation a) if F(s)=L[f(t)]

  27. Example Laplace transform properties g(t) f(t) • Translation b) if g(t) = f(t-a) for t>a = 0 for t<a t a

  28. Example Laplace transform properties •Change of time scale

  29. Laplace transform properties • Derivatives

  30. •If discontinuity in a Laplace transform properties • Derivatives

  31. Laplace transform properties • Derivatives examples

  32. Remarques sur la dérivation Deux cas à prévoir a) En intégrant par parties b) Si f(t) et toutes ses dérivées sont nulles pour t<0, alors on peut ne pas tenir compte des valeurs initiales pour étudier le comportement

  33. Laplace transform properties • Integral

  34. More general Laplace transform properties Multiplication by t Leibnitz’s rule

  35. Laplace transform properties Division by t

  36. Laplace transform properties • Periodic function

  37. Hint

  38. Laplace transform properties Sine and cosine are periodic functions

  39. Laplace transform properties Example f(t) 1 t 3 1 0 2 -1

  40. Laplace transform properties Periodic function

  41. Laplace transform properties Example 1 t 0 1 2 3

  42. Laplace transform properties

  43. Exponential order Laplace transform properties •Limit behaviour Initial value

  44. Laplace transform properties •Limit behaviour Final value

  45. Equation describing the circuit RC circuit R Laplace transform v(t) e0.(t) C Laplace transform applications

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