Ditemukan oleh Piere Simon Maequis de Laplace tahun ( ) 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 =  + j2  f atau  + j   = frekuensi neper = neper/detik  = frekuensi radian = radian/detik

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

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

Contoh: Transpormasi Laplace 1. f(t) = A –Jawab

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

Contoh 3 f(t) = e -at jawab

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

5.f(t) = Sin(  t) 6.f(t) = Cos (  t) 7.f(t) = Sin(  t+  ) 8.f(t) = e -at. Sin(  t)

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

Contoh 10;

f(t) L (f)f(t) L (f) 1 1 1/s 7 cos  t 2 t 1/s 2 8 sin  t 3 t2t2 2!/s 3 9 cosh at 4 t n (n=0, 1,…) 10 sinh at 5 t a (a positive) 11 e at cos  t 6 e at 12 e at sin  t

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

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

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

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

Transpormasi Laplace

Pembalikan transpormasi laplace Lihat tabel

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

Persamaan rangkaian Transpormasi Laplace

Transpormasi dari cos  t

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

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

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

Laplace transform Examples f(t) Ramp t

Laplace transform properties Linearity

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

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

Laplace transform properties Change of time scale Example

Derivatives Laplace transform properties

Derivatives Laplace transform properties If discontinuity in a

Derivatives examples Laplace transform properties

Remarques sur la dérivation Deux cas à prévoir En intégrant par parties a) 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

Laplace transform properties Integral

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

Laplace transform properties Division by t

Periodic function Laplace transform properties

Hint

Laplace transform properties Sine and cosine are periodic functions

Laplace transform properties Example t f(t)

Laplace transform properties Periodic function

Laplace transform properties Example 1 t 0123

Laplace transform properties

Limit behaviour Initial value Laplace transform properties Exponential order

Limit behaviour Final value Laplace transform properties

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

Laplace transform applications Impulse function Impulse response

Laplace transform applications Step function e0

Laplace transform applications Step function and initial conditions v(0)  0

Laplace transform applications Ramp function

(Heaviside) Laplace transform properties a t

at

Limits Initial value Final value

e(t) E(s) v(t) V(s) R C Harmonic analysis Laplace transform properties

Forced Transient