Laplace Transform (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University

Laplace Transform (1) - Hany Ferdinando2 Overview Introduction Laplace Transform Convergence of Laplace Transform Properties of Laplace Transform Using table Inverse of Laplace Transform

Laplace Transform (1) - Hany Ferdinando3 Introduction It was discovered by Pierre-Simon Laplace, French Mathematician ( )

Laplace Transform (1) - Hany Ferdinando4 Introduction It transforms signal/system from time- domain to s-domain for continuous- time LTI system It is analogous to Z Transform in discrete-time LTI system It is similar to Fourier Transform, but ‘j  ’ is substituted by s

Laplace Transform (1) - Hany Ferdinando5 Introduction Laplace Transform is continuous sum of exponential function of the form e st, where s =  + j  is complex frequency Therefore, Fourier can be viewed as a special case in which s = j 

Laplace Transform (1) - Hany Ferdinando6 Laplace Transform

Laplace Transform (1) - Hany Ferdinando7 Laplace Transform For h(t) = e -at, find H(s)  What is your assumption in finishing the integration?  If you do not have that assumption, then what you can do?  Is it important to have that assumption?

Laplace Transform (1) - Hany Ferdinando8 Convergence… The two-sided Laplace Transform exists if is finite Therefore, is finite

Laplace Transform (1) - Hany Ferdinando9 Convergence… Suppose there exists a real positive number R so that for some real  and  we know that f(t) 0, and f(t) 0

Laplace Transform (1) - Hany Ferdinando10 Convergence…

Laplace Transform (1) - Hany Ferdinando11 Convergence… How did you make your assumption in order to solve the equation? Can you solve it without that assumption? The negative portion converges for  

Laplace Transform (1) - Hany Ferdinando12 Region of Convergence (RoC)

Laplace Transform (1) - Hany Ferdinando13 Region of Convergence (RoC)

Laplace Transform (1) - Hany Ferdinando14 Region of Convergence (RoC)

Laplace Transform (1) - Hany Ferdinando15 Region of Convergence (RoC)

Laplace Transform (1) - Hany Ferdinando16 Properties Linearity Scaling Time shift Frequency shift

Laplace Transform (1) - Hany Ferdinando17 Properties Time convolution Frequency convolution Time differentiation

Laplace Transform (1) - Hany Ferdinando18 Properties Time integration Frequency differentation

Laplace Transform (1) - Hany Ferdinando19 Properties One-sided time differentiation One-sided time integration

Laplace Transform (1) - Hany Ferdinando20 Using Standard Table Use table from books both for transform and for its inverse No RoC is needed Find the general form of the equation Properties of Laplace transform are helpful You use that table also to find the inverse

Laplace Transform (1) - Hany Ferdinando21 Exercise

Laplace Transform (1) - Hany Ferdinando22 Next… Signals and Linear Systems by Alan V. Oppenheim, chapter 9, p Signals and System by Robert A. Gabel, chapter 6, p The Laplace Transform is already discussed. It transforms continuous-time LTI system from time- domain to s-domain. There are two types, one-sided (unilateral) and two-sided Next, we will study the application of Laplace Transform in Electrical Engineering. Read the Electric Circuit handout to prepare yourself!