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

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

of 22

  • date post

    25-Dec-2015
  • Category

    Documents

  • view

    217
  • download

    4

Embed Size (px)

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

  • Slide 1
  • Laplace Transform (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University
  • Slide 2
  • Laplace Transform (1) - Hany Ferdinando2 Overview Introduction Laplace Transform Convergence of Laplace Transform Properties of Laplace Transform Using table Inverse of Laplace Transform
  • Slide 3
  • Laplace Transform (1) - Hany Ferdinando3 Introduction It was discovered by Pierre-Simon Laplace, French Mathematician (1749- 1827)
  • Slide 4
  • 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
  • Slide 5
  • 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
  • Slide 6
  • Laplace Transform (1) - Hany Ferdinando6 Laplace Transform
  • Slide 7
  • 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?
  • Slide 8
  • Laplace Transform (1) - Hany Ferdinando8 Convergence The two-sided Laplace Transform exists if is finite Therefore, is finite
  • Slide 9
  • 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
  • Slide 10
  • Laplace Transform (1) - Hany Ferdinando10 Convergence
  • Slide 11
  • 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
  • Slide 12
  • Laplace Transform (1) - Hany Ferdinando12 Region of Convergence (RoC)
  • Slide 13
  • Laplace Transform (1) - Hany Ferdinando13 Region of Convergence (RoC)
  • Slide 14
  • Laplace Transform (1) - Hany Ferdinando14 Region of Convergence (RoC)
  • Slide 15
  • Laplace Transform (1) - Hany Ferdinando15 Region of Convergence (RoC)
  • Slide 16
  • Laplace Transform (1) - Hany Ferdinando16 Properties Linearity Scaling Time shift Frequency shift
  • Slide 17
  • Laplace Transform (1) - Hany Ferdinando17 Properties Time convolution Frequency convolution Time differentiation
  • Slide 18
  • Laplace Transform (1) - Hany Ferdinando18 Properties Time integration Frequency differentation
  • Slide 19
  • Laplace Transform (1) - Hany Ferdinando19 Properties One-sided time differentiation One-sided time integration
  • Slide 20
  • 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
  • Slide 21
  • Laplace Transform (1) - Hany Ferdinando21 Exercise
  • Slide 22
  • Laplace Transform (1) - Hany Ferdinando22 Next Signals and Linear Systems by Alan V. Oppenheim, chapter 9, p 603-616 Signals and System by Robert A. Gabel, chapter 6, p 373-394 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!