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lt1 - Boise State University Department of Electrical and...

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Boise State University Department of Electrical and Computer Engineering ECE225 – Circuit Analysis and Design The Laplace Transform I Reading Assignment : Read Sections 15.2-15.3 Lecture Objectives : 1. To define the Laplace transform. 2. To review some properties of Laplace transforms. 3. To derive the Laplace transforms of elementary time functions. Example of a Transform: The Logarithm Transform Real Domain -→ Real Domain a ln -→ x = ln a b ln -→ y = ln b c = ab = e ln z e = ln - 1 ←- z = x + y = ln a + ln b = ln ab Definition of the (One-Sided) Laplace Transform : L { f ( t ) } = Z 0 - f ( t ) e - st dt = F ( s ) Notes : 1. The Laplace integral is integrated over time starting shortly time t = 0. (This is indicated by the notation 0 - which means 0 - ² , ² being an arbitrarily small number.) The reason for using 0 - instead of 0 will become apparent later on. 2. The variable s is a complex variable. Thus the Laplace domain (or s-domain) represents functions of a complex number s = σ + . 3. The Laplace integral defined above will converge for a particular s if Z 0 - | f ( t ) e - st | dt = Z 0 - | f ( t ) | e - σt dt < 4. In particular, a function that does not grow faster than an exponential,
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