Laplace Transform

Laplace Transform Chairul Hudaya, ST, M.Sc Electric Power & Energy Studies (EPES) Department of Electrical Engineering University of Indonesia http://www.ee.ui.ac.id/epes Depok, October, 2009 Electric Circuit

Introduction • Laplace transform is another method to transform a signal from time domain to frequency domain (s-domain) • The basic idea of Laplace transform comes from the Fourier transform • As we have seen in the previous chapter, not many functions have their Fourier transform such as t, t2, et etc.

The Fourier transform formula: • The Laplace transform formula is the modification of the above formula, that is, the term jw is replaced by s • s is equal to s+jw, where s is a large positive real number • The Laplace transform formula: • However, the Laplace transform only support the function f(t) which domain t ≥ 0

Example 1 Using definition, find the Laplace transform of (a) (b) (c) (d)

Solution (a) (b)

(c)

(d)

Properties of L-transform • Linearity L{af(t) ± bg(t)} = aL{f(t)} ± bL{g(t)} • First shift theorem L{e−atf(t)} = F(s + a) • Second shift thorem L{f(t − d) u(t − d)} = e−dsF(s) • Time scaling

Properties of L-transform (cont.) • Time derivatives • Time integral

Example 2 Determine the Laplace transform of (a) (b)

Solution (a) (b)

Example 3 Determine the Laplace transform of (a) (b)

Solution Let (a) then Therefore

(b) Let then Also Therefore

Laplace Transform