TABLE OF CONTENTS This chapter begins with an introduction to the Laplace transformation, definitions, and properties of the Laplace transformation. The initial value and final value theorems are also discussed and proved. It continues with the derivation of the Laplace transform of common functions of time, and concludes with the derivation of the Laplace transforms of common wave-forms.
The two-sided or bilateral Laplace Transform pair is defined as
where L{f(t)} denotes the Laplace transform of the time function f(t), L ?1{ F (s)} denotes the Inverse Laplace transform, and s is a complex variable whose real part is ?, and imaginary part ?, that is, ? = s+j ?.
In most problems, we are concerned with values of time t greater than some reference time, say t = t 0= 0, and since the initial conditions are generally known, the two-sided Laplace transform pair of (2.1) and (2.2) simplifies to the unilateral or one-sided Laplace transform defined as
The Laplace Transform of (2.3) has meaning only if the integral converges (reaches a limit), that is, if
To determine the conditions that will ensure us that the integral of (2.3) converges, we rewrite (2.5) as
The term e ?j?t in the integral of (2.6) has magnitude of unity, i.e., e ?j?t = 1, and thus the condition for convergence becomes
Fortunately, in most engineering applications the functions f(t) are of exponential order [1]. Then, we...
