The
Laplace transform of a function f(t), defined for all
real numbers t ≥ 0, is the function F(s),
defined by:
where the parameter s is in general
complex:
s = σ + jw
The most
significant advantage of Laplace transformation is that it turns
differentiation and integration in the time (t) domain into simple
multiplication and division operations, respectively, in the
frequency (s) domain. Laplace
transforms thus allow integral equations and differential equations
to become polynomial equations, which are much easier to solve.
To apply Laplace
transforms in facilitating the process of solving integro-differential
equations, the following
steps
are followed: 1) find the corresponding
Laplace transform
of the integro-differential equation being solved; 2)
manipulate the Laplace transform algebraically to reach the solution
in the s domain, which is also known as the
'revised transform';
and 3) perform an
inverse Laplace transformation
to get the solution back in the time domain.
To
perform an inverse Laplace transformation, i.e., find f(t) from F(s),
the following complex inversion integral is used:
which is
a contour integral whose path of integration, known as Bromwich
path, is along the vertical line s=
γ
from -j∞ to +j∞.
The
Laplace transforms of basic time-domain functions may be found in
this Table of Basic Laplace Transforms.
See Also:
Table of Basic
Laplace Transforms;
Math Used
in ECE
