Given $ F(s) $, how do we transform it back to the time domain and obtain the corresponding $ f(t) $ ? By matching entries in Table 15.2, we avoid using Eq. (15.5) to find $ f(t) $. Suppose $ F(s) $ has the general form of
where $ N(s) $ is the numerator polynomial and $ D(s) $ is the denominator polynomial. The roots of $ N(s)=0 $ are called the zeros of $ F(s) $, while the roots of $ D(s)=0 $ are the poles of $ F(s) $. Although Eq. (1) is similar in form to Transfer Function
, here $ F(s) $ is the Laplace transform of a function, which is not necessarily a transfer function. We use partial fraction expansion to break $ F(s) $ down into simple terms whose inverse transform we obtain from Table 2
. Thus, finding the inverse Laplace transform of $ F(s) $ involves two steps.
Steps to Find the Laplace Transform:
1. Decompose $ F(s) $ into simple terms using partial fraction expansion.
2. Find the inverse of each term by matching entries in Table 2
Let us consider the three possible forms F(s) may take and how to apply
the two steps to each form.