Page 268 - Dynamics and Control of Nuclear Reactors

270 APPENDIX D Laplace transforms and transfer functions

of functions is large and is an important class of functions encountered in engineering
applications. It should be noted that it is not necessary to specify numerical values of
s to evaluate Laplace transforms. It is only necessary to guarantee that some value of
s exists which insures convergence of the integral.

D.3 Calculating Laplace transforms
A few examples of Laplace transforms of time functions are derived in this section in
order to illustrate the general procedure for obtaining Laplace transforms.

Example D.1
-at
Consider the exponential function, f(t) ¼e , shown in Fig. D.1.
The Laplace transform is
ð ∞
e
FsðÞ ¼ e at st dt (D.2)
0
1 ∞
FsðÞ ¼ exp s + aÞtŠ | (D.3)
ð
½
s + a 0
If the real part of (s+a) is positive, then the function vanishes at the upper limit. We require that
the real part of s is large enough to insure the real part of (s+a) is positive. Thus, the integral
become
1
FsðÞ ¼ (D.4)
s + a
The function, f(t), and its transform, F(s), are called a transform pair.
f(t)

1

0 Time t
FIG. D.1
An exponential function.