Page 267 - Dynamics and Control of Nuclear Reactors

APPENDIX

Laplace transforms

and transfer functions D

D.1 Introduction
Laplace transforms have an important role in the analysis of dynamic systems. They
permit conversion of a linear differential equation into an algebraic equation. This
algebraic equation may then be manipulated so that the solution to the original dif-
ferential equation is obtained by inspection using Laplace transform tables and/or
some simple rules.
The important concept of a system transfer function is developed in this chapter.
Transformation to the Laplace domain facilitates the simplification of large systems
with several dynamic subsystems and thus solve for the overall system response.
The transfer function representation of a dynamic system may be used to develop
the frequency response of the system.

D.2 Defining the Laplace transform
The Laplace transform of a time function, f(t), is defined as follows [1, 2]:
ð ∞
f
FsðÞ ¼ Lf tðÞg ¼ ftðÞe st dt (D.1)
0
The Laplace transform exists when the above integral is finite (<∞)
f(t)¼some function of time, t
F(s)¼Laplace transform of f(t)
s¼Laplace variable. It is, in general, a complex quantity, which has an arbitrary
value except for the requirement that there must be values of s that will insure
convergence of the integral.
L¼Laplace transform operator (this is just a notation).

The requirement for the parameter, s, to have a value that ensures convergence of the
integral restricts the number of functions that have Laplace transforms. The integral
st
is said to be of exponential order if e f(t) is bounded for large t. Thus, Laplace trans-
forms exist only for functions, which are of exponential order. Fortunately, this class
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