Page 94 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Basic Concepts in Process Analysis 69

For the time being, assume that s and s are different and real,
1 2
that is, assume that

Expanding Eq. (3-45) using partial fractions gives

(gks+ gl)S,
'f
(3-46)

The details of the partial fraction expansion and the inversion are
carried out in App. F but Eq. (3-46) shows that Y(t) will have three
terms: two exponentials from the poles at s and s and one constant
1 2
from the pole at zero (or at s ). After the inversion is complete, the
3
result is

(3-47)

Therefore, starting with a Laplace transform, partial frac-
tions allowed the transform to be broken down into three sim-
ple terms, each of which had a known time domain function as its
inverse.

Question 3·11 If Eq. (3-44) had yielded complex poles, how would the
development of the partial fraction expansion have changed?

Answer First, One has to remember that s and s are now complex conjugates.
1 2
Second, one has to figure out how to use Euler's fonnula to present the result. So,
there is no major difference other than a lot more algebra that includes complex
numbers. If you are energetic you might try it.
Poles and Time Domain Exponential Terms
The development of Eq. (3-47) suggests that a nonzero pole in the
Laplace transform of a quantity relates directly to an exponential
term in the time domain. In fact, this is always true and it is a good
reason for being so interested in poles. That is, a factor in the Laplace
transform having the form showing a pole at s = p, as in

1
s-p