Page 85 - Modern Control Systems

Section 2.4 The Laplace Transform 59

The transformation integrals have been employed to derive tables of Laplace trans-
forms that are used for the great majority of problems. A table of important Laplace
transform pairs is given in Table 2.3, and a more complete list of Laplace transform
pairs can be found at the MCS website.

Table 2.3 Important Laplace Transform Pairs
m F(s)

Step function, u(t)

s + a
bi
sin cot
S 2 + bi 2
s
COS bit
S 2 + bi 2
nl
t"
k
d f(t) k _, k 2
s F{s) - 5* /(0") - s - f'(Q-)
dt k
- . . . - / ^ - ^ ( 0 - )
I f(t)dt s 1 sj -oo j

Impulse function 8(t) 1
bi
e sin cut 2
(s + a) + bi 2
s + a
e "' cos bit
(s + af + bi"
s + a
2 V2
-[(a - a) 2 + bi ] e- al sin(bit + cf>).
+ bi
(s + af 2 a. ,.2
—
<j> = +or*~l-
tan
2
^=e _ f a , n' sin bi nVl - £ t, £ < 1
Vl-£ 2 S + 2£b) nS + bi n
1 1 1
at
e sm(bit - ¢), 2 2
a 1 + to" bi\/a 2 + bi' s[(s + a) + bi ]
_i bi l
<j> = tan —
—a
{
1 . * e" V sin(<«>„Vl - eft + <f>),
V l - C 2 S(S + 2£b) nS + bi n)
_1
<f> = cos £, £ < 1
a 1 (a - a) 2 + to 2 1/2 e~ %m.{bit + <f>). s + a
al
2
a 1 + b/ (o a 2 + co 2 s[(s + a) 2 + bi ]
- 1
(f) — = t o n 1 tan —
tan
a — a —a