1 (43)

Prévia do material em texto

Problem 3.07PP
Find the time function corresponding to each of the following Laplace transforms using partial- 
fraction expansions:
(a) F(s) -
(b) f W — s(s+j)(s+5)
(c) F ( l) = 3i+2
i '+ i+ l O
(d) F(s) = 3F+6S+6
'W + t(i+ l)( i^+ 6 i+ 1 0 )
(j+D d^+da+lO )
(e) F(s) _ 1
? + l6
(f) F(s) = 2(i+3)
(J+1)(J^+16)
(g) m = ^
(h) F(s) =
(i) F(s) = 4
? 4 4
(i) F(4) = e-‘
7 ^
Step-by-step solution
Step 1 of 16
(a)
Consider the Laplace transform equation.
Take partial fraction expansion and rewrite F (.r)-
F ( s ) . ? - + J - (1)
S J + 1
Use the cover up method and find a and b ■
* 4 u , = - i
Substitute i for a and _ ] for b equation (1).
F ( i ) = i — L
S 5 + 1
Take Inverse Laplace Transform.
i - ( F ( 5 ) ) = r ' ( i ) - r ' ( ^ ]
Thus, the time function of ^ is
Step 2 of 16
(b)
Consider the Laplace transform equation
^ 5{5 + 1)(j + 5)
Take partial fraction expansion and rewrite F ( ^ ) • 
Use the cover up method and find a, ^and c.
(5 + 1)(5 + 5) 
• 1
u
_ _ 5
"~ 4
- _ 5_ |
s ( 5 + \ y ^
I
"4
Step 3 of 16
Substitute i for a , —— for b ^nd — for c in equation (2). 
4 4
F ( f ) s -------2 _ + _ l_
s s+1 f+ 5
Take Inverse Laplace Transform.
L - ' ( F ( 5 ) ) = r ' i - r '
Thus, the time function of
' 5 ' f j '
_4_ + r ‘ _ L
5 + 1 5+5
«(j + I)(5+5)
Step 4 of 16
(c)
Consider the Laplace transform equation 
F ( 5 ) .
' ' 5’ +25 + 10
Take partial fraction expansion and rewrite F ( j ) -
3(5 + 1) 1
~(5+1)’ +3 ' (5+1)'+3*
Take Inverse Laplace Transform.
/ ( / ) = 3e"'cos3f l ( / ) - i c ‘ 's in 3 r l( / )
Thus, the time function of 3j + 2 j 
s ^ * 2 s * l0
3e"'cos 3 /1 ( / ) - —c"* sin 3 /1 ( /)
Step 5 of 16
(d)
Consider the Laplace transform equation.
= ___
' ' (5 + l)(5 ’ + 6 5 + 1 0 )
Take partial fraction expansion and rewrite F (s )-
= ,3)
s * \ j +6^+10
Find the value of a.
(35’ + 6 s+ 6 )| 3
‘ ' ” (5"+65 + 1 0 ) ' '~ '“ 5
Substitute — for a in equation (3) and equate numerators.
fa + e 3s*+6s+ 6
(s + i) (s *+ 6 s + 10) (s + 2 )(s *+ 6 f+ 1 0 )
^ 6 + | js * + ^ ^ + c + 6 js + ( c + 6 ) = 3s*+6s+ 6
Step 6 of 16
Equate powers of s to find ^ and c.
“ y
c+6-6
CmO
3 12Substitute j ^ ^ in equation (3).
F (5 ) =
3
5
12
y '
( 5 + 1) (5^ + 65+ 10)
3
r ' ( F ( 5 ) ) = r '
(5 + 1) [ 5 (5 ^+6 5 + 10) ]
=[|e'''(0-ye'*s‘»(')l(0+y«'* r 4 . '1
17 - r ' 17^
(5 + 1) (5»+4»)
 ̂ /
+ C i f 19 4
[3 4 (5 '+ 4 = )J
/ W = f | e ' ' ' ( ' ) - ^ « » 4