§ 15-6 The partial-fraction-expansion method
n
nn
m
mm
bsbsb
asasa
)s(D
)s(N)s(F
???
?????
?
?
?
?
1
10
1
10
1.n>m
(1)The roots of D(s) are unequal:
Assume Eq,D(s)=0,to be n simple-roots,p1,p2,…,pn.
n
n
ps
k
ps
k
ps
k)s(F
??????? ?2
2
1
1
)ps kps k)(ps(k)s(F)ps(
n
n
???????? ?2
2
111
Let s=p1 ? ?
1
11 )()( pssFpsk ???
In a similar manner we may obtain
? ?
2
22 )()( pssFpsk ??? ? ?
n
nn ps)s(F)ps(k ???
? ? )t(u)ekekek()s(FL)t(f tpntptp n?????? ? ?21 211
Multiply two sides by (s-p1)
Example 1:
).(,65 54)( 2 tfF i n dss ssFIf ?? ??
Solution,N(s)=4s+5 and D(s)=s2+5s+6=(s+2)(s+3)
We have p1= -2,p2= -3
32)3( 542)3)(2( 54)2(1 ?????
?
?
??
?
?
??
????
?
??
?
??
???
ss
s
sss
ssk
73)2( 543)3)(2( 54)3(2 ????
?
?
??
?
?
??
????
?
??
?
??
???
ss
s
sss
ssk
)()73()( 32 tueetf tt ?? ????
By means of Nopital's rule may determine k,that is
n,.,,,i
)p('D
)p(N
)s(D
)s(N
l i m
)s('D
)s(N)s('N)ps(
l i m
)s(D
)s(N)ps(
l i mk
i
i
'ps
i
ps
i
psi iii
21?
??
??
?
?
?
???
? ? )t(u)ekekek()s(FL)t(f tpntptp n?????? ? ?21 211
3265
54)(,21
2 ??????
??
s
k
s
k
ss
ssFIf
3252 541 ??????? sssk 7352 542 ?????? sssk
)()73()( 32 tueetf tt ?? ????
(2)The roots of D(s) are conjugate complex,
Assume Eq.,D(s)=0,to contain conjugate roots
?? jp ??1 ?? jp ??2
? ? ?? jssD sNpssFpsk ?????? )( )()()(( '
1
11
? ? ?? jssD sNpssFpsk ?????? )( )()()(( '
2
22
k1,k2 must be conjugate complex number,
111 ?jekk ?
112 ?jekk ??
tjjtjj eekeektf )(1)(1 11)( ?????? ??? ??
? ?)()(1 11 ????? ??? ?? tjtjt eeek
)co s (2 11 ??? ?? tek t
Example 2,).(,
1050
33268.0)(
52 tff i n dss
ssFIf
??
??
N(s)=0.268s+33 and D(s)=s2+50s+105
from D(s)=0,we have
??
?
???
???
31 525
31 525
2
1
jp
jp
?3.1714.0
630
4.843.26 ?????
j
j
31525)(
)(
'1 jssD
sNk
???? 50)31525(2
33)31525(268.0
???
????
j
j
)3.173 1 5c o s (28.0)( 25 ???? ? tetf t
Solution:
(3)The roots of D(s) are equal:
In D(s) there are factors (s-p1)2.
?
? ?
?????
n
i i
i
ps
k
ps
k
ps
ksF
2
2
1
11
1
12
)()()()(
1
2
111 )()( pssFpsk ??? ? ?
1
2
112 )()( pssFpsds
dk
???
? ?
1
11
1
1 )()()!1(
1
pssFpsds
d
mk
m
m
m
m ???? ?
?
})!1()( 1{ 1
1
tp
m
m em
t
s ???
?
?
?
?
???
n
i
tp
i
tptp iektekek)t(f
2
1112
11
Example 3:
).(,)1()2( 4)( 3 tff i n dss ssFIf ?? ??
Solution:
)1()2()2()2()(
2
3
11
2
1213
???????? s
k
s
k
s
k
s
ksF
22142)()2( 311 ??????????? sssssFsk
? ? 31 32)1( )4(12)()2( 2312 ??????? ???????? ss ssssFsdsdk
32)1( )1(2)3(2121421 42
2
13 ?????
????
????
?
??
?
?
??
ss
s
ss
s
ds
dk
3131)()1(2 ?????? ssFsk
)()333()( 2222 tueetteetf tttt ???? ?????
mn ?.2
Example 4:
).(,65 11156)( 2
23
tff i n dss ssssFIf ?? ????
3
7
2
31
???
????
sss
)()73()()()( 32' tueetttf tt ?? ????? ??
? ? sdt tdLtL ???????? )()(' ??
Solution:
65
541)(
2 ??
????
ss
sssF