Fraccoes Simples

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•
H(s) =
N(s)
D(s)
.
•
4s� 1
s2 + s� 2
3s2 + 4s� 1
s2 + s� 2 ,
•
A
(s� p)q ,
s A p q
4s� 1
s2 + s� 2 =
1
s� 1 +
3
s+ 2
.
3s2 + 4s� 1
s2 + s� 2
N(s)
(s� p1)(s� p2) . . . (s� pn) =
A1
s� p1 +
A2
s� p2 + · · · +
An
s� pn .
N(s)
(s� p1)2(s� p2)3 =
A11
s� p1 +
A12
(s� p1)2 +
A21
s� p2 +
A22
(s� p2)2 +
A23
(s� p2)3 .
4s� 1
s2 + s� 2 =
4s� 1
(s� 1)(s+ 2) .
4s� 1
(s� 1)(s+ 2) =
A1
s� 1 +
A2
s+ 2
.
s� 1
4s� 1
s+ 2
= A1 + (s� 1) A2
s+ 2
.
s = 1
3
3
= A1 + 0
A2
3
,
A1 = 1
A1 =
4s� 1
⇠⇠⇠
⇠(s� 1)(s+ 2)
����
s=1
,
s = 1
A2 =
4s� 1
(s� 1)⇠⇠⇠⇠(s+ 2)
����
s=�2
=
�9
�3 = 3.
4s� 1
s2 + s� 2 =
1
s� 1 +
3
s+ 2
.
1
s3 � 2s2 + s =
1
s(s� 1)2 =
A1
s
+
A21
s� 1 +
A22
(s� 1)2
A22 =
1
s⇠⇠⇠
⇠⇠(s� 1)2
����
s=1
= 1.
A1
A1 =
1
�s (s� 1)2
����
s=0
= 1.
A21
1
s(s� 1)2 =
1
s
+
A21
s� 1 +
1
(s� 1)2
=
(A21 + 1)s2 � (A21 + 1)s+ 1
s(s� 1)2 .
A21 = �1
1
s3 � 2s2 + s =
1
s(s� 1)2 =
1
s
� 1
s� 1 +
1
(s� 1)2 .
N(s)
D(s)
= Q(s) +
R(s)
D(s)
.
R(s)
6s4 + 11s3 + 2s2 � 4s� 4 3s2 + s� 2
•
6s4 + 11s3 + 2s2 � 4s � 4 |3s2 + s� 2
•
3s2 6s4
2s2
6s4 + 11s3 + 2s2 � 4s � 4 |3s2 + s� 2
2s2
•
6s4 + 11s3 + 2s2 � 4s � 4 |3s2 + s� 2
6s4 + 2s3 � 4s2 2s2
•
��6s4 + 11s3 + 2s2 � 4s � 4 |3s2 + s� 2
��6s4 + 2s3 � 4s2 2s2
9s3 + 6s2
•
��6s4 + 11s3 + 2s2 � 4s � 4 |3s2 + s� 2
��6s4 + 2s3 � 4s2 2s2
9s3 + 6s2 � 4s
•
��6s4 + 11s3 + 2s2 � 4s � 4 |3s2 + s� 2
��6s4 + 2s3 � 4s2 2s2 + 3s
9s3 + 6s2 � 4s
9s3 + 3s2 � 6s
•
��6s4 + 11s3 + 2s2 � 4s � 4 |3s2 + s� 2
��6s4 + 2s3 � 4s2 2s2 + 3s
��9s3 + 6s2 � 4s
��9s3 + 3s2 � 6s
3s2 + 2s � 4
•
��6s4 + 11s3 + 2s2 � 4s � 4 |3s2 + s� 2
��6s4 + 2s3 � 4s2 2s2 + 3s+ 1
��9s3 + 6s2 � 4s
��9s3 + 3s2 � 6s
��3s2 + 2s � 4
��3s2 + s � 2
s � 2
•
2s2 + 3s + 1
s� 2
6s4 + 11s3 + 2s2 � 4s� 4
3s2 + s� 2 = 2s
2 + 3s+ 1 +
s� 2
3s2 + s� 2 ,
��3s4 + 10s3 � 2s2 + 2s + 6 |s3 + 3s2
��3s4 + 9s3 3s+ 1
��s3 + 2s2
��s3 + 3s2
�5s2 + 2s
��3s4 + 10s3 � 2s2 + 2s + 6 |s3 + 3s2
��3s4 + 9s3 3s+ 1
��s3 + 2s2
��s3 + 3s2
�5s2 + 2s + 6
3s+ 1 �5s2 + 2s+ 6
��3s4 + 10s3 � 2s2 + 2s + 6 |s3 + 3s2 + 0s+ 0
��3s4 + 9s3 + 0s2 + 0s 3s+ 1
��s3 + 2s2 + 2s
��s3 + 3s2 + 0s + 0
�5s2 + 2s + 6
5s+ 1
s2 + 2s� 15 =
5s+ 1
(s� 3)(s+ 5) =
A
s� 3 +
B
s+ 5
.
A =
5s+ 1
⇠⇠⇠
⇠(s� 3)(s+ 5)
����
s=3
= 2.
B =
5s+ 1
(s� 3)⇠⇠⇠⇠(s+ 5)
����
s=�5
= 3.
s+ 1
s2 � 2s+ 1 =
s+ 1
(s� 1)2 =
A
s� 1 +
B
(s� 1)2 .
B =
s+ 1
⇠⇠⇠
⇠⇠(s� 1)2
����
s=1
= 2.
A
s+ 1
(s� 1)2 =
A
s� 1 +
2
(s� 1)2
=
As+ (2� A)
(s� 1)2 ,
A = 1
3s� 4
2s2 � 6s+ 4 =
3s� 4
2(s� 1)(s� 2) =
A
s� 1 +
B
s� 2 .
(2s� 2)(s� 2)
A =
3s� 4
2⇠⇠⇠
⇠(s� 1)(s� 2)
����
s=1
= 1/2.
B =
3s� 4
2(s� 1)⇠⇠⇠⇠(s� 2)
����
s=2
= 1.
6s4 + 11s3 + 2s2 � 4s� 4
3s2 + s� 2 = 2s
2 + 3s+ 1 +
s� 2
3s2 + s� 2
= 2s2 + 3s+ 1 +
s� 2
3
�
s� 23
�
(s+ 1)
= 2s2 + 3s+ 1 +
A�
s� 23
� + B
s+ 1
,
A =
s� 2
3⇠⇠⇠
⇠�s� 23�(s+ 1)
����
s= 23
= � 4
15
,
B =
s� 2
3
�
s� 23
�
⇠⇠⇠
⇠(s+ 1)
�����
s=�1
=
3
5
.