Method of Partial Fractions for Inverting Laplace Transforms
13.13
The System Function
H s
V s
V s
s
s
s
o
s
( )
( )
( )
=
=
+ +
+
1
2
1
3
2
Laplace transform of
e
-
1.5
t
cos2
t
u
(
t
) is
s
s
+
+
+
1 5
1 5
4
2
.
(
. )
Therefore,
V
s
(
s
)
=
s
s
+
+
+
1 5
1 5
4
2
.
(
. )
and
V
o
(
s
)
=
1
2
1
10
1 5
1 5
4
3
2
2
s
s
s
s
s
+ +
+
×
+
+
+
(
. )
(
. )
We observe that the denominator polynomial is the same as the left side of the characteristic equation
of the governing differential equation. This will always be so.
Hence poles of System Function
(
they
are also called ‘system poles’
)
will be the same as the natural frequencies of the circuit for any linear
time-invariant circuit.
Therefore the system poles are
p
1
= -
0.2151
+
j
1.307,
p
2
= -
0.2151
-
j
1.307 and
p
3
= -
0.5698.
The numerator polynomial of System Function in this case is trivial and there are no ‘system zeros’.
The
excitation poles are at
p
e
1
= -
1.5
+
j
2 and
p
e
1
= -
1.5
-
j
2 and excitation zero is at
z
e
1
= -
1.5.
The pole-zero plots are shown in Fig. 13.5-2.
2
x
(–1.5, 2)
o
(–1.5, 0)
x
(–1.5, –2)
(b)
–2
Re(
s
)
–1
Im(
s
)
2
x
(–0.2151, 1.307)
(–0.57, 0)
x
(–0.2151, –1.307)
(a)
–2
Re(
s
)
–1
Im(
s
)
x
o
(–1.5, 0)
x
(–1.5, –2)
(–0.57, 0)
x
(–1.5, 2)
2
x
(–0.2151, –1.307)
x
(–0.2151, 1.307)
(c)
–2
Re(
s
)
–1
Im(
s
)
x
Fig. 13.5-2
Pole-zero plots in Example: 13.5-1 (a) for System Function (b) for excitation
function (c) for output function
13.6
method of partIal fractIons for InvertIng laplace transforms
Any Laplace transform can be inverted by evaluating the synthesis integral in Eqn. 13.2-2 on a suitably
selected vertical line extending from
-∞
to
∞
in the
s
-plane within the ROC of the transform being
inverted. However, simpler methods based on Residue Theorem in Complex Analysis exist for special
Laplace transforms. We do not take up the detailed analysis based on Residue Theorem here. However,
the reader has to bear in mind the fact that the ‘
method of partial fractions
’ for inverting certain special
types of Laplace transforms is based on Residue Theorem in Complex Analysis.
Linear time-invariant circuits are described by linear constant-coefficient ordinary differential
equations. All the coefficients are real. Such a circuit will have only real-valued natural frequencies or
complex-conjugate natural frequencies. Thus, the impulse response of such a circuit will contain only
complex exponential functions. Each complex exponential function will have a Laplace transform of
the form
k
s s
o
-
where
s
o
is the complex frequency of that particular term. Laplace transformation is a
13.14
Analysis of Dynamic Circuits by Laplace Transforms
linear operation. Hence, Laplace transform of the sum of impulse response terms will be the sum of
Laplace transform of impulse response terms. Therefore, Laplace transform of impulse response of a
linear time-invariant circuit will be the sum of finite number of terms of the
k
s s
o
-
type. Such a sum
will finally become a ratio of rational polynomials in
s
. The order of denominator polynomial will be
the same as the number
of first-order terms of
k
s s
o
-
type that entered the sum.
Many of the normally employed excitation functions in linear time-invariant circuits are also of
complex exponential nature. Input functions that can be expressed as linear combinations of complex
exponential functions will have Laplace transforms that are ratios of rational polynomials in
s
as
explained in the last paragraph.
Product of Laplace transforms that are ratios of rational polynomials in
s
will result in a new
Laplace transform which will also be a ratio of rational polynomials in
s
. Hence, the Laplace transform
of output of a linear time-invariant circuit excited by an input source function, that can be expressed
as a linear combination of complex exponential functions, will be a ratio of rational polynomials in
s
.
A Laplace transform that is in the form of a ratio of rational polynomials in s can be inverted by
the method of partial fractions.
Let
Y
(
s
)
=
Q
(
s
)/
P
(
s
) be such a Laplace transform. Let the degree of denominator polynomial be
n
and that of numerator be
m
. The degree of numerator polynomial will usually be less than
n
. If
the Laplace transform of output of a linear time-invariant
circuit shows
n
≤
m
, it usually implies
that the circuit model
employed to model physical processes has been idealised too much. We
assume that
m
<
n
in this section. If
m
is equal to
n
or more than
n,
then
Y
(
s
)
can be written as
Y s
k s
k
Q s
P s
m n
m n
( )
( )
( )
=
+ +
+ ′
−
−
1
, and we employ method of partial fractions on
′
Q s
P s
( )
( )
only.
Let
p
1
,
p
2
,…,
p
n
be the
n
roots of denominator polynomial. They may be real or complex. If there is
a complex root, the conjugate of that root will also be a root of the polynomial. We identify two cases.
In the first case all the
n
roots (
i.e.,
poles of
Y
(
s
)) are distinct.
Case-1 All the
n
roots of
P
(
s
) are distinct
Then
we can express
Y
(
s
) as a sum of first-order factors as below.
Y s
A
s p
A
s p
A
s p
n
n
( )
(
) (
)
(
)
=
−
+
−
+ +
−
1
1
2
2
(13.6-1)
Each term in this expansion is a partial fraction. The value of
A
i
appearing in the numerator
of
i
th
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