Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

13.14 
summary
• Let 
v
(
t
) be a right-sided function that is bounded by 
Me
a
 
t
with some finite value of 
M
and 
a
. Then 
the Laplace transform pair is defined as 
V s
v t e dt
st
( )
( )
=
−
−
∞
−
∫
0
the analysis equation
v t
j
V s e ds
st
j
j
( )
( )
=
−
− ∞
+ ∞
∫
1
2
p
s
s
the synthesis equation
where 
s
= 
s
+ 
j
w
is the complex frequency variable standing for the complex exponential function 
e
st
with 
s
value 
≥
a
. The ROC of 
V
(
s
) is the entire plane to the right of Re(
s
) 
= 
a
line.

---

Summary 
13.59
table 13.14-1
Some Impotant Properties of Laplace Transforms
Signal / Property
Laplace Transform
v
(
t
) 
= 
f 
(
t
)
 u
(
t
)
V
(
s
) 
- 
Definition
av
1
(
t
) 
+ 
bv
2
(
t
)
 
aV
1
(
s
) 
+ 
bV
2
(
s
) 
- 
Linearity
v
(
t
-
t
d
)
 
e
jst
d
-
V
(
s
) – Time-shifting
e
s t
o
v
(
t
)
, s
o
is a complex number 
V
(
s
– 
s
o
) – Multiplication by a complex exponential
In time-domain – Frequency Shifting
dv t
dt
( )
sV
(
s
) – 
v
(0
-
) Time-domain differentiation
v t dt
t
( )
0
∫
1
s
V s
( )
- 
Time-domain integration
-
t v(t) 
dV s
ds
( )
- 
Frequency-domain differentiation
v t
t
( )
V s ds
s
( )
∞
∫
v
1
(t)*v
2
(t) 
V
1
(
s
)
V
2
(
s
)
 
- 
Time-domain convolution
v
( )
0
+
lim
( )
s
sV s
→∞
= 
v
( )
0
+
, if the limit exists.
v
( )
∞
lim
( )
s
sV s
→
0
= 
v
( )
∞
, if the limit exists and all poles of 
sV(s) are in the left-half of 
s
-plane.
• Laplace transform expands a transient right-sided time-function in terms of infinitely many 
complex exponential functions of infinitesimal amplitudes. The ROC of such a Laplace transform 
will include right-half of 
s
-plane and hence the time-domain waveform gets constructed by 
growing complex exponential functions.
• For a linear time-invariant circuit described by 
d y
dt
a
d
y
dt
a
dy
dt
a y
b
d x
dt
n
n
n
n
n
m
m
m
+
+ +
+
=
+
−
−
−
1
1
1
1
0
b
d
x
dt
m
m
m
−
−
−
1
1
1
++ +
+
b
dx
dt
b x
1
0
, the ratio of rational polynomials in 
s
defined as 
H s
b s
b
s
b s b
s
a s
a s a
m
m
m
m
o
n
n
n
o
( )
=
+
+ +
+
+
+ +
+
−
−
−
−
1
1
1
1
1
1
has three interpretations.
1. It may be viewed as a generalised frequency response function. Its magnitude gives the ratio 
between the amplitude of output complex exponential function and input complex exponential 
function when input is of the form 
Ae
st
. Its angle gives the phase angle by which the output 
complex exponential function leads the input complex exponential function.
2. It is also the ratio of Laplace transform of zero-state response to Laplace transform of input 
source function called ‘the 
s
-domain System Function’.
3. Further, it is the Laplace transform of Impulse Response 

---

13.60
Analysis of Dynamic Circuits by Laplace Transforms
• Laplace transforms can be inverted by the method of partial fractions.
• Laplace transformation of both sides of a linear constant-coefficient ordinary differential equation 
converts it into an algebraic equation on transforms. Thus Laplace transform affords a convenient 
way to solve such differential equations with initial conditions.
• All circuit elements have 
s
-domain equivalents. The 
s
-domain equivalent of the complete circuit 
can be constructed by replacing each element with its 
s
-domain equivalent. KVL and KCL are 
directly applicable to the transformed quantities.
• s
-domain equivalent circuits may be analysed by nodal analysis or mesh analysis procedures. 
All circuit theorems developed in the context of memoryless circuits are applicable to 
s
-domain 
equivalent circuits.
• The 
s
-domain System Function is also called the network function. Immittance functions, transfer 
functions and transfer immittance functions are three classes of network functions usually 
employed in circuit analysis. Those complex frequency values at which a network function goes 
to 
∞
are called its 
poles
and those complex frequency values at which the network function goes 
to zero are called its 
zeros.
• Pole-zero plot along with a gain factor 
K
will specify a network function uniquely. The impulse 
response of the network function may be obtained from its pole-zero plot. The poles decide the 
number of terms in impulse response and their complex frequencies. The zeros along with the 
poles and gain factor 
K
decide the amplitude of each impulse response term.
• A network function is a stable one if its impulse response decays to zero with time. This is 
equivalent to stating that its impulse response must be absolutely integrable, 
i.e.,
| ( ) |
h t dt
0
∞
∫
must 
be finite. Therefore, a network function is stable if all the impulse response terms are damped ones. 
That is, all the poles must have negative real values or complex values with negative real parts. 
Therefore, a network function is stable if and only if all its poles are in the left-half of 
s
-plane 
excluding the 
j
w
-
axis.
• Sinusoidal steady-state frequency response function 
H
(
j
w
) can be obtained by evaluating 
H
(
s
) on 
j
w
-axis. This evaluation can be carried out in a graphical manner too.
• The DC steady-state gain of a network function is 
H
(0).