Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

The three faces of
 
H
(
s
)
H 
(
s
), the network function, is a 
Laplace transform
if we 
invert
it to find the impulse 
response. 
H 
(
s
), the network function, is a 
complex gain
if we 
evaluate
it at a particular 
value of 
s
. In that case, it gives the complex amplitude of the forced response with an 
input of 
e
st
with the value of 
s
same as the value at which 
H 
(
s
) was evaluated. 
H 
(
s
), 
the network function, functions as a 
ratio of Laplace transforms
when we multiply it by 
the Laplace transform of input source function and invert the product to determine the 
zero-state response in time-domain.
13.11.3 
poles and Zeros of 
H
(
s
) and natural frequencies of the circuit
A network function goes to infinite magnitude at certain values of 
s
. These values are obviously the values 
of 
s
at which the denominator polynomial evaluates to zero, 
i.e.,
at the roots of denominator polynomial. 
These values of 
s
are called 
poles
of the network function. Thus, 
poles
are roots of denominator 
polynomial of a network function. Similarly, a network function attains zero magnitude at certain values 
of
 s
. They are roots of numerator polynomial. They are called 
zeros 
of the network function. 
A diagram showing the 
pole
points by ‘
×
’ marking and 
zero
points by ‘o’ marking in complex 
signal plane (
i.e.,
s
-plane) is called the 
pole-zero plot 
of the network function.
We note from the discussion in the previous subsection that the denominator polynomial of a 
network function apparently has the same order and same coefficients as that of the characteristic 
polynomial of differential equation describing the linear time-invariant circuit. The roots of the 
characteristic polynomial have been defined as the natural frequencies of the circuit. Does this mean 
that (i) the degree of denominator polynomial in a network function is the same as the degree of 
characteristic polynomial (ii) the poles and natural frequencies are the same?
The order of a differential equation is the order of highest derivative of dependent variable. The 
order of a circuit and order of the describing differential equation are the same. It will also be equal 

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13.46
Analysis of Dynamic Circuits by Laplace Transforms
to the total number of independent inductors and capacitors – (number of all-capacitor-voltage source 
loops 
+ 
number of all-inductor-current source nodes).
The order of a network function is the degree of denominator polynomial, 
i.e.,
the highest power of 
s
appearing in the denominator polynomial. 
Thus we are raising the question – is the order of a network function in a linear time-invariant 
circuit same as the order of the circuit?
The characteristic polynomial of a differential equation is quite independent of right-hand side of 
differential equation. But, a network function is very much dependent on the right-hand side of the 
differential equation. Therefore, there exists a possibility of cancellation of some of the denominator 
factors by numerator factors in the case of a network function. Therefore, the order of a network 
function can be lower than the order of the circuit. It cannot, however, be higher. This will also 
imply that the order of two network functions defined within the same network need not be the
same.
For instance, let the differential equation describing a linear time-invariant circuit be
d y
dt
dy
dt
y
dx
dt
x
2
2
3
2
+
+
=
+
The characteristic equation is 
s
s
2
3
2 0
+ + =
and the order of circuit is 2. The natural frequencies 
are 
s
= -
1 and 
s
= -
2. The zero-input response 
can 
contain 
e
-
t
and 
e
-
2
t
terms. But it may contain only 
one of them for certain combination of initial conditions. Consider 
y
(0)
= 
1 and 
y
′
(0) 
= -
1. Then 
y
(
t
) 
= 
e
-
t
and it will not contain 
e
-
2
t
. 
Therefore, not all natural response terms need be present in all circuit 
variables under all initial conditions.
Now consider the network function. It is 
Y s
X s
s
s
s
s
s
s
s
( )
( )
(
)
(
)
(
)
(
)(
)
(
)
=
+
+ +
=
+
+
+
=
+
1
3
2
1
2
1
1
2
2
The order of network function is 1. It has one pole at 
s
= -
2. Therefore, zero-state response to 
any 
input
will not contain
e
-
t
term. This is the effect that a pole-zero cancellation in a network function 
has on circuit response. But, note that the same circuit may have other network functions that may not 
involve such pole-zero cancellation. It is only this particular circuit variable denoted by 
y
that refuses 
to have anything to do with the natural response term 
e
-
t
.
Therefore, we conclude the following:
• The order of a network function and the order of the circuit can be different due to possible pole-
zero cancellations in a particular network function. 
• Poles of any network function defined in a linear time-invariant circuit will be natural frequencies 
of the circuit. 
• However, all natural frequencies need not be present as poles in all network functions defined in 
that circuit. 
• However, all natural frequencies will appear as poles in some network function or other. 
• Thus, poles of a network function is a sub-set of natural frequencies of the circuit and natural 
frequencies will be a union-set of poles of all possible network functions in the circuit.
• A complex frequency that is not a natural frequency of the circuit cannot appear as a pole in any 
network function in that circuit. 
• Both the denominator polynomial and the numerator polynomial of a network function in a linear 
time-invariant circuit have real coefficients. Therefore, poles and zeros of a network function 
either will be real-valued or will occur in complex conjugate pairs.

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Network Functions and Pole-Zero Plots 
13.47