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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
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.
Network Functions and Pole-Zero Plots
13.47
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