L3023.V5
31
Drexel University
Electrical and Computer Engr. Dept.
Electrical Engineering Laboratory II, ECEL 302
E. L. Gerber
TRANSFER
FUNCTION
ANALYSIS
I
Objective
The object of this experiment is to learn how to determine and characterize the transfer
function of a circuit. And then identify the various types of transfer functions and graph
their gain and phase response by inspection and with a computer. You will investigate
polezero diagrams of transfer functions. The Bode Approximation method will be
employed to quickly obtain transfer function response curves.
Introduction
Two and four terminal networks (amplifiers and filters) are often characterized by
determining the ratio of two network signals. For example, the ratio of the output
voltage to the input voltage is defined as the voltage gain transfer function. The current
gain would be the ratio of the output current to the input current. Usually the network
contains frequency dependent elements (L and C); hence, the transfer function is
frequency dependent. In general, the transfer function is expressed in terms of the
complex frequency variable s (=
σ
+
j
ω
). Where s is also the operator, s = d /dt.
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L3023.V5
32
Theory
1)
The Transfer Function
:
When we solve a linear circuit for its transfer function we can write it as T(s) or H(s).
After simplifying the function we can express it in the
polynomial form
,
T
(
s
)
=
H
(
s
)
=
N
(
s
)
D
(
s
)
=
a
m
s
m
+
a
m
−
1
s
m
−
1
+
• • •
+
a
o
(
)
b
n
s
n
+
b
n
−
1
s
n
−
1
+
• • •
+
b
o
(
)
where N(s) is the numerator polynomial and D(s) is the denominator of the transfer
function. When expressed in this form it may be possible to solve for the roots of the two
polynomials. The roots of the numerator are called the zeros, ‘z’, and the roots of the
denominator are called the poles, ‘p’. If the polynomial N(s) is a
m
th order polynomial
then there are ‘m’ roots and ‘m’ zeros in the function. If the polynomial D(s) is an
n
th
order polynomial then there are ‘n’ roots and ‘n’ poles in the function. In the
factored
form
, the poles and zeros are expressed explicitly. The transfer function can be expressed
as follows,
H(s)
=
K(s
+
z
1
)(s
+
z
2
) ... (s
+
z
m
)
(s
+
p
1
)(s
+
p
2
) ... (s
+
p
n
)
A zero is that value of s that makes the transfer function equal to zero; a pole is that
value of s that makes the transfer function equal to infinity. K is a constant of the system.
The poles and zeros of a transfer function are important characteristics of the function
and it will be necessary to solve for their values. They are constants of the system and
may be complex values. They have the same dimensions as frequency, rad/sec. We often
plot their location on the complex frequency plane, the
splane
. We call the plot the ‘pole
zero diagram’ of the system.
2) Logarithmic Gain:
We usually graph the magnitude of the transfer function and its phase against the
frequency,
ω
. It is common to use logarithmic plots instead of linear plots because of the
large range of values. The logarithmic gain is defined as the logarithm of the magnitude
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 Spring '09
 Decibel, H.W. Bode

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