1
© Bob York 2009
1
Frequency Response
and Bode Plots
1.1
Preliminaries
The steadystate sinusoidal frequencyresponse of a circuit is described by the phasor transfer
function
(
)
H
j
.
A
Bode plot
is a graph of the magnitude (in dB) or phase of the transfer
function versus frequency.
Of course we can easily program the transfer function into a
computer to make such plots, and for very complicated transfer functions this may be our
only recourse.
But in many cases the key features of the plot can be quickly sketched by
hand using some simple rules that identify the impact of the poles and zeroes in shaping the
frequency response.
The advantage of this approach is the insight it provides on how the
circuit elements influence the frequency response.
This is especially important in the
design
of frequencyselective circuits.
We will first consider how to generate Bode plots for simple
poles, and then discuss how to handle the general secondorder response.
Before doing this,
however, it may be helpful to review some properties of transfer functions, the decibel scale,
and properties of the log function.
Poles, Zeroes, and Stability
The
s
domain transfer function is always a rational polynomial function of the form
1
2
1
2
1
0
1
2
1
2
1
0
( )
( )
( )
m
m
m
m
m
n
n
n
n
n
s
a
s
a
s
a s
a
N s
H s
K
K
D s
s
b
s
b
s
b s
b
(1.1)
As we have seen already, the polynomials in the numerator and denominator are factored to
find the poles and zeroes; these are the values of
s
that make the numerator or denominator
zero.
If we write the zeroes as
1
2
3
,
,
z
z
z
etc., and similarly write the poles as
1
2
3
,
,
p
p
p
,
then
( )
H s
can be written in factored form as
1
2
1
2
(
)(
)
(
)
( )
(
)(
)
(
)
m
n
s
z
s
z
s
z
H s
K
s
p
s
p
s
p
(1.2)
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Frequency Response and Bode Plots
© Bob York 2009
The pole and zero locations can be real or complex.
When the roots are real they are called
simple poles
or
simple zeros
.
When the roots are complex they always occur in pairs that are
complex conjugates of each other.
Another important observation is that stable networks must always have poles and zeroes
in the lefthalf of the complex splane, such that the real parts of the poles/zeroes will be
negative
. As an example, lets assume a stable network with simple poles at
1
1
p
and
2
10
p
.
The transfer function would then be
1
2
1
1
( )
(
)(
)
(
1)(
10)
H s
s
p
s
p
s
s
(1.3)
Thus for stable networks we
always
will find terms of the form (
)
s
a
in the denominator,
where
a
is a
positive
number.
Students sometimes get confused by the use of (
)
s
p
or
(
)
s
a
to represent the same pole location; just remember that the poles are the values of
s
that make the denominator zero,
i.e.
s
p
or
s
a
in this example; clearly these will
represent the same pole if
p
a
, and will represent a
stable
pole if Re{ }
0
a
or
Re{ }
0
p
.
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 Winter '07
 York
 Frequency, Decibel, Complex number, Bode Plots, Bob York

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