Signals and Systems with MATLAB
Computing and Simulink
Modeling, Fourth Edition
11
−
21
Copyright
©
Orchard Publications
LowPass Analog Filter Prototypes
Figure 11.18.
VCVS low
−
pass filter (Courtesy Reference Data for Engineers Handbook)
The transfer function of the second order VCVS low
−
pass filter of Figure 11.18 is given as
(11.37)
This is referred to as a second order
all
−
pole
*
approximation to the ideal low
−
pass filter with cut
off frequency
, where
is the gain, and the coefficients
and
are provided by tables.
For a non

inverting positive gain
, the circuit of Figure 11.18 satisfies the transfer function of
(11.37) with the conditions that
(11.38)
(11.39)
(11.40)
(11.41)
From (11.40) and (11.41), we observe that
.
A fourth

order all
−
pole low
−
pass filter transfer function is a ratio of a constant to a fourth degree
polynomial. A practical method of obtaining a fourth order transfer function, is to factor it into
two second
−
order transfer functions of the form of relation (11.37), i.e.,
*
The terminology “all
−
pole” stems from the fact that the s
−
plane contains poles only and the zeros are at
, that is, the
s
−
plane is all poles and no zeros.
C
1
R
3
R
4
R
1
R
2
C
2
v
in
v
out
G s
( )
Kb
ω
C
2
s
2
a
ω
C
s
b
ω
C
2
+
+

=
∞
±
ω
C
K
a
b
K
R
1
2
aC
2
a
2
4b K
1
–
(
)
+
[
]
C
2
2
4bC
1
C
2
–
(
)
+
ω
C

=
R
2
1
bC
1
C
2
R
1
ω
C
2

=
R
3
K R
1
R
2
+
(
)
K
1
–
(
)

K
1
≠
=
R
4
K R
1
R
2
+
(
)
=
K
1
R
4
R
3
⁄
+
=
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Chapter 11
Analog and Digital Filters
11
−
22
Signals and Systems with MATLAB
Computing and Simulink
Modeling, Fourth Edition
Copyright
©
Orchard Publications
(11.42)
Each factor in (11.42) can be realized by a stage (circuit). Then, the two stages can be cascaded
as shown in Figure 11.19.
Figure 11.19.
Cascaded stages
Table 11.3 lists the Butterworth low
−
pass coefficients for second and fourth
−
order designs, where
and
apply to the transfer functions of (11.37) and (11.42) respectively.
For a practical design of a second
−
order VCVS circuit, we select standard values for capacitors
and
for the circuit of Figure 11.18, we substitute the appropriate values for the coefficients
and
from Table 11.3, we choose desired values for the gain
and cutoff frequency
, and
we substitute these in relations (11.38) through (11.41) to find the values of the resistors
through
.
Example 11.6
Design a second
−
order VCVS Butterworth low
−
pass filter with gain
and cutoff frequency
.
TABLE 11.3
Coefficients for Butterworth low
−
pass filter designs
Coefficients for Second and Fourth Order Butterworth Low
−
Pass Filter Designs
Order
a
1.41421
2
b
1.0000
a
1
0.76537
b
1
1.0000
4
a
2
1.84776
b
2
1.0000
G s
( )
K
1
b
1
ω
C
2
s
2
a
1
ω
C
s
b
1
ω
C
2
+
+

K
2
b
2
ω
C
2
s
2
a
2
ω
C
s
b
2
ω
C
2
+
+

⋅
=
Stage 1
Stage 2
v
in
v
out
a
b
C
1
C
2
a
b
K
ω
C
R
1
R
4
K
2
=
f
C
1 KHz
=