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V
in
2
V
in
3
þ
C
1
C
2
s
2
1
Ds
1
R
2
1
R
2
V
02
¼
þ
sC
1
sC
2
ð
V
in
4
Þ
ðÞ
1
ð
12
:
8
Þ
R
1
V
in
1
1
R
2
þ
sC
2
þ
V
in
3
þ
V
in
1
C
1
C
2
s
2
1
Ds
1
R
1
1
R
1
s
2
C
1
C
2
V
in
2
þ
V
03
¼
þ
sC
2
sC
2
ðÞ
1
R
2
V
in
4
1
R
1
þ
sC
1
þ
sC
2
ð
:
Þ
12
9
C
1
1
R
2
C
2
1
R
1
1
R
1
R
2
s
2
C
1
C
2
þ
where D
ðÞ¼
s
ð
C
2
Þ þ
þ
ð
12
:
10
Þ
From equations (
12.7
)to(
12.10
), the authors have described as many as thirteen
cases however, it is obvious that because the circuit is minimal (i.e., employment of
only two resistors and two capacitors) the requirement of conditions/constraints
cannot be eliminated to realize the various filter responses. In view of this, there-
fore, we present in the following five exemplary special cases that lead to realiza-
tion of five standard filter functions.
LPF, HPF and Notch can be realized at V
01
,V
02
and V
03
respectively for
V
in1
¼
V
in3
¼
0
(grounded), V
in2
¼
V
in4
¼
input
voltage
and C
1
¼
C
2
. f
V
in1
¼
V
in2
¼
0 (grounded), V
in3
¼
V
in4
¼
input voltage, a BPF is realized at V
02,
and if V
in2
¼
V
in4
¼
0 (grounded), V
in1
¼
V
in3
¼
input voltage and 2C
2
/R
2
¼
2C
2
/
R
1
+C
1
/R
2
, an APF is realized at V
01
and a HPF is realized at V
03
.
Chen-Shen biquad A versatile MIMO-type VM biquadratic filter circuit was
proposed by Chen and Shen [
6
] employing two DVCCs with six passive elements
and is shown in Fig.
12.5
. The circuit has the flexibility to be used as either a three-
input two-output universal filter or single-input five output filter without altering the
circuit topology. The filter configuration also offers orthogonal control of
ˉ
0
and Q
0
without any conditions/constraints.
Assuming ideal DVCCs, a routine circuit analysis of Fig.
12.5
yields the
following transfer functions:
h
R
1
R
3
V
i
1
þ
R
1
R
2
V
i
3
i
sC
2
1
ð
V
i
2
Þ þ
sC
2
þ
1
R
3
1
V
01
¼
ð
12
:
11
Þ
Ds
ðÞ
h
R
1
R
2
R
3
V
i
1
þ
R
3
R
2
V
i
3
i
1
1
ð
V
i
2
Þ
sC
1
V
02
¼
ð
12
:
12
Þ
Ds
ðÞ
h
n
R
3
o
V
i
ðÞ
R
1
R
2
V
i
2
þ
R
2
V
i
3
i
s
2
C
1
C
2
1
1
R
1
R
2
R
3
1
s
2
C
1
C
2
1
þ
sC
2
V
03
¼
ð
:
Þ
12
13
Ds
ðÞ
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