ACTIVE_FILTERS_Theory_and_Design

Sallen–Key Filters 29
v i
R 1 R 2
C 1 C 2
v o
v i
+ K
v o
(a)
(b)
FIGURE 2.7 (a) Passive RC network, (b) active device (op-amp).
RC network is best achieved in a feedback configuration, although there are several
ways to choose (Figure 2.8a).
From Figure 2.8a, for node v 1 , we have:
1 ⎛ 1 1 ⎞ 1
− + ⎜ + +
1⎟ 1− 2
−
1 0
= 0
R V sC V
⎝ R R ⎠ R V sCV
i
1 1 2
2
(2.15)
node
−
1 ⎛ 1
1
+ ⎜
R V
⎝ R
+
v 2
2
2
⎞
sC2⎟ V2
= 0
⎠
(2.16)
but
V
2
V0
=
K
(2.17)
jω
R 1 R 2
C 1
v 2
v 1
v i
C 2
(a)
+ K
v o
K = 0
K = 1 j K = 2
j0.5
−1.5 0.5
−j0.5
1 1.5 σ
K = 1
−j K = 2
(b)
FIGURE 2.8 (a) An RC passive network and an op-amp are combined to form a general
active filter. (b) When the gain K is varied throughout the s-plane, the response with respect
to pole position is obtained.