Active Filters
1. Second order active low pass filter.
C1
R2
R1
Vi
LF356
V0
C2
Fig.1 2nd order unity gain Sallen-Key active low pass filter circuit.
𝑉0(𝑠)
𝑉𝑖 (𝑠)
=
𝐻 (𝑠 ) =
𝑄=
𝜔0
2𝛼
1
𝑅1 𝑅2 𝐶1 𝐶2
𝑅 𝐶 +𝑅 𝐶
1
𝑠 2 + 1 2 2 2 𝑠+
𝑅1 𝑅2 𝐶1 𝐶2
𝑅1 𝑅2 𝐶1 𝐶2
𝜔0 2
Eq. (2)
𝑠 2 + 2 𝑠+ 𝜔0 2
=
√𝑅1 𝑅2 𝐶1 𝐶2
(𝑅1 +𝑅2 )𝐶1
;
Eq. (1).
=
1
2𝑄
; 𝑓𝑐
𝜔0
=
2𝜋
where: Q = quality factor
 = attenuation factor
 = damping factor
0 = cut-off freq.(rad/s)
f0 = cut-off freq.(Hz)
2. Second order active high pass filter.
R1
Vi
C1
C2
LF356
R2
V0
Fig.2 . 2nd order unity gain Sallen-Key active high pass filter circuit.
𝑉0(𝑠)
𝑉𝑖 (𝑠)
=
𝑠2
𝑅 𝐶 +𝑅 𝐶
𝑠 2 + 1 1 1 2 𝑠+
𝑅1 𝑅2 𝐶1 𝐶2
1
𝑅1 𝑅2 𝐶1 𝐶2
Eq. (3).
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𝐻 (𝑠 ) =
𝑄=
𝑠2
𝑠 2 + 2 𝑠+ 𝜔0 2
𝜔0
√𝑅 𝑅 𝐶 𝐶
= 1 2 1 2; 
2𝛼
(𝐶1 +𝐶2 )𝑅1
Eq. (4)
=
1
2𝑄
; 𝑓𝑐
=
𝜔0
2𝜋
where: Q = quality factor
 = attenuation factor
 = damping factor
0 = cut-off freq.(rad/s)
f0 = cut-off freq.(Hz)
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