Formula Sheets Active Filters - Analog and Digital Electronics - Electrical Engineering

 Page 1


Formula Sheet for Active Filters (Analog and Digital
Electronics) – GATE
1. Basic Concepts
• Active Filter: Uses active components (e.g., Op-Amps) with resistors and capac-
itors to ?lter signals.
• Types: Low-pass, High-pass, Band-pass, Band-stop (Notch), All-pass.
• Advantages: High input impedance, low output impedance, adjustable gain, no
inductors.
• Key Parameters: Cut-o? frequency (f
c
), passband gain (A), quality factor (Q).
2. Low-Pass Filter
• First-Order Low-Pass:
H(s) =
A
1+
s
?c
, ?
c
=
1
RC
• Cut-o? Frequency:
f
c
=
1
2pRC
• Gain at DC: A = 1+
R
f
R
1
(non-inverting), A =-
R
f
R
1
(inverting).
• Roll-o? Rate: -20dB/decade.
3. High-Pass Filter
• First-Order High-Pass:
H(s) =
A·
s
?c
1+
s
?c
, ?
c
=
1
RC
• Cut-o? Frequency:
f
c
=
1
2pRC
• Gain at High Frequencies: A = 1+
R
f
R
1
(non-inverting), A =-
R
f
R
1
(inverting).
• Roll-o? Rate: -20dB/decade.
4. Band-Pass Filter
• Transfer Function:
H(s) =
A·
s
?
0
1+
s
Q?
0
+
s
2
?
2
0
1
Page 2


Formula Sheet for Active Filters (Analog and Digital
Electronics) – GATE
1. Basic Concepts
• Active Filter: Uses active components (e.g., Op-Amps) with resistors and capac-
itors to ?lter signals.
• Types: Low-pass, High-pass, Band-pass, Band-stop (Notch), All-pass.
• Advantages: High input impedance, low output impedance, adjustable gain, no
inductors.
• Key Parameters: Cut-o? frequency (f
c
), passband gain (A), quality factor (Q).
2. Low-Pass Filter
• First-Order Low-Pass:
H(s) =
A
1+
s
?c
, ?
c
=
1
RC
• Cut-o? Frequency:
f
c
=
1
2pRC
• Gain at DC: A = 1+
R
f
R
1
(non-inverting), A =-
R
f
R
1
(inverting).
• Roll-o? Rate: -20dB/decade.
3. High-Pass Filter
• First-Order High-Pass:
H(s) =
A·
s
?c
1+
s
?c
, ?
c
=
1
RC
• Cut-o? Frequency:
f
c
=
1
2pRC
• Gain at High Frequencies: A = 1+
R
f
R
1
(non-inverting), A =-
R
f
R
1
(inverting).
• Roll-o? Rate: -20dB/decade.
4. Band-Pass Filter
• Transfer Function:
H(s) =
A·
s
?
0
1+
s
Q?
0
+
s
2
?
2
0
1
• Center Frequency:
f
0
=
1
2p
v
R
1
R
2
C
1
C
2
• Bandwidth:
BW =f
H
-f
L
=
f
0
Q
• Quality Factor:
Q =
Q =
f
0
? f
• Gain: A
0
(typically set by resistors).
5. Band-Stop (Notch) Filter
– Transfer Function:
H(s) =A·
1+
s
2
?
2
0
1+
s
Q?
0
+
s
2
?
2
0
– Center Frequency:
f
0
=
1
2p
v
LC
– Bandwidth:
BW =
f
0
Q
– Characteristics: Rejects a speci?c frequency band, passes others.
6. All-Pass Filter
– Transfer Function:
H(s) =
s-?
0
s+?
0
– Phase Shift:
? =-2tan
-1
(
f
f
0
)
, f
0
=
1
2pRC
– Gain: |H(f)| = 1 (constant amplitude, phase shift varies).
– Application: Phase correction, delay equalization.
2
Page 3


Formula Sheet for Active Filters (Analog and Digital
Electronics) – GATE
1. Basic Concepts
• Active Filter: Uses active components (e.g., Op-Amps) with resistors and capac-
itors to ?lter signals.
• Types: Low-pass, High-pass, Band-pass, Band-stop (Notch), All-pass.
• Advantages: High input impedance, low output impedance, adjustable gain, no
inductors.
• Key Parameters: Cut-o? frequency (f
c
), passband gain (A), quality factor (Q).
2. Low-Pass Filter
• First-Order Low-Pass:
H(s) =
A
1+
s
?c
, ?
c
=
1
RC
• Cut-o? Frequency:
f
c
=
1
2pRC
• Gain at DC: A = 1+
R
f
R
1
(non-inverting), A =-
R
f
R
1
(inverting).
• Roll-o? Rate: -20dB/decade.
3. High-Pass Filter
• First-Order High-Pass:
H(s) =
A·
s
?c
1+
s
?c
, ?
c
=
1
RC
• Cut-o? Frequency:
f
c
=
1
2pRC
• Gain at High Frequencies: A = 1+
R
f
R
1
(non-inverting), A =-
R
f
R
1
(inverting).
• Roll-o? Rate: -20dB/decade.
4. Band-Pass Filter
• Transfer Function:
H(s) =
A·
s
?
0
1+
s
Q?
0
+
s
2
?
2
0
1
• Center Frequency:
f
0
=
1
2p
v
R
1
R
2
C
1
C
2
• Bandwidth:
BW =f
H
-f
L
=
f
0
Q
• Quality Factor:
Q =
Q =
f
0
? f
• Gain: A
0
(typically set by resistors).
5. Band-Stop (Notch) Filter
– Transfer Function:
H(s) =A·
1+
s
2
?
2
0
1+
s
Q?
0
+
s
2
?
2
0
– Center Frequency:
f
0
=
1
2p
v
LC
– Bandwidth:
BW =
f
0
Q
– Characteristics: Rejects a speci?c frequency band, passes others.
6. All-Pass Filter
– Transfer Function:
H(s) =
s-?
0
s+?
0
– Phase Shift:
? =-2tan
-1
(
f
f
0
)
, f
0
=
1
2pRC
– Gain: |H(f)| = 1 (constant amplitude, phase shift varies).
– Application: Phase correction, delay equalization.
2
7. Second-OrderFilters(Butterworth,Chebyshev,
etc.)
– Butterworth Filter:
|H(f)| =
A
v
1+
(
f
fc
)
2n
where n: Filter order.
– Cut-o? Frequency:
f
c
=
1
2p
v
R
1
R
2
C
1
C
2
– Quality Factor for Band-Pass:
Q =
v
R
1
R
2
C
1
C
2
R
1
C
2
+R
2
C
1
+R
1
C
1
(1-A)
– Characteristics: Butterworth (?at passband), Chebyshev (steeper roll-o?,
passband ripple).
8. Sallen-Key Topology
– Low-Pass:
f
c
=
1
2p
v
R
1
R
2
C
1
C
2
– High-Pass:
f
c
=
1
2p
v
R
1
R
2
C
1
C
2
– Quality Factor:
Q =
v
R
1
R
2
C
1
C
2
R
1
(C
1
+C
2
)
– Gain: A = 1+
R
f
R
1
(non-inverting con?guration).
9. Frequency Response Parameters
– Gain in dB:
A
dB
= 20log
10
|H(f)|
– 3 dB Frequency:
f
3dB
=f
c
(where gain drops to
A
0
v
2
)
– Roll-o? Rate: -20ndB/decade for n-th order ?lter.
3
Page 4


Formula Sheet for Active Filters (Analog and Digital
Electronics) – GATE
1. Basic Concepts
• Active Filter: Uses active components (e.g., Op-Amps) with resistors and capac-
itors to ?lter signals.
• Types: Low-pass, High-pass, Band-pass, Band-stop (Notch), All-pass.
• Advantages: High input impedance, low output impedance, adjustable gain, no
inductors.
• Key Parameters: Cut-o? frequency (f
c
), passband gain (A), quality factor (Q).
2. Low-Pass Filter
• First-Order Low-Pass:
H(s) =
A
1+
s
?c
, ?
c
=
1
RC
• Cut-o? Frequency:
f
c
=
1
2pRC
• Gain at DC: A = 1+
R
f
R
1
(non-inverting), A =-
R
f
R
1
(inverting).
• Roll-o? Rate: -20dB/decade.
3. High-Pass Filter
• First-Order High-Pass:
H(s) =
A·
s
?c
1+
s
?c
, ?
c
=
1
RC
• Cut-o? Frequency:
f
c
=
1
2pRC
• Gain at High Frequencies: A = 1+
R
f
R
1
(non-inverting), A =-
R
f
R
1
(inverting).
• Roll-o? Rate: -20dB/decade.
4. Band-Pass Filter
• Transfer Function:
H(s) =
A·
s
?
0
1+
s
Q?
0
+
s
2
?
2
0
1
• Center Frequency:
f
0
=
1
2p
v
R
1
R
2
C
1
C
2
• Bandwidth:
BW =f
H
-f
L
=
f
0
Q
• Quality Factor:
Q =
Q =
f
0
? f
• Gain: A
0
(typically set by resistors).
5. Band-Stop (Notch) Filter
– Transfer Function:
H(s) =A·
1+
s
2
?
2
0
1+
s
Q?
0
+
s
2
?
2
0
– Center Frequency:
f
0
=
1
2p
v
LC
– Bandwidth:
BW =
f
0
Q
– Characteristics: Rejects a speci?c frequency band, passes others.
6. All-Pass Filter
– Transfer Function:
H(s) =
s-?
0
s+?
0
– Phase Shift:
? =-2tan
-1
(
f
f
0
)
, f
0
=
1
2pRC
– Gain: |H(f)| = 1 (constant amplitude, phase shift varies).
– Application: Phase correction, delay equalization.
2
7. Second-OrderFilters(Butterworth,Chebyshev,
etc.)
– Butterworth Filter:
|H(f)| =
A
v
1+
(
f
fc
)
2n
where n: Filter order.
– Cut-o? Frequency:
f
c
=
1
2p
v
R
1
R
2
C
1
C
2
– Quality Factor for Band-Pass:
Q =
v
R
1
R
2
C
1
C
2
R
1
C
2
+R
2
C
1
+R
1
C
1
(1-A)
– Characteristics: Butterworth (?at passband), Chebyshev (steeper roll-o?,
passband ripple).
8. Sallen-Key Topology
– Low-Pass:
f
c
=
1
2p
v
R
1
R
2
C
1
C
2
– High-Pass:
f
c
=
1
2p
v
R
1
R
2
C
1
C
2
– Quality Factor:
Q =
v
R
1
R
2
C
1
C
2
R
1
(C
1
+C
2
)
– Gain: A = 1+
R
f
R
1
(non-inverting con?guration).
9. Frequency Response Parameters
– Gain in dB:
A
dB
= 20log
10
|H(f)|
– 3 dB Frequency:
f
3dB
=f
c
(where gain drops to
A
0
v
2
)
– Roll-o? Rate: -20ndB/decade for n-th order ?lter.
3
10. Design Considerations
– Op-Amp Selection: High slew rate, low noise, high gain-bandwidth prod-
uct.
– Component Tolerance: A?ects f
c
, Q; use precision resistors/capacitors.
– Applications: Audio processing, signal conditioning, noise ?ltering.
– Stability: Ensure Op-Amp unity-gain bandwidth »f
c
.
4