13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
13: Filters
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 1 / 13
Filters
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
A filter is a circuit whose gain varies with frequency. Often a filter aims to
allow some frequencies to pass while blocking others.
• Radio/TV: a “tuning” filter blocks all frequencies
except the wanted channel
• Loudspeaker: “crossover” filters send the right
frequencies to different drive units
• Sampling: an “anti-aliasing filter” eliminates all
frequencies above half the sampling rate
◦ Phones: Sample rate = 8 kHz : filter
eliminates frequencies above 3.4 kHz.
• Computer cables: filter eliminates interference
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 2 / 13
Filters
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
A filter is a circuit whose gain varies with frequency. Often a filter aims to
allow some frequencies to pass while blocking others.
• Radio/TV: a “tuning” filter blocks all frequencies
except the wanted channel
• Loudspeaker: “crossover” filters send the right
frequencies to different drive units
• Sampling: an “anti-aliasing filter” eliminates all
frequencies above half the sampling rate
◦ Phones: Sample rate = 8 kHz : filter
eliminates frequencies above 3.4 kHz.
• Computer cables: filter eliminates interference
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 2 / 13
Filters
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
A filter is a circuit whose gain varies with frequency. Often a filter aims to
allow some frequencies to pass while blocking others.
• Radio/TV: a “tuning” filter blocks all frequencies
except the wanted channel
• Loudspeaker: “crossover” filters send the right
frequencies to different drive units
• Sampling: an “anti-aliasing filter” eliminates all
frequencies above half the sampling rate
◦ Phones: Sample rate = 8 kHz : filter
eliminates frequencies above 3.4 kHz.
[Wikipedia]
• Computer cables: filter eliminates interference
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 2 / 13
Filters
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
A filter is a circuit whose gain varies with frequency. Often a filter aims to
allow some frequencies to pass while blocking others.
• Radio/TV: a “tuning” filter blocks all frequencies
except the wanted channel
• Loudspeaker: “crossover” filters send the right
frequencies to different drive units
• Sampling: an “anti-aliasing filter” eliminates all
frequencies above half the sampling rate
◦ Phones: Sample rate = 8 kHz : filter
eliminates frequencies above 3.4 kHz.
[Wikipedia]
• Computer cables: filter eliminates interference
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 2 / 13
Filters
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
A filter is a circuit whose gain varies with frequency. Often a filter aims to
allow some frequencies to pass while blocking others.
• Radio/TV: a “tuning” filter blocks all frequencies
except the wanted channel
• Loudspeaker: “crossover” filters send the right
frequencies to different drive units
• Sampling: an “anti-aliasing filter” eliminates all
frequencies above half the sampling rate
◦ Phones: Sample rate = 8 kHz : filter
eliminates frequencies above 3.4 kHz.
[Wikipedia]
• Computer cables: filter eliminates interference
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 2 / 13
Filters
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
A filter is a circuit whose gain varies with frequency. Often a filter aims to
allow some frequencies to pass while blocking others.
• Radio/TV: a “tuning” filter blocks all frequencies
except the wanted channel
• Loudspeaker: “crossover” filters send the right
frequencies to different drive units
• Sampling: an “anti-aliasing filter” eliminates all
frequencies above half the sampling rate
◦ Phones: Sample rate = 8 kHz : filter
eliminates frequencies above 3.4 kHz.
[Wikipedia]
• Computer cables: filter eliminates interference
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 2 / 13
1st Order Low-Pass Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 3 / 13
1st Order Low-Pass Filter
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
=
b
Corner frequency: p = a =
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
1
jω
p +1
1
RC
0
|Gain| (dB)
13: Filters
-10
-20
-30
E1.1 Analysis of Circuits (2015-7256)
0.1p
p
ω (rad/s)
10p
Filters: 13 – 3 / 13
1st Order Low-Pass Filter
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
=
b
Corner frequency: p = a =
1
jω
p +1
1
RC
p
Asymptotes: 1 and jω
• Summary
0
|Gain| (dB)
13: Filters
-10
-20
-30
E1.1 Analysis of Circuits (2015-7256)
0.1p
p
ω (rad/s)
10p
Filters: 13 – 3 / 13
1st Order Low-Pass Filter
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
=
b
Corner frequency: p = a =
1
jω
p +1
1
RC
p
Asymptotes: 1 and jω
Very low ω : Capacitor = open circuit
• Summary
0
|Gain| (dB)
13: Filters
-10
-20
-30
E1.1 Analysis of Circuits (2015-7256)
0.1p
p
ω (rad/s)
10p
Filters: 13 – 3 / 13
1st Order Low-Pass Filter
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
=
b
Corner frequency: p = a =
1
jω
p +1
1
RC
p
Asymptotes: 1 and jω
Very low ω : Capacitor = open circuit
Very high ω : Capacitor short circuit
0
|Gain| (dB)
13: Filters
-10
-20
-30
E1.1 Analysis of Circuits (2015-7256)
0.1p
p
ω (rad/s)
10p
Filters: 13 – 3 / 13
1st Order Low-Pass Filter
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
=
b
Corner frequency: p = a =
1
jω
p +1
1
RC
p
Asymptotes: 1 and jω
Very low ω : Capacitor = open circuit
Very high ω : Capacitor short circuit
0
|Gain| (dB)
13: Filters
-10
-20
-30
0.1p
p
ω (rad/s)
10p
A low-pass filter because it allows low frequencies to pass but attenuates
(makes smaller) high frequencies.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 3 / 13
1st Order Low-Pass Filter
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
=
b
Corner frequency: p = a =
1
jω
p +1
1
RC
p
Asymptotes: 1 and jω
Very low ω : Capacitor = open circuit
Very high ω : Capacitor short circuit
0
|Gain| (dB)
13: Filters
-10
-20
-30
0.1p
p
ω (rad/s)
10p
A low-pass filter because it allows low frequencies to pass but attenuates
(makes smaller) high frequencies.
The order of a filter: highest power of jω in the denominator.
Almost always equals the total number of L and/or C .
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 3 / 13
Low-Pass with Gain Floor
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
R+1/jωC
4R+1/jωC
=
jωRC+1
jω4RC+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 4 / 13
Low-Pass with Gain Floor
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
R+1/jωC
4R+1/jωC
=
jωRC+1
jω4RC+1
=
jω
q +1
jω
p +1
1
1
, q = RC
Corner frequencies: p = 4RC
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
0
-5
-10
0.1q
E1.1 Analysis of Circuits (2015-7256)
p
ω
q
10q
Filters: 13 – 4 / 13
Low-Pass with Gain Floor
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
R+1/jωC
4R+1/jωC
=
jωRC+1
jω4RC+1
=
jω
q +1
jω
p +1
1
1
, q = RC
Corner frequencies: p = 4RC
Asymptotes: 1 and 14
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
0
-5
-10
0.1q
E1.1 Analysis of Circuits (2015-7256)
p
ω
q
10q
Filters: 13 – 4 / 13
Low-Pass with Gain Floor
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
=
R+1/jωC
4R+1/jωC
=
jωRC+1
jω4RC+1
=
jω
q +1
jω
p +1
1
1
, q = RC
Corner frequencies: p = 4RC
Asymptotes: 1 and 14
Very low ω :
Capacitor = open circuit
Resistor R unattached. Gain = 1
0
-5
-10
0.1q
E1.1 Analysis of Circuits (2015-7256)
p
ω
q
10q
Filters: 13 – 4 / 13
Low-Pass with Gain Floor
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
=
R+1/jωC
4R+1/jωC
=
jωRC+1
jω4RC+1
=
jω
q +1
jω
p +1
1
1
, q = RC
Corner frequencies: p = 4RC
Asymptotes: 1 and 14
Very low ω :
Capacitor = open circuit
Resistor R unattached. Gain = 1
0
-5
-10
Very high ω :
Capacitor short circuit
0.1q
p
ω
q
10q
Circuit is potential divider with gain 20 log10 14 = −12 dB.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 4 / 13
Opamp filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Inverting amplifier so
Y
X
/jωC )
= − 3R||(R+
R
1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 5 / 13
Opamp filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Inverting amplifier so
Y
X
/jωC )
3R(R+ /jωC )
= − 3R||(R+
=
−
R
R×(3R+R+1/jωC )
1
1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 5 / 13
Opamp filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Inverting amplifier so
Y
X
/jωC )
3R(R+ /jωC )
= − 3R||(R+
=
−
R
R×(3R+R+1/jωC )
1
= −3 ×
1
R+1/jωC
4R+1/jωC
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 5 / 13
Opamp filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Inverting amplifier so
Y
X
/jωC )
3R(R+ /jωC )
= − 3R||(R+
=
−
R
R×(3R+R+1/jωC )
1
= −3 ×
R+1/jωC
4R+1/jωC
1
= −3 ×
jωRC+1
jω4RC+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 5 / 13
Opamp filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Inverting amplifier so
Y
X
/jωC )
3R(R+ /jωC )
= − 3R||(R+
=
−
R
R×(3R+R+1/jωC )
1
= −3 ×
R+1/jωC
4R+1/jωC
1
= −3 ×
jωRC+1
jω4RC+1
Same transfer function as before except × − 3 = +9.5 dB.
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
10
5
0
0.1q
E1.1 Analysis of Circuits (2015-7256)
p
ω
q
10q
Filters: 13 – 5 / 13
Opamp filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
Inverting amplifier so
Y
X
/jωC )
3R(R+ /jωC )
= − 3R||(R+
=
−
R
R×(3R+R+1/jωC )
1
= −3 ×
R+1/jωC
4R+1/jωC
1
= −3 ×
jωRC+1
jω4RC+1
Same transfer function as before except × − 3 = +9.5 dB.
Advantages of op-amp crcuit:
• Summary
10
5
0
0.1q
E1.1 Analysis of Circuits (2015-7256)
p
ω
q
10q
Filters: 13 – 5 / 13
Opamp filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Inverting amplifier so
Y
X
/jωC )
3R(R+ /jωC )
= − 3R||(R+
=
−
R
R×(3R+R+1/jωC )
1
= −3 ×
R+1/jωC
4R+1/jωC
1
= −3 ×
jωRC+1
jω4RC+1
Same transfer function as before except × − 3 = +9.5 dB.
Advantages of op-amp crcuit:
1. Can have gain > 1.
10
5
0
0.1q
E1.1 Analysis of Circuits (2015-7256)
p
ω
q
10q
Filters: 13 – 5 / 13
Opamp filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Inverting amplifier so
Y
X
/jωC )
3R(R+ /jωC )
= − 3R||(R+
=
−
R
R×(3R+R+1/jωC )
1
= −3 ×
R+1/jωC
4R+1/jωC
1
= −3 ×
jωRC+1
jω4RC+1
Same transfer function as before except × − 3 = +9.5 dB.
Advantages of op-amp crcuit:
1. Can have gain > 1.
10
5
0
2. Low output impedance - loading
does not affect filter
E1.1 Analysis of Circuits (2015-7256)
0.1q
p
ω
q
10q
Filters: 13 – 5 / 13
Opamp filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Inverting amplifier so
Y
X
/jωC )
3R(R+ /jωC )
= − 3R||(R+
=
−
R
R×(3R+R+1/jωC )
1
= −3 ×
R+1/jωC
4R+1/jωC
1
= −3 ×
jωRC+1
jω4RC+1
Same transfer function as before except × − 3 = +9.5 dB.
Advantages of op-amp crcuit:
1. Can have gain > 1.
10
5
0
2. Low output impedance - loading
does not affect filter
0.1q
p
ω
q
10q
3. Resistive input impedance - does
not vary with frequency
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 5 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=−
1/jωC
R
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=−
Transformations (A)
• Conformal Filter
Transformations (B)
1/jωC
R
1
= − jωRC
20
0
• Summary
-20
0.1
E1.1 Analysis of Circuits (2015-7256)
1
ω RC
10
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=−
1/jωC
R
1
= − jωRC
C
Capacitor: i = C dv
dt
Transformations (A)
• Conformal Filter
Transformations (B)
20
0
• Summary
-20
0.1
E1.1 Analysis of Circuits (2015-7256)
1
ω RC
10
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=−
1/jωC
R
1
= − jωRC
C
Capacitor: i = C dv
dt
i=
x
R
Transformations (A)
• Conformal Filter
Transformations (B)
= −C dy
dt
20
0
• Summary
-20
0.1
E1.1 Analysis of Circuits (2015-7256)
1
ω RC
10
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=−
1/jωC
R
1
= − jωRC
C
Capacitor: i = C dv
dt
i=
dy
dt
x
R
=
Transformations (A)
• Conformal Filter
Transformations (B)
= −C dy
dt
−1
RC x
20
0
• Summary
-20
0.1
E1.1 Analysis of Circuits (2015-7256)
1
ω RC
10
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
Y
X
=−
1/jωC
R
1
= − jωRC
C
Capacitor: i = C dv
dt
i=
dy
dt
x
R
=
Rt
= −C dy
dt
−1
RC x
dy
dt
0 dt
=
−1
RC
Rt
0
xdt
20
0
• Summary
-20
0.1
E1.1 Analysis of Circuits (2015-7256)
1
ω RC
10
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
1/jωC
=−
R
1
= − jωRC
C
Capacitor: i = C dv
dt
i=
dy
dt
= −C dy
dt
x
R
=
−1
RC x
Rt
dy
dt
0 dt
y(t) =
E1.1 Analysis of Circuits (2015-7256)
=
−1
RC
−1
RC
Rt
0
Rt
0
xdt
xdt + y(0)
20
0
-20
0.1
1
ω RC
10
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
1/jωC
=−
R
1
= − jωRC
C
Capacitor: i = C dv
dt
i=
dy
dt
= −C dy
dt
x
R
=
−1
RC x
Rt
dy
dt
0 dt
y(t) =
=
−1
RC
−1
RC
Rt
0
Rt
0
xdt
0
xdt + y(0)
ωt
RNote: if x(t) = cos
cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝
E1.1 Analysis of Circuits (2015-7256)
20
-20
0.1
1
ω RC
10
1
ω.
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
1/jωC
=−
R
1
= − jωRC
C
Capacitor: i = C dv
dt
i=
dy
dt
= −C dy
dt
x
R
=
−1
RC x
Rt
dy
dt
0 dt
y(t) =
=
−1
RC
−1
RC
Rt
0
Rt
0
xdt
20
0
xdt + y(0)
ωt
RNote: if x(t) = cos
cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝
-20
0.1
1
ω RC
10
1
ω.
We can limit the LF gain to 20 dB:
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
1/jωC
=−
1
= − jωRC
R
C
Capacitor: i = C dv
dt
i=
dy
dt
= −C dy
dt
x
R
=
−1
RC x
Rt
dy
dt
0 dt
y(t) =
=
−1
RC
−1
RC
Rt
0
Rt
0
xdt
20
0
xdt + y(0)
ωt
RNote: if x(t) = cos
cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝
-20
0.1
1
ω RC
10
1
ω.
We can limit the LF gain to 20 dB:
Y
X
= − 10R||R/jωC
E1.1 Analysis of Circuits (2015-7256)
1
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
1/jωC
=−
1
= − jωRC
R
C
Capacitor: i = C dv
dt
i=
dy
dt
= −C dy
dt
x
R
=
−1
RC x
Rt
dy
dt
0 dt
y(t) =
=
−1
RC
−1
RC
Rt
0
Rt
0
xdt
20
0
xdt + y(0)
-20
0.1
ωt
RNote: if x(t) = cos
cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝
1
ω RC
10
1
ω.
We can limit the LF gain to 20 dB:
Y
X
10R× /jωC
= − 10R||R/jωC = − R(10R+
1/jωC )
E1.1 Analysis of Circuits (2015-7256)
1
1
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
1/jωC
=−
1
= − jωRC
R
C
Capacitor: i = C dv
dt
i=
dy
dt
= −C dy
dt
x
R
=
−1
RC x
Rt
dy
dt
0 dt
y(t) =
=
−1
RC
−1
RC
Rt
0
Rt
0
xdt
20
0
xdt + y(0)
-20
0.1
ωt
RNote: if x(t) = cos
cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝
1
ω RC
10
1
ω.
We can limit the LF gain to 20 dB:
Y
X
10R× /jωC
= − 10R||R/jωC = − R(10R+
1/jωC )
1
1
10
= − jω10RC+1
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 6 / 13
Integrator
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
1/jωC
=−
1
= − jωRC
R
C
Capacitor: i = C dv
dt
i=
dy
dt
= −C dy
dt
x
R
=
−1
RC x
Rt
dy
dt
0 dt
y(t) =
=
−1
RC
−1
RC
Rt
0
Rt
0
xdt
20
0
xdt + y(0)
-20
0.1
ωt
RNote: if x(t) = cos
cos(ωt)dt = ω1 sin(ωt) ⇒ gain ∝
1
ω RC
10
1
ω.
We can limit the LF gain to 20 dB:
Y
X
10R× /jωC
= − 10R||R/jωC = − R(10R+
1/jωC )
0.1
10
= − jω10RC+1
ωc = RC
E1.1 Analysis of Circuits (2015-7256)
1
1
Filters: 13 – 6 / 13
High Pass Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
R
R+1/jωC
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 7 / 13
High Pass Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
R
R+1/jωC
=
jωRC
jωRC+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 7 / 13
High Pass Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
R
R+1/jωC
=
jωRC
jωRC+1
1
Corner Freq: p = RC
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 7 / 13
High Pass Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
R
R+1/jωC
=
jωRC
jωRC+1
1
Corner Freq: p = RC
Asymptotes: jωRC and 1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 7 / 13
High Pass Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
R
R+1/jωC
=
jωRC
jωRC+1
1
Corner Freq: p = RC
Asymptotes: jωRC and 1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
0
-10
-20
-30
0.1p
E1.1 Analysis of Circuits (2015-7256)
ω
p
10p
Filters: 13 – 7 / 13
High Pass Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
Y
X
=
R
R+1/jωC
=
jωRC
jωRC+1
1
Corner Freq: p = RC
Asymptotes: jωRC and 1
Very low ω : C open circuit: gain = 0
Very high ω : C short circuit: gain = 1
• Summary
0
-10
-20
-30
0.1p
E1.1 Analysis of Circuits (2015-7256)
ω
p
10p
Filters: 13 – 7 / 13
High Pass Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
=
R
R+1/jωC
=
jωRC
jωRC+1
1
Corner Freq: p = RC
Asymptotes: jωRC and 1
Very low ω : C open circuit: gain = 0
Very high ω : C short circuit: gain = 1
We can add an op-amp to give a
low-impedance output.
0
-10
-20
-30
0.1p
E1.1 Analysis of Circuits (2015-7256)
ω
p
10p
Filters: 13 – 7 / 13
High Pass Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Y
X
=
R
R+1/jωC
=
jωRC
jωRC+1
1
Corner Freq: p = RC
Asymptotes: jωRC and 1
Very low ω : C open circuit: gain = 0
Very high ω : C short circuit: gain = 1
We can add an op-amp to give a
low-impedance output. Or add gain:
Z
X
= 1+
RB
RA
×
jωRC
jωRC+1
0
-10
-20
-30
0.1p
E1.1 Analysis of Circuits (2015-7256)
ω
p
10p
Filters: 13 – 7 / 13
2nd order filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
=
R2 +jωL
1/jωC +R1 +R2 +jωL
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 8 / 13
2nd order filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
R2 +jωL
1/jωC +R1 +R2 +jωL
=
=
LC(jω)2 +R2 Cjω
LC(jω)2 +(R1 +R2 )Cjω+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 8 / 13
2nd order filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
R2 +jωL
1/jωC +R1 +R2 +jωL
=
=
LC(jω)2 +R2 Cjω
LC(jω)2 +(R1 +R2 )Cjω+1
=
jωC(jωL+R2 )
LC(jω)2 +(R1 +R2 )Cjω+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 8 / 13
2nd order filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
R2 +jωL
1/jωC +R1 +R2 +jωL
=
=
LC(jω)2 +R2 Cjω
LC(jω)2 +(R1 +R2 )Cjω+1
=
jωC(jωL+R2 )
LC(jω)2 +(R1 +R2 )Cjω+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Asymptotes: jωR2 C and 1
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 8 / 13
2nd order filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
R2 +jωL
1/jωC +R1 +R2 +jωL
=
=
LC(jω)2 +R2 Cjω
LC(jω)2 +(R1 +R2 )Cjω+1
=
jωC(jωL+R2 )
LC(jω)2 +(R1 +R2 )Cjω+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Asymptotes: jωR2 C and 1
Corner frequencies:
+20 dB/dec at p = RL2 = 100 rad/s
−40 dB/dec at
q=
√1
LC
E1.1 Analysis of Circuits (2015-7256)
= 1000 rad/s
Filters: 13 – 8 / 13
2nd order filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
R2 +jωL
1/jωC +R1 +R2 +jωL
=
=
LC(jω)2 +R2 Cjω
LC(jω)2 +(R1 +R2 )Cjω+1
=
jωC(jωL+R2 )
LC(jω)2 +(R1 +R2 )Cjω+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Asymptotes: jωR2 C and 1
Corner frequencies:
+20 dB/dec at p = RL2 = 100 rad/s
−40 dB/dec at
q=
√1
LC
E1.1 Analysis of Circuits (2015-7256)
= 1000 rad/s
0
-20
-40
p
100
q
ω
1k
10k
Filters: 13 – 8 / 13
2nd order filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
R2 +jωL
1/jωC +R1 +R2 +jωL
=
=
LC(jω)2 +R2 Cjω
LC(jω)2 +(R1 +R2 )Cjω+1
=
jωC(jωL+R2 )
LC(jω)2 +(R1 +R2 )Cjω+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Asymptotes: jωR2 C and 1
Corner frequencies:
+20 dB/dec at p = RL2 = 100 rad/s
−40 dB/dec at
q=
√1
LC
= 1000 rad/s
0
-20
-40
p
100
q
ω
1k
10k
q
=
2 (R1 + R2 ) C = 0.6.
4ac
= Q = 0.83 = −1.6 dB (+0.04 dB due to p)
Damping factor: ζ = √ b
1
Gain error at q is 2ζ
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 8 / 13
2nd order filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Y
X
R2 +jωL
1/jωC +R1 +R2 +jωL
=
=
LC(jω)2 +R2 Cjω
LC(jω)2 +(R1 +R2 )Cjω+1
=
jωC(jωL+R2 )
LC(jω)2 +(R1 +R2 )Cjω+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Asymptotes: jωR2 C and 1
Corner frequencies:
+20 dB/dec at p = RL2 = 100 rad/s
−40 dB/dec at
q=
√1
LC
= 1000 rad/s
0
-20
-40
p
100
q
ω
1k
10k
q
=
2 (R1 + R2 ) C = 0.6.
4ac
= Q = 0.83 = −1.6 dB (+0.04 dB due to p)
Damping factor: ζ = √ b
1
Gain error at q is 2ζ
Compare with 1st order:
2nd order filter attenuates more rapidly than a 1st order filter.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 8 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
• Summary
E1.1 Analysis of Circuits (2015-7256)
[assume V+ = V− = Z ]
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC
Sub Y : Z
E1.1 Analysis of Circuits (2015-7256)
(1+jωmRC)
jωmRC
(1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC
(1+jωmRC)
(1 + 2jωRC)
jωmRC
m(jωRC)2
Z
X = m(jωRC)2 +2jωRC+1
Sub Y : Z
⇒
E1.1 Analysis of Circuits (2015-7256)
− Z (1 + jωRC) = XjωRC
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC
(1+jωmRC)
(1 + 2jωRC) − Z (1 + jωRC) =
jωmRC
m(jωRC)2
(jω/p)2
Z
X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1
Sub Y : Z
⇒
XjωRC
1
= 996 rad/s
Corner freq: p = √mRC
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC
(1+jωmRC)
(1 + 2jωRC) − Z (1 + jωRC) =
jωmRC
m(jωRC)2
(jω/p)2
Z
X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1
Sub Y : Z
⇒
XjωRC
1
1
= pRC = √1m = 0.6
= 996 rad/s, ζ = 2Q
Corner freq: p = √mRC
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
0
-20
-40
100
ω
1k
10k
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC
(1+jωmRC)
(1 + 2jωRC) − Z (1 + jωRC) =
jωmRC
m(jωRC)2
(jω/p)2
Z
X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1
Sub Y : Z
⇒
XjωRC
1
1
= pRC = √1m = 0.6
= 996 rad/s, ζ = 2Q
Corner freq: p = √mRC
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
0
-20
-40
100
Asymptotes:
ω
1k
jω
p
2
10k
and 1
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC
(1+jωmRC)
(1 + 2jωRC) − Z (1 + jωRC) =
jωmRC
m(jωRC)2
(jω/p)2
Z
X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1
Sub Y : Z
⇒
XjωRC
1
1
= pRC = √1m = 0.6
= 996 rad/s, ζ = 2Q
Corner freq: p = √mRC
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
0
-20
-40
100
Asymptotes:
ω
1k
jω
p
2
10k
and 1
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC
(1+jωmRC)
(1 + 2jωRC) − Z (1 + jωRC) =
jωmRC
m(jωRC)2
(jω/p)2
Z
X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1
Sub Y : Z
⇒
XjωRC
1
1
= pRC = √1m = 0.6
= 996 rad/s, ζ = 2Q
Corner freq: p = √mRC
Sallen-Key: 2nd order filter without inductors. Can easily have gain >1.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 9 / 13
Sallen-Key Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
0
-20
-40
100
Asymptotes:
ω
1k
jω
p
2
10k
and 1
−X
Y −Z
Y −Z
KCL @ Y : Y
1/jωC + 1/jωC +
R =0
[assume V+ = V− = Z ]
⇒ Y (1 + 2jωRC) − Z (1 + jωRC) = XjωRC
Z
+ 1Z−Y
KCL @ V+ : mR
/jωC = 0 ⇒ Z(1 + jωmRC) = Y jωmRC
(1+jωmRC)
(1 + 2jωRC) − Z (1 + jωRC) =
jωmRC
m(jωRC)2
(jω/p)2
Z
X = m(jωRC)2 +2jωRC+1 = (jω/p)2 +2ζ(jω/p)+1
Sub Y : Z
⇒
XjωRC
1
1
= pRC = √1m = 0.6
= 996 rad/s, ζ = 2Q
Corner freq: p = √mRC
Sallen-Key: 2nd order filter without inductors. Can easily have gain >1.
ζ
Designing: Choose m = ζ −2 ; C any convenient value; R = pC .
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 9 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
After much algebra:
Z
X
=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
After much algebra:
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
• Summary
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
Very low ω : C open circuit
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
Very low ω : C open circuit
Z
Non-inverting amp, X
=1+m
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
Very low ω : C open circuit
Z
Non-inverting amp, X
=1+m
Very high ω : C short circuit
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
Very low ω : C open circuit
Z
Non-inverting amp, X
=1+m
Very high ω : C short circuit
Z
Non-inverting amp, X
=1+m
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
Very low ω : C open circuit
Z
Non-inverting amp, X
=1+m
Very high ω : C short circuit
Z
Non-inverting amp, X
=1+m
At ω = p,
jω
p
2
= −1: numerator = zero resulting in infinite attenuation.
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
Very low ω : C open circuit
Z
Non-inverting amp, X
=1+m
Very high ω : C short circuit
Z
Non-inverting amp, X
=1+m
At ω = p,
jω
p
2
m+1 = 5.6dB
2ζ p
0
-20
-40
200
300
ω (rad/s)
500
= −1: numerator = zero resulting in infinite attenuation.
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
Very low ω : C open circuit
Z
Non-inverting amp, X
=1+m
Very high ω : C short circuit
Z
Non-inverting amp, X
=1+m
At ω = p,
jω
p
m+1 = 5.6dB
2ζ p
0
-20
-40
200
300
ω (rad/s)
500
2
= −1: numerator = zero resulting in infinite attenuation.
The 3 dB notch width is approximately 2ζp = 2(1 − m)p.
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Twin-T Notch Filter
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
After much algebra:
p=
(1+m)((2jωRC)2 +1)
(2jωRC)2 +4(1−m)jωRC+1
Z
X
=
=
(1+m)((jω/p)2 +1)
(jω/p)2 +2ζ(jω/p)+1
1
2RC
= 314, ζ = 1 − m = 0.1
Very low ω : C open circuit
Z
Non-inverting amp, X
=1+m
Very high ω : C short circuit
Z
Non-inverting amp, X
=1+m
At ω = p,
jω
p
m+1 = 5.6dB
2ζ p
0
-20
-40
200
300
ω (rad/s)
500
2
= −1: numerator = zero resulting in infinite attenuation.
The 3 dB notch width is approximately 2ζp = 2(1 − m)p.
Used to remove one specific frequency (e.g. mains hum @ 50 Hz)
Do not try to memorize this circuit
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 10 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
= −ω 2 LC .
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
Transformations (A)
• Conformal Filter
Transformations (B)
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
= −ω 2 LC .
0
♠
-10
-20
-30
10
100
1k
ω rad/s
10k
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
Impedance scaling:
ZL
ZR
=
jωL ZL
R , ZC
0
♠
-10
Scale all impedances by k :
Transformations (A)
• Conformal Filter
Transformations (B)
= −ω 2 LC .
-20
-30
10
100
1k
ω rad/s
10k
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
= −ω 2 LC .
0
Impedance scaling:
♠
-10
Scale all impedances by k :
′
′
R = kR, C = k
−1
′
C, L = kL
-20
-30
10
100
1k
ω rad/s
10k
• Summary
k = 20
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
= −ω 2 LC .
0
Impedance scaling:
♠
-10
Scale all impedances by k :
′
′
R = kR, C = k
−1
′
C, L = kL
Impedance ratios are unchanged
so graph stays the same.
-20
-30
10
100
1k
ω rad/s
10k
k = 20
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
= −ω 2 LC .
0
Impedance scaling:
♠
-10
Scale all impedances by k :
′
′
R = kR, C = k
−1
′
C, L = kL
Impedance ratios are unchanged
so graph stays the same.
(k is arbitrary)
-20
-30
10
100
1k
ω rad/s
10k
k = 20
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
= −ω 2 LC .
0
Impedance scaling:
♠
-10
Scale all impedances by k :
′
′
R = kR, C = k
−1
′
C, L = kL
Impedance ratios are unchanged
so graph stays the same.
(k is arbitrary)
Frequency Shift:
Scale reactive components by k :
E1.1 Analysis of Circuits (2015-7256)
-20
-30
10
100
1k
ω rad/s
10k
k = 20
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
= −ω 2 LC .
0
Impedance scaling:
♠
-10
Scale all impedances by k :
′
′
R = kR, C = k
−1
′
C, L = kL
Impedance ratios are unchanged
so graph stays the same.
(k is arbitrary)
Frequency Shift:
Scale reactive components by k :
-20
-30
10
100
1k
ω rad/s
10k
k = 20
R′ = R, C ′ = kC, L′ = kL
k=5
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
= −ω 2 LC .
0
Impedance scaling:
♠
-10
Scale all impedances by k :
′
′
R = kR, C = k
−1
′
C, L = kL
Impedance ratios are unchanged
so graph stays the same.
(k is arbitrary)
Frequency Shift:
Scale reactive components by k :
-20
-30
10
100
1k
ω rad/s
10k
k = 20
R′ = R, C ′ = kC, L′ = kL
⇒ Z ′ (k−1 ω) ≡ Z(ω)
k=5
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
0
Impedance scaling:
♥
-10
Scale all impedances by k :
′
′
R = kR, C = k
−1
′
C, L = kL
Impedance ratios are unchanged
so graph stays the same.
(k is arbitrary)
Frequency Shift:
Scale reactive components by k :
R′ = R, C ′ = kC, L′ = kL
⇒ Z ′ (k−1 ω) ≡ Z(ω)
Graph shifts left by a factor of k .
E1.1 Analysis of Circuits (2015-7256)
= −ω 2 LC .
♠
-20
-30
10
100
1k
ω rad/s
10k
k = 20
k=5
Filters: 13 – 11 / 13
Conformal Filter Transformations (A)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
VY
A dimensionless gain, V
, can always be written using dimensionless
X
impedance ratio terms:
ZR
ZC
= jωRC,
ZL
ZR
=
jωL ZL
R , ZC
0
Impedance scaling:
♥
-10
Scale all impedances by k :
′
′
R = kR, C = k
= −ω 2 LC .
−1
′
C, L = kL
Impedance ratios are unchanged
so graph stays the same.
(k is arbitrary)
Frequency Shift:
Scale reactive components by k :
R′ = R, C ′ = kC, L′ = kL
⇒ Z ′ (k−1 ω) ≡ Z(ω)
Graph shifts left by a factor of k .
♠
-20
-30
10
100
1k
ω rad/s
10k
k = 20
k=5
Must scale all reactive components in the circuit by the same factor.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 11 / 13
Conformal Filter Transformations (B)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Change LR circuit to RC:
0
♦
-10
-20
-30
1k
10k
ω rad/s
100k
1M
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 12 / 13
Conformal Filter Transformations (B)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
0
Change LR circuit to RC:
♦
-10
′
′
Change R = kL, C =
1
kR
-20
-30
1k
10k
ω rad/s
100k
1M
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
k = 106
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 12 / 13
Conformal Filter Transformations (B)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
0
Change LR circuit to RC:
♦
-10
′
Change R = kL, C
⇒
ZR′
ZC ′
′
= jωR′ C ′ =
1
= kR
jωL
ZL
=
R
ZR
-20
-30
1k
10k
ω rad/s
100k
1M
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
k = 106
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 12 / 13
Conformal Filter Transformations (B)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
0
Change LR circuit to RC:
♦
-10
′
Change R = kL, C
⇒
ZR′
ZC ′
′
= jωR′ C ′ =
1
= kR
jωL
ZL
=
R
ZR
-20
-30
1k
Impedance ratios are unchanged
at all ω so graph stays the same.
10k
ω rad/s
100k
1M
• Summary
k = 106
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 12 / 13
Conformal Filter Transformations (B)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
0
Change LR circuit to RC:
♦
-10
′
Change R = kL, C
⇒
ZR′
ZC ′
′
= jωR′ C ′ =
1
= kR
jωL
ZL
=
R
ZR
-20
-30
1k
Impedance ratios are unchanged
at all ω so graph stays the same.
(k is arbitrary)
10k
ω rad/s
100k
1M
• Summary
k = 106
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 12 / 13
Conformal Filter Transformations (B)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
0
Change LR circuit to RC:
♦
-10
′
Change R = kL, C
⇒
ZR′
ZC ′
′
= jωR′ C ′ =
1
= kR
jωL
ZL
=
R
ZR
-20
-30
1k
Impedance ratios are unchanged
at all ω so graph stays the same.
(k is arbitrary)
10k
ω rad/s
100k
1M
• Summary
Reflect frequency axis around ωm :
k = 106
Change R′ = ω k C , C ′ = ω 1kR
m
m
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 12 / 13
Conformal Filter Transformations (B)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
0
Change LR circuit to RC:
♦
-10
′
Change R = kL, C
⇒
ZR′
ZC ′
′
= jωR′ C ′ =
1
= kR
jωL
ZL
=
R
ZR
-20
-30
1k
Impedance ratios are unchanged
at all ω so graph stays the same.
(k is arbitrary)
10k
ω rad/s
100k
1M
• Summary
Reflect frequency axis around ωm :
k = 106
Change R′ = ω k C , C ′ = ω 1kR
∗m
2 m
⇒
ZR′
ZC ′
ωm
ω
=
ZC
ZR (ω)
k = 0.1, ωm = 20 k
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 12 / 13
Conformal Filter Transformations (B)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
0
Change LR circuit to RC:
♣
-10
′
Change R = kL, C
⇒
ZR′
ZC ′
′
= jωR′ C ′ =
1
= kR
jωL
ZL
=
R
ZR
-20
-30
1k
Impedance ratios are unchanged
at all ω so graph stays the same.
(k is arbitrary)
♦
10k
ω rad/s
100k
1M
• Summary
Reflect frequency axis around ωm :
k = 106
Change R′ = ω k C , C ′ = ω 1kR
∗m
2 m
⇒
ZR′
ZC ′
ωm
ω
=
ZC
ZR (ω)
(a) Magnitude graph flips
E1.1 Analysis of Circuits (2015-7256)
k = 0.1, ωm = 20 k
Filters: 13 – 12 / 13
Conformal Filter Transformations (B)
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
0
Change LR circuit to RC:
♣
-10
′
Change R = kL, C
⇒
ZR′
ZC ′
′
= jωR′ C ′ =
1
= kR
jωL
ZL
=
R
ZR
-20
-30
1k
Impedance ratios are unchanged
at all ω so graph stays the same.
(k is arbitrary)
♦
10k
ω rad/s
100k
1M
• Summary
Reflect frequency axis around ωm :
k = 106
Change R′ = ω k C , C ′ = ω 1kR
∗m
2 m
⇒
ZR′
ZC ′
ωm
ω
=
ZC
ZR (ω)
(a) Magnitude graph flips
k = 0.1, ωm = 20 k
(b) Phase graph flips and negates since ∠z ∗ = −∠z .
(k is arbitrary)
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 12 / 13
Summary
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
• The order of a filter is the highest power of jω in the transfer function
denominator.
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 13 / 13
Summary
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
• The order of a filter is the highest power of jω in the transfer function
denominator.
• Active filters use op-amps and usually avoid the need for inductors.
◦ Sallen-Key design for high-pass and low-pass.
◦ Twin-T design for notch filter: gain = 0 at notch.
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 13 / 13
Summary
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
• The order of a filter is the highest power of jω in the transfer function
denominator.
• Active filters use op-amps and usually avoid the need for inductors.
◦ Sallen-Key design for high-pass and low-pass.
◦ Twin-T design for notch filter: gain = 0 at notch.
• For filters using R and C only:
◦ Scale R and C : Substituting R′ = kR and C ′ = pC scales
−1
frequency by (pk) .
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 13 / 13
Summary
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
• The order of a filter is the highest power of jω in the transfer function
denominator.
• Active filters use op-amps and usually avoid the need for inductors.
◦ Sallen-Key design for high-pass and low-pass.
◦ Twin-T design for notch filter: gain = 0 at notch.
• For filters using R and C only:
◦ Scale R and C : Substituting R′ = kR and C ′ = pC scales
−1
frequency by (pk) .
◦ Interchange R and C : Substituting R′ = ω0kC and C ′ = kω10 R
flips
the frequency response around ω0 (∀k ).
Changes a low-pass filter to high pass and vice-versa.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 13 / 13
Summary
13: Filters
• Filters
• 1st Order Low-Pass Filter
• Low-Pass with Gain Floor
• Opamp filter
• Integrator
• High Pass Filter
• 2nd order filter
• Sallen-Key Filter
• Twin-T Notch Filter
• Conformal Filter
Transformations (A)
• Conformal Filter
Transformations (B)
• Summary
• The order of a filter is the highest power of jω in the transfer function
denominator.
• Active filters use op-amps and usually avoid the need for inductors.
◦ Sallen-Key design for high-pass and low-pass.
◦ Twin-T design for notch filter: gain = 0 at notch.
• For filters using R and C only:
◦ Scale R and C : Substituting R′ = kR and C ′ = pC scales
−1
frequency by (pk) .
◦ Interchange R and C : Substituting R′ = ω0kC and C ′ = kω10 R
flips
the frequency response around ω0 (∀k ).
Changes a low-pass filter to high pass and vice-versa.
For further details see Hayt et al. Chapter 16.
E1.1 Analysis of Circuits (2015-7256)
Filters: 13 – 13 / 13