Active filters
Active filters are a special family of filters. They take their name from the fact
that, aside from passive components, they also contain active elements such
as transistors or operational amplifiers. Just like passive filters, depending on
the design, they retain or eliminate a specific portion of a signal.
These types of filters have an advantage over passive filters, mainly because
bulky inductors at low frequencies can be avoided and higher quality factors
can be obtained.
Active filters can be implemented with different topologies:
1.
2.
3.
4.
5.
6.
7.
8.
Akerberg-Mossberg
Biquadratic
Dual Amplifier BandPass (DABP)
Fliege
Multiple feedback
Voltage-Controlled Voltage-Source (VCVS) and Sallen/Key
State variable
Wien
Active filters also come in different varieties:
1. Butterworth
2. Linkwitz-Riley
3. Bessel
4. Chebyshev (2 types)
5. Elliptic or Cauer
6. Synchronous
7. Gaussian
8. Legendre-Paupolis
9. Butterworth-Thomson or Linear phase
10. Transitional or Paynter
The Butterworth filter has the flattest response in the pass-band. The
Chebyshev Type II filter has the steepest cutoff. The Linkwitz-Riley filter is
often used in audio applications (crossovers).
Note: Cauer is the name of an active filter but it’s also the name of a passive
topology. The two are different concepts.
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Sallen/Key topology
The Sallen/Key topology was invented by R. P. Sallen and E. L. Key at MIT
Lincoln Laboratory in 1955.
It is a degenerate form of a Voltage-Controlled Voltage-Source (VCVS) filter
topology. It features an extremely high input impedance (practically infinite)
and an extremely low output impedance (practically zero). These two
characteristics are provided by the op-amp and they are often desired in
circuit design for signal integrity.
The network for the Sallen/Key topology includes an op-amp, often in a buffer
configuration, and a set of resistors and capacitors. The op-amp can
sometimes be substituted by an emitter follower or a source follower circuit
since both circuits produce unity gain. Cascading two or more stages will
produce higher-order filters.
Sallen/Key generic configuration for 0dB gain (unity-gain)
Unlike RC passive filters, where only one pole is present in the circuit so that
the gain drops by 6bB/octave or 20dB/decade past the cutoff frequency, the
Sallen/Key circuit topology has two poles so the gain drops by 12bB/octave or
40dB/decade past the cutoff frequency.
Considering two signals with magnitudes A1 and A2, the definitions for octave
and decade are
A
20 log10  1
 A2

1
  20 log10    6dB / octave
2

and
A
20 log10  1
 A2

1
  20 log10    20dB / decade
 10 

where octave means twice the magnitude and decade means ten times the
magnitude.
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Low-pass
The low-pass filter blocks high-frequency signals while leaving low-frequency
signals untouched.
C1
-797.3uV
1n
V
10k
10k
U1
+
10.00V
OS2
OUT
2
C2
V1
-
OS1
-10.00V
V2
10Vdc
6
1
V3
V
10Vdc
AD741
1n
0V
V-
10.00V
5
-10.00V
0V
4
1Vac
0Vdc
3
-1.595mV
V+
R2
V-
R1
0V
V+
7
V+
0V
0
V-
0
0V
-1.575mV
0
0
Sallen/Key low-pass filter
1.0V
(15.910K,500.000m)
0.5V
0V
1.0Hz
V(R1:1)
10Hz
V(U1:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1mHz to 1MHz
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 10k  10k  1nF  1nF
 15.915kHz
The quality factor is
Q
R1 R2 C1C 2
C 2 R1  R2 

10k  10k  1nF  1nF
 0.5
1nF  10k  10k 
This circuit is critically damped.
3
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-0
(16.094K,-6.1371)
-20
-40
(160.941K,-40.508)
-60
1.0Hz
10Hz
20*LOG10(V(U1:OUT)/V(V1:+))
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 15.915kHz and it decreases by –40dB/decade.
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High-pass
The high-pass filter blocks low-frequency signals while leaving high-frequency
signals untouched.
R1
-778.0uV
10k
C2
-797.3uV
+
10.00V
220n
R2
V1
-
AD741
10k
0V
V3
10Vdc
1
OS1
-10.00V
V2
6
OUT
2
V-
10.00V
5
OS2
V
10Vdc
-10.00V
0V
4
1Vac
0Vdc
U1
V-
V
220n
3
V+
C1
0V
V+
7
V+
0V
0
V-
0
0V
-778.0uV
0
0
Sallen/Key high-pass filter
1.0V
(72.334,500.000m)
0.5V
0V
1.0Hz
V(C1:1)
10Hz
V(R1:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
10MHz
100MHz
Frequency
AC sweep from 1mHz to 100MHz
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 10k  10k  220nF  220nF
 72.34 Hz
The quality factor is
Q
R1 R2 C1C 2
R1 C1  C 2 

10k  10k  220nF  220nF
 0.5
10k220nF  220nF 
This circuit is critically damped.
5
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-0
(71.969,-6.1226)
-20
-40
(7.2718,-40.000)
-60
-80
1.0Hz
10Hz
100Hz
20*LOG10(V(R1:2)/V(C1:1))
1.0KHz
10KHz
100KHz
1.0MHz
10MHz
100MHz
Frequency
Bode plot from 1Hz to 100MHz
The gain drops to –6dB at 72Hz and it decreases by –40dB/decade.
Note: the gain starts to roll off at 1MHz which is the bandwidth of the AD741
op-amp when it’s connected in a buffer/non-inverting configuration.
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Band-pass
The band-pass filter blocks low and high frequency signals. It peaks at the
so-called center frequency. Ra and Rb provide gain. C1-R1 form a low-pass
filter and C2-R2 form a high-pass filter.
R3
-782.6uV
-1.565mV
10k
V
10k
220n
V1
U1 10.00V
+
OS2
OUT
2
R2
C1
3
V+
-1.595mV
-
AD741
OS1
20k
220n
-10.00V
V2
6
V3
10Vdc
1
-10.00V
V-
10.00V
5
V
Ra
4
1Vac
0Vdc
C2
V-
R1
0V
V+
7
V+
10k
V-
10Vdc
0V
0V
0
0
0V
0V
-1.575mV
0
0
0
Rb
20k
0
Sallen/Key band-pass filter
1.0V
(72.287,1.0000)
0.5V
0V
1.0Hz
V(R1:1)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1mHz to 1MHz
The center frequency is
fc 
1
2
R3  R1
1

C1C 2 R1 R2 R3 2
10k  10k
 72.34 Hz
220nF  220nF  10k  20k  10k
The gains are
G  1
Ra
10k
 1
 1  0 .5  1 .5
Rb
20k
A
G
1.5

1
3  G 3  1.5
where G is the internal gain and A is the external gain.
The value of G should be below 3 to avoid oscillation.
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The previous is a Sallen/Key circuit as long as the value of Rb is twice the
value of Ra. If A is more or less than unity, the circuit provides amplification
and becomes a VCVS filter.
R3
-285.5uV
10k
V
V1
7
10.00V
+
OS2
OUT
2
R2
C1
3
V+
-
AD741
OS1
5
20k
220n
-10.00V
V3
10Vdc
1
-10.00V
V-
V2
6
V
Ra
4
1Vac
0Vdc
-1.595mV
220n
V+
U1 10.00V
V+
0V
C2
V-
R1
-571.0uV
10k
1Meg
V-
10Vdc
0V
0V
0
0
0V
0V
-1.575mV
0
0
0
Rb
20k
0
VCVS band-pass filter I
1.2V
(66.069,1.0625)
0.8V
0.4V
0V
1.0Hz
V(R1:1)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1mHz to 1MHz
The center frequency is
fc 
1
2
R3  R1
1

C1C 2 R1 R2 R3 2
10k  10k
 72.34 Hz
220nF  220nF  10k  20k  10k
However, if Ra>>Rb, the frequency response is rather flat.
G  1
Ra
1M
 1
 1  50  51
Rb
20k
A
G
51

 1.06
3  G 3  51
This circuit will oscillate because G is greater than 3.
Note: the magnitude of the output is 6% higher than the input and this is an
indication of amplification.
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R3
-748.5uV
V
-1.595mV
220n
V1
7
10.00V
+
OS2
OUT
2
R2
C1
3
V+
-
AD741
OS1
5
20k
220n
-10.00V
V2
1
V3
V
Ra
-10.00V
V-
10Vdc
6
4
1Vac
0Vdc
10k
V+
U1 10.00V
V+
0V
C2
V-
R1
-1.497mV
10k
V-
1k
10Vdc
0V
0V
0
0
0V
0V
-1.575mV
0
0
0
Rb
1Meg
0
VCVS band-pass filter II
1.0V
(72.287,500.734m)
0.5V
0V
1.0Hz
V(R1:1)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1mHz to 1MHz
The center frequency is
fc 
1
2
R3  R1
1

C1C 2 R1 R2 R3 2
10k  10k
 72.34 Hz
220nF  220nF  10k  20k  10k
If Ra<<Rb, the gain drops to ½ at the center frequency.
G  1
Ra
1k
 1
 1 0  1
Rb
1M
A
G
1

 0.5
3  G 3 1
The minimum gain attainable by this circuit is 0.5.
Connecting the op-amp in the buffer configuration would produce the same
frequency response.
Note: the magnitude of the output at the center frequency is 50% lower than
the input.
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-0
(72.444,-718.738u)
-20
(1.0000,-33.667)
-40
(100.000m,-53.667)
(10.000K,-39.293)
-60
(100.000K,-59.369)
-80
1.0mHz
10mHz
100mHz
20*LOG10(V(R3:2)/V(R1:1))
1.0Hz
10Hz
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Sallen/Key band-pass filter: Bode plot from 1mHz to 1MHz
The center frequency is at 72Hz. The gain decreases by –20dB/decade to the
left and right of the center frequency.
0
(66.069,526.810m)
-40
-80
1.0mHz
10mHz
100mHz
20*LOG10(V(R3:2)/V(R1:1))
1.0Hz
10Hz
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
VCVS band-pass filter I: Bode plot from 1mHz to 1MHz
The response is rather flat.
-0
(72.444,-6.0081)
-20
(1.0000,-37.181)
-40
(100.000m,-57.180)
(10.000K,-42.805)
-60
(100.000K,-62.833)
-80
1.0mHz
10mHz
100mHz
1.0Hz
V(R1:1)
20*LOG10(V(R3:2)/V(R1:1))
10Hz
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
VCVS band-pass filter II: Bode plot from 1mHz to 1MHz
The center frequency is at 72Hz with –6dB. The gain decreases by
–20dB/decade to the left and right of the center frequency.
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Biquadratic topology
The biquadratic topology takes its name from the fact its transfer function is
the ratio of two quadratic functions. It can be implemented in two ways:
single-amplifier biquadratic or two-integrator-loop.
An example of this filter topology is the so-called Tow-Thomas circuit. This
circuit consists of three op-amps and it can be used as a low-pass or
band-pass filter, depending on where its output is taken.
R2
1k
R3
V+
1k
C2
C1
R6
V2
1u
1u
1k
5Vdc
V-
+
OS2
5
4
-
V
OS1
OUT
3
band-pass
+
OS2
1
6
5
uA741
U3
4
2
1k
V-
R4
R5
2
1k
V-
3
V1
6
V+
OS1
OUT
0
V-
U2
-
OS1
OUT
7
3
V+
4
-
V3
-5Vdc
0
uA741
1
7
1Vac
0Vdc
1k
V-
U1
V+
V
2
V-
uA741
R1
V-
+
OS2
1
6
5
low-pass
V
V+
7
0
V+
0
0
V+
0
Biquadratic filter
1.2V
(158.489,1.0007)
0.8V
0.4V
0V
1.0Hz
V(R1:1)
10Hz
V(R2:2)
100Hz
V(U1:OUT)
1.0KHz
10KHz
100KHz
1.0MHz
10MHz
Frequency
AC sweep from 1Hz to 10MHz
The band-pass gain is G BP  
The low-pass gain is G LP  
R4
.
R2
R2
.
R1
The output for the band-pass option is shown in red.
The output for the low-pass option is shown in blue.
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The center/cutoff frequency is
fc 
1
2 R2 R4 C1C 2

1
2 1k  1k  1F  1F
 159.15Hz
The quality factor is
Q
R32C1
1k3  1F

 0.03
R2 R4C2
1k  1k  1F
The circuit is overdamped.
The bandwidth is approximately given by
BW 
2f c 2  159.15Hz

 33.33kHz
Q
0.03
-0
-50
-100
1.0Hz 3.0Hz
10Hz
30Hz
100Hz 300Hz
1.0KHz 3.0KHz
10KHz
V(R1:1) 20*log10(V(R4:1)/V(R1:1)) 20*log10(V(R6:2)/V(R1:1))
Frequency
30KHz
100KHz
300KHz
1.0MHz
3.0MHz 10MHz
Bode plots from 1Hz to 10MHz
The center frequency for band-pass filter is at 159Hz and 0dB. The gain
decreases by –20dB/decade to the left and right.
The cutoff frequency for low-pass filter is at 159Hz. The gain decreases
linearly by –40dB/decade to the right of it.
The bandwidth for both plots is about 33kHz.
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Multiple feedback topology
The multiple feedback topology takes its name from the fact it has positive
and negative feedback.
R3
C2
1k
1n
V+
V-
1k
1k
2
-
V
OS1
+
1n
V+
OUT
3
C1
V1
OS2
AD741
1
V2
5Vdc
6
5
0
V+
0
V3
-5Vdc
V
7
1Vac
0Vdc
U1
4
R2
V-
R1
V-
0
0
0
Multiple feedback filter
1.0V
0.5V
(151.741K,333.257m)
0V
1.0Hz
V(R1:1)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
(159.224K,316.568m)
100KHz
1.0MHz
10MHz
Frequency
AC sweep from 1Hz to 10MHz
The transfer function for the circuit is
V
1
H s   o   2

Vi
As  Bs  C
K o2
s2 
o
Q
s   o2
where
A  R1 R2 C1C 2  1k  1k  1nF  1nF  1 ps
RRC
1k  1k  1nF
B  R2 C 2  R1C 2  1 2 2  1k  1nF  1k  1nF 
 3s
R3
1k
R
1k
C 1 
1
R3 1k
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www.ice77.net
K 
Q
fc 
R3
1k

 1
R1
1k
R2 R3C1C2
1k  1k  1nF  1nF

 0.333
R3  R2  K R2  C2 1k  1k   1  1k  1nF
1
2 R2 R3 C1C 2

1
2 1k  1k  1nF  1nF
 159.15kHz
 o  2f c  2  159.15kHz  999.968kHz
where K is the DC voltage gain, Q is the quality factor, fc is the cutoff
frequency and ωo is the angular frequency.
The circuit is overdamped.
40
(160.456K,-10.060)
0
(1.6046M,-38.113)
(16.046M,-61.581)
-40
-80
1.0Hz
V(R1:1)
10Hz
100Hz
1.0KHz
20*LOG10(V(U1:OUT)/V(R1:1))
10KHz
100KHz
1.0MHz
10MHz
100MHz
Frequency
Bode plot from 1Hz to 100MHz
The gain drops to –10.060dB at 160.456kHz and then it decreases in a
nonlinear fashion to the right of the cutoff frequency.
Several stages can be cascaded to obtain high-order multiple feedback filters.
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First-order filters
First-order filters are very simple. They have only one capacitor which in turn
produces a single pole. The –3dB frequency is given by
f 3dB 
1
2RC
First-older filters come in combinations of non-inverting, inverting,
low-pass and high-pass. All the filters presented here have unity-gain.
Non-inverting low-pass
This circuit lets the low-frequency signals through without inverting the input.
V+
OS2
OUT
2
V1
-
V-
OS1
5
V2
3.6Vdc
6
1
V3
-3.6Vdc
V
AD741
C1
4
1Vac
0Vdc
+
V+
3
2k
V-
V
U1
7
V+
R1
47n
0
0
V-
0
0
First-order non-inverting low-pass filter
The –3dB frequency is given by
f 3dB 
1
1

 1.693kHz
2RC 2  2k  47 nF
1.0V
(1.6930K,707.101m)
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The –3dB frequency for the filter is at 1.693kHz.
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10
(1.6853K,-2.9899)
-0
(16.853K,-20.004)
-20
(168.526K,-40.039)
-40
-60
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(U1:OUT)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1.693kHz and then it decreases by –20dB/decade.
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Inverting low-pass
This circuit lets the low-frequency signals through while inverting the input.
C1
220n
R1
723.4
VV+
U1
R2
V
2
-
723.4
VOS1
OUT
3
1Vac
0Vdc
4
V1
+ 7
OS2
V+
1AD741
V-
V2
6
3.6Vdc
5
V3
-3.6Vdc
V
0
0
V+
0
0
First-order inverting low-pass filter
The –3dB frequency is given by
f 3dB 
1
1

 1kHz
2RC 2  723.4  220nF
1.0V
(1.0005K,706.625m)
0.5V
0V
1.0Hz
3.0Hz
V(R2:1) V(C1:1)
10Hz
30Hz
100Hz
300Hz
1.0KHz
3.0KHz
10KHz
30KHz
100KHz
300KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The –3dB frequency for the filter is at 1kHz.
17
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10
(1.0000K,-3.0140)
0
(10.115K,-20.095)
-20
(128.825K,-36.092)
-40
1.0Hz
3.0Hz
10Hz
30Hz
V(R2:1) 20*LOG10(V(C1:1)/V(R2:1))
100Hz
300Hz
1.0KHz
3.0KHz
10KHz
30KHz
100KHz
300KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –20dB/decade.
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Non-inverting high-pass
This circuit lets the high-frequency signals through without inverting the input.
V+
5
OS2
-
V-
V2
3.6Vdc
6
OUT
2
V1
1
OS1
V3
-3.6Vdc
V
AD741
R1
4
1Vac
0Vdc
+
V+
3
47n
V-
V
U1
7
V+
C1
0
2k
0
0
V-
0
First-order non-inverting high-pass filter
The –3dB frequency is given by
f 3dB 
1
1

 1.693kHz
2RC 2  2k  47 nF
1.0V
(1.6933K,707.101m)
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The –3dB frequency for the filter is at 1.693kHz.
0
(1.6853K,-3.0304)
(168.526,-20.084)
-40
(16.853,-40.041)
-80
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(U1:OUT)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1.693kHz and then it decreases by –20dB/decade.
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Inverting high-pass
This circuit lets the high-frequency signals through while inverting the input.
C2
100p
R2
723.4
VV+
U1
C1
V
R1
220n
2
-
723.4
VOS1
OUT
3
V1
1Vac
0Vdc
4
+ 7
OS2
V+
1AD741
V-
V2
6
3.6Vdc
5
V3
-3.6Vdc
V
0
0
V+
0
0
First-order inverting high-pass filter
The 100pF capacitor ensures stability at high frequency.
The –3dB frequency is given by
f 3dB 
1
1

 1kHz
2RC1 2  723.4  220nF
1.0V
(1.0000K,707.461m)
0.5V
0V
1.0Hz
3.0Hz
V(C1:2) V(C2:1)
10Hz
30Hz
100Hz
300Hz
1.0KHz
3.0KHz
10KHz
30KHz
100KHz
300KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The –3dB frequency for the filter is at 1kHz.
20
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(1.0000K,-3.0060)
-0
(100.000,-20.044)
-20
(10.000,-40.001)
-40
-60
1.0Hz
3.0Hz
10Hz
30Hz
V(C1:2) 20*LOG10(V(C2:1)/V(C1:2))
100Hz
300Hz
1.0KHz
3.0KHz
10KHz
30KHz
100KHz
300KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –20dB/decade.
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Second-order filters
Second-order filters have two poles which are given by two capacitors.
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2
Depending on low-pass or high-pass configurations, the quality factors are
R1 R2 C1C 2
Q
C 2 R1  R2 
Q
R1 R2 C1C 2
R1 C1  C 2 
low-pass
high-pass
where C2 is the shunt capacitor for the low-pass configuration and R1 is the
bridging resistor in the high-pass configuration (consistency is important).
The quality factors for 2nd-order and higher filters are summarized here:
Butterworth
Linkwitz-Riley
Bessel
Chebyshev (0.5dB)
Chebyshev (1dB)
Chebyshev (2dB)
Chebyshev (3dB)
Transitional or Paynter
Butterworth-Thomson or Linear phase
0.7071
0.5
0.577
0.8637
0.9565
1.1286
1.3049
0.639
0.6304
Every type of filter is designed to retain a portion of a signal for a specific
range of frequencies while suppressing the same signal at other undesired
frequencies. The major difference among filters is in the mathematical
framework that lies behind them. Every filter has different ratios among
resistors and capacitors and this produces a different response.
Filters of second and higher orders come in unity-gain or gain versions. If they
are in the unity-gain configuration, the op-amp is in buffer mode. If gain is
needed, then two additional resistors are used to provide the proper gain.
The trick behind designing 2nd-order filters is to set up a system of two
equations (fc and Q) in two unknowns (C and R). The first equation sets the
frequency. The second equation sets the quality factor. It is necessary to
choose specific values for frequency and quality factor while leaving two
passive components unknowns. By using software like Maple it is possible to
force a convergence and calculate the two unknowns (C and R).
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Butterworth filter
The Butterworth filter was originally proposed by Stephen Butterworth in 1930.
This filter can be implemented with different orders. For every order, the gain
of the filter will drop by –6dB/octave or –20dB/decade past the cutoff
frequency. Increasing the order of the filter will produce a sharper cutoff.
The Butterworth filter has a very flat response and does not present ripples in
the pass-band. It can be arranged for low-pass, high-pass, band-pass and
band-stop/notch purposes.
A band-pass Butterworth filter is obtained by placing an inductor in parallel
with each capacitor to form resonant circuits. The value of each additional
component must be selected to resonate with the other component at the
frequency of interest.
A band-stop/notch Butterworth filter is obtained by placing an inductor in
series with each capacitor to form resonant circuits. The value of each
additional component must be selected to resonate with the other component
at the frequency to be rejected.
The Butterworth filter can be implemented with different topologies, including
Cauer (passive) and Sallen/Key (active).
For a Butterworth filter, the quality factor must be 0.707.
Second-order low-pass
In the following example, the circuit is implemented with the Sallen/Key
topology. The capacitor values do not match here. This is a second-order filter
because it has two capacitors.
C1
200n
V+
V+
V-
U1
3
V1
1k
OUT
C2
2
100n
-
V-
1Vac
0Vdc
1k
V-
0
OS2
OS1
LM741
5
V2
V3
6
1
V
5Vdc
-5Vdc
4
V
+
V+
7
R2
R1
0
0
0
Butterworth filter (low-pass) (2nd-order)
Note: R1=R2 and C1=2xC2.
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1.0V
(1.1253K,707.101m)
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(U1:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 1k  1k  200nF  100nF
 1.125kHz
The quality factor is
Q
R1 R2 C1C 2
C 2 R1  R2 

1k  1k  200nF  100nF
 0.707
100nF  1k  1k 
20
-0
(1.1307K,-3.0528)
-20
-40
(11.307K,-40.148)
-60
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(U1:-)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1.125kHz and then it decreases by –40dB/decade.
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Second-order high-pass
In the following example, the circuit is implemented with the Sallen/Key
topology. The resistor values do not match here. This is a second-order filter
because it has two capacitors.
R1
1k
V+
V+
V-
V
10n
+
V+
3
10n
V1
OS2
OUT
2
R2
-
OS1
LM741
5
V2
V3
6
1
V
5Vdc
-5Vdc
4
1Vac
0Vdc
C2
V-
C1
7
U1
2k
V-
0
0
0
0
Butterworth filter (high-pass) (2nd-order)
Note: C1=C2 and R2=2xR1.
1.0V
(11.247K,707.101m)
0.5V
0V
1.0Hz
V(C1:1)
10Hz
V(U1:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 1k  2k  10nF  10nF
 11.253kHz
The quality factor is
Q
25
R1 R2 C1C 2
R1 C1  C 2 

1k  2k  10nF  10nF
 0.707
1k  10nF  10nF 
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40
0
(11.142K,-3.0893)
(1.1142K,-40.169)
-100
(111.424,-80.170)
-200
1.0Hz
V(C1:1)
10Hz
100Hz
20*LOG10(V(U1:-)/V(C1:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 10Hz to 1MHz
The gain drops to –3dB at about 11.253kHz and then it decreases by
–40dB/decade.
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Third-order low-pass (same capacitor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The capacitor values match here and the circuit provides gain by
means of two additional resistors. This is a third-order filter because it has
three capacitors.
V+
V+
V-
U1
1Vac
0Vdc
C1
V1
7
476
C2
150n
+
V+
3
1059
2294
OS2
OUT
C3
2
150n
150n
-
V-
V
R3
R2
OS1
LM741
5
V2
1
0
5Vdc
V
-5Vdc
R4
1.2k
V-
0
V3
6
4
R1
0
0
0
R5
1k
0
Butterworth filter (low-pass) (3rd-order)
Note: C1=C2=C3.
3.0V
(10.000,2.1999)
2.0V
(1.0029K,1.5710)
1.0V
0V
1.0Hz
V(V1:+)
10Hz
V(C2:1)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
The gain is given by the gain resistors:
A
27
R4  R5 1.2k  1k

 2 .2
R5
1k
or
+6.848dB
www.ice77.net
50
(10.000,6.8481)
(1.0000K,3.9619)
0
(10.000K,-53.006)
-50
-100
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(C2:1)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is flat at +6.8481dB in the lower frequency range and then it drops to
+3.9619dB at 1kHz. It eventually decreases to –53.006dB one decade later.
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Third-order low-pass (cascaded)
This circuit is implemented with the Sallen/Key topology by cascading a
simple low-pass filter after a second-order filter. For the second-order block
the quality factor must be 1 whereas the quality factor for the low-pass filter is
not defined. This is a third-order filter because it has three capacitors. The
overall quality factor is 0.707.
C1
400n
V+
V+
V-
U1
3
795
OUT
C2
V1
OS2
2
-
OS1
LM741
5
R3
V2
V3
6
1
V
1.59k
V
5Vdc
-5Vdc
C3
100n
4
1Vac
0Vdc
795
V-
V
+
V+
7
R2
R1
100n
V-
0
0
0
0
0
Butterworth filter (low-pass) (3rd-order)
Note: R1=R2 and C1=4xC2.
1.2V
(1.0000K,1.0009)
(1.0000K,708.091m)
0.8V
0.4V
0V
1.0Hz
V(V1:+)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
V(R3:2)
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
f c1 
f c2
1

1
2 R1 R2 C1C 2 2 795  795  400nF  100nF
1
1


 1kHz
2R3 C 3 2  1.59k  100nF
 1kHz
The quality factor of the second-order block is
Q1 
29
R1 R2 C1C 2
C 2 R1  R2 

795  795  400nF  100nF
1
100nF  795  795 
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20
(1.0000K,-2.9982)
0
-50
(10.000K,-60.083)
-100
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(U1:OUT)/ V(V1:+))
1.0KHz
20*LOG10(V(R3:2)/ V(V1:+))
Frequency
10KHz
100KHz
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –60dB/decade.
30
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Third-order high-pass (same resistor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a third-order filter because it has three capacitors.
V+
V+
V-
V1
47n
R1
+
V+
3
47n
R2
OS2
OUT
2
R3
-
OS1
LM741
5
V2
V3
6
1
V
5Vdc
-5Vdc
4
1Vac
0Vdc
47n
C3
V-
V
C2
7
U1
C1
2431
954
16.73k
0
V-
0
0
0
0
Butterworth filter (high-pass) (3rd-order)
Note: C1=C2=C3.
1.0V
(1.0000K,707.639m)
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(R2:1)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
100
(1.0000K,-3.0038)
0
(100.000,-60.007)
-100
(10.000,-120.007)
-200
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(R2:1)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –60dB/decade.
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Third-order high-pass (cascaded)
This circuit is implemented with the Sallen/Key topology by cascading a
simple high-pass filter after a second-order filter. For the second-order block
the quality factor must be 1 whereas the quality factor for the low-pass filter is
not defined. This is a third-order filter because it has three capacitors. The
overall quality factor is 0.707.
R1
750
V+
V+
V-
3
106n
V
106n
OUT
R2
V1
OS2
2
-
OS1
LM741
5
V2
C3
V3
6
1
V
100n
V
5Vdc
-5Vdc
R3
3k
4
1Vac
0Vdc
+
V+
C2
V-
C1
7
U1
1.59k
V-
0
0
0
0
0
Butterworth filter (high-pass) (3rd-order)
Note: C1=C2 and R2=4xR1.
1.2V
(1.0000K,0.9991)
(1.0000K,706.096m)
0.8V
0.4V
0V
1.0Hz
V(C1:2)
10Hz
V(R1:2)
V(R3:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
f c1 
f c2
1

1
2 R1 R2 C1C 2 2 750  3k  106nF  106nF
1
1


 1kHz
2R3 C 3 2  1.59k  100nF
 1kHz
The quality factor of the second-order block is
Q1 
32
R1 R2 C1C 2
R1 C1  C 2 

750  3k  106nF  106nF
1
750  106nF  106nF 
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100
(1.0000K,-3.0227)
0
(100.000,-60.025)
-100
-200
1.0Hz
3.0Hz
10Hz
30Hz
V(C1:2) 20*log10(V(U1:-)/V(C1:2))
100Hz
300Hz
20*log10(V(R3:2)/V(C1:2))
1.0KHz
3.0KHz
10KHz
30KHz
100KHz
300KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –60dB/decade.
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Fourth-order low-pass (same resistor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The resistor values match here and the circuit provides gain by
means of two additional resistors. This is a fourth-order filter because it has
four capacitors.
V+
V+
V-
U1
V
1k
V1
1k
C1
3
+
V+
7
R4
R3
1k
1k
C2
C3
LM741
5
OS2
C4
2
-
V2
V3
6
OUT
1
OS1
5Vdc
V
-5Vdc
R5
212n
296n
214n
47n
4
1Vac
0Vdc
R2
V-
R1
V-
0
0
430
0
0
0
R6
1k
0
Butterworth filter (low-pass) (4th-order)
Note: all resistor values match.
(10.000,1.4299)
1.5V
(1.0009K,1.0385)
1.0V
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(C1:1)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
20
(10.000,3.1064)
(1.0000K,343.421m)
0
-40
(9.3325K,-61.319)
-80
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(C1:1)/V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is flat at +3.1064dB in the lower frequency range and then it drops to
+0.343dB at 1kHz. It eventually decreases to –61.319dB at 9.3325kHz.
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Fourth-order low-pass (same capacitor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The capacitor values match here and the circuit provides gain by
means of two additional resistors. This is a fourth-order filter because it has
four capacitors.
V+
V+
V-
U1
V
495
V1
2.65k
C1
3
195
1.08k
C2
+
V+
7
R4
R3
C3
OS2
OUT
C4
2
-
OS1
LM741
5
V2
V3
6
1
5Vdc
V
-5Vdc
R5
220n
220n
220n
220n
4
1Vac
0Vdc
R2
V-
R1
V-
0
0
1.2k
0
0
0
R6
1k
0
Butterworth filter (low-pass) (4th-order)
Note: all capacitor values match.
2.6V
(10.000,2.1999)
2.0V
(1.0005K,1.5311)
1.0V
0V
1.0Hz
V(V1:+)
10Hz
V(R5:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
20
(10.000,6.8481)
(1.0000K,3.7091)
-0
-20
-40
(8.7096K,-50.861)
-60
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(R5:2)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is flat at +6.8481dB in the lower frequency range and then it drops to
+3.7091dB at 1kHz. It eventually decreases to –50.861dB at 8.7096kHz.
35
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Fourth-order low-pass (cascaded)
This circuit is implemented with the Sallen/Key topology by cascading two
second-order filters. The quality factors for the first and the second block must
be 0.5412 and 1.3065 respectively. This is a fourth-order filter because it has
four capacitors. The overall quality factor is 0.707.
C1
C3
120n
685n
V+
U1
C2
2
V-
-
100n
OS1
6
1
U2
V
7
R4
R3
3
610
610
OS2
OUT
C4
2
100n
V-
0
+
V+
OUT
LM741
5
-
V-
OS2
OS1
V+
LM741
5
V-
6
1
V2
V
5Vdc
4
+
1.45k
1.45k
V1
1Vac
0Vdc
V+
3
4
V
V+
7
R2
R1
V3
-5Vdc
V-
0
0
0
0
th
Butterworth filter (low-pass) (4 -order)
1.2V
(1.0063K,706.509m)
0.8V
(1.0000K,548.870m)
0.4V
0V
1.0Hz
V(V1:+)
10Hz
V(C1:1)
V(C3:1)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
f c1 
f c2
1
1
 1001Hz
2 1.45k  1.45k  120nF  100nF
1


 997 Hz
2 R1 R2 C1C 2 2 610  610  685nF  100nF
2 R1 R2 C1C 2
1

The quality factors are
Q1 
R1 R2 C1C 2
C 2 R1  R2 

1.45k  1.45k  120nF  100nF
 0.5477
100nF  1.45k  1.45k 
Q2 
R3 R4 C 3 C 4

1k  1k  685nF  100nF
 1.3086
100nF  1k  1k 
C 4  R3  R 4 
The quality factors are reasonably close to 0.5412 and 1.3065.
36
www.ice77.net
40
(1.0000K,-2.9035)
-0
(1.0000K,-5.2106)
-40
-80
(10.000K,-80.232)
-120
1.0Hz
V(V1:+)
10Hz
100Hz
1.0KHz
20*LOG10(V(C1:1)/ V(V1:+))
20*LOG10(V(C3:1)/ V(V1:+))
Frequency
10KHz
100KHz
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –2.9035dB at 1kHz and then it decreases by
–80dB/decade.
37
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Fourth-order high-pass (same capacitor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The capacitor values match here and the circuit provides gain by
means of two additional resistors. This is a fourth-order filter because it has
four capacitors.
V+
V+
V-
V1
100n
R1
100n
R2
3
+
V+
C4
100n
R3
OS2
OUT
R4
2
-
OS1
LM741
5
V2
V3
6
1
5Vdc
V
-5Vdc
R5
1195
850
1.2k
5.28k
4
1Vac
0Vdc
100n
C3
V-
V
C2
7
U1
C1
V-
0
0
440
0
0
0
R6
1k
0
Butterworth filter (high-pass) (4th-order)
Note: all capacitor values match.
1.8V
(10.000K,1.4233)
1.5V
(1.0000K,1.0184)
1.0V
0.5V
0V
1.0Hz
V(C1:2)
10Hz
V(R3:1)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
40
(10.000K,3.0658)
0
(1.0000K,158.734m)
-100
(100.000,-76.834)
-200
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(R5:2)/V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is flat at +3.6058dB in the higher frequency range and then it drops
to +0.158dB at 1kHz. It eventually decreases to –76.834 at 100Hz.
38
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Fourth-order high-pass (cascaded)
This circuit is implemented with the Sallen/Key topology by cascading two
second-order filters. The quality factors for the first and the second block must
be 0.5412 and 1.3065 respectively. This is a fourth-order filter because it has
four capacitors. The overall quality factor is 0.707.
R1
R3
1k
1k
V+
U1
V+
7
OUT
2
-
1.3k
OS1
6
1
U2
C3
V
7
LM741
5
C4
23n
3
100n
OS2
OUT
R4
2
11k
V-
0
+
V+
V+
OS2
-
V-
1Vac
0Vdc
100n
R2
V1
+
OS1
V+
LM741
5
V-
6
1
V2
V
5Vdc
4
192n
3
V-
V
C2
4
C1
V3
-5Vdc
V-
0
0
0
0
Butterworth filter (high-pass) (4th-order)
1.2V
(1.0094K,706.509m)
0.8V
(1.0000K,537.091m)
0.4V
0V
1.0Hz
V(V1:+)
10Hz
V(R1:2)
V(U2:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
f c1 
f c2 
1
2 R1 R2 C1C 2
1
2 R1 R 2 C1C 2


1
2 1k  1.3k  192nF  100nF
1
2 1k  11k  23nF  100nF
 1007 Hz
 1kHz
The quality factors are
Q1 
R1 R2 C1C 2
R1 C1  C 2 

1k  1.3k  192nF  100nF
 0.5411
1k  192nF  100nF 
Q2 
R1 R2 C1C 2

1k  11k  23nF  100nF
 1.2932
1k  23nF  100nF 
R1 C1  C 2 
The quality factors are reasonably close to 0.5412 and 1.3065.
39
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60
(1.0000K,-3.1741)
-0
(1.0000K,-5.3991)
-100
(100.000,-80.138)
-200
-300
1.0Hz
V(V1:+)
10Hz
100Hz
1.0KHz
20*LOG10(V(R1:2)/ V(V1:+))
20*LOG10(V(U2:-)/V(V1:+))
Frequency
10KHz
100KHz
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –80dB/decade.
40
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Fifth-order low-pass (same resistor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The resistor values match here and the circuit provides gain by
means of two additional resistors. This is a fifth-order filter because it has five
capacitors.
V+
V+
V-
U1
V
1k
1k
1k
C2
C4
154.56n
182.49n
+
1k
1k
C3
7
3
OS2
OUT
C5
2
83.06n
-
OS1
LM741
5
V2
V3
6
1
5Vdc
V
34.57n
-5Vdc
R6
4
246.83n
R5
R4
V+
R3
C1
V1
1Vac
0Vdc
R2
V-
R1
1k
V-
0
0
0
0
0
0
R7
1k
0
Butterworth filter (low-pass) (5th-order)
Note: all resistor values match.
3.0V
(1.0000K,2.5885)
(10.000,2.0000)
2.0V
1.0V
0V
1.0Hz
V(V1:+)
10Hz
V(C2:1)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
50
(10.000,6.0207)
(1.0000K,8.2609)
0
-50
(10.000K,-75.288)
-100
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(C2:1)/V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is flat at +6dB in the lower frequency range and then it peaks to
+8.2609dB at 1kHz. It eventually decreases to –75.288dB at 10kHz.
41
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Fifth-order low-pass (cascaded)
This circuit is implemented with the Sallen/Key topology by cascading a
simple low-pass filter after two second-order filters. The quality factors for the
first and the second block must be 0.6180 and 1.6181 whereas the quality
factor for the low-pass filter is not defined. This is a fifth-order filter because it
has five capacitors. The overall quality factor is 0.707.
C1
C3
153n
1050n
V+
U1
C2
V1
1Vac
0Vdc
2
-
OS1
3
1
V
490
+
490
OS2
OUT
4
2
-
OS1
LM741
5
R5
V2
V3
6
1
V
1.59k
5Vdc
V
-5Vdc
C5
100n
V-
0
V-
7
6
C4
100n
V+
U2
R4
R3
V+
OUT
LM741
5
V-
1.285k
OS2
4
1.285k
+
V+
3
V-
V
V+
7
R2
R1
100n
V-
0
0
0
0
0
Butterworth filter (low-pass) (5th-order)
1.3V
1.0V
(1.0033K,705.621m)
0.5V
(1.0033K,617.409m)
0V
1.0Hz
V(R1:1)
10Hz
V(U1:-)
V(U2:-)
100Hz
V(R5:2)
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
f c1 
f c2
f c3
42
1
1
 1001Hz
2 1.285k  1.285k  153nF  100nF
1


 1002 Hz
2 R3 R4 C 3 C 4 2 490  490  1050nF  100nF
1
1


 1kHz
2R5 C 5 2  1.59k  100nF
2 R1 R2 C1C 2
1

www.ice77.net
The quality factors are
Q1 
R1 R2 C1C 2
C 2 R1  R2 

Q2 
R3 R 4 C 3 C 4

C 4  R3  R 4 
1.285k  1.285k  153nF  100nF
 0.6185
100nF  1.285k  1.285k 
490  490  1050nF  100nF
 1.6202
100nF  490  490 
The quality factors are reasonably close to 0.6180 and 1.6181.
50
(1.0000K,-2.9558)
-0
-50
-100
(10.000K,-100.244)
-150
1.0Hz
V(R1:1)
10Hz
100Hz
1.0KHz
20*LOG10(V(U1:-)/V(R1:1))
20*LOG10(V(U2:-)/V(R1:1))
Frequency
10KHz
100KHz
20*LOG10(V(R5:2)/V(R1:1))
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –100dB/decade.
43
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Fifth-order high-pass (cascaded)
This circuit is implemented with the Sallen/Key topology by cascading a
simple low-pass filter after two second-order filters. The quality factors for the
first and the second block must be 0.6180 and 1.6181 whereas the quality
factor for the high-pass filter is not defined. This is a fifth-order filter because it
has five capacitors. The overall quality factor is 0.707.
R1
R3
1.285k
1k
V+
U1
V+
7
OUT
R2
2
-
OS1
LM741
5
C4
6
3
1
V
48n
OS2
OUT
4
2
10.475k
V-
0
+
50n
R4
2.44k
V+
V-
U2
C3
7
OS2
V+
V+
50n
V1
1Vac
0Vdc
+
-
V-
207n
3
OS1
LM741
5
1
V2
C5
6
V
100n
V3
-5Vdc
R5
1.59k
V-
0
5Vdc
V
4
V
C2
V-
C1
0
0
0
0
Butterworth filter (high-pass) (5th-order)
1.3V
1.0V
(1.0000K,703.322m)
(1.0000K,617.445m)
0.5V
0V
1.0Hz
V(C1:2)
10Hz
V(U1:OUT)
V(R3:2)
100Hz
V(R5:2)
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
f c1 
f c2 
f c3
44
1
2 R1 R2 C1C 2
1

1
2 1k  2.44k  207 nF  50nF
1
 1001Hz

 1003Hz
2 R3 R4 C 3C 4 2 1k  10.475k  48nF  50nF
1
1


 1kHz
2R5 C 5 2  1.59k  100nF
www.ice77.net
The quality factors are
Q1 
R1 R2 C1C 2
R1 C1  C 2 

1k  2.44k  207nF  50nF
 0.6183
1k  207nF  50nF 
Q2 
R1 R2 C1C 2

1k  10.475k  48nF  50nF
 1.6179
1k  48nF  50nF 
R1 C1  C 2 
The quality factors are reasonably close to 0.6180 and 1.6181.
100
(1.0000K,-3.0569)
0
(100.000,-100.100)
-200
-400
1.0Hz
V(C1:2)
10Hz
100Hz
20*LOG10(V(U1:OUT)/V(C1:2))
1.0KHz
20*LOG10(V(R3:2)/V(C1:2))
Frequency
10KHz
100KHz
20*LOG10(V(R5:2)/V(C1:2))
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –100dB/decade.
45
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Sixth-order low-pass (same resistor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The resistor values match here and the circuit provides gain by
means of two additional resistors. This is a sixth-order filter because it has six
capacitors.
V+
V+
V-
1k
1k
1k
C2
C3
205n
465n
R6
1k
C4
3
212n
+
1k
OS2
OUT
C5
218n
V+
R5
C6
2
104n
36n
V-
1Vac
0Vdc
1k
C1
V1
R4
R3
-
OS1
LM741
5
V2
1
0
0
5Vdc
V
-5Vdc
R7
1k
V-
0
V3
6
4
V
R2
7
U1
R1
0
0
0
R8
1k
0
Butterworth filter (low-pass) (6th-order)
Note: all resistor values match.
2.6V
(10.000,1.9999)
2.0V
(1.0000K,1.4347)
1.0V
0V
1.0Hz
V(R1:1)
10Hz
V(C5:1)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
30
(10.000,6.0202)
(1.0000K,3.1353)
0
-40
(3.8019K,-65.177)
-80
1.0Hz
V(R1:1)
10Hz
100Hz
20*LOG10(V(C5:1)/V(R1:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is flat at +6dB in the lower frequency range and then it drops to
+3.1353dB at 1kHz. It eventually decreases to –65.177dB at 3.8019kHz.
46
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Sixth-order low-pass (cascaded)
For sake of illustration, a sixth-order low-pass Butterworth filter is shown
below. The circuit is nothing more than a cascade of 3 second-order blocks
like the ones previously shown. The stages must have Q1=0.5177, Q2=1.7071
and Q3=1.9320.
C1
C3
100n
OS1
1
V
165
C4
2
100n
-
OS1
LM741
5
6
3
1
V
619
+
204
OS2
OUT
C6
2
100n
V-
0
V+
V-
U3
R6
R5
7
OS2
OUT
V-
0
+
V+
3
767
V+
7
R4
R3
6
V+
-
2000n
U2
V-
2
LM741
5
4
OS2
OUT
C2
V1
1Vac
0Vdc
+
490
V-
2.584k
V+
3
4
V
V+
7
R2
R1
C5
2000n
U1
-
V-
V+
OS1
LM741
5
V2
6
-5Vdc
V
0
V-
0
V3
5Vdc
1
4
200n
0
0
Butterworth filter (low-pass) (6th-order)
3.0V
(1.0000K,1.7122)
2.0V
1.0V
0V
1.0Hz
V(R1:1)
10Hz
V(U1:-)
V(U2:OUT)
100Hz
V(U3:-)
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The plot shown above clearly demonstrates that increasing the order of the
filter will produce a sharper response. Increasing the order of the filter will
produce a roughly vertical drop at the cutoff frequency.
The cutoff frequencies are
f c1 
1
1
 1kHz
2 R1 R2 C1C 2 2 2.584k  490  200nF  100nF
1
1
f c2 

 1kHz
2 R1 R2 C1C 2 2 767  165  2000nF  100nF
1
1
f c3 

 1.001kHz
2 R1 R2 C1C 2 2 619  204  2000nF  100nF
47

www.ice77.net
The quality factors are
Q1 
R1 R2 C1C 2
C 2 R1  R2 

2.584k  490  200nF  100nF
 0.5177
100nF  2.584k  490 
Q2 
R1 R2 C1C 2
C 2 R1  R2 

767  165  2000nF  100nF
 1.7070
100nF  767  165 
Q3 
R1 R2 C1C 2

619  204  2000nF  100nF
 1.9310
100nF  619  204 
C 2 R1  R2 
The quality factors are reasonably close to 0.5177, 1.7071 and 1.9320.
100
(1.0000K,-5.7160)
(1.0000K,-1.0514)
0
-100
(1.0000K,4.6711)
(10.000K,-121.211)
-200
1.0Hz
V(R1:1)
10Hz
100Hz
1.0KHz
20*LOG10(V(U1:-)/V(R1:1))
20*LOG10(V(U2:OUT)/V(R1:1))
Frequency
10KHz
100KHz
20*LOG10(V(U3:-)/V(R1:1))
1.0MHz
Bode plot from 1Hz to 1MHz
The gain peaks to +4.6711dB at 1 kHz and then it decreases by about
–120dB/decade.
48
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Sixth-order high-pass (cascaded)
For sake of illustration, a sixth-order high-pass Butterworth filter is shown
below. The circuit is nothing more than a cascade of 3 second-order blocks
like the ones previously shown. The stages must have Q1=0.5177, Q2=1.7071
and Q3=1.9320 respectively.
R1
R3
615
R5
V+
186
U1
V+
7
0
165
100n
OUT
2
R4
V-
-
OS1
3.396k
V+
V+
LM741
5
V-
U3
C5
6
1
V
7
OS2
C6
400n
3
+
100n
OS2
OUT
2
R6
4
1.03k
+
V+
400n
3
V-
V-
OS1
V
C4
-
OS1
LM741
5
V2
6
V3
5Vdc
1
-5Vdc
V
4
-
1
C3
4
R2
6
U2
7
V+
OS2
OUT
2
V1
1Vac
0Vdc
+
100n
LM741
5
V+
400n
3
V-
V
C2
V-
C1
3.844k
0
0
V-
0
0
0
Butterworth filter (high-pass) (6th-order)
3.0V
(1.0000K,1.7095)
2.0V
1.0V
0V
1.0Hz
V(C1:1)
10Hz
V(C3:1)
V(U2:-)
100Hz
V(U3:-)
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The plot shown above clearly demonstrates that increasing the order of the
filter will produce a sharper response. Increasing the order of the filter will
produce a roughly vertical drop at the cutoff frequency.
The cutoff frequencies are
f c1 
f c2 
f c3 
49
1
2 R1 R2 C1C 2
1
2 R1 R2 C1C 2
1
2 R1 R2 C1C 2



1
2 615  1.03k  400nF  100nF
1
2 186  3.396k  400nF  100nF
1
2 165  3.844k  400nF  100nF
 999.9 Hz
 1001Hz
 999.2 Hz
www.ice77.net
The quality factors are
Q1 
R1 R2 C1C 2
R1 C1  C 2 

615  1.03k  400nF  100nF
 0.5177
615  400nF  100nF 
Q2 
R1 R2 C1C 2
R1 C1  C 2 

186  3.396k  400nF  100nF
 1.7092
186  400nF  100nF 
Q3 
R1 R2 C1C 2

165  3.844k  400nF  100nF
 1.9307
165  400nF  100nF 
R1 C1  C 2 
The quality factors are reasonably close to 0.5177, 1.7071 and 1.9320.
200
(1.0000K,-1.0675)
(1.0000K,-5.7166)
(1.0000K,-1.0675)
0
(1.0000K,4.6575)
-200
-400
1.0Hz
V(C1:1)
(100.000,-119.930)
10Hz
100Hz
1.0KHz
20*LOG10(V(C3:1)/V(C1:1))
20*LOG10(V(U2:-)/V(C1:1))
Frequency
10KHz
100KHz
20*LOG10(V(U3:-)/V(C1:1))
1.0MHz
Bode plot from 1Hz to 1MHz
The gain peaks at +4.6575dB at 1kHz and then it decreases by about
–120dB/decade.
50
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Linkwitz-Riley filter
The Linkwitz-Riley filter was invented by Siegfried Linkwitz and Russ Riley in
1978. This filter is alternatively called Butterworth squared filter (squared
because for the Linkwitz-Riley filter Q=0.5, for the Butterworth filter Q=0.707
and the square of 0.707 is 0.5). This filter is used in audio crossovers.
The Linkwitz-Riley filter can be implemented with different orders. For every
order, the gain of the filter will drop by –6dB/octave or –20dB/decade past the
cutoff frequency. Increasing the order of the filter will produce a sharper cutoff.
A 2nth-order Linkwitz-Riley filter can be obtained from cascading 2 nth-order
Butterworth filters (2 2nd-order Butterworth filters will produce a 4th-order
Linkwitz-Riley filter).
In a way, the Linkwitz-Riley filter is a superset of the Butterworth filter which in
turn exploits the Sallen/Key topology.
For a Linkwitz-Riley filter, the quality factor must be 0.5.
Second-order low-pass
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors.
C1
100n
V+
V+
V-
U1
1k
V+
+
1k
V1
C2
2
100n
-
V-
0
OS2
OUT
V-
1Vac
0Vdc
3
OS1
LM741
5
V2
V3
6
1
V
5Vdc
-5Vdc
4
V
7
R2
R1
0
0
0
Linkwitz-Riley filter (low-pass) (2nd-order)
Note: R1=R2 and C1=C2.
51
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1.0V
(1.5915K,500.000m)
0.5V
0V
1.0Hz
V(R1:1)
10Hz
V(C1:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 1k  1k  100nF  100nF
 1.5915kHz
The quality factor is
Q
R1 R2 C1C 2
C 2 R1  R2 

1k  1k  100nF  100nF
 0.5
100nF  1k  1k 
20
-0
(1.5849K,-5.9848)
-20
-40
(15.849K,-40.062)
-60
1.0Hz
V(R1:1)
10Hz
100Hz
20*LOG10(V(C1:2)/V(R1:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 1.5915kHz and then it decreases by
–40dB/decade.
52
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Second-order high-pass
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors.
R1
1.5k
V+
V+
V-
V
10n
3
10n
V1
OS2
OUT
2
R2
-
OS1
LM741
5
V2
V3
6
1
V
5Vdc
-5Vdc
4
1Vac
0Vdc
+
V+
C2
V-
C1
7
U1
1.5k
0
V-
0
0
0
Linkwitz-Riley filter (high-pass) (2nd-order)
Note: C1=C2 and R1=R2.
1.0V
(10.635K,501.385m)
0.5V
0V
1.0Hz
V(C1:1)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 1.55k  1.65k  10nF  10nF
 10.61kHz
The quality factor is
Q
53
R1 R 2 C1C 2
R1 C1  C 2 

1.5k 1.5k 10nF 10nF
 0 .5
1.5k  10nF  10nF 
www.ice77.net
40
(10.540K,-6.0746)
-0
(1.0540K,-40.197)
-40
(105.404,-80.113)
-80
-120
10Hz
V(C1:1)
100Hz
20*LOG10(V(U1:OUT)/V(C1:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 10Hz to 1MHz
The gain drops to –6dB at about 10.61kHz and then it decreases by
–40dB/decade.
54
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Fourth-order low-pass (cascaded)
This circuit is implemented with the Sallen/Key topology by cascading two
identical second-order filters. The quality factors for the blocks must be 0.707
(equivalent to two cascaded Butterworth stages). This is a fourth-order filter
because it has four capacitors. The overall quality factor is 0.5.
C1
C3
200n
200n
V+
U1
2
100n
-
OS1
U2
6
1
V
7
R4
R3
3
1k
1k
OS2
OUT
C4
2
V-
0
+
V+
OUT
LM741
5
100n
-
V-
OS2
OS1
V+
LM741
5
V-
6
1
V2
V
5Vdc
4
1k
C2
V1
1Vac
0Vdc
+
V-
1k
V+
3
4
V
V+
7
R2
R1
V3
-5Vdc
V-
0
0
0
0
Linkwitz-Riley filter (low-pass) (4th-order)
1.0V
(1.1254K,707.134m)
(1.1254K,500.000m)
0.5V
0V
1.0Hz
V(R1:1)
10Hz
V(R3:1)
V(U2:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
f c1  f c 2 
1
2 R1 R2 C1C 2

1
2 1k  1k  200nF  100nF
 1.125kHz
The quality factors are
Q1  Q2 
55
R1 R2 C1C 2
C 2 R1  R2 

1k  1k  200nF  100nF
 0.707
100nF  1k  1k 
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30
(1.1316K,-3.0594)
-0
(1.1316K,-6.1196)
-40
-80
(11.316K,-80.323)
-120
1.0Hz
V(R1:1)
10Hz
100Hz
1.0KHz
20*LOG10(V(R3:1)/V(R1:1))
20*LOG10(V(U2:-)/V(R1:1))
Frequency
10KHz
100KHz
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 1.125kHz and then it decreases by –80dB/decade.
56
www.ice77.net
Fourth-order high-pass (same capacitor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The capacitor values match here and the circuit provides gain by
means of two additional resistors. This is a fourth-order filter because it has
four capacitors.
V+
V+
V-
V1
100n
R1
100n
R2
3
+
V+
C4
100n
R3
OS2
OUT
R4
2
-
OS1
LM741
5
V2
V3
6
1
5Vdc
V
-5Vdc
R5
1.125k
906
1.12k
5.623k
4
1Vac
0Vdc
100n
C3
V-
V
C2
7
U1
C1
V-
0
0
320
0
0
0
R6
1k
0
Linkwitz-Riley filter (high-pass) (4th-order)
Note: all capacitor values match.
1.5V
(10.000K,1.3061)
1.0V
(1.0000K,662.007m)
0.5V
0V
1.0Hz
V(C1:2)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
60
(1.0000K,-3.5828)
(10.000K,2.3198)
0
-100
(100.000,-77.614)
-200
1.0Hz
V(C1:2)
10Hz
100Hz
20*LOG10(V(U1:OUT)/V(C1:2))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is flat at +2.3198dB in the higher frequency range and then it drops
to –3.5828dB at 1kHz. It eventually decreases by –80dB/decade.
57
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Fourth-order high-pass (cascaded)
This circuit is implemented with the Sallen/Key topology by cascading two
identical second-order filters. The quality factors for the blocks must be 0.707
(equivalent to two cascaded Butterworth stages). This is a fourth-order filter
because it has four capacitors. The overall quality factor is 0.5.
R1
R3
1k
V+
1k
U1
10n
OUT
2
-
OS1
C3
6
1
V
C4
10n
3
2k
0
+
10n
OS2
OUT
2
R4
4
R2
V-
U2
7
OS2
V+
V+
7
V+
+
V+
10n
V1
1Vac
0Vdc
3
LM741
5
-
OS1
LM741
5
V2
6
5Vdc
1
V3
-5Vdc
V
4
V-
V-
V
C2
V-
C1
2k
0
V-
0
0
0
Linkwitz-Riley filter (high-pass) (4th-order)
1.0V
(11.245K,707.211m)
(11.245K,500.000m)
0.5V
0V
1.0Hz
V(C1:1)
10Hz
V(R1:2)
V(U2:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
f c1  f c 2 
1
2 R1 R2 C1C 2

1
2 1k  2k  10nF  10nF
 11.253kHz
The quality factors are
Q1  Q2 
58
R1 R2 C1C 2
R1 C1  C 2 

1k  2k  10nF  10nF
 0.707
1k  10nF  10nF 
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60
(11.167K,-6.1380)
0
(11.167K,-3.0696)
(1.1167K,-80.261)
-200
-350
1.0Hz
V(C1:1)
10Hz
100Hz
1.0KHz
20*LOG10(V(R1:2)/V(C1:1))
20*LOG10(V(U2:-)/V(C1:1))
Frequency
10KHz
100KHz
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 11.253kHz and then it decreases by
–80dB/decade.
59
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Sixth-order low-pass (cascaded)
For sake of illustration, a 6th-order low-pass Linkwitz-Riley filter is shown
below. The circuit is nothing more than a cascade of 3 2nd-order blocks like
the ones previously shown. All blocks cut off at the same frequency. However,
the first stage must have Q=0.5, the second and the third stages must have
Q=1.
C1
C3
100n
OS1
1
1k
1k
V
OS2
OUT
C4
2
50n
-
OS1
V-
U3
R6
R5
6
3
1
V
1k
+
1k
OS2
OUT
C6
2
50n
V-
0
V+
V+
LM741
5
7
+
V+
3
V-
0
200n
7
R4
R3
V+
-
V-
2
6
U2
V-
OS2
OUT
C2
V1
1Vac
0Vdc
+
1k
LM741
5
4
1k
V+
3
4
V
V+
7
R2
R1
C5
200n
U1
-
V-
V+
OS1
LM741
5
V2
6
-5Vdc
V
0
V-
0
V3
5Vdc
1
4
100n
0
0
Linkwitz-Riley filter (low-pass) (6th-order)
1.2V
0.8V
(1.5895K,501.939m)
0.4V
0V
1.0Hz
V(V1:+)
10Hz
V(C1:2)
V(R5:1)
100Hz
V(C5:2)
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The plot shown above clearly demonstrates that increasing the order of the
filter will produce a sharper response. Increasing the order of the filter will
produce a roughly vertical drop at the cutoff frequency.
The cutoff frequencies are
f c1 
f c2
60
1

1
 1.591kHz
2 R1 R2 C1C 2 2 1k  1k  100nF  100nF
1
1
 f c3 

 1.591kHz
2 R1 R2 C1C 2 2 1k  1k  200nF  50nF
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The quality factors are
Q1 
R1 R2 C1C 2
C 2 R1  R2 
Q2  Q3 

R1 R2 C1C 2
C 2 R1  R2 
1k  1k  100nF  100nF
 0.5
100nF  1k  1k 

1k  1k  200nF  50nF
1
50nF  1k  1k 
50
(1.5898K,-5.9938)
0
-100
(15.898K,-120.259)
-200
1.0Hz
V(R1:1)
10Hz
100Hz
1.0KHz
20*LOG10(V(U1:-)/V(R1:1))
20*LOG10(V(U2:OUT)/V(R1:1))
Frequency
10KHz
100KHz
20*LOG10(V(U3:-)/V(R1:1))
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 1.591kHz and then it decreases by
–120dB/decade.
61
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Sixth-order high-pass (cascaded)
For sake of illustration, a 6th-order high-pass Linkwitz-Riley filter is shown
below. The circuit is nothing more than a cascade of 3 2nd-order blocks like
the ones previously shown. All blocks cut off at the same frequency. However,
the first stage must have Q=0.5, the second and the third stages must have
Q=1.
R1
R3
1.5k
R5
V+
750
-
OS1
V
10n
0
+
750
10n
OS2
OUT
2
R4
V-
-
OS1
3k
V+
V+
LM741
5
V-
U3
C5
6
1
V
C6
10n
3
+
10n
OS2
OUT
2
R6
4
1.5k
3
7
C4
4
R2
1
C3
V-
-
OS1
LM741
5
V2
6
V3
5Vdc
1
-5Vdc
V
4
2
6
7
OS2
OUT
U2
V+
7
+
V+
3
10n
V1
1Vac
0Vdc
V+
V+
10n
LM741
5
V-
V
C2
V-
C1
V-
U1
3k
0
0
V-
0
0
0
Linkwitz-Riley filter (high-pass) (6th-order)
1.2V
0.8V
(10.616K,501.939m)
0.4V
0V
1.0Hz
V(C1:1)
10Hz
V(U1:-)
V(U2:-)
100Hz
V(R5:2)
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The plot shown above clearly demonstrates that increasing the order of the
filter will produce a sharper response. Increasing the order of the filter will
produce a roughly vertical drop at the cutoff frequency.
The cutoff frequencies are
f c1 
f c2
62
1

1
 10.61kHz
2 R1 R2 C1C 2 2 1.5k  1.5k  10nF  10nF
1
1
 f c3 

 10.61kHz
2 R1 R2 C1C 2 2 750  3k  10nF  10nF
www.ice77.net
The quality factors are
Q1 
R1 R2 C1C 2
R1 C1  C 2 
Q2  Q3 

R1 R2 C1C 2
1.5k  1.5k  10nF  10nF
 0.5
1.5k  10nF  10nF 
R1 C1  C 2 

750  3k  10nF  10nF
1
750  10nF  10nF 
100
(10.638K,-5.9403)
0
-200
(1.0638K,-119.853)
-400
-600
1.0Hz
V(V1:+)
10Hz
100Hz
1.0KHz
20*LOG10(V(R1:2)/V(V1:+))
20*LOG10(V(C5:1)/V(V1:+))
Frequency
10KHz
100KHz
20*LOG10(V(R5:2)/V(V1:+))
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 10.61kHz and then it decreases by
–120dB/decade.
63
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Bessel filter
The Bessel filter is named after Friedrich Bessel, a German mathematician
who studied the mathematics behind the filter before it was implemented. This
is also known as Bessel-Thomson filter. W. E. Thomson was responsible for
actually using the theory and putting it to work.
For a second-order Bessel filter, the quality factor must be 0.577.
Second-order low-pass
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors.
C1
133n
V+
V+
U1
1k
OS2
2
100n
-
V-
OUT
C2
V1
1Vac
0Vdc
+
V+
3
1k
OS1
LM741
5
V2
6
1
V
5Vdc
V3
-5Vdc
4
V
V-
7
R2
R1
V-
0
0
0
0
Bessel filter (low-pass) (2nd-order)
Note: R1=R2 and C1=1.336xC2.
1.0V
(1.3791K,576.923m)
0.5V
0V
1.0Hz
V(R1:1)
10Hz
V(C1:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is
fc 
64
1
2 R1 R2 C1C 2

1
2 1k  1k  133nF  100nF
 1.38kHz
www.ice77.net
The quality factor is
Q
R1 R2 C1C 2
C 2 R1  R2 

1k  1k  133nF  100nF
 0.577
100nF  1k  1k 
20
-0
(1.3804K,-4.7854)
-20
(13.971K,-40.118)
-40
-60
1.0Hz
V(R1:1)
10Hz
100Hz
20*LOG10(V(C1:2)/ V(R1:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –4.78dB at 1.38kHz and then it decreases by
–40dB/decade.
65
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Second-order high-pass
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors.
R1
1k
V+
V+
3
OS2
OUT
R2
V1
1Vac
0Vdc
+
100n
2
1.33k
-
V-
100n
V+
C2
OS1
LM741
5
V2
6
1
V
5Vdc
V3
-5Vdc
4
V
V-
7
U1
C1
V-
0
0
0
0
Bessel filter (high-pass) (2nd-order)
Note: C1=C2 and R2=R1x1.336.
1.0V
(1.3807K,576.923m)
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(R1:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 1k  1.33k  100nF  100nF
 1.38kHz
The quality factor is
Q
66
R1R2C1C2
1k 1.33k 100nF  100nF
 0.577

1k  100nF  100nF 
R1 C1  C2 
www.ice77.net
40
-0
(1.3834K,-4.7606)
-50
(138.337,-40.002)
-100
-150
1.0Hz
V(V1:+)
(13.834,-79.958)
10Hz
100Hz
20*LOG10(V(R1:2)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –4.78dB at 1.38kHz and then it decreases by
–40dB/decade.
67
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Third-order low-pass (same resistor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The resistor values match here. This is a third-order filter because it
has three capacitors.
V+
V+
V-
U1
1Vac
0Vdc
V1
C2
218n
+
712
712
C1
7
3
V+
R2
712
OS2
OUT
C3
2
314n
-
V-
V
R3
56n
OS1
LM741
5
V2
1
V
5Vdc
V-
0
0
V3
6
-5Vdc
4
R1
0
0
0
Bessel filter (low-pass) (3rd-order)
Note: R1=R2=R3.
1.0V
(1.0006K,715.976m)
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(U1:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
20
(1.0000K,-2.8984)
0
(10.034K,-51.006)
-50
-100
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(U1:-)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –2.8984dB at 1kHz and then it decreases to –51.006dB a
decade later.
68
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Third-order low-pass (same capacitor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The capacitor values match here and the circuit provides gain by
means of two additional resistors. This is a third-order filter because it has
three capacitors.
V+
V+
V-
U1
1Vac
0Vdc
V1
C1
7
+
V+
3
902
3.05k
520
C2
100n
LM741
5
OS2
C3
2
100n
100n
-
V2
1
OS1
V
0
5Vdc
-5Vdc
R4
1.2k
V-
0
V3
6
OUT
V-
V
R3
R2
4
R1
0
0
0
R5
1k
0
Bessel filter (low-pass) (3rd-order)
Note: C1=C2=C3.
3.0V
(10.000,2.1998)
2.0V
(1.0000K,1.5651)
1.0V
0V
1.0Hz
V(V1:+)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
30
(10.000,6.8478)
(1.0000K,3.8909)
0
(10.000K,-44.399)
-50
-100
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(U1:OUT)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is flat at +6.8478dB in the lower frequency range and then it drops to
+3.8909dB at 1kHz. It eventually decreases to –44.399dB one decade later.
69
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Fourth-order low-pass (same capacitor values)
In the following example, the circuit is implemented with the Sallen/Key
topology. The capacitor values match here and the circuit provides gain by
means of two additional resistors. This is a fourth-order filter because it has
four capacitors.
V+
V+
V-
U1
V
640
V1
4.41k
C1
3
333
1.265k
C2
+
V+
7
R4
R3
C3
OS2
OUT
C4
2
-
OS1
LM741
5
V2
V3
6
1
V
5Vdc
-5Vdc
R5
100n
100n
100n
100n
4
1Vac
0Vdc
R2
V-
R1
V-
0
0
1.2k
0
0
0
R6
1k
0
Bessel filter (low-pass) (4th-order)
Note: all capacitor values match.
2.8V
(10.000,2.1998)
2.0V
(1.0000K,1.5627)
1.0V
0V
1.0Hz
V(V1:+)
10Hz
V(C3:1)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
20
(10.000,6.8478)
(1.0000K,3.8777)
-0
-20
-40
(12.589K,-50.168)
-60
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(C3:1)/V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is flat at +6.8478dB in the lower frequency range and then it drops to
+3.8777dB at 1kHz. It eventually decreases to –50.168dB at 12.689kHz.
70
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Chebyshev filter
The Chebyshev filter bears the name of Pafnuty Chebyshev, a Russian
mathematician who developed the theory behind the Chebyshev polynomials.
Chebyshev filters come in two variants: if the ripple is present in the
passband, they are called Type I otherwise, if the ripple is present in the
stopband, they are called Type II (also known as Inverted).
The filter can be designed to have different ripples that can vary from 0.5dB to
3dB and intermediate values (typically in 0.5dB increments).
Type I
This is the Chebyshev filter with the ripple in the passband. It’s probably the
most common version of the filter.
For a second-order Chebyshev Type I filter, the quality factors must be 0.864,
0.956, 1.129 and 1.305 for 0.5dB, 1dB, 2dB and 3dB versions of the filter
respectively.
Second-order low-pass (0.5dB ripple)
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors. This
specific filter is designed to have a 0.5dB ripple in the passband.
C1
298n
V+
V+
U1
V+
OS2
OUT
C2
2
100n
-
V-
1Vac
0Vdc
V1
+
1k
OS1
LM741
5
V2
6
1
V
5Vdc
V3
-5Vdc
4
V
3
1k
V-
7
R2
R1
V-
0
0
0
0
Chebyshev 0.5dB ripple filter (low-pass) (2nd-order)
Note: for a 0.5dB ripple, R1=R2 and C1=2.986xC2.
71
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1.2V
(922.189,862.722m)
0.8V
0.4V
0V
1.0Hz
V(V1:+)
10Hz
V(U1:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable.
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 1k  1k  298nF  100nF
 921.96 Hz
The quality factor is
Q
R1 R2 C1C 2
C 2 R1  R2 

1k  1k  298nF  100nF
 0.863
100nF  1k  1k 
20
(926.119,-1.3203)
-0
-20
(9.2612K,-40.123)
-40
-60
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(U1:OUT)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –1.3203dB at 926Hz and then it decreases by
–40dB/decade.
72
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2.0
1.0
(524.808,499.849m)
0
-1.0
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(U1:OUT)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Close-up of 0.5dB ripple
The frequency response of the circuit peaks at 524Hz with a 0.5dB overshoot
and then it decays rapidly.
73
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Second-order low-pass (1dB ripple)
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors. This
specific filter is designed to have a 1dB ripple in the passband.
C3
366n
V+
V+
V-
U2
1k
LM741
5
OS2
C4
2
100n
-
V2
1
OS1
V
V-
0
V3
6
OUT
V-
1k
V4
1Vac
0Vdc
+
V+
3
5Vdc
-5Vdc
4
V
7
R4
R3
0
0
0
Chebyshev 1dB ripple filter (low-pass) (2nd-order)
Note: for a 1dB ripple, R3=R4 and C3=3.663xC4.
1.2V
(830.016,958.580m)
0.8V
0.4V
0V
1.0Hz
V(V4:+)
10Hz
V(U2:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable.
The cutoff frequency is
fc 
1
2 R2 R3C 2 C 3

1
2 1k  1k  366nF  100nF
 831.92 Hz
The quality factor is
Q
74
R 2 R3 C 2 C 3
C 3 R 2  R 3 

1k 1k  366nF 100nF
 0.957
100nF  1k  1k 
www.ice77.net
50
(830.736,-863.103m)
-0
-50
(8.3074K,-92.129)
-100
-150
1.0Hz
V(V4:+)
10Hz
100Hz
20*LOG(V(U2:-)/ V(V4:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –0.863dB at 830Hz and then it decreases to –92dB a
decade past the cutoff frequency (much faster roll-off).
2.0
(562.341,1.0039)
1.0
0
-1.0
1.0Hz
V(V4:+)
10Hz
100Hz
20*LOG10(V(U2:-)/ V(V4:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Close-up of 1dB ripple
The frequency response of the circuit peaks at 562Hz with a 1dB overshoot
and then it decays rapidly.
75
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Second-order low-pass (2dB ripple)
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors. This
specific filter is designed to have a 2dB ripple in the passband.
C5
509n
V+
V+
V-
U3
1k
OS2
OUT
C6
2
100n
-
V-
1k
V5
1Vac
0Vdc
+
V+
3
OS1
LM741
5
V2
1
5Vdc
V
V-
0
V3
6
-5Vdc
4
V
7
R6
R5
0
0
0
Chebyshev 2dB ripple filter (low-pass) (2nd-order)
Note: for a 1dB ripple, R5=R6 and C5=5.098xC6.
1.5V
(705.774,1.1272)
1.0V
0.5V
0V
1.0Hz
V(V5:+)
10Hz
V(C5:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable.
The cutoff frequency is
fc 
1
2 R5 R6 C 5 C 6

1
2 1k  1k  509nF  100nF
 705.44 Hz
The quality factor is
Q
76
R5 R 6 C 5 C 6
C 6 R5  R 6 

1k 1k  509nF 100nF
 1.128
100nF  1k  1k 
www.ice77.net
20
(702.533,1.0863)
-0
-20
(6.9970K,-39.882)
-40
-60
1.0Hz
V(V5:+)
10Hz
100Hz
20*LOG10(V(C5:2)/ V(V5:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is +1.0863dB at 702Hz and then it decreases by –40dB/decade.
3.0
(549.541,2.0010)
2.0
1.0
0
-1.0
1.0Hz
V(V5:+)
10Hz
100Hz
20*LOG10(V(C5:2)/ V(V5:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Close-up of 2dB ripple
The frequency response of the circuit peaks at 549Hz with a 2dB overshoot
and then it decays rapidly.
77
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Second-order low-pass (3dB ripple)
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors. This
specific filter is designed to have a 3dB ripple in the passband.
C7
681n
V+
V+
V-
U4
1k
OS2
OUT
C8
2
100n
-
V-
1k
V6
1Vac
0Vdc
+
V+
3
OS1
LM741
5
V2
1
V
V-
0
V3
6
5Vdc
-5Vdc
4
V
7
R8
R7
0
0
0
Chebyshev 3dB ripple filter (low-pass) (2nd-order)
Note: for a 1dB ripple, R7=R8 and C7=6.812xC8.
1.5V
(609.575,1.3047)
1.0V
0.5V
0V
1.0Hz
V(V6:+)
10Hz
V(U4:OUT)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable.
The cutoff frequency is
fc 
1
2 R7 R8 C 7 C8

1
2 1k  1k  681nF  100nF
 609.88 Hz
The quality factor is
Q
78
R 7 R8 C 7 C 8
C 8  R 7  R8 

1k 1k  681nF 100nF
 1.305
100nF  1k  1k 
www.ice77.net
20
(611.881,1.8548)
-0
-20
(6.1188K,-37.523)
-40
-60
1.0Hz
V(V6:+)
10Hz
100Hz
20*LOG10(V(U4:OUT)/ V(V6:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is +1.8548dB at 611Hz and then it decreases by –40dB/decade.
4.0
(512.861,3.0052)
2.0
0
-1.0
1.0Hz
V(V6:+)
10Hz
100Hz
20*LOG10(V(U4:OUT)/ V(V6:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Close-up of 3dB ripple
The frequency response of the circuit peaks at 512Hz with a 3dB overshoot
and then it decays rapidly.
79
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Second-order low-pass ripple comparison
1.5V
1.0V
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(U1:OUT)
V(V4:+)
100Hz
V(U2:OUT)
1.0KHz
V(V5:+)
V(C5:2)
Frequency
10KHz
V(V6:+)
V(U4:OUT)
100KHz
1.0MHz
0.5dB, 1dB, 2dB and 3dB circuit response comparison
Above is a comparison of the response for the low-pass circuits previously
presented. The peaks in the response are clearly not matching because the
circuits have been designed with the same resistor values so the cutoff
frequency moves to the left of the plot as the magnitude of the ripple
increases.
4.0
(512.861,3.0052)
(549.541,2.0010)
2.5
(562.341,1.0039)
(524.808,499.849m)
0
-2.5
-5.0
100Hz
300Hz
1.0KHz
V(V1:+)
20*LOG10(V(U1:OUT)/ V(V1:+))
V(V4:+)
20*LOG10(V(U2:OUT)/ V(V4:+))
V(V5:+)
20*LOG10(V(U3:OUT)/ V(V5:+))
V(V6:+)
20*LOG10(V(U4:OUT)/ V(V6:+))
Frequency
3.0KHz
0.5dB, 1dB, 2dB and 3dB ripple comparison
The plot shown above compares the ripples for every circuit previously
presented. It is clear that a higher amount of ripple will produce a sharper
cutoff.
80
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Second-order high-pass (0.5dB ripple)
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors. This
specific filter is designed to have a 0.5dB ripple in the passband.
R1
1k
V+
V+
3
OS2
OUT
R2
V1
1Vac
0Vdc
+
100n
2
-
2.98k
V-
100n
V+
C2
OS1
LM741
5
V2
6
1
V
5Vdc
V3
-5Vdc
4
V
V-
7
U1
C1
V-
0
0
0
0
Chebyshev 0.5dB ripple filter (high-pass) (2nd-order)
Note: for a 0.5dB ripple, C1=C2 and R2=2.986xR1.
1.2V
0.8V
(921.668,862.722m)
0.4V
0V
1.0Hz
V(C1:1)
10Hz
V(R1:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable.
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 1k  2.98k  100nF  100nF
 921.96 Hz
The quality factor is
Q
81
R1 R2 C1C 2
R1 C1  C 2 

1k  2.98k  100nF  100nF
 0.863
1k  100nF  100nF 
www.ice77.net
40
(925.642,-1.2459)
-0
(92.564,-39.902)
-40
-80
-120
1.0Hz
V(C1:1)
10Hz
100Hz
20*LOG10(V(R1:2)/ V(C1:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –1.2459dB at 925Hz and then it decreases by
–40dB/decade.
2.0
1.0
(1.6218K,486.213m)
0
-1.0
1.0Hz
V(C1:1)
10Hz
100Hz
20*LOG10(V(R1:2)/ V(C1:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Close-up of 0.5dB ripple
The frequency response of the circuit peaks at 1.62kHz with a 0.5dB
overshoot and then it decays rapidly.
82
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Second-order high-pass (1dB ripple)
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors. This
specific filter is designed to have a 1dB ripple in the passband.
R3
1k
V+
V+
3
OS2
OUT
R4
V4
1Vac
0Vdc
+
100n
2
-
3.66k
V-
100n
V+
C4
OS1
LM741
5
V2
6
1
V
5Vdc
V3
-5Vdc
4
V
V-
7
U2
C3
V-
0
0
0
0
Chebyshev 1dB ripple filter (high-pass) (2nd-order)
Note: for a 1dB ripple, C3=C4 and R4=3.663xR3.
1.2V
0.8V
(833.787,958.580m)
0.4V
0V
1.0Hz
V(V4:+)
10Hz
V(R3:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable.
The cutoff frequency is
fc 
1
2 R3 R4 C 3C 4

1
2 1k  3.66k  100nF  100nF
 831.92 Hz
The quality factor is
Q
83
R3 R 4 C 3 C 4
R3 C 3  C 4 

1k  3.66k  100nF  100nF
 0.957
1k  100nF  100nF 
www.ice77.net
40
(830.736,-398.059m)
-0
(83.074,-39.986)
-40
-80
-120
1.0Hz
V(V4:+)
10Hz
100Hz
20*LOG10(V(R3:2)/ V(V4:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –0.398dB at 830Hz and then it decreases by
–40dB/decade.
2.0
(1.2303K,990.336m)
1.0
0
-1.0
1.0Hz
V(V4:+)
10Hz
100Hz
20*LOG10(V(R3:2)/ V(V4:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Close-up of 1dB ripple
The frequency response of the circuit peaks at 1.23kHz with a 1dB overshoot
and then it decays rapidly.
84
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Second-order high-pass (2dB ripple)
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors. This
specific filter is designed to have a 2dB ripple in the passband.
R5
1k
V+
V+
V-
100n
100n
OS2
OUT
R6
V5
1Vac
0Vdc
+
V+
3
2
-
5.09k
V-
V
C6
OS1
LM741
5
V2
1
5Vdc
V
V-
0
V3
6
-5Vdc
4
C5
7
U3
0
0
0
Chebyshev 2dB ripple filter (high-pass) (2nd-order)
Note: for a 2dB ripple, C5=C6 and R6=5.098xR5.
1.5V
(705.088,1.1272)
1.0V
0.5V
0V
1.0Hz
V(C5:1)
10Hz
V(R5:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable.
The cutoff frequency is
fc 
1
2 R5 R6 C 5 C 6

1
2 1k  5.09k  100nF  100nF
 705.44 Hz
The quality factor is
Q
85
R5 R 6 C 5 C 6
R5 C 5  C 6 

1k  5.09k 100nF 100nF
 1.128
1k  100nF  100nF 
www.ice77.net
40
(700.870,985.793m)
-0
(70.087,-40.061)
-40
-80
-120
1.0Hz
V(C5:1)
10Hz
100Hz
20*LOG10(V(R5:2)/ V(C5:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is +0.985dB at 700Hz and then it decreases by –40dB/decade.
3.0
(912.011,1.9866)
2.0
1.0
0
-1.0
1.0Hz
V(C5:1)
10Hz
100Hz
20*LOG10(V(R5:2)/ V(C5:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Close-up of 2dB ripple
The frequency response of the circuit peaks at 912Hz with a 2dB overshoot
and then it decays rapidly.
86
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Second-order high-pass (3dB ripple)
In the following example, the circuit is implemented with the Sallen/Key
topology. This is a second-order filter because it has two capacitors. This
specific filter is designed to have a 3dB ripple in the passband.
R7
1k
V+
V+
V-
+
100n
R8
V6
OS2
OUT
2
6.81k
-
V-
1Vac
0Vdc
100n
3
OS1
LM741
5
V2
1
5Vdc
V
V-
0
V3
6
-5Vdc
4
V
C8
V+
C7
7
U4
0
0
0
Chebyshev 3dB ripple filter (high-pass) (2nd-order)
Note: for a 3dB ripple, C7=C8 and R8=6.812xR7.
1.5V
(610.162,1.3047)
1.0V
0.5V
0V
1.0Hz
V(C7:1)
10Hz
V(U4:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable.
The cutoff frequency is
fc 
1
2 R7 R8 C 7 C8

1
2 1k  6.81k  100nF  100nF
 609.88 Hz
The quality factor is
Q
87
R7 R8 C 7 C8
R7 C 7  C8 

1k  6.81k  100nF  100nF
 1.305
1k  100nF  100nF 
www.ice77.net
40
(609.867,2.3057)
-0
(60.987,-39.939)
-40
-80
-120
1.0Hz
V(C7:1)
10Hz
100Hz
20*LOG10(V(U4:-)/ V(C7:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is +2.3057dB at 609Hz and then it decreases by –40dB/decade.
4.0
(724.436,2.9916)
2.0
0
-1.0
1.0Hz
V(C7:1)
10Hz
100Hz
20*LOG10(V(U4:-)/ V(C7:1))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Close-up of 3dB ripple
The frequency response of the circuit peaks at 724Hz with a 3dB overshoot
and then it decays rapidly.
88
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Second-order high-pass ripple comparison
1.5V
1.0V
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(R1:2)
V(C3:1)
100Hz
V(U2:OUT)
V(C5:1)
1.0KHz
V(R5:2)
Frequency
V(C7:1)
10KHz
V(U4:-)
100KHz
1.0MHz
0.5dB, 1dB, 2dB and 3dB circuit response comparison
Above is a comparison of the response for the high-pass circuits previously
presented. The peaks in the response are clearly not matching because the
circuits have been designed with the same capacitor value so the cutoff
frequency moves to the left of the plot as the magnitude of the ripple
increases.
4.0
(724.436,2.9916)
(912.011,1.9866)
2.0
(1.2303K,990.336m)
(1.6218K,486.213m)
0
-2.0
300Hz
V(V1:+)
20*LOG10(V(R1:2)/V(V1:+))
V(C7:1)
20*LOG10(V(U4:-)/V(C7:1))
1.0KHz
3.0KHz
V(C3:1)
20*LOG10(V(U2:OUT)/V(C3:1))
V(C5:1)
10KHz
20*LOG10(V(R5:2)/V(C5:1))
Frequency
0.5dB, 1dB, 2dB and 3dB ripple comparison
The plot shown above compares the ripples for every circuit previously
presented. It is clear that a higher amount of ripple will produce a sharper
cutoff.
89
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Type II
This is the Chebyshev filter with the ripple in the stopband. This filter has a
very sharp cutoff.
Fifth-order low-pass
This circuit is implemented with two notch filter blocks and a simple RC filter.
This is a fifth-order filter because the circuit contains five capacitors.
C1
C3
10n
10n
R3
R8
92.3k
22.86k
V-
V+
U1
OS1
OUT
3
R2
+
OS2
R6
4
-
10n
C4
6
5
2
V
6184
10n
3
7
OS1
OUT
R7
337k
+
OS2
V2
LM741
1
R11
6
5
V
5Vdc
777
23.36k
100n
V+
0
0
R4
V3
-5Vdc
V
C5
7
V+
0
-
V-
2398
V+
V
V1
1Vac
0Vdc
U2
LM741
1
V+
2
V-
V-
4
C2
V-
R1
0
0
0
R9
670
R5
1.04k
R10
10k
1k
0
0
Inverted Chebyshev filter (low-pass) (5th-order)
1.5V
(10.000,1.1579)
(1.0004K,807.692m)
1.0V
0.5V
0V
1.0Hz
V(R1:1)
10Hz
V(R3:2)
V(C3:2)
100Hz
V(R11:2)
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The gain in the passband boosts the input from 1V to 1.15V. The filter drops
rapidly right before 1kHz. The ripple in the stopband is noticeable.
90
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0
(299.358,1.2741)
(1.0002K,-1.8478)
(10.000K,-21.079)
-20
-40
100Hz
V(R1:1)
300Hz
1.0KHz
3.0KHz
20*LOG10(V(R3:2)/V(R1:1))
20*LOG10(V(C3:2)/V(R1:1))
Frequency
10KHz
30KHz
20*LOG10(V(R11:2)/V(R1:1))
100KHz
Bode plot from 100Hz to 100kHz
The gain is +1.2741dB in the passband and –1.8478dB at 1kHz. Then it drops
to –21.079dB a decade later. The ripple in the stopband is noticeable.
91
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Elliptic filter
The Elliptic filter, also known as Cauer filter, has a sharp cutoff. This filter is
named after Wilheld Cauer, a German mathematician who developed the
theory behind the filter.
Third-order low-pass
This circuit is implemented with an asymmetrical twin-T notch filter (R1, R2, R3,
C2, C3, C4). This is a third-order filter.
R6
VU1
2
C1
-
V-
4
4.42k
OS1
20n
3
+
OS2
6
R7
5
2358
1k
V
C5
7
R5
V+
OUT
uA741
1
100n
V+
0
R1
V
1Vac
0Vdc
0
R2
6242
6242
C2
C3
10n
10n
C4
V+
R4
V-
V1
44n
47.94k
5Vdc
R3
3111
0
0
V2
5Vdc
V3
0
0
0
0
rd
Elliptic filter (low-pass) (3 -order)
6.0V
(10.000,4.2998)
4.0V
(1.0011K,2.8047)
2.0V
0V
1.0Hz
V(C2:2)
10Hz
V(R7:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The gain in the passband boosts the input from 1V to 4.3V. The ripple in the
passband is barely noticeable. The gain drops rapidly right before 1kHz.
92
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50
(10.000,12.669)
(1.0098K,8.7278)
0
(10.000K,-29.254)
-50
-100
1.0Hz
V(C2:2)
10Hz
100Hz
20*LOG10(V(R7:2)/V(C2:2))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain is +12.669dB in the passband and +8.7278dB at 1kHz. Then it drops
to –29.254dB a decade later.
93
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Synchronous filter
Synchronous filters are made up by a series of filters cascaded after each
other. Each capacitor introduces a pole. Resistors and capacitors usually
have all the same values and they are tuned to a specific cutoff frequency.
Second-order low-pass
In the following example, the circuit is implemented with the Sallen/Key
topology. This circuit is equivalent to the Butterworth circuit discussed
previously (Q=0.707). This is a second-order filter because it has two
capacitors.
C1
200n
V+
V+
V-
U1
1125
OS2
OUT
C2
2
100n
-
V-
1125
V1
1Vac
0Vdc
+
OS1
LM741
5
V2
1
V
V-
0
V3
6
5Vdc
-5Vdc
4
V
3
V+
R1
7
R2
0
0
0
Synchronous filter (low-pass) (2nd-order)
Note: R1=R2 and C1=2xC2.
1.0V
(1.0003K,707.101m)
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(U1:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is
fc 
94
1
2 R1 R2 C1C 2

1
2 1125  1125  200nF  100nF
 1kHz
www.ice77.net
The quality factor is
Q
R1 R2 C1C 2
C 2 R1  R2 

1125  1125  200nF  100nF
 0.707
100nF  1125  1125 
20
-0
(1.0000K,-3.0079)
-20
(10.000K,-40.051)
-40
-60
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(U1:-)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –40dB/decade.
95
www.ice77.net
Second-order high-pass
In the following example, the circuit is implemented with the Sallen/Key
topology. This circuit is equivalent to the Butterworth circuit discussed
previously (Q=0.707). This is a second-order filter because it has two
capacitors.
R1
1125
V+
V+
V-
C2
3
100n
R2
V1
1Vac
0Vdc
OUT
2
-
2.25k
V-
100n
OS2
OS1
LM741
5
V2
1
5Vdc
V
V-
0
V3
6
-5Vdc
4
V
+
V+
C1
7
U1
0
0
0
Synchronous filter (high-pass) (2nd-order)
Note: C1=C2 and R2=2xR1.
1.2V
(0.9995K,706.509m)
0.8V
0.4V
0V
1.0Hz
V(C1:2)
10Hz
V(U1:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is
fc 
1
2 R1 R2 C1C 2

1
2 1125  2.25k  100nF  100nF
 1kHz
The quality factor is
Q
96
R1 R2 C1C 2
R1 C1  C 2 

1125  2.25k  100nF  100nF
 0.707
1125  100nF  100nF 
www.ice77.net
50
-0
(1.0000K,-3.0135)
-50
(100.000,-40.007)
-100
-150
1.0Hz
V(C1:2)
(10.000,-80.006)
10Hz
100Hz
20*LOG10(V(U1:-)/ V(C1:2))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –40dB/decade.
97
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Third-order low-pass
In the following example, the circuit is implemented with the Sallen/Key
topology by cascading a simple low-pass filter after a second-order filter. This
is a third-order filter because it has three capacitors.
C1
100n
V+
V+
V-
U1
V
810
810
OS2
OUT
C2
V1
1Vac
0Vdc
+
V+
3
2
-
V-
R1
7
R2
OS1
LM741
5
R3
V2
1
810
V
5Vdc
-5Vdc
C3
4
100n
100n
V-
0
V3
6
0
0
0
0
Synchronous filter (low-pass) (3rd-order)
Note: all resistor and capacitor values match.
1.0V
(1.0021K,707.101m)
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(C3:2)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
20
(1.0000K,-2.9990)
0
(10.000K,-42.932)
-40
-80
1.0Hz
V(V1:+)
10Hz
100Hz
20*LOG10(V(C3:2)/ V(V1:+))
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –40dB/decade.
98
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Fourth-order low-pass
In the following example, the circuit is implemented with the Sallen/Key
topology by cascading two identical second-order filters. This is a fourth-order
filter because it has four capacitors.
C1
C3
100n
100n
V+
U1
2
100n
-
OS1
6
U2
1
V
7
R4
R3
3
691
691
OS2
OUT
C4
2
100n
V-
0
+
V+
LM741
5
-
V-
OUT
C2
V1
V+
691
OS2
OS1
V+
LM741
5
V-
6
1
V2
V
5Vdc
4
691
+
V-
1Vac
0Vdc
3
4
V
V+
7
R2
R1
V3
-5Vdc
V-
0
0
0
0
Synchronous filter (low-pass) (4th-order)
Note: all resistor and capacitor values match.
1.0V
(1.0026K,840.896m)
(1.0026K,707.101m)
0.5V
0V
1.0Hz
V(V1:+)
10Hz
V(C1:1)
V(U2:-)
100Hz
1.0KHz
10KHz
100KHz
1.0MHz
Frequency
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
20
(1.0000K,-1.4979)
0
(1.0000K,-2.9961)
-50
(10.000K,-51.990)
-100
1.0Hz
V(V1:+)
10Hz
100Hz
1.0KHz
20*LOG10(V(C1:1)/V(V1:+))
20*LOG10(V(U2:-)/V(V1:+))
Frequency
10KHz
100KHz
1.0MHz
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases to –51.99dB a decade
later.
99
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Linkwitz-Riley crossover
The Linkwitz-Riley crossover is an audio application that stems from the work
of Linkwitz and Riley.
The crossover can be implemented with different orders. For every order, the
gain of the filter will drop by –6dB/octave or –20dB/decade past the cutoff
frequency. Increasing the order of the filter will produce a sharper cutoff.
The crossover can be designed to split the audible spectrum in 2, 3 or 4 ways.
A 2-way audio crossover splits the audible spectrum in two parts, it has a
single cutoff frequency and it’s implemented by cascoding two Butterworth
filters (low-pass and high-pass). For a 3-way crossover, there will be three
regions with a low cutoff frequency and a high cutoff frequency. This is
arguably the most popular crossover configuration in the market. The reason
why the audible spectrum is divided into 3 sections is explained by the need
for audio systems to handle each section more effectively through speakers
for proper sound reproduction. A 3-way system has 6 speakers (2 for each
channel).
A 3-way 4th-order Linkwitz-Riley crossover can be designed with the following
expression:
f 
1
2 2 RC
First of all the designer needs to choose cutoff frequencies for the specific
regions of the spectrum. At that point, with a set frequency, a value for
capacitance (C) or resistance (R) is chosen and the other one is derived.
Assuming that the desired low cutoff frequency is 340Hz then C can be
chosen to be 100nF and R can be chosen to be 3.3kΩ.
fL 
1
1

 341.029 Hz
2 2 RC 2 2  3.3k  100nF
Assuming that the desired high cutoff frequency is 3.5kHz then C can be
chosen to be 6.8nF and R can be chosen to be 4.7kΩ.
fH 
100
1
1

 3.521kHz
2 2 RC 2 2  4.7 k  6.8nF
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E.S.P. Linkwitz-Riley Crossover Calculator screenshots for low pass and high pass
The values for the two sections of the crossover need to be arranged just like
shown above. The values of capacitance or resistance double depending on
the configuration of the specific section of the filter.
R5
R4
100k
2
-
OS1
C3
6
6.8n
1
R3
9.4k
-VCC
7
6.8n
1
OS1
C7
-VCC
U3
3
+
6.8n
R21
9.4k
7
V+
-
VCC
C6
6
uA741OUT
2
6.8n
5
OS2
V+
C2
+
5
OS2
2
-
R6
HIGH
6
uA741OUT
1
OS1
100
V
R22
4
R2
V-
uA741OUT
V3
1Vac
0Vdc
2.2k
5
4.7k
VCC
U2
3
OS2
V-
1u
V
+
4
3
V+
7
U1 VCC
R1
V-
4.7k
C1
0
4
-VCC
0
1000k
0
0
0
C5
C4
-
OS1
R8
+U5
1
4.7k
V+
7
uA741OUT
2
R7
OS2
uA741OUT
6
2
4.7k
C9
V-
OS2
3
5
-
6.8n
OS1
5
6
1
-VCC
4
6.8n
7
+
4.7k
C10
V+
U4
3
4.7k
13.6n VCC
VCC
13.6n
R9
V-
R10
4
-VCC
0
0
R11
OS2
uA741OUT
2
-
OS1
C13
VCC
C14
6
U7
3
100n
1
+
100n
7
+
V+
100n
R19
6.6k
5
OS2
uA741OUT
R20
4
-VCC
2
-
V-
100n
V+
U6
3
3.3k
7
3.3k
VCC
C12
V-
C11
R12
OS1
5
MID
R13
6
1
100
V
6.6k
0
R23
4
-VCC
100k
0
0
C15
C16
200n
C18
OS2
uA741OUT
2
100n
-
OS1
R16
200n
VCC
R15
U9
6
1
3
3.3k
+
3.3k
OS2
uA741OUT
C17
2
4
-VCC
5
100n
-
VCC
GND
-VCC
7
+U8
V+
3
V-
3.3k
7
3.3k
VCC
V+
R17
V-
R18
OS1
V1
R14
V2
15Vdc
LOW
-15Vdc
6
1
100
R24
V
100k
0
0
0
4
-VCC
5
0
0
0
3-way Linkwitz-Riley crossover
101
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The circuit previously shown is a cascode of 3 sections. The top provides the
high frequencies, the bottom provides the low frequencies and the central part
provides the mid frequencies. High and low sections are made up by a
cascade of 2 2nd-order Butterworth filters. The middle section is a high-pass
section followed by a low-pass section.
1.0V
High cut-off frequency
(3.5445K,501.188m)
0.5V
(340.041,503.457m)
Low cut-off frequency
0V
10Hz
30Hz
V(R22:2)
V(R13:2)
100Hz
V(R24:2)
300Hz
V(V3:+)
1.0KHz
3.0KHz
10KHz
30KHz
100KHz
Frequency
AC sweep from 20Hz to 20kHz
The frequency response for the 3-way Linkwitz-Riley crossover is shown
above. The low cutoff frequency is 340Hz. The high cutoff frequency is
3.5kHz.
A 3-way 4th-order crossover’s gain will drop by –24dB/octave or
–80dB/decade past the cutoff frequency.
102
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