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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. Akerberg-Mossberg 2. Biquadratic 3. Dual Amplifier BandPass (DABP) 4. Fliege 5. Multiple feedback 6. Voltage-Controlled Voltage-Source (VCVS) and Sallen/Key 7. State variable 8. 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
octavedBA
A/6
2
1log20log20 10
2
110
and
decadedBA
A/20
10
1log20log20 10
2
110
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.
R1
10kV
-797.3uV
-10.00V
-1.575mV
V
0
0
R2
10k
10.00V
0V
0V
-1.595mV
V11Vac0Vdc
V-
10.00V
0V
V3
10Vdc
0
0V V210Vdc
0
C2
1n
C1
1n
-10.00V
0V
V-
V+V+
U1
AD741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
Sallen/Key low-pass filter
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(U1:-)
0V
0.5V
1.0V
(15.910K,500.000m)
AC sweep from 1mHz to 1MHz
The cutoff frequency is
kHznFnFkkCCRR
fc 915.151110102
1
2
1
2121
The quality factor is
5.010101
111010
212
2121
kknF
nFnFkk
RRC
CCRRQ
This circuit is critically damped.
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Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHz20*LOG10(V(U1:OUT)/V(V1:+))
-60
-40
-20
-0
(160.941K,-40.508)
(16.094K,-6.1371)
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.
V210Vdc
0
V
0
0V
V-
-778.0uV
0
10.00V
0V
0
R1
10k
0V
V+
V11Vac0Vdc
R2
10k
V+
0V
10.00V
-10.00V
C1
220n
V
-797.3uV
V-
U1
AD741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V3
10Vdc
-778.0uV
C2
220n
0V
-10.00V
Sallen/Key high-pass filter
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHz 10MHz 100MHzV(C1:1) V(R1:2)
0V
0.5V
1.0V
(72.334,500.000m)
AC sweep from 1mHz to 100MHz
The cutoff frequency is
HznFnFkkCCRR
fc 34.7222022010102
1
2
1
2121
The quality factor is
5.022022010
2202201010
211
2121
nFnFk
nFnFkk
CCR
CCRRQ
This circuit is critically damped.
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Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHz 10MHz 100MHz20*LOG10(V(R1:2)/V(C1:1))
-80
-60
-40
-20
-0
(7.2718,-40.000)
(71.969,-6.1226)
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.
10.00V
V+
0
-1.595mV
-1.575mV
0V
-10.00V
Ra
10k 0
0
-1.565mV
0
V210Vdc
0V
V3
10Vdc
0V
C2
220n0V
V-V+
V
C1
220n
V
R1
10k
-782.6uV
U1
AD741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V-
-10.00V
R2
20k
Rb
20k
10.00V
0
R3
10k
0
0V
V11Vac0Vdc
Sallen/Key band-pass filter
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(U1:OUT)
0V
0.5V
1.0V
(72.287,1.0000)
AC sweep from 1mHz to 1MHz
The center frequency is
HzkkknFnF
kk
RRRCC
RRfc 34.72
102010220220
1010
2
1
2
1
32121
13
The gains are
5.15.0120
1011
k
k
R
RG
b
a 15.13
5.1
3
G
GA
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.
V3
10Vdc
-10.00V
0 Rb
20k
V11Vac0Vdc
V
-1.575mV
0VRa
1Meg
C2
220n
10.00V
V-
V-
0
U1
AD741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R1
10k
0
V+
-10.00V
C1
220n
-1.595mV0V
10.00V
0V
0
V
V+
0V
0
R2
20k
V210Vdc
R3
10k
0
0V
-285.5uV -571.0uV
VCVS band-pass filter I
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(U1:OUT)
0V
0.4V
0.8V
1.2V(66.069,1.0625)
AC sweep from 1mHz to 1MHz
The center frequency is
HzkkknFnF
kk
RRRCC
RRfc 34.72
102010220220
1010
2
1
2
1
32121
13
However, if Ra>>Rb, the frequency response is rather flat.
5150120
111
k
M
R
RG
b
a 06.1513
51
3
G
GA
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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-748.5uV -1.497mV
-10.00V
0
V
V-
Rb
1Meg
10.00V-10.00V
10.00V
V210Vdc
0
R3
10k
0
-1.575mV
0V
V+
V V3
10Vdc
C2
220n
0
R2
20k0
0V
R1
10k
Ra
1k
U1
AD741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V+
-1.595mV
0V
0
0V
0V
C1
220n V-
V11Vac0Vdc
VCVS band-pass filter II
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(U1:OUT)
0V
0.5V
1.0V
(72.287,500.734m)
AC sweep from 1mHz to 1MHz
The center frequency is
HzkkknFnF
kk
RRRCC
RRfc 34.72
102010220220
1010
2
1
2
1
32121
13
If Ra<<Rb, the gain drops to ½ at the center frequency.
1011
111
M
k
R
RG
b
a 5.013
1
3
G
GA
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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Frequency
1.0mHz 10mHz 100mHz 1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHz20*LOG10(V(R3:2)/V(R1:1))
-80
-60
-40
-20
-0
(10.000K,-39.293)
(100.000K,-59.369)
(100.000m,-53.667)
(1.0000,-33.667)
(72.444,-718.738u)
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.
Frequency
1.0mHz 10mHz 100mHz 1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHz20*LOG10(V(R3:2)/V(R1:1))
-80
-40
0
(66.069,526.810m)
VCVS band-pass filter I: Bode plot from 1mHz to 1MHz
The response is rather flat.
Frequency
1.0mHz 10mHz 100mHz 1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) 20*LOG10(V(R3:2)/V(R1:1))
-80
-60
-40
-20
-0
(100.000K,-62.833)
(10.000K,-42.805)
(1.0000,-37.181)
(100.000m,-57.180)
(72.444,-6.0081)
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.
0
V
R6
1k
V+
0
V-
V+
R1
1k
low-pass
R2
1k
C1
1u
V11Vac0Vdc
0
C2
1u
0
R4
1k
U1uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R3
1k
V3
-5Vdc
V
V-
0
V+
V-
U3uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V2
5Vdc
V+
U2uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
V-
R5
1k
V
band-pass
Biquadratic filter
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHz 10MHzV(R1:1) V(R2:2) V(U1:OUT)
0V
0.4V
0.8V
1.2V
(158.489,1.0007)
AC sweep from 1Hz to 10MHz
The band-pass gain is 2
4
R
RGBP .
The low-pass gain is 1
2
R
RGLP .
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
HzFFkkCCRR
fc 15.15911112
1
2
1
2142
The quality factor is
03.0111
11 3
242
123
Fkk
Fk
CRR
CRQ
The circuit is overdamped. The bandwidth is approximately given by
kHzHz
Q
fBW c 33.33
03.0
15.15922
Frequency
1.0Hz 3.0Hz 10Hz 30Hz 100Hz 300Hz 1.0KHz 3.0KHz 10KHz 30KHz 100KHz 300KHz 1.0MHz 3.0MHz 10MHzV(R1:1) 20*log10(V(R4:1)/V(R1:1)) 20*log10(V(R6:2)/V(R1:1))
-100
-50
-0
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.
R2
1k
0
V11Vac0Vdc
V-
V25Vdc
V
0
0
R1
1k
V-
V
C2
1n
V3-5Vdc
V+
U1
AD741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2C1
1n0
R3
1k
0
V+
Multiple feedback filter
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHz 10MHzV(R1:1) V(U1:OUT)
0V
0.5V
1.0V
(151.741K,333.257m) (159.224K,316.568m)
AC sweep from 1Hz to 10MHz
The transfer function for the circuit is
22
2
2
1
oo
o
i
o
sQ
s
K
CBsAsV
VsH
where
psnFnFkkCCRRA 111112121
sk
nFkknFknFk
R
CRRCRCRB 3
1
1111111
3
2212122
11
1
3
1
k
k
R
RC
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11
1
1
3
k
k
R
RK
333.011111
1111
2223
2132
nFkkk
nFnFkk
CRKRR
CCRRQ
kHznFnFkkCCRR
fc 15.15911112
1
2
1
2132
kHzkHzfco 968.99915.15922
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.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHz 10MHz 100MHzV(R1:1) 20*LOG10(V(U1:OUT)/V(R1:1))
-80
-40
0
40
(16.046M,-61.581)
(1.6046M,-38.113)
(160.456K,-10.060)
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
RCf dB 2
13
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.
0
V-
V+
V11Vac0Vdc
V3-3.6Vdc
V-
R1
2kV2
3.6Vdc
0
0
C1
47n
V
U1
AD741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
V
V+
First-order non-inverting low-pass filter
The –3dB frequency is given by
kHznFkRC
f dB 693.14722
1
2
13
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(U1:OUT)
0V
0.5V
1.0V
(1.6930K,707.101m)
AC sweep from 1Hz to 1MHz
The –3dB frequency for the filter is at 1.693kHz.
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Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(U1:OUT)/ V(V1:+))
-60
-40
-20
-0
10
(168.526K,-40.039)
(16.853K,-20.004)
(1.6853K,-2.9899)
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.
U1
AD741
+3
-2
V+7
V-4
OUT6
OS11
OS25
C1
220n
R1
723.4
V11Vac0Vdc
0
V23.6Vdc
V3-3.6Vdc
V-
V+
0
V+
0
V-
R2
723.4
0
V
V
First-order inverting low-pass filter
The –3dB frequency is given by
kHznFRC
f dB 12204.7232
1
2
13
Frequency
1.0Hz 3.0Hz 10Hz 30Hz 100Hz 300Hz 1.0KHz 3.0KHz 10KHz 30KHz 100KHz 300KHz 1.0MHzV(R2:1) V(C1:1)
0V
0.5V
1.0V
(1.0005K,706.625m)
AC sweep from 1Hz to 1MHz The –3dB frequency for the filter is at 1kHz.
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Frequency
1.0Hz 3.0Hz 10Hz 30Hz 100Hz 300Hz 1.0KHz 3.0KHz 10KHz 30KHz 100KHz 300KHz 1.0MHzV(R2:1) 20*LOG10(V(C1:1)/V(R2:1))
-40
-20
0
10
(128.825K,-36.092)
(10.115K,-20.095)
(1.0000K,-3.0140)
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.
0
0
V
C1
47nV2
3.6Vdc
0
V-
V
0
V3-3.6Vdc
U1
AD741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V+
R1
2k
V11Vac0Vdc
V+ V-
First-order non-inverting high-pass filter
The –3dB frequency is given by
kHznFkRC
f dB 693.14722
1
2
13
F r e q u e n c y
1 . 0 H z 1 0 H z 1 0 0 H z 1 . 0 K H z 1 0 K H z 1 0 0 K H z 1 . 0 M H zV ( V 1 : + ) V ( U 1 : O U T )
0 V
0 . 5 V
1 . 0 V
( 1 . 6 9 3 3 K , 7 0 7 . 1 0 1 m )
AC sweep from 1Hz to 1MHz
The –3dB frequency for the filter is at 1.693kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(U1:OUT)/ V(V1:+))
-80
-40
0
(16.853,-40.041)
(168.526,-20.084)
(1.6853K,-3.0304)
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.
U1
AD741
+3
-2
V+7
V-4
OUT6
OS11
OS25
R2
723.4
V11Vac0Vdc
0
V23.6Vdc
V3-3.6Vdc
V-
V+
0
V+
0
V-
R1
723.4
C1
220n
0
C2
100p
V
V
First-order inverting high-pass filter
The 100pF capacitor ensures stability at high frequency. The –3dB frequency is given by
kHznFRC
f dB 12204.7232
1
2
1
13
Frequency
1.0Hz 3.0Hz 10Hz 30Hz 100Hz 300Hz 1.0KHz 3.0KHz 10KHz 30KHz 100KHz 300KHz 1.0MHzV(C1:2) V(C2:1)
0V
0.5V
1.0V
(1.0000K,707.461m)
AC sweep from 1Hz to 1MHz
The –3dB frequency for the filter is at 1kHz.
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Frequency
1.0Hz 3.0Hz 10Hz 30Hz 100Hz 300Hz 1.0KHz 3.0KHz 10KHz 30KHz 100KHz 300KHz 1.0MHzV(C1:2) 20*LOG10(V(C2:1)/V(C1:2))
-60
-40
-20
-0
(10.000,-40.001)
(100.000,-20.044)
(1.0000K,-3.0060)
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
21212
1
CCRRfc
Depending on low-pass or high-pass configurations, the quality factors are
212
2121
RRC
CCRRQ
low-pass
211
2121
CCR
CCRRQ
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 0.7071 Linkwitz-Riley 0.5 Bessel 0.577 Chebyshev (0.5dB) 0.8637 Chebyshev (1dB) 0.9565 Chebyshev (2dB) 1.1286 Chebyshev (3dB) 1.3049 Transitional or Paynter 0.639 Butterworth-Thomson or Linear phase 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.
R1
1k
V-
C2
100n
V+
V
C1
200n
V2
5Vdc
0
V
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0 0
V11Vac0Vdc
V+
R2
1k
0
V-
V3
-5Vdc
Butterworth filter (low-pass) (2nd-order) Note: R1=R2 and C1=2xC2.
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Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(U1:-)
0V
0.5V
1.0V
(1.1253K,707.101m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is
kHznFnFkkCCRR
fc 125.1100200112
1
2
1
2121
The quality factor is
707.011100
10020011
212
2121
kknF
nFnFkk
RRC
CCRRQ
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(U1:-)/ V(V1:+))
-60
-40
-20
-0
20
(11.307K,-40.148)
(1.1307K,-3.0528)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1.125kHz and then it decreases by –40dB/decade.
25 www.ice77.net
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.
V2
5VdcV
0
R2
2k
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
00
C1
10n
V11Vac0Vdc
V-V+
R1
1k
V-
V
V+
V3
-5Vdc
C2
10n
0 Butterworth filter (high-pass) (2nd-order)
Note: C1=C2 and R2=2xR1.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) V(U1:-)
0V
0.5V
1.0V
(11.247K,707.101m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is
kHznFnFkkCCRR
fc 253.111010212
1
2
1
2121
The quality factor is
707.010101
101021
211
2121
nFnFk
nFnFkk
CCR
CCRRQ
26 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) 20*LOG10(V(U1:-)/V(C1:1))
-200
-100
0
40
(111.424,-80.170)
(1.1142K,-40.169)
(11.142K,-3.0893)
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+
R3
476
0 0
V3
-5Vdc
R4
1.2k0
R1
2294V2
5VdcC1
150n
V
0
V
0
V-0
V+ V-
V11Vac0Vdc
C3
150n
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
C2
150n
R2
1059
R5
1k
Butterworth filter (low-pass) (3rd-order) Note: C1=C2=C3.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C2:1)
0V
1.0V
2.0V
3.0V
(1.0029K,1.5710)
(10.000,2.1999)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz. The gain is given by the gain resistors:
2.21
12.1
5
54
k
kk
R
RRA or +6.848dB
28 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(C2:1)/ V(V1:+))
-100
-50
0
50
(10.000K,-53.006)
(1.0000K,3.9619)(10.000,6.8481)
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.
29 www.ice77.net
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.
V
0V-
0
0
V
R3
1.59k
V2
5Vdc
0
C2
100n
VV11Vac0Vdc C3
100n
V-
V3
-5Vdc
R2
795
R1
795
V+
0
C1
400n
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V+
Butterworth filter (low-pass) (3rd-order) Note: R1=R2 and C1=4xC2.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(U1:OUT) V(R3:2)
0V
0.4V
0.8V
1.2V
(1.0000K,1.0009)
(1.0000K,708.091m)
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
kHznFnFCCRR
fc 11004007957952
1
2
1
2121
1
kHznFkCR
fc 110059.12
1
2
1
332
The quality factor of the second-order block is
1795795100
100400795795
212
21211
nF
nFnF
RRC
CCRRQ
30 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(U1:OUT)/ V(V1:+)) 20*LOG10(V(R3:2)/ V(V1:+))
-100
-50
0
20
(10.000K,-60.083)
(1.0000K,-2.9982)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –60dB/decade.
31 www.ice77.net
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.
R2
954
C1
47n
0
0
V
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V11Vac0Vdc
V-
C3
47nV
R3
16.73k
V3
-5Vdc
C2
47n
0
V+
R1
2431
0
V2
5Vdc
0
V-
V+
Butterworth filter (high-pass) (3rd-order) Note: C1=C2=C3.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(R2:1)
0V
0.5V
1.0V
(1.0000K,707.639m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(R2:1)/ V(V1:+))
-200
-100
0
100
(10.000,-120.007)
(100.000,-60.007)
(1.0000K,-3.0038)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –60dB/decade.
32 www.ice77.net
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.
V-V+
V
C3
100n
0
V-
C2
106n
V11Vac0Vdc
C1
106nV3
-5VdcR2
3k
0
V
V+
R1
750
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2 V2
5Vdc
V
0
0
0
R3
1.59k
Butterworth filter (high-pass) (3rd-order) Note: C1=C2 and R2=4xR1.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:2) V(R1:2) V(R3:2)
0V
0.4V
0.8V
1.2V
(1.0000K,0.9991)
(1.0000K,706.096m)
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
kHznFnFkCCRR
fc 110610637502
1
2
1
2121
1
kHznFkCR
fc 110059.12
1
2
1
332
The quality factor of the second-order block is
1106106750
1061063750
211
21211
nFnF
nFnFk
CCR
CCRRQ
33 www.ice77.net
Frequency
1.0Hz 3.0Hz 10Hz 30Hz 100Hz 300Hz 1.0KHz 3.0KHz 10KHz 30KHz 100KHz 300KHz 1.0MHzV(C1:2) 20*log10(V(U1:-)/V(C1:2)) 20*log10(V(R3:2)/V(C1:2))
-200
-100
0
100
(100.000,-60.025)
(1.0000K,-3.0227)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –60dB/decade.
34 www.ice77.net
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
R6
1k
0
R5
430
V+ V-
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
C4
47n
V+
R1
1k
R2
1k
0
R3
1k
V11Vac0Vdc
00
V3
-5VdcC1
212n
V
V-
R4
1k
0
0
C2
296n
C3
214n
V2
5Vdc
Butterworth filter (low-pass) (4th-order)
Note: all resistor values match.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C1:1)
0V
0.5V
1.0V
1.5V
(10.000,1.4299)
(1.0009K,1.0385)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(C1:1)/V(V1:+))
-80
-40
0
20
(9.3325K,-61.319)
(1.0000K,343.421m)(10.000,3.1064)
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.
35 www.ice77.net
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.
R1
495
C3
220n
V3
-5Vdc
0 0
V-
V
R4
195
R2
2.65k
0
V+
R6
1k
C4
220n
0
R3
1.08k
0
C1
220n
V11Vac0Vdc
V-
R5
1.2k
V2
5Vdc
V+
0
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V
C2
220n
Butterworth filter (low-pass) (4th-order)
Note: all capacitor values match.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(R5:2)
0V
1.0V
2.0V
2.6V
(10.000,2.1999)
(1.0005K,1.5311)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(R5:2)/ V(V1:+))
-60
-40
-20
-0
20
(8.7096K,-50.861)
(1.0000K,3.7091)(10.000,6.8481)
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.
36 www.ice77.net
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.
V
C1
120n
0
R3
610V
V+
V-
R1
1.45k
C4
100n
V2
5Vdc
R4
610V11Vac0Vdc
0
V
C2
100n
0
C3
685n
V+
V3
-5Vdc
00
R2
1.45k
V-
V-
V+
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
Butterworth filter (low-pass) (4th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C1:1) V(C3:1)
0V
0.4V
0.8V
1.2V
(1.0000K,548.870m)
(1.0063K,706.509m)
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
HznFnFkkCCRR
fc 100110012045.145.12
1
2
1
2121
1
HznFnFCCRR
fc 9971006856106102
1
2
1
2121
2
The quality factors are
5477.045.145.1100
10012045.145.1
212
21211
kknF
nFnFkk
RRC
CCRRQ
3086.111100
10068511
434
43432
kknF
nFnFkk
RRC
CCRRQ
The quality factors are reasonably close to 0.5412 and 1.3065.
37 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(C1:1)/ V(V1:+)) 20*LOG10(V(C3:1)/ V(V1:+))
-120
-80
-40
-0
40
(10.000K,-80.232)
(1.0000K,-2.9035)
(1.0000K,-5.2106)
Bode plot from 1Hz to 1MHz
The gain drops to –2.9035dB at 1kHz and then it decreases by –80dB/decade.
38 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.
R3
1.2k
V3
-5Vdc
V+
V
R2
850
V-
V2
5Vdc
R5
4400
0
C4
100n
V
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
C3
100n
V11Vac0Vdc
R6
1k
V-
0
R1
1195
C2
100n
0 0
C1
100n
V+
R4
5.28k
0
Butterworth filter (high-pass) (4th-order)
Note: all capacitor values match.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:2) V(R3:1)
0V
0.5V
1.0V
1.5V
1.8V
(1.0000K,1.0184)
(10.000K,1.4233)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(R5:2)/V(V1:+))
-200
-100
0
40
(100.000,-76.834)
(1.0000K,158.734m)
(10.000K,3.0658)
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.
39 www.ice77.net
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.
00
V+
C1
192n
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
C3
23n
V2
5Vdc
V+
V
V
V-
V+
R2
1.3k
V
V-
V3
-5Vdc
C4
100n
R1
1k
R3
1k
V11Vac0Vdc
0
C2
100n
V-
0 0
U1
LM7413
27
4
6
1
5+
-V
+V
-
OUT
OS1
OS2
R4
11k
Butterworth filter (high-pass) (4th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(R1:2) V(U2:-)
0V
0.4V
0.8V
1.2V
(1.0000K,537.091m)
(1.0094K,706.509m)
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
HznFnFkkCCRR
fc 10071001923.112
1
2
1
2121
1
kHznFnFkkCCRR
f c 1100231112
1
2
1
2121
2
The quality factors are
5411.01001921
1001923.11
211
21211
nFnFk
nFnFkk
CCR
CCRRQ
2932.1100231
10023111
211
21212
nFnFk
nFnFkk
CCR
CCRRQ
The quality factors are reasonably close to 0.5412 and 1.3065.
40 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(R1:2)/ V(V1:+)) 20*LOG10(V(U2:-)/V(V1:+))
-300
-200
-100
-0
60
(1.0000K,-5.3991)
(100.000,-80.138)
(1.0000K,-3.1741)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –80dB/decade.
41 www.ice77.net
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-
R1
1k
R5
1k
R4
1k
0 0
V
0
R3
1k
V+
V11Vac0Vdc
V+
V3
-5Vdc
R6
1k
0
0
V-
C2
154.56n
V2
5Vdc
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
VC4
83.06n
R7
1k
0
R2
1k
0
C3
182.49n
C1
246.83n
C5
34.57n
Butterworth filter (low-pass) (5th-order)
Note: all resistor values match.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C2:1)
0V
1.0V
2.0V
3.0V(1.0000K,2.5885)
(10.000,2.0000)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(C2:1)/V(V1:+))
-100
-50
0
50
(10.000K,-75.288)
(1.0000K,8.2609)(10.000,6.0207)
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.
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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.
V+
0
0
C5
100n0
C2
100nV
V-
V3
-5Vdc
R1
1.285k
0
V-
R3
490
R4
490
V
C3
1050n
V2
5Vdc
0
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2 R5
1.59k
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V
R2
1.285k
C4
100n
V-
C1
153n
V
V+
V11Vac0Vdc
0
V+
Butterworth filter (low-pass) (5th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(U1:-) V(U2:-) V(R5:2)
0V
0.5V
1.0V
1.3V
(1.0033K,705.621m)
(1.0033K,617.409m)
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
HznFnFkkCCRR
fc 1001100153285.1285.12
1
2
1
2121
1
HznFnFCCRR
fc 100210010504904902
1
2
1
4343
2
kHznFkCR
fc 110059.12
1
2
1
553
43 www.ice77.net
The quality factors are
6185.0285.1285.1100
100153285.1285.1
212
21211
kknF
nFnFkk
RRC
CCRRQ
6202.1490490100
1001050490490
434
43432
nF
nFnF
RRC
CCRRQ
The quality factors are reasonably close to 0.6180 and 1.6181.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) 20*LOG10(V(U1:-)/V(R1:1)) 20*LOG10(V(U2:-)/V(R1:1)) 20*LOG10(V(R5:2)/V(R1:1))
-150
-100
-50
-0
50
(10.000K,-100.244)
(1.0000K,-2.9558)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –100dB/decade.
44 www.ice77.net
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.
C4
50n
R3
1k
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V-
V-
V11Vac0Vdc
V+
V
V+
V
0
R2
2.44k
V3
-5Vdc
C1
207n
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R4
10.475k
0
0
0
C2
50nC3
48n
V
V+
V
0
R5
1.59k0
V2
5Vdc
R1
1.285k
V-
C5
100n
Butterworth filter (high-pass) (5th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:2) V(U1:OUT) V(R3:2) V(R5:2)
0V
0.5V
1.0V
1.3V
(1.0000K,703.322m)
(1.0000K,617.445m)
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
HznFnFkkCCRR
fc 10015020744.212
1
2
1
2121
1
HznFnFkkCCRR
fc 10035048475.1012
1
2
1
4343
2
kHznFkCR
fc 110059.12
1
2
1
553
45 www.ice77.net
The quality factors are
6183.0502071
5020744.21
211
21211
nFnFk
nFnFkk
CCR
CCRRQ
6179.150481
5048475.101
211
21212
nFnFk
nFnFkk
CCR
CCRRQ
The quality factors are reasonably close to 0.6180 and 1.6181.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:2) 20*LOG10(V(U1:OUT)/V(C1:2)) 20*LOG10(V(R3:2)/V(C1:2)) 20*LOG10(V(R5:2)/V(C1:2))
-400
-200
0
100
(100.000,-100.100)
(1.0000K,-3.0569)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –100dB/decade.
46 www.ice77.net
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.
C2
465n
C4
212n
V
R3
1kV2
5VdcC6
36n
R6
1k
R1
1kV3
-5Vdc
V-
R7
1k
R2
1k
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
C5
104n
0
0
V11Vac0Vdc
V+
0
V+
0
0
R8
1k
0
C1
205n
V
V-
R5
1k
0
R4
1k
C3
218n
Butterworth filter (low-pass) (6th-order)
Note: all resistor values match.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(C5:1)
0V
1.0V
2.0V
2.6V
(10.000,1.9999)
(1.0000K,1.4347)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) 20*LOG10(V(C5:1)/V(R1:1))
-80
-40
0
30
(3.8019K,-65.177)
(1.0000K,3.1353)(10.000,6.0202)
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.
47 www.ice77.net
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.
U3
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R2
490
V-
V+
0
V
V3
-5Vdc
V R6
204V
V2
5VdcC4
100n
C5
2000n
V-
V+
0
V-
C1
200n
0
C2
100n
R1
2.584k
R4
165
V+
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V-
R5
619
V11Vac0Vdc
C6
100n
V+
0 0
C3
2000n
V
0
R3
767
U1
LM7413
27
4
6
1
5+
-V
+V
-
OUT
OS1
OS2
Butterworth filter (low-pass) (6th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(U1:-) V(U2:OUT) V(U3:-)
0V
1.0V
2.0V
3.0V
(1.0000K,1.7122)
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
kHznFnFkCCRR
fc 1100200490584.22
1
2
1
2121
1
kHznFnFCCRR
fc 110020001657672
1
2
1
2121
2
kHznFnFCCRR
fc 001.110020002046192
1
2
1
2121
3
48 www.ice77.net
The quality factors are
5177.0490584.2100
100200490584.2
212
21211
knF
nFnFk
RRC
CCRRQ
7070.1165767100
1002000165767
212
21212
nF
nFnF
RRC
CCRRQ
9310.1204619100
1002000204619
212
21213
nF
nFnF
RRC
CCRRQ
The quality factors are reasonably close to 0.5177, 1.7071 and 1.9320.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) 20*LOG10(V(U1:-)/V(R1:1)) 20*LOG10(V(U2:OUT)/V(R1:1)) 20*LOG10(V(U3:-)/V(R1:1))
-200
-100
0
100
(10.000K,-121.211)
(1.0000K,4.6711)
(1.0000K,-1.0514)
(1.0000K,-5.7160)
Bode plot from 1Hz to 1MHz
The gain peaks to +4.6711dB at 1 kHz and then it decreases by about –120dB/decade.
49 www.ice77.net
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.
0
C4
100n
0
R6
3.844kV-
V
V-
V
R5
165C1
400n
V2
5Vdc
R3
186
0
R4
3.396k
C6
100n
V+
0
V11Vac0Vdc
V-
U1
LM7413
27
4
6
1
5+
-V
+V
-
OUT
OS1
OS2
V+U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V+
0
U3
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V-
0
V
C5
400nV
V+
R2
1.03k
R1
615
V3
-5Vdc
C2
100nC3
400n
Butterworth filter (high-pass) (6th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) V(C3:1) V(U2:-) V(U3:-)
0V
1.0V
2.0V
3.0V
(1.0000K,1.7095)
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
HznFnFkCCRR
fc 9.99910040003.16152
1
2
1
2121
1
HznFnFkCCRR
fc 1001100400396.31862
1
2
1
2121
2
HznFnFkCCRR
fc 2.999100400844.31652
1
2
1
2121
3
50 www.ice77.net
The quality factors are
5177.0100400615
10040003.1615
211
21211
nFnF
nFnFk
CCR
CCRRQ
7092.1100400186
100400396.3186
211
21212
nFnF
nFnFk
CCR
CCRRQ
9307.1100400165
100400844.3165
211
21213
nFnF
nFnFk
CCR
CCRRQ
The quality factors are reasonably close to 0.5177, 1.7071 and 1.9320.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) 20*LOG10(V(C3:1)/V(C1:1)) 20*LOG10(V(U2:-)/V(C1:1)) 20*LOG10(V(U3:-)/V(C1:1))
-400
-200
0
200
(100.000,-119.930)
(1.0000K,4.6575)
(1.0000K,-1.0675)
(1.0000K,-1.0675) (1.0000K,-5.7166)
Bode plot from 1Hz to 1MHz
The gain peaks at +4.6575dB at 1kHz and then it decreases by about –120dB/decade.
51 www.ice77.net
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.
0
V+
V2
5Vdc
R2
1k
R1
1k
0
V
C1
100n
V
V-
0
V-V+
V11Vac0Vdc
0
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2 V3
-5VdcC2
100n
Linkwitz-Riley filter (low-pass) (2nd-order)
Note: R1=R2 and C1=C2.
52 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(C1:2)
0V
0.5V
1.0V
(1.5915K,500.000m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is
kHznFnFkkCCRR
fc 5915.1100100112
1
2
1
2121
The quality factor is
5.011100
10010011
212
2121
kknF
nFnFkk
RRC
CCRRQ
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) 20*LOG10(V(C1:2)/V(R1:1))
-60
-40
-20
-0
20
(15.849K,-40.062)
(1.5849K,-5.9848)
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 1.5915kHz and then it decreases by –40dB/decade.
53 www.ice77.net
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.
V2
5Vdc
R1
1.5k
R2
1.5k
C1
10n
0
0
V+
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V-V+
V
0
V11Vac0Vdc
V
0
V3
-5Vdc
V-
C2
10n
Linkwitz-Riley filter (high-pass) (2nd-order)
Note: C1=C2 and R1=R2.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) V(U1:OUT)
0V
0.5V
1.0V
(10.635K,501.385m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is
kHznFnFkkCCRR
fc 61.10101065.155.12
1
2
1
2121
The quality factor is
5.010105.1
10105.15.1
211
2121
nFnFk
nFnFkk
CCR
CCRRQ
54 www.ice77.net
Frequency
10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) 20*LOG10(V(U1:OUT)/V(C1:1))
-120
-80
-40
-0
40
(105.404,-80.113)
(1.0540K,-40.197)
(10.540K,-6.0746)
Bode plot from 10Hz to 1MHz
The gain drops to –6dB at about 10.61kHz and then it decreases by –40dB/decade.
55 www.ice77.net
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.
00
C3
200n
R2
1k
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V11Vac0Vdc
V+
0V-
V
V-
C1
200n
V-
R1
1k
C2
100n
R3
1k
V
0
0
V
V3
-5Vdc
V+
R4
1k
V2
5Vdc
V+
C4
100n
Linkwitz-Riley filter (low-pass) (4th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(R3:1) V(U2:-)
0V
0.5V
1.0V
(1.1254K,707.134m)
(1.1254K,500.000m)
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
kHznFnFkkCCRR
ff cc 125.1100200112
1
2
1
2121
21
The quality factors are
707.011100
10020011
212
212121
kknF
nFnFkk
RRC
CCRRQQ
56 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) 20*LOG10(V(R3:1)/V(R1:1)) 20*LOG10(V(U2:-)/V(R1:1))
-120
-80
-40
-0
30
(11.316K,-80.323)
(1.1316K,-3.0594)
(1.1316K,-6.1196)
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 1.125kHz and then it decreases by –80dB/decade.
57 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.
0
R2
906
V
V-
R3
1.12k
V3
-5Vdc
0
V
V11Vac0Vdc
0
0
R1
1.125k
R4
5.623k
C4
100n
V+
V-
C1
100n
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
V+
R5
320
R6
1k
C3
100nV2
5Vdc
C2
100n
0
Linkwitz-Riley filter (high-pass) (4th-order)
Note: all capacitor values match.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:2) V(U1:OUT)
0V
0.5V
1.0V
1.5V(10.000K,1.3061)
(1.0000K,662.007m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:2) 20*LOG10(V(U1:OUT)/V(C1:2))
-200
-100
0
60
(100.000,-77.614)
(1.0000K,-3.5828)(10.000K,2.3198)
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.
58 www.ice77.net
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.
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V
V+
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V+
R4
2k
C2
10nC3
10n
0
V+
R2
2kV-
V-
V-
0
R3
1k
0
V3
-5Vdc
V
C1
10nV
00
C4
10n
V2
5Vdc
R1
1k
V11Vac0Vdc
Linkwitz-Riley filter (high-pass) (4th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) V(R1:2) V(U2:-)
0V
0.5V
1.0V
(11.245K,707.211m)
(11.245K,500.000m)
AC sweep from 1Hz to 1MHz
The cutoff frequencies are
kHznFnFkkCCRR
ff cc 253.111010212
1
2
1
2121
21
The quality factors are
707.010101
101021
211
212121
nFnFk
nFnFkk
CCR
CCRRQQ
59 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) 20*LOG10(V(R1:2)/V(C1:1)) 20*LOG10(V(U2:-)/V(C1:1))
-200
0
-350
60
(11.167K,-3.0696)
(1.1167K,-80.261)
(11.167K,-6.1380)
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 11.253kHz and then it decreases by –80dB/decade.
60 www.ice77.net
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.
V
V
R5
1k
V-
V+
U3
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V-
0
V
0
V3
-5VdcC6
50n
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V
V+
0
R6
1k
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
V2
5Vdc
C2
100n
R1
1k
V-
0
V11Vac0Vdc
R3
1k
C5
200nV+
C1
100n
C4
50n
0
V+
R4
1k
R2
1k
V-
C3
200n
Linkwitz-Riley filter (low-pass) (6th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C1:2) V(R5:1) V(C5:2)
0V
0.4V
0.8V
1.2V
(1.5895K,501.939m)
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
kHznFnFkkCCRR
fc 591.1100100112
1
2
1
2121
1
kHznFnFkkCCRR
ff cc 591.150200112
1
2
1
2121
32
61 www.ice77.net
The quality factors are
5.011100
10010011
212
21211
kknF
nFnFkk
RRC
CCRRQ
11150
5020011
212
212132
kknF
nFnFkk
RRC
CCRRQQ
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) 20*LOG10(V(U1:-)/V(R1:1)) 20*LOG10(V(U2:OUT)/V(R1:1)) 20*LOG10(V(U3:-)/V(R1:1))
-200
-100
0
50
(15.898K,-120.259)
(1.5898K,-5.9938)
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 1.591kHz and then it decreases by –120dB/decade.
62 www.ice77.net
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.
V11Vac0Vdc
R5
750
V
V
V-
0
V+
0
0
C2
10n
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
V-V+
V2
5Vdc
V-
C3
10n
V
R3
750
V
V3
-5Vdc
C6
10n
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R2
1.5k
C5
10n
C4
10n
R1
1.5k
V+C1
10n
R6
3kV-
U3
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V+
0
R4
3k
0
Linkwitz-Riley filter (high-pass) (6th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) V(U1:-) V(U2:-) V(R5:2)
0V
0.4V
0.8V
1.2V
(10.616K,501.939m)
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
kHznFnFkkCCRR
fc 61.1010105.15.12
1
2
1
2121
1
kHznFnFkCCRR
ff cc 61.10101037502
1
2
1
2121
32
63 www.ice77.net
The quality factors are
5.010105.1
10105.15.1
211
21211
nFnFk
nFnFkk
CCR
CCRRQ
11010750
10103750
211
212132
nFnF
nFnFk
CCR
CCRRQQ
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(R1:2)/V(V1:+)) 20*LOG10(V(C5:1)/V(V1:+)) 20*LOG10(V(R5:2)/V(V1:+))
-600
-400
-200
0
100
(1.0638K,-119.853)
(10.638K,-5.9403)
Bode plot from 1Hz to 1MHz
The gain drops to –6dB at 10.61kHz and then it decreases by –120dB/decade.
64 www.ice77.net
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.
V+
0
R1
1k
V11Vac0Vdc
V3
-5VdcC2
100n
0
V
C1
133n
00
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2V2
5Vdc
V-
V
V+
R2
1k
V-
Bessel filter (low-pass) (2nd-order)
Note: R1=R2 and C1=1.336xC2.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(C1:2)
0V
0.5V
1.0V
(1.3791K,576.923m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is
kHznFnFkkCCRR
fc 38.1100133112
1
2
1
2121
65 www.ice77.net
The quality factor is
577.011100
10013311
212
2121
kknF
nFnFkk
RRC
CCRRQ
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) 20*LOG10(V(C1:2)/ V(R1:1))
-60
-40
-20
-0
20
(13.971K,-40.118)
(1.3804K,-4.7854)
Bode plot from 1Hz to 1MHz
The gain drops to –4.78dB at 1.38kHz and then it decreases by –40dB/decade.
66 www.ice77.net
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.
V-
V
R2
1.33k
C2
100n
0
V11Vac0Vdc
V+U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2V3
-5Vdc
0
V-
V2
5Vdc
V+
R1
1k
0
V
C1
100n
0
Bessel filter (high-pass) (2nd-order)
Note: C1=C2 and R2=R1x1.336.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(R1:2)
0V
0.5V
1.0V
(1.3807K,576.923m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is
kHznFnFkkCCRR
fc 38.110010033.112
1
2
1
2121
The quality factor is
577.01001001
10010033.11
211
2121
nFnFk
nFnFkk
CCR
CCRRQ
67 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(R1:2)/ V(V1:+))
-150
-100
-50
-0
40
(13.834,-79.958)
(138.337,-40.002)
(1.3834K,-4.7606)
Bode plot from 1Hz to 1MHz
The gain drops to –4.78dB at 1.38kHz and then it decreases by –40dB/decade.
68 www.ice77.net
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.
R3
712V3
-5Vdc
V-
C1
218n
V
V+
00
V11Vac0Vdc
V-
C2
314n
V2
5VdcV
V+
0 0
C3
56n
0
R1
712
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R2
712
Bessel filter (low-pass) (3rd-order) Note: R1=R2=R3.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(U1:-)
0V
0.5V
1.0V
(1.0006K,715.976m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(U1:-)/ V(V1:+))
-100
-50
0
20
(10.034K,-51.006)
(1.0000K,-2.8984)
Bode plot from 1Hz to 1MHz
The gain drops to –2.8984dB at 1kHz and then it decreases to –51.006dB a decade later.
69 www.ice77.net
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.
V11Vac0Vdc
C3
100n
0
R5
1k
V
C2
100n
V-
V-V+
00
0
V3
-5Vdc
R1
3.05k
V
0
R3
520
V+
C1
100nR4
1.2k
R2
902
0
V2
5Vdc
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
Bessel filter (low-pass) (3rd-order)
Note: C1=C2=C3.
F r e q u e n c y
1 . 0 H z 1 0 H z 1 0 0 H z 1 . 0 K H z 1 0 K H z 1 0 0 K H z 1 . 0 M H zV ( V 1 : + ) V ( U 1 : O U T )
0 V
1 . 0 V
2 . 0 V
3 . 0 V
( 1 . 0 0 0 0 K , 1 . 5 6 5 1 )
( 1 0 . 0 0 0 , 2 . 1 9 9 8 )
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(U1:OUT)/ V(V1:+))
-100
-50
0
30
(10.000K,-44.399)
(1.0000K,3.8909)(10.000,6.8478)
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.
70 www.ice77.net
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.
R6
1k
0V-
R2
4.41kV
R5
1.2k0
V+ V-
0
V3
-5VdcC3
100n
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
V+
V
R1
640
C4
100n
0
0
C2
100n
V11Vac0Vdc
V2
5Vdc
R4
333
C1
100n
R3
1.265k
Bessel filter (low-pass) (4th-order)
Note: all capacitor values match.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C3:1)
0V
1.0V
2.0V
2.8V
(10.000,2.1998)
(1.0000K,1.5627)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(C3:1)/V(V1:+))
-60
-40
-20
-0
20
(12.589K,-50.168)
(10.000,6.8478)(1.0000K,3.8777)
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.
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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.
V+
C1
298n
V2
5VdcC2
100n
R2
1k
V11Vac0Vdc
V-
V3
-5Vdc
0
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
V-
V
0
R1
1k
0
V
V+
Chebyshev 0.5dB ripple filter (low-pass) (2nd-order)
Note: for a 0.5dB ripple, R1=R2 and C1=2.986xC2.
72 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(U1:OUT)
0V
0.4V
0.8V
1.2V
(922.189,862.722m)
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable. The cutoff frequency is
HznFnFkkCCRR
fc 96.921100298112
1
2
1
2121
The quality factor is
863.011100
10029811
212
2121
kknF
nFnFkk
RRC
CCRRQ
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(U1:OUT)/ V(V1:+))
-60
-40
-20
-0
20
(9.2612K,-40.123)
(926.119,-1.3203)
Bode plot from 1Hz to 1MHz
The gain drops to –1.3203dB at 926Hz and then it decreases by –40dB/decade.
73 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(U1:OUT)/ V(V1:+))
-1.0
0
1.0
2.0
(524.808,499.849m)
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.
74 www.ice77.net
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.
V
R3
1k
0
C3
366n
V2
5VdcV41Vac0Vdc
V3
-5Vdc
V-
V
V+
0
C4
100n
R4
1k
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V-
0 0
V+
Chebyshev 1dB ripple filter (low-pass) (2nd-order)
Note: for a 1dB ripple, R3=R4 and C3=3.663xC4.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V4:+) V(U2:-)
0V
0.4V
0.8V
1.2V
(830.016,958.580m)
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable. The cutoff frequency is
HznFnFkkCCRR
fc 92.831100366112
1
2
1
3232
The quality factor is
957.011100
10036611
323
3232
kknF
nFnFkk
RRC
CCRRQ
75 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V4:+) 20*LOG(V(U2:-)/ V(V4:+))
-150
-100
-50
-0
50
(8.3074K,-92.129)
(830.736,-863.103m)
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).
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V4:+) 20*LOG10(V(U2:-)/ V(V4:+))
-1.0
0
1.0
2.0
(562.341,1.0039)
Close-up of 1dB ripple
The frequency response of the circuit peaks at 562Hz with a 1dB overshoot and then it decays rapidly.
76 www.ice77.net
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.
V
V-
V+ V-
U3
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V+
V3
-5Vdc
0
V51Vac0Vdc
C6
100n
V
V2
5Vdc
00
R6
1k
0
R5
1k
C5
509n
Chebyshev 2dB ripple filter (low-pass) (2nd-order)
Note: for a 1dB ripple, R5=R6 and C5=5.098xC6.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V5:+) V(C5:2)
0V
0.5V
1.0V
1.5V
(705.774,1.1272)
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable. The cutoff frequency is
HznFnFkkCCRR
fc 44.705100509112
1
2
1
6565
The quality factor is
128.111100
10050911
656
6565
kknF
nFnFkk
RRC
CCRRQ
77 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V5:+) 20*LOG10(V(C5:2)/ V(V5:+))
-60
-40
-20
-0
20
(6.9970K,-39.882)
(702.533,1.0863)
Bode plot from 1Hz to 1MHz
The gain is +1.0863dB at 702Hz and then it decreases by –40dB/decade.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V5:+) 20*LOG10(V(C5:2)/ V(V5:+))
-1.0
0
1.0
2.0
3.0
(549.541,2.0010)
Close-up of 2dB ripple
The frequency response of the circuit peaks at 549Hz with a 2dB overshoot and then it decays rapidly.
78 www.ice77.net
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.
V
0
V-
U4
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R8
1kV3
-5Vdc
R7
1k
V-
C8
100n
V
V+
0
V+
00
V2
5Vdc
C7
681n
V61Vac0Vdc
Chebyshev 3dB ripple filter (low-pass) (2nd-order)
Note: for a 1dB ripple, R7=R8 and C7=6.812xC8.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V6:+) V(U4:OUT)
0V
0.5V
1.0V
1.5V(609.575,1.3047)
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable. The cutoff frequency is
HznFnFkkCCRR
fc 88.609100681112
1
2
1
8787
The quality factor is
305.111100
10068111
878
8787
kknF
nFnFkk
RRC
CCRRQ
79 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V6:+) 20*LOG10(V(U4:OUT)/ V(V6:+))
-60
-40
-20
-0
20
(6.1188K,-37.523)
(611.881,1.8548)
Bode plot from 1Hz to 1MHz
The gain is +1.8548dB at 611Hz and then it decreases by –40dB/decade.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V6:+) 20*LOG10(V(U4:OUT)/ V(V6:+))
0
2.0
4.0
-1.0
(512.861,3.0052)
Close-up of 3dB ripple
The frequency response of the circuit peaks at 512Hz with a 3dB overshoot and then it decays rapidly.
80 www.ice77.net
Second-order low-pass ripple comparison
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(U1:OUT) V(V4:+) V(U2:OUT) V(V5:+) V(C5:2) V(V6:+) V(U4:OUT)
0V
0.5V
1.0V
1.5V
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.
Frequency
100Hz 300Hz 1.0KHz 3.0KHzV(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:+))
-5.0
-2.5
0
2.5
4.0
(524.808,499.849m)
(562.341,1.0039)
(549.541,2.0010)
(512.861,3.0052)
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.
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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.
V
V+
0
C2
100n
R2
2.98k
R1
1k
0
C1
100n
V11Vac0Vdc
V
V+
0
V-
V2
5Vdc
V3
-5Vdc
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
V-
Chebyshev 0.5dB ripple filter (high-pass) (2nd-order)
Note: for a 0.5dB ripple, C1=C2 and R2=2.986xR1.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) V(R1:2)
0V
0.4V
0.8V
1.2V
(921.668,862.722m)
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable. The cutoff frequency is
HznFnFkkCCRR
fc 96.92110010098.212
1
2
1
2121
The quality factor is
863.01001001
10010098.21
211
2121
nFnFk
nFnFkk
CCR
CCRRQ
82 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) 20*LOG10(V(R1:2)/ V(C1:1))
-120
-80
-40
-0
40
(92.564,-39.902)
(925.642,-1.2459)
Bode plot from 1Hz to 1MHz
The gain drops to –1.2459dB at 925Hz and then it decreases by –40dB/decade.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:1) 20*LOG10(V(R1:2)/ V(C1:1))
-1.0
0
1.0
2.0
(1.6218K,486.213m)
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.
83 www.ice77.net
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.
V41Vac0Vdc
V
0
V3
-5Vdc
R3
1k
R4
3.66k
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V
V2
5Vdc
V-
0
V-
V+
C4
100n
V+
00
C3
100n
Chebyshev 1dB ripple filter (high-pass) (2nd-order)
Note: for a 1dB ripple, C3=C4 and R4=3.663xR3.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V4:+) V(R3:2)
0V
0.4V
0.8V
1.2V
(833.787,958.580m)
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable. The cutoff frequency is
HznFnFkkCCRR
fc 92.83110010066.312
1
2
1
4343
The quality factor is
957.01001001
10010066.31
433
4343
nFnFk
nFnFkk
CCR
CCRRQ
84 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V4:+) 20*LOG10(V(R3:2)/ V(V4:+))
-120
-80
-40
-0
40
(83.074,-39.986)
(830.736,-398.059m)
Bode plot from 1Hz to 1MHz
The gain drops to –0.398dB at 830Hz and then it decreases by –40dB/decade.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V4:+) 20*LOG10(V(R3:2)/ V(V4:+))
-1.0
0
1.0
2.0
(1.2303K,990.336m)
Close-up of 1dB ripple
The frequency response of the circuit peaks at 1.23kHz with a 1dB overshoot and then it decays rapidly.
85 www.ice77.net
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.
0
V3
-5Vdc
0
C6
100n
V
0
V
V-
V+
V2
5VdcR6
5.09k
C5
100n
V51Vac0Vdc
V+ V-
U3
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R5
1k
0 Chebyshev 2dB ripple filter (high-pass) (2nd-order)
Note: for a 2dB ripple, C5=C6 and R6=5.098xR5.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C5:1) V(R5:2)
0V
0.5V
1.0V
1.5V
(705.088,1.1272)
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable. The cutoff frequency is
HznFnFkkCCRR
fc 44.70510010009.512
1
2
1
6565
The quality factor is
128.11001001
10010009.51
655
6565
nFnFk
nFnFkk
CCR
CCRRQ
86 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C5:1) 20*LOG10(V(R5:2)/ V(C5:1))
-120
-80
-40
-0
40
(70.087,-40.061)
(700.870,985.793m)
Bode plot from 1Hz to 1MHz
The gain is +0.985dB at 700Hz and then it decreases by –40dB/decade.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C5:1) 20*LOG10(V(R5:2)/ V(C5:1))
-1.0
0
1.0
2.0
3.0
(912.011,1.9866)
Close-up of 2dB ripple
The frequency response of the circuit peaks at 912Hz with a 2dB overshoot and then it decays rapidly.
87 www.ice77.net
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.
V
V+
U4
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
C8
100n
C7
100n
V-
0
0
V
V3
-5Vdc
V2
5Vdc
V+
R7
1k
R8
6.81k
V61Vac0Vdc
V-
0
0
Chebyshev 3dB ripple filter (high-pass) (2nd-order)
Note: for a 3dB ripple, C7=C8 and R8=6.812xR7.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C7:1) V(U4:-)
0V
0.5V
1.0V
1.5V
(610.162,1.3047)
AC sweep from 1Hz to 1MHz
The ripple in the passband is noticeable. The cutoff frequency is
HznFnFkkCCRR
fc 88.60910010081.612
1
2
1
8787
The quality factor is
305.11001001
10010081.61
877
8787
nFnFk
nFnFkk
CCR
CCRRQ
88 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C7:1) 20*LOG10(V(U4:-)/ V(C7:1))
-120
-80
-40
-0
40
(60.987,-39.939)
(609.867,2.3057)
Bode plot from 1Hz to 1MHz
The gain is +2.3057dB at 609Hz and then it decreases by –40dB/decade.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C7:1) 20*LOG10(V(U4:-)/ V(C7:1))
0
2.0
4.0
-1.0
(724.436,2.9916)
Close-up of 3dB ripple
The frequency response of the circuit peaks at 724Hz with a 3dB overshoot and then it decays rapidly.
89 www.ice77.net
Second-order high-pass ripple comparison
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(R1:2) V(C3:1) V(U2:OUT) V(C5:1) V(R5:2) V(C7:1) V(U4:-)
0V
0.5V
1.0V
1.5V
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.
Frequency
300Hz 1.0KHz 3.0KHz 10KHzV(V1:+) 20*LOG10(V(R1:2)/V(V1:+)) V(C3:1) 20*LOG10(V(U2:OUT)/V(C3:1)) V(C5:1) 20*LOG10(V(R5:2)/V(C5:1))V(C7:1) 20*LOG10(V(U4:-)/V(C7:1))
-2.0
0
2.0
4.0(724.436,2.9916)
(912.011,1.9866)
(1.2303K,990.336m)
(1.6218K,486.213m)
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.
90 www.ice77.net
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.
V+
V+
V
R3
92.3k
U2
LM741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
C1
10n
V2
5Vdc
V-
R4
670
V
0
R1
2398
R2
337kR7
23.36k
00
V
R6
6184
C2
10n
0
R9
1.04kR5
10k
C3
10n
R10
1k
0
V11Vac0Vdc
V+
V-
0
V3
-5Vdc
C5
100n
C4
10n
0
V-
R8
22.86k
V
R11
777
U1
LM741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
Inverted Chebyshev filter (low-pass) (5th-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(R1:1) V(R3:2) V(C3:2) V(R11:2)
0V
0.5V
1.0V
1.5V
(10.000,1.1579)(1.0004K,807.692m)
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.
91 www.ice77.net
Frequency
100Hz 300Hz 1.0KHz 3.0KHz 10KHz 30KHz 100KHzV(R1:1) 20*LOG10(V(R3:2)/V(R1:1)) 20*LOG10(V(C3:2)/V(R1:1)) 20*LOG10(V(R11:2)/V(R1:1))
-40
-20
0
(299.358,1.2741)
(10.000K,-21.079)
(1.0002K,-1.8478)
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.
92 www.ice77.net
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.
V-
C1
20n
V+
V+
0
R6
4.42k
0
V11Vac0Vdc
R4
47.94k
R5
1k
C2
10n
R3
3111
V25Vdc
C4
44n
V35Vdc
0
0 0
0
C3
10n
U1
uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V
0
R2
6242
C5
100n
V-
R1
6242
V
R7
2358
0
Elliptic filter (low-pass) (3rd-order)
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C2:2) V(R7:2)
0V
2.0V
4.0V
6.0V
(10.000,4.2998)
(1.0011K,2.8047)
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.
93 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C2:2) 20*LOG10(V(R7:2)/V(C2:2))
-100
-50
0
50
(10.000K,-29.254)
(10.000,12.669) (1.0098K,8.7278)
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.
94 www.ice77.net
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.
0
V+
R1
1125
V-
V-
R2
1125
C2
100n
V11Vac0Vdc
V
0
V
V+
C1
200n
V3
-5Vdc
V2
5Vdc
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
0 Synchronous filter (low-pass) (2nd-order)
Note: R1=R2 and C1=2xC2.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(U1:-)
0V
0.5V
1.0V
(1.0003K,707.101m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is
kHznFnFCCRR
fc 1100200112511252
1
2
1
2121
95 www.ice77.net
The quality factor is
707.011251125100
10020011251125
212
2121
nF
nFnF
RRC
CCRRQ
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(U1:-)/ V(V1:+))
-60
-40
-20
-0
20
(10.000K,-40.051)
(1.0000K,-3.0079)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –40dB/decade.
96 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.
C1
100n
V11Vac0Vdc
V
V3
-5Vdc
V+
V-
V+
R1
1125
00
V
V-
V2
5Vdc
0
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
0
C2
100n
R2
2.25k
Synchronous filter (high-pass) (2nd-order)
Note: C1=C2 and R2=2xR1.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:2) V(U1:-)
0V
0.4V
0.8V
1.2V
(0.9995K,706.509m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is
kHznFnFkCCRR
fc 110010025.211252
1
2
1
2121
The quality factor is
707.01001001125
10010025.21125
211
2121
nFnF
nFnFk
CCR
CCRRQ
97 www.ice77.net
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(C1:2) 20*LOG10(V(U1:-)/ V(C1:2))
-150
-100
-50
-0
50
(10.000,-80.006)
(100.000,-40.007)
(1.0000K,-3.0135)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –40dB/decade.
98 www.ice77.net
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.
R1
810
C1
100n
V3
-5Vdc
0
R3
810
V-
R2
810
V11Vac0Vdc
V
V+
V-
0
VC2
100n
V2
5Vdc
00
0
C3
100n
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
V+
Synchronous filter (low-pass) (3rd-order)
Note: all resistor and capacitor values match.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C3:2)
0V
0.5V
1.0V
(1.0021K,707.101m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(C3:2)/ V(V1:+))
-80
-40
0
20
(10.000K,-42.932)
(1.0000K,-2.9990)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases by –40dB/decade.
99 www.ice77.net
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.
V
V+
R2
691
0
V-
0
C3
100n
V3
-5Vdc
C4
100n
R3
691
V+R1
691
V11Vac0Vdc
C2
100n
V-
0
V
V-
C1
100n
V
0
U2
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
U1
LM7413
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2 V+R4
691
0
V2
5Vdc
Synchronous filter (low-pass) (4th-order)
Note: all resistor and capacitor values match.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(C1:1) V(U2:-)
0V
0.5V
1.0V
(1.0026K,840.896m)
(1.0026K,707.101m)
AC sweep from 1Hz to 1MHz
The cutoff frequency is 1kHz.
Frequency
1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) 20*LOG10(V(C1:1)/V(V1:+)) 20*LOG10(V(U2:-)/V(V1:+))
-100
-50
0
20
(10.000K,-51.990)
(1.0000K,-2.9961)
(1.0000K,-1.4979)
Bode plot from 1Hz to 1MHz
The gain drops to –3dB at 1kHz and then it decreases to –51.99dB a decade later.
100 www.ice77.net
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:
RCf
22
1
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.
HznFkRC
fL 029.3411003.322
1
22
1
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.
kHznFkRC
fH 521.38.67.422
1
22
1
101 www.ice77.net
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.
0
V
R8
4.7k
C5
13.6n
V31Vac0Vdc
R9
4.7k
C7
6.8n
R22
1000k
U1
uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R16
3.3k
0
0
R7
4.7k
U2
uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
C13
100n
VCC
C17
100n
VCC
R10
4.7k
-VCC
R219.4k-VCC
-VCC
-VCC
R18
3.3k V115Vdc
C1
1u
R12
3.3k
R13
100
R14
100
VCC
0
C16
200n
R1
2.2k
0
R2 100k
C15
200n
R196.6k
U9
uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R24 100k
0
0
VCC
U6
uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
VCC
C6
6.8n
C14
100n
V
C4
13.6n
R4
4.7k
VCC
R5
4.7k
0
GND
0
U8
uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
C18
100n
R39.4k
VCC
MID
-VCC
0R23
100k
-VCC
R20
6.6k
LOW
R17
3.3k
0
U7
uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
U3
uA741
3
27
46
1
5+
-V
+V
-OUT
OS1
OS2
R15
3.3k
0
C2
6.8n
V2-15Vdc
C12
100n
VCC
0
VCC
C10 6.8n
HIGH
U5
uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
R11
3.3k
-VCC
C3
6.8n
U4
uA741
3
2
74
6
1
5+
-
V+
V-
OUT
OS1
OS2
C9 6.8n
V
V
0
R6
100
0
-VCC
-VCC
-VCC
0
C11
100n
VCC
3-way Linkwitz-Riley crossover
102 www.ice77.net
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.
Frequency
10Hz 30Hz 100Hz 300Hz 1.0KHz 3.0KHz 10KHz 30KHz 100KHzV(R22:2) V(R13:2) V(R24:2) V(V3:+)
0V
0.5V
1.0V
High cut-off frequency
Low cut-off frequency
(3.5445K,501.188m)
(340.041,503.457m)
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.