106-190 - active filters - ice77 filters.pdf · active filters active filters are a special family...

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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.

4 www.ice77.net

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)



HznFnFkkCCRR

fc 34.7222022010102

1

2

1

2121


5.022022010

2202201010

211

2121

nFnFk

nFnFkk

CCR

CCRRQ

This circuit is critically damped.

6 www.ice77.net

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)


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)


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


0V

0.4V

0.8V

1.2V(66.069,1.0625)



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.

9 www.ice77.net

-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


0V

0.5V

1.0V

(72.287,500.734m)



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.

12 www.ice77.net

The center/cutoff frequency is

HzFFkkCCRR

fc 15.15911112

1

2

1

2142


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)


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

14 www.ice77.net

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)


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.

15 www.ice77.net

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)


The –3dB frequency for the filter is at 1.693kHz.

16 www.ice77.net

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)


The gain drops to –3dB at 1.693kHz and then it decreases by –20dB/decade.

17 www.ice77.net

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


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.

18 www.ice77.net

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)


The gain drops to –3dB at 1kHz and then it decreases by –20dB/decade.

19 www.ice77.net

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


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 )


The –3dB frequency for the filter is at 1.693kHz.

Frequency


-80

-40

0

(16.853,-40.041)

(168.526,-20.084)

(1.6853K,-3.0304)



20 www.ice77.net

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)


The –3dB frequency for the filter is at 1kHz.

21 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(C2:1)/V(C1:2))

-60

-40

-20

-0

(10.000,-40.001)

(100.000,-20.044)

(1.0000K,-3.0060)



22 www.ice77.net

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).

23 www.ice77.net

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.

24 www.ice77.net

Frequency

1.0Hz 10Hz 100Hz 1.0KHz 10KHz 100KHz 1.0MHzV(V1:+) V(U1:-)

0V

0.5V

1.0V

(1.1253K,707.101m)



kHznFnFkkCCRR

fc 125.1100200112

1

2

1

2121


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)



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)



kHznFnFkkCCRR

fc 253.111010212

1

2

1

2121


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)


The gain drops to –3dB at about 11.253kHz and then it decreases by –40dB/decade.

27 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.

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)


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)


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)


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)



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)


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)



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)



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)



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


0V

0.5V

1.0V

1.5V

(10.000,1.4299)

(1.0009K,1.0385)



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)


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


Note: all capacitor values match.

Frequency


0V

1.0V

2.0V

2.6V

(10.000,2.1999)

(1.0005K,1.5311)



Frequency


-60

-40

-20

-0

20

(8.7096K,-50.861)

(1.0000K,3.7091)(10.000,6.8481)



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


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)



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)


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)


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)



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)


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


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)



HznFnFkkCCRR

fc 10071001923.112

1

2

1

2121

1

kHznFnFkkCCRR

f c 1100231112

1

2

1

2121

2


5411.01001921

1001923.11

211

21211

nFnFk

nFnFkk

CCR

CCRRQ

2932.1100231

10023111

211

21212

nFnFk

nFnFkk

CCR

CCRRQ


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)



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



Frequency


0V

1.0V

2.0V

3.0V(1.0000K,2.5885)

(10.000,2.0000)



Frequency


-100

-50

0

50

(10.000K,-75.288)

(1.0000K,8.2609)(10.000,6.0207)


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.

42 www.ice77.net

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+


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)



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


6185.0285.1285.1100

100153285.1285.1

212

21211

kknF

nFnFkk

RRC

CCRRQ

6202.1490490100

1001050490490

434

43432

nF

nFnF

RRC

CCRRQ


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)



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


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)



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


6183.0502071

5020744.21

211

21211

nFnFk

nFnFkk

CCR

CCRRQ

6179.150481

5048475.101

211

21212

nFnFk

nFnFkk

CCR

CCRRQ


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)



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



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)



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)


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


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)


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


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)


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


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)



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


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)


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


0V

0.5V

1.0V

(1.5915K,500.000m)



kHznFnFkkCCRR

fc 5915.1100100112

1

2

1

2121


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)



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)



kHznFnFkkCCRR

fc 61.10101065.155.12

1

2

1

2121


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)


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)



kHznFnFkkCCRR

ff cc 125.1100200112

1

2

1

2121

21


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)



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)


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)



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)


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


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)



kHznFnFkkCCRR

ff cc 253.111010212

1

2

1

2121

21


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)



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)



kHznFnFkkCCRR

fc 591.1100100112

1

2

1

2121

1

kHznFnFkkCCRR

ff cc 591.150200112

1

2

1

2121

32

61 www.ice77.net


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)



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


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)



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


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)



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


0V

0.5V

1.0V

(1.3791K,576.923m)



kHznFnFkkCCRR

fc 38.1100133112

1

2

1

2121

65 www.ice77.net


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)


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


0V

0.5V

1.0V

(1.3807K,576.923m)



kHznFnFkkCCRR

fc 38.110010033.112

1

2

1

2121


577.01001001

10010033.11

211

2121

nFnFk

nFnFkk

CCR

CCRRQ

67 www.ice77.net

Frequency


-150

-100

-50

-0

40

(13.834,-79.958)

(138.337,-40.002)

(1.3834K,-4.7606)


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


0V

0.5V

1.0V

(1.0006K,715.976m)



Frequency


-100

-50

0

20

(10.034K,-51.006)

(1.0000K,-2.8984)


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 )



Frequency


-100

-50

0

30

(10.000K,-44.399)

(1.0000K,3.8909)(10.000,6.8478)


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)


Frequency


0V

1.0V

2.0V

2.8V

(10.000,2.1998)

(1.0000K,1.5627)



Frequency


-60

-40

-20

-0

20

(12.589K,-50.168)

(10.000,6.8478)(1.0000K,3.8777)



71 www.ice77.net

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


0V

0.4V

0.8V

1.2V

(922.189,862.722m)


The ripple in the passband is noticeable. The cutoff frequency is

HznFnFkkCCRR

fc 96.921100298112

1

2

1

2121


863.011100

10029811

212

2121

kknF

nFnFkk

RRC

CCRRQ

Frequency


-60

-40

-20

-0

20

(9.2612K,-40.123)

(926.119,-1.3203)


The gain drops to –1.3203dB at 926Hz and then it decreases by –40dB/decade.

73 www.ice77.net

Frequency


-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


0V

0.4V

0.8V

1.2V

(830.016,958.580m)



HznFnFkkCCRR

fc 92.831100366112

1

2

1

3232


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)


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


-1.0

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


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



Frequency


0V

0.5V

1.0V

1.5V

(705.774,1.1272)



HznFnFkkCCRR

fc 44.705100509112

1

2

1

6565


128.111100

10050911

656

6565

kknF

nFnFkk

RRC

CCRRQ

77 www.ice77.net

Frequency


-60

-40

-20

-0

20

(6.9970K,-39.882)

(702.533,1.0863)


The gain is +1.0863dB at 702Hz and then it decreases by –40dB/decade.

Frequency


-1.0

0

1.0

2.0

3.0

(549.541,2.0010)



78 www.ice77.net


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



Frequency


0V

0.5V

1.0V

1.5V(609.575,1.3047)



HznFnFkkCCRR

fc 88.609100681112

1

2

1

8787


305.111100

10068111

878

8787

kknF

nFnFkk

RRC

CCRRQ

79 www.ice77.net

Frequency


-60

-40

-20

-0

20

(6.1188K,-37.523)

(611.881,1.8548)



Frequency


0

2.0

4.0

-1.0

(512.861,3.0052)



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.

81 www.ice77.net

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


0V

0.4V

0.8V

1.2V

(921.668,862.722m)



HznFnFkkCCRR

fc 96.92110010098.212

1

2

1

2121


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)



Frequency


-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


0V

0.4V

0.8V

1.2V

(833.787,958.580m)



HznFnFkkCCRR

fc 92.83110010066.312

1

2

1

4343


957.01001001

10010066.31

433

4343

nFnFk

nFnFkk

CCR

CCRRQ

84 www.ice77.net

Frequency


-120

-80

-40

-0

40

(83.074,-39.986)

(830.736,-398.059m)



Frequency


-1.0

0

1.0

2.0

(1.2303K,990.336m)


The frequency response of the circuit peaks at 1.23kHz with a 1dB overshoot and then it decays rapidly.

85 www.ice77.net


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)


Frequency


0V

0.5V

1.0V

1.5V

(705.088,1.1272)



HznFnFkkCCRR

fc 44.70510010009.512

1

2

1

6565


128.11001001

10010009.51

655

6565

nFnFk

nFnFkk

CCR

CCRRQ

86 www.ice77.net

Frequency


-120

-80

-40

-0

40

(70.087,-40.061)

(700.870,985.793m)



Frequency


-1.0

0

1.0

2.0

3.0

(912.011,1.9866)



87 www.ice77.net


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)


Frequency


0V

0.5V

1.0V

1.5V

(610.162,1.3047)



HznFnFkkCCRR

fc 88.60910010081.612

1

2

1

8787


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)



Frequency


0

2.0

4.0

-1.0

(724.436,2.9916)



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)


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


0V

2.0V

4.0V

6.0V

(10.000,4.2998)

(1.0011K,2.8047)


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)


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


0V

0.5V

1.0V

(1.0003K,707.101m)



kHznFnFCCRR

fc 1100200112511252

1

2

1

2121

95 www.ice77.net


707.011251125100

10020011251125

212

2121

nF

nFnF

RRC

CCRRQ

Frequency


-60

-40

-20

-0

20

(10.000K,-40.051)

(1.0000K,-3.0079)



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


0V

0.4V

0.8V

1.2V

(0.9995K,706.509m)



kHznFnFkCCRR

fc 110010025.211252

1

2

1

2121


707.01001001125

10010025.21125

211

2121

nFnF

nFnFk

CCR

CCRRQ

97 www.ice77.net

Frequency


-150

-100

-50

-0

50

(10.000,-80.006)

(100.000,-40.007)

(1.0000K,-3.0135)



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


0V

0.5V

1.0V

(1.0021K,707.101m)



Frequency


-80

-40

0

20

(10.000K,-42.932)

(1.0000K,-2.9990)



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)



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)


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.

106-190 - active filters - ice77 filters.pdf · active filters active filters are a special family...

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