A Sallen-Key 3-Pole Chebyshev Type I Active Lowpass Filter - Design Sheet (DS3)

by John-Paul Bedinger

 

      Of the various topologies you can select for making active filters, the Sallen-Key uses the least number of filter components. Furthermore, a 3-pole response (18 dB/oct) is possible using only 1 op-amp. Below is a brief mathematical description on how to compute the component values for a Chebyshev Type I (equiripple in passband) 3-pole lowpass filter with selectable output gain.

Diagram 1: A 3-pole Sallen-Key lowpass filter with output gain.

 

Circuit Analysis:

 

Looking at node 3b (where v3 = v3a = v3b) , the node voltage equation can be re-written to be:

 

  (Eq. 1)

 

Let:           Then:   (Eq. 2)

 

The rest of the node voltages can be written:

 

(Eq. 3)

 

(Eq. 4)

 

(Eq. 5)

 

When solved for the filter transfer function H(s) = Vout/Vin, we get:  H(s)=

 

  (Eq. 6)

 

Note that the general form of a 3-pole lowpass filter at cutoff frequency 1 rad/sec is: 

 

H(s) =   (Eq. 7a)

 

 

The form of a 3-pole prototype Chebyshev Type I Filter for a 1 rad/sec cutoff frequency (at magnitude of gain less ripple) and unity gain is:

H(s) =       (Eq. 7b) 

 

where rdB is the permissible ripple in the passband. The coefficients a2, a1, and a0 of Eq. 7a are calculated by first solving for the 3 roots in the denominator of Eq. 7b that have negative real parts. These 3 roots(p1, p2, and p3) are then re-factored back together and expanded to give the final transfer function:

 

1/H(s) = (s-p1)(s-p2)(s-p3) =        (Eq. 7c)

 

To help simplify programming, you may wish to use the following table for a0,a1,a2 at different values of permissible ripple:

 

rdB

a2

a1

a0

0.1dB

1.9388

2.6295

1.6380

0.2dB

1.6293

2.0773

1.1516

0.5dB

1.2530

1.5349

0.7157

1dB

0.9883

1.2384

0.4913

2dB

0.7378

1.0222

0.3269

 Table 1: Prototype Chebyshev filter coefficients vs. permissible passband ripple. fc = 1 rad/sec.

 

Next, equating like terms in Equation Eq. 6 and Eq. 7c gives the solve block:

 

  (Eq. 8)

 

      (Eq. 9)

 

    (Eq. 10)

 

  (Eq. 11)

 

where Kac = M.

 

The traditional half-power(gain-3dB) filter cutoff(ωc) is a function of both the order of the filter and of the ripple(rdB), and is found in the 3-pole case by solving for the positive real root of:

 

     (Eq. 12)

 

The following table tabulates various solutions of Eq. 12 for a 3-pole prototype Chebyshev:

 

rdB

ωc

0.1 dB

1.3890

0.2 dB

1.2835

0.5 dB

1.1675

1 dB

1.0949

2 dB

1.0327

3 dB

1.000

 Table 2: Half-power cutoff frequency(ωc) vs. rdB for a 3-pole Chebyshev prototype filter.

 

To begin the prototype filter design we choose our output gain and permissible ripple:  

Kac = 3 (9.5 dB), rdB = 0.2dB

 

Also, we choose values for components: C1=3000 uF, C2=1000uF,  C3=1000uF, R5=20 k-ohms

 

Using Table 1 for a2...a0 and solving the solve block for R1, R2, R3,and R4 gives:

R1=801.11 ohms, R2=508.6 ohms, R3=710.28 ohms, R4=20k ohms

 

Rounding R1-R4 to standard EIA 1% tolerance decade values gives:

R1=806 ohms, R2=511 ohms, R3=715 ohms, R4=20k ohms

 

Diagram 2: A 3-pole Sallen-Key Cheybshev lowpass filter with cutoff(gain-ripple) at 1 rad/sec and a gain of 3. 

Practical Notes:

·        For different gain and passband ripple values you can refer to my table below, or resolve the solve block with different values of rdB of Kac. Use solutions only with all real positive roots. If you have Mathcad(TM), you can download this MathCAD worksheet to help you.

·        R1, R2, and R3 from the table can be scaled together by a factor x, which should be done so that R1 is much greater than the source impedance at Vin. This will set the cutoff to 1/x rad/sec.

·        C1, C2, and C3 from the table can be scaled together by a factor y, which will set the cutoff frequency(at magnitude of gain less ripple) to 1/(x*y) rad/sec, or:

 

Fc(gain-ripple) = 1/(2*3.1416*x*y) Hz       (Eq. 13)

 

·        For the traditional half-power cutoff frequency (at magnitude of gain less 3dB), you must modify this formula to:

 

Fc(gain-3dB) = Wc/(2*3.1416*x*y) Hz       (Eq. 14)

 

where Wc = ωc is found from Table 2.

 

rdB=0.1dB

0dB

6dB

12dB

18dB

24dB

30dB

36dB

R1

1032

13331

1500.3

3581.3

2745.7

1397.1

2747.3

R2

55228

16826

3610.3

1547.7

1889.1

13280

6185.7

R3

1071.2

2721.7

11270

11014

1177

32904

35924

R4

0

10000

30000

70000

15000

31000

63000

R5

infinite

10000

10000

10000

1000

1000

1000

C1

1.00E-03

1.00E-04

1.00E-03

1.00E-03

1.00E-03

1.00E-03

1.00E-03

C2

1.00E-03

1.00E-04

1.00E-04

1.00E-04

1.00E-04

1.00E-05

1.00E-05

C3

1.00E-05

1.00E-04

1.00E-04

1.00E-04

1.00E-03

1.00E-04

1.00E-04

 

rdB=0.2dB

0dB

6dB

12dB

18dB

24dB

30dB

36dB

R1

1499.4

15267

1751

4078

3142.6

1646.5

3153.9

R2

2162.2

20528

4073

1745.5

2129.8

14695

6910.7

R3

26786

2770.7

12176

12200

1297.4

35890

39841

R4

0

10000

30000

70000

15000

31000

63000

R5

infinite

10000

10000

10000

1000

1000

1000

C1

1.00E-03

1.00E-04

1.00E-03

1.00E-03

1.00E-03

1.00E-03

1.00E-03

C2

1.00E-03

1.00E-04

1.00E-04

1.00E-04

1.00E-04

1.00E-05

1.00E-05

C3

1.00E-05

1.00E-04

1.00E-04

1.00E-04

1.00E-03

1.00E-04

1.00E-04

 

rdB=0.5dB

0dB

6dB

12dB

18dB

24dB

30dB

36dB

R1

1842

18766

2214.4

4967.2

3861.2

2112.3

3900

R2

2981.9

27777

4761

2037.8

2484.7

16652

7933.4

R3

25438

26810

13253

13804

1456.4

39728

45166

R4

0

10000

30000

70000

15000

31000

63000

R5

infinite

10000

10000

10000

1000

1000

1000

C1

1.00E-03

1.00E-04

1.00E-03

1.00E-03

1.00E-03

1.00E-03

1.00E-03

C2

1.00E-03

1.00E-04

1.00E-04

1.00E-04

1.00E-04

1.00E-05

1.00E-05

C3

1.00E-05

1.00E-04

1.00E-04

1.00E-04

1.00E-03

1.00E-04

1.00E-04

 

rdB=1dB

0dB

6dB

12dB

18dB

24dB

30dB

36dB

R1

2231.8

22752

2746.3

5977.4

4681

2649.8

4760.5

R2

4031.7

36444

5331.9

2276.6

2774.3

18136

8719.1

R3

22620

2454.9

13900

14959

1567.3

42355

49049

R4

0

10000

30000

70000

15000

31000

63000

R5

infinite

10000

10000

10000

1000

1000

1000

C1

1.00E-03

1.00E-04

1.00E-03

1.00E-03

1.00E-03

1.00E-03

1.00E-03

C2

1.00E-03

1.00E-04

1.00E-04

1.00E-04

1.00E-04

1.00E-05

1.00E-05

C3

1.00E-05

1.00E-04

1.00E-04

1.00E-04

1.00E-03

1.00E-04

1.00E-04

 Table 3: Prototype component values for a Chebyshev filter response at 1 rad/sec.

 

 

Example:

 

We want Fc(gain-3dB) = 1000 Hz, C1 = 0.1uF, C2= 0.1uF, and C3= 0.1uF, a passband ripple of 0.2 dB, and a gain of 6 dB. The source resistance is 10 ohms.

 

Find y for the correct capacitor range: The scale factor y is 0.1uF/100uF, or y = 0.001.

 

Find x using Eq. 7 and Table 2: 1000 Hz = 1.2835/(2*3.1416*x*0.001). Solving for x gives: x = 0.2043

 

Use Table 3 for 6dB gain and rdB= 0.2dB prototype values: Scaling R1, R2, R3, and R4 by x gives:

 

R1= 3118.7 ohms, R2= 4193.4 ohms, R3=   565.99 ohms, R4= 10k ohms, R5=10k ohms

 

Finally, round R1-R4 to standard EIA 1% tolerance decade values:

R1= 3.09k ohms, R2= 4.22k ohms, R3= 562 ohms, R4= 10k ohms, R5= 10k ohms

 

Since 3.09k ohms >> 10 ohms source resistance, the value for R1 should work well.

 

 

Diagram 3: A 3-pole Sallen-Key Chebyshev Type I lowpass filter with cutoff(gain-3dB) at 1000 Hz, passband ripple of 0.2dB, and a gain of 6dB.

 

Change Log:

v.1.14. Removed some legal rambling at the bottom of this change log – that’s all in the About This Site and General Disclaimer now. Reordered change Log, clarified Wc = ωc, fixed spellings, added more equation references, and put back in some missing equations and pictures.

v.1.13 Made MathCad file a zip file. Note some equations are still missing.    

v1.12 Some equations and diagrams may have been lost during the move. I will fix these as I get time.

v1.11 Corrected HTML title, misplacement of 'k's in example. Fixed various small text formatting issues. Corrected H0..H3 to H0...H2 in MathCad worksheet.

v.1.1 Added traditional half-power frequency equation for Fc, and changed example to use it. Corrected capacitor value C3 in Table 2 for rdB=0.2dB, gain = 0dB to 1E-5. was 1E-4

v.1.0 Initial release. Table values not verified by simulation.