Make your own free website on Tripod.com

A Sallen-Key 3-Pole Butterworth Active Highpass Filter - Design Sheet (DS2)

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 Butterworth (steepest response with no ripple) 3-pole highpass filter with selectable output gain.

 

 

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

 

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

 

   where         and M defines the AC gain of the circuit in the passband (Eq. 1 and 2).

 

The rest of the node voltage equations can be written:

 

                                                    (Eq. 3)

            (Eq. 4)

                            (Eq. 5)

 

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

 

(Eq. 6)

 

Note that the standard form of a 3-pole Butterworth highpass filter at cutoff frequency 1 rad/sec is: 

 

             (Eq.7)   where  Kac is the AC gain of the filter, the same as our M.

 

Equating like terms in Equations 7 and 6 gives the solve block:

 

                                                                     (Eq.8) 

                                                     (Eq.9) 

 

      (Eq.10) 

 

               (Eq.11) 

 

We choose our output gain: Kac = M = 4 (12 dB)

 

Also, we choose values for components:  C1= 100 uF, C2= 100uF,  C3= 100uF, R5= 10 k-ohms

 

Solving now for R1, R2, R3,and R4 gives (exact):

R1= 5 k-ohms, R2= 20 k-ohms, R3= 10 k-ohms, R4= 30 k-ohms

 

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

R1= 4.99 k-ohms, R2= 20 k-ohms, R3= 10 k-ohms, R4= 30 k-ohms

 

Diagram 2: A 3-pole Sallen-Key Butterworth highpass filter with cutoff at 1 rad/sec and a gain of 4 (12dB).

 


Practical Notes:

Fc = 1/(2*3.1416*x*y) Hz        (Eq. 12)

 

M 0dB 6dB 12dB 18dB 24dB 30dB 36dB
R1(ohms) 7180.57 5862.23 5000 4223 3531.2 2933.18 2425.61
R2(ohms) 2819.43 11533.14 20000 31821.84 49433.83 76312.84 117808
R3(ohms) 49394.66 14790.74 10000 7441.38 5728.67 4467.49 3499.48
R4(ohms) 0 10000 30000 70000 15000 31000 63000
R5(ohms) infinite 10000 10000 10000 1000 1000 1000
C1(Farads) 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6
C2(Farads) 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6
C3(Farads) 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6

 

M

42dB 48dB 54dB 60dB 66dB 72dB 78dB
R1(ohms) 1998.55 1640.85 1342.31 1094.1 888.6151 719.3089 580.4752
R2(ohms) 182295.4 282964.2 440633.6 688213.9 1077780 1691800 2660920
R3(ohms) 2744.79 2153.77 1690.71 1328.07 1044.13 821.7427 647.4178
R4(ohms) 127000 25500 51100 102300 20470 40950 81910
R5(ohms) 1000 100 100 100 10 10 10
C1(Farads) 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6
C2(Farads) 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6
C3(Farads) 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6 100.E-6

 

Table 1: Prototype component values for a Butterworth highpass filter response at 1 rad/sec.

 


Example:

 

We want Fc= 80 Hz, C1= 0.1uF, C2= 0.1uF, and C3= 0.1uF, and a gain of 30 dB. The source resistance is 220 ohms or less.

 

Use Table 1 for 30dB prototype values, then scale y for the correct capacitor range:

 

The scale factor y is 0.1uF/100uF, or y = 0.001. Thus, 80 Hz = 1/(2*3.1416*x*0.001). Solving for x gives: x = 1.989

 

Scaling R1, R2, R3,and R4 by x gives:

R1= 5834.1 ohms, R2= 151786 ohms, R3= 8885.8 ohms, R4= 31 k-ohms,  R5=1 k-ohm

 

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

R1= 5.9 k-ohms, R2= 150 k-ohms, R3= 8.87 k-ohms, R4= 30.9 k-ohms, R5=1 k-ohm

 

The magnitude of (C1 + Rin) at 80Hz is:

 (1/(2*3.14*80*0.1e-6)^2+220^2)^.5 = 19.9 k-ohm at 80 Hz.

 

Since the combined impedance of 19.9 k-ohm at the cutoff frequency is much greater than the 220 ohm source resistance by itself, the value for C1 should work well.

 


Change Log:

v.1.0.1 Made MathCad file a zip file.

v.1.0 Initial release.


Questions or feedback? E-mail me at jpbedinger@hotmail.com

(c)2005 John-Paul Bedinger. All rights reserved. Revision: 1.0.1

Do not duplicate, distribute, or modify without my expressed written permission.

Disclaimer: The author is not responsible for any damages resulting from the content or application of this document. Use at your own risk.