A Sallen-Key 3-Pole Butterworth Active Lowpass Filter - Design Sheet (DS1)

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 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:   When solved for the filter transfer function H(s) = Vout/Vin, we get:  H(s)= (Eq. 3)

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

H(s) = (Eq. 4)

Equating like terms in Equations 3 and 4 gives the solve block:    where Kac = M.

We choose our output gain:   Kac = 3 (9.5 dB)

Also, we choose values for components:  C1= 3000 uF, C2= 1000uF,  C3= 1000uF, R5= 10 k-ohms

Solving now for R1, R2, R3,and R4 gives:

R1= 816.46 ohms, R2= 481.26 ohms, R3= 848.33 ohms, R4= 10 kohms

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

R1= 825 ohms, R2= 487 ohms, R3= 845 ohms, R4= 20 k-ohms Diagram 2: A 3-pole Sallen-Key Butterworth lowpass filter with cutoff at 1 rad/sec and a gain of 3.

Practical Notes:

• For different gain values you can refer to my table below, or resolve the solve block with a different value of Kac. Use solutions only with all real positive roots. If you have Mathcad(TM), you can download this zipped 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 to 1/(x*y) rad/sec, or:

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

 M(Kac) 0dB 6dB 12dB 18dB 24dB 30dB 36dB R1(ohms) 1292 15652 1624 4305 3246 1437 3234 R2(ohms) 2093 14694 4067 1750 2134 16260 7198 R3(ohms) 3698 4348 15144 13276 1444 42794 42950 R4(ohms) 0 10000 30000 70000 15000 31000 63000 R5(ohms) infinite 10000 10000 10000 1000 1000 1000 C1(farads) 1.E-3 100.E-6 1.E-3 1.E-3 1.E-3 1.E-3 1.E-3 C2(farads) 1.E-3 100.E-6 100.E-6 100.E-6 100.E-6 10.E-6 10.E-6 C3(farads) 100.E-6 100.E-6 100.E-6 100.E-6 1.E-3 100.E-6 100.E-6

 M(Kac) 42dB 48dB 54dB 60dB 66dB 72dB 78dB R1(ohms) 1640 1242 2243 1030 1137 1700 6053 R2(ohms) 13615 69066 32123 185004 285242 136533 47723 R3(ohms) 4479 116556 138815 5249 308473 430832 346170 R4(ohms) 127000 25500 51100 102300 20470 40950 81910 R5(ohms) 1000 100 100 100 10 10 10 C1(farads) 1.E-3 1.E-3 1.E-3 1.E-3 1.E-3 1.E-3 1.E-3 C2(farads) 10.E-6 1.E-6 1.E-6 1.E-6 100.E-9 100.E-9 100.E-9 C3(farads) 1.E-3 100.E-6 100.E-6 1.E-3 100.E-6 100.E-6 100.E-6

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

Example:

We want Fc= 1000 Hz, C1= 0.1uF, C2= 0.1uF, and C3= 0.1uF, and a gain of 6 dB. The source resistance is 10 ohms.

Use Table 1 for 6dB prototype values, then scale y for the correct capacitor range:

The scale factor y is 0.1uF/100uF, or y = 0.001. Thus, 1000 Hz = 1/(2*3.1416*x*0.001). Solving for x gives: x = 0.159

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

R1= 2.489 kohms, R2= 2.336 kohms, R3= 691.3 ohms, R4= 10 kohms,  R5=10 kohms

Rounding R1-R4 to standard EIA 5% tolerance decade values gives:

R1= 2.4 kohms, R2= 2.4 kohms, R3= 680 ohms, R4= 10 kohms, R5=10 kohms

Since 2.4 kohms >> 10 ohms source resistance, the value for R1 should work well.

Change Log:

v.1.3 Changed 66db gain resistors in Table 1. (Older values were ok as well)

Moved design sheet to new location on server.

Changed title and included design sheet identifier (DS1)

v1.2  Made R5=10k in solve block constants to match pictured results.

Corrected component values for gains above 36dB in Table 1.

Made bigger JPEG for diagram 1.

Made various small text formatting changes, and switched to Arial font.

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