SLOA049D July   2000  – February 2023

 

  1.   Abstract
  2.   Trademarks
  3. 1 Introduction
  4. 2 Filter Characteristics
  5. 3 Second-Order Low-Pass Filter Standard Form
  6. 4 Math Review
  7. 5 Examples
    1. 5.1 Second-Order Low-Pass Butterworth Filter
    2. 5.2 Second-Order Low-Pass Bessel Filter
    3. 5.3 Second-Order Low-Pass Chebyshev Filter with 3-dB Ripple
  8. 6 Low-Pass Sallen-Key Architecture
  9. 7 Low-Pass Multiple Feedback (MFB) Architecture
  10. 8 Cascading Filter Stages
  11. 9 Filter Tables
  12. 10Example Circuit Simulated Results
  13. 11Non-ideal Circuit Operation
    1. 11.1 Non-ideal Circuit Operation: Sallen-Key
    2. 11.2 Non-ideal Circuit Operation: MFB
  14. 12Comments About Component Selection
  15. 13Conclusion
  16.   A Filter Design Specifications
    1.     A.1 Sallen-Key Design Simplifications
      1.      A.1.1 Sallen-Key Simplification 1: Set Filter Components as Ratios
      2.      A.1.2 Sallen-Key Simplification 2: Set Filter Components as Ratios and Gain = 1
      3.      A.1.3 Sallen-Key Simplification 3: Set Resistors as Ratios and Capacitors Equal
      4.      A.1.4 Sallen-Key Simplification 4: Set Filter Components Equal
    2.     A.2 MFB Design Simplifications
      1.      A.2.1 MFB Simplification 1: Set Filter Components as Ratios
      2.      A.2.2 MFB Simplification 2: Set Filter Components as Ratios and Gain = –1
  17.   B Higher-Order Filters
    1.     B.1 Fifth-Order Low-Pass Butterworth Filter
    2.     B.2 Sixth-Order Low-Pass Bessel Filter
  18.   C Revision History

10 Example Circuit Simulated Results

To further show how to use the previous information, component values are calculated and the filter circuits are simulated in PSpice.

Table 10-1 and Table 10-2 show typical component values computed for the three different filters discussed using the Sallen-Key architecture and the MFB architecture. The equivalent simplification (see Appendix A) is used for each circuit: setting the filter components as ratios, the gain equal to 1 for the Sallen-Key, and the gain equal to –1 for the MFB. The circuits and simplifications are shown for convenience. A corner frequency fc of 1 kHz is chosen. The values used for m and n are shown. The values shown for R1, R2, C1, and C2 are chosen to be standard values or the nearest standard values to those computed by using the formulas given.

Figure 10-1 Sallen-Key Circuit and Component Values (fc = 1 kHz)

R1=mR, R2=R, C1=C, C2=nC, and K=1 result in FSF×fc=12πRCmn and Q=mnm+1.

Table 10-1 Sallen-Key Circuit and Component Values (fc = 1 kHz)
Filter TypenmC1C2R1R2
Butterworth3.30.2290.01 μF33 nF4.22 kΩ18.4 kΩ
Bessel1.50.50.01 μF15 nF7.23 kΩ14.5 kΩ
3-dB Chebyshev6.81.00.01 μF68 nF7.32 kΩ7.32 kΩ

R1=R2=R, R3=mR, C1=C, C2=nC, and K=1 result in FSF×fc=12πRCmn and Q=mn1+2m.

Table 10-2 MFB Circuit and Component Values (fc = 1 kHz)
Filter TypenmC1C2R1 and R2R3

Butterworth

4.7

0.222

0.01 μF

47 nF

15.4 kΩ

3.48 kΩ

Bessel

3.3

0.195

0.01 μF

33 nF

15.4 kΩ

3.01 kΩ

3-dB Chebyshev

15

0.267

0.01 μF

150 nF

9.42 kΩ

2.52 kΩ

The circuits are simulated using a TLV9062 operational amplifier, 1%-tolerance resistors, and 2%-tolerance capacitors. Figure 10-2 through Figure 10-5 show the frequency response of the circuits.

Figure 10-2 compares the frequency response of Sallen-Key and MFB second-order Butterworth filters. The frequency response of the filters is almost identical from 10 Hz to about 80 kHz.

Figure 10-2 Second-Order Butterworth Filter Frequency Response

Figure 10-3 compares the frequency response of Sallen-Key and MFB second-order Bessel filters. The frequency response of the filters is almost identical from 10 Hz to about 90 kHz. Above 90 kHz, the MFB has excellent performance.

Figure 10-3 Second-Order Bessel Filter Frequency Response

Figure 10-4 compares the frequency response of Sallen-Key and MFB second-order 3-dB Chebyshev filters. The frequency response of the filters is almost identical from 10 Hz to about 60 kHz. Above 60 kHz, the MFB shows better performance.

Figure 10-4 Second-Order 3-dB Chebyshev Filter Frequency Response

Figure 10-5 is an expanded view of the frequency response of the three filters in the MFB topology, near fc (the Sallen-Key circuits are almost identical). The graph shows the increased rate of attenuation near the cutoff frequency, with the Bessel filter being the slowest and the 3-dB Chebyshev filter being the fastest.

Figure 10-5 Second-Order Butterworth, Bessel, and 3-dB Chebyshev Filter Frequency Response