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

5.2 Second-Order Low-Pass Bessel Filter

Referring to a table listing the zeros of the second-order Bessel polynomial:

Equation 13. z 1 = – 1 . 103 + j 0 . 6368
Equation 14. z 1 * = – 1 . 103 – j 0 . 6368

A table of coefficients provides a 0 = 1 . 622 and a 1 = 2 . 206 .

Again, coefficients directly appear in standard form, so the realization of a second-order low-pass Bessel filter is made by a circuit with the transfer function:

Equation 15. HLPf=K-ffc2+2.206jffc+1.622

Normalize Equation 15 so that it is in standard form. Dividing both the numerator and denominator by 1 . 622 scales the gain factor K (which is arbitrary) and gives the normalized form:

Equation 16. HLPf=K-f1.274fc2+1.360jffc+1

Equation 22 is the same as Equation 17 with F S F = 1 . 274 and Q = 1 1 . 360 × 1 . 274 = 0 . 577.