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

5.3 Second-Order Low-Pass Chebyshev Filter with 3-dB Ripple

Referring to a table listing the zeros of the 3-dB second-order Chebyshev polynomial:

Equation 17. z 1 = – 0 . 3224 + j 0 . 7772
Equation 18. z 1 * = – 0 . 3224 – j 0 . 7772

A table of coefficients provides a 0 = 0 . 7080 and a 1 = 0 . 6448 .

Again, coefficients directly appear in standard form, so the realization of a second-order low-pass Chebyshev filter with 3-dB ripple is made by a circuit with the transfer function:

Equation 19. H LP f = K - f f c 2 + 0 . 6448 j f f c + 0 . 7080

Again, normalize Equation 19 so that it is in standard form by dividing both the numerator and denominator by 0 . 7080 to get:

Equation 20. H LP f = K - f 0 . 8414 f c 2 + 0 . 9107 j f f c + 1

Equation 20 is the same as Equation 17 with F S F = 0 . 8414 and Q = 1 0 . 8414 × 0 . 9107 = 1 . 3050.

The previous work is the first step in designing any of the filters. The next step is to determine which circuit topology to use to implement these filters.