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

B Higher-Order Filters

This application note previously stated that higher-order filters can be constructed by cascading second-order stages for even-order, and adding a first-order stage for odd-order. To show how this is accomplished, two examples are considered: constructing a fifth-order Butterworth filter and a sixth-order Bessel filer.

By breaking higher than second-order filters into complex-conjugate zero pairs, second-order stages are constructed that, when cascaded, realize the overall polynomial. For example, a sixth-order filter has three complex-zero pairs and can be written as:

Equation 23. P 6 th s = s + z 1 s + z 1 * s + z 2 s + z 2 * s + z 2 s + z 3 *

Each of the complex-conjugate-zero pairs can be multiplied out and written as:

Equation 24. s + z 1 s + z 1 * = s 2 + a 1 , 1 s + a 0 , 1
Equation 25. s + z 2 s + z 2 * = s 2 + a 1 , 2 s + a 0 , 2
Equation 26. s + z 3 s + z 3 * = s 2 + a 1 , 3 s + a 0 , 3

The overall polynomial is then reconstructed in the following form:

Equation 27. P 6 th s = s 2 + a 1 , 1 s + a 0 , 1 s 2 + a 1 , 2 s + a 0 , 2 s 2 + a 1 , 3 s + a 0 , 3

The circuit implementation consists of three second-order stages cascaded to form the overall response.