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

1 Introduction

There are many books that provide information on popular filter types like the Butterworth, Bessel, and Chebyshev filters. This application note examines how to implement these three types of filters.

The mathematics used to transform standard filter table data into the transfer functions required to build filter circuits is examined. Using the same method, filter tables are developed that enable the designer to skip straight to the calculation of the required circuit component values. Actual filter implementation is shown for two circuit topologies: the Sallen-Key and the Multiple Feedback (MFB). The Sallen-Key circuit is sometimes referred to as a voltage-controlled voltage source (VCVS) filter.

Circuits are often referred to as Butterworth filters, Bessel filters, or a Chebyshev filters because their transfer function has the same coefficients as the Butterworth, Bessel, or the Chebyshev polynomial. The MFB or Sallen-Key circuits are also often referred to as filters. The difference is that the Butterworth filter defines a transfer function that can be realized by many different circuit topologies (both active and passive), while the MFB or Sallen-Key circuit defines an architecture or a circuit topology that can be used to realize various second-order transfer functions.

The choice of circuit topology depends on performance requirements. The MFB is generally preferred because the MFB has better sensitivity to component variations and better high-frequency behavior. The unity-gain Sallen-Key inherently has the best gain accuracy because the gain is not dependent on component values.

Table 1-1 and Table 1-2 give a brief summary of the overall trade-offs.

Table 1-1 Summary of Filter Type Trade-Offs
Filter TypeAdvantagesDisadvantages
ButterworthMaximum pass-band flatnessSlight overshoot in response to pulse input and moderate rate of attenuation above fc
BesselConstant group delay – no overshoot with pulse inputSlow rate of attenuation above fc
3-dB ChebyshevFast rate of attenuation above fcLarge overshoot and ringing in response to pulse input
Table 1-2 Summary of Architecture Trade-Offs
ArchitectureAdvantagesDisadvantages
Sallen-KeyNot sensitive to component variation at unity gainHigh-frequency response limited by the frequency response of the amplifier
MFBLess sensitive to component variations and excellent high- frequency responseLess simplifications available to ease design