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

5.1 Second-Order Low-Pass Butterworth Filter

The Butterworth polynomial requires the least work out of the three types of filters because the frequency scaling factor is always equal to one.

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

Equation 9. z 1 = – 0 . 707 + j 0 . 707
Equation 10. z 1 * = – 0 . 707 – j 0 . 707

This is used with the factored form of the polynomial. Alternatively, the coefficients of the polynomial a 0 = 1 and a 1 = 1 . 414 can be found. The formula in Equation 19 can be confirmed:

Equation 11. ( s + 0 . 707 + j 0 . 707 )   ( s + 0 . 707 – j 0 . 707 ) = s 2 + 1 . 414 s + 1

To translate the polynomial into standard form, use the coefficient form of the polynomial in the denominator of the transfer function. The realization of a second-order low-pass Butterworth filter is made by a circuit with the following transfer function:

Equation 12. H LP f = K - f f c 2 + 1 . 414 j f f c + 1

Equation 20 is the same as Equation 17 with F S F = 1 and Q = 1 1 . 414 = 0 . 707.

Figure 5-1 is an example of a Second-Order Butterworth Low-Pass Filter using the Sallen-Key topology and TLV9062 created with the Filter Design Tool. This circuit has a gain of 1 V/V and a pass-band frequency at 1 kHz. The circuit was built in PSpice and then constructed and measured with a gain-phase analyzer.

Figure 5-1 Butterworth Low-Pass Filter With TLV9062 Circuit

Figure 5-2 and Figure 5-3 show the Butterworth low-pass filter with TLV9062 PSpice and Measured results, respectively.

Figure 5-2 Butterworth Low-Pass Filter With TLV9062 PSpice® Results
Figure 5-3 Butterworth Low-Pass Filter With TLV9062 Measured Results