# Fixed-Gain Op Amps Simplify Filter Design

### Abstract

Use less component count, money, and board space when designing Sallen-Key filters. Simplify band pass filters with fixed gain amplfiers.

Simple second-order filters meet many filtering requirements. A low-order low-pass filter, for example, is often adequate for antialiasing in ADC applications or for eliminating high-frequency noise in audio applications. Similarly, a low-order high-pass filter can easily remove power-supply noise. When you design such filters with built-in gain, fixed-gain op amps can save space, cost, and time. Figure 1 illustrates the use of fixed-gain op amps in building second-order low-pass and high-pass Sallen-Key filters. Filter "cookbooks" are useful in designing these filters, but the cookbook procedures usually break down for a given response, such as Butterworth, if the gain set by R_{F} and R_{G} is greater than unity. What's more, the cookbook component-value formulas can yield unrealistic values for the capacitors and the resistors.

Butterworth filters, for example, offer the flattest passband. They also provide a fast initial falloff and reasonable overshoot. You can easily design such filters using the table below with the following equations: R_{2} = 1/(2πf_{C} √) and R_{1} = XR_{2}.

Gain | Low-Pass X | High-Pass X |

1.25 | * | 1.372 |

1.5 | 2 | 1.072 |

2 | 0.5 | 0.764 |

2.25 | 0.404 | 0.672 |

2.5 | 0.343 | 0.602 |

3 | 0.268 | 0.5 |

3.5 | 0.222 | 0.429 |

4 | 0.191 | 0.377 |

5 | 0.15 | 0.305 |

6 | 0.125 | 0.257 |

7 | 0.107 | 0.222 |

9 | 0.084 | 0.176 |

10 | 0.076 | 0.159 |

11 | 0.07 | 0.146 |

13.5 | 0.057 | 0.121 |

16 | 0.049 | 0.103 |

21 | 0.038 | 0.08 |

25 | 0.032 | 0.068 |

26 | 0.031 | 0.066 |

31 | 0.026 | 0.056 |

41 | 0.02 | 0.043 |

50 | 0.017 | 0.035 |

51 | 0.017 | 0.035 |

61 | 0.014 | 0.029 |

81 | 0.011 | 0.022 |

100 | 0.009 | 0.018 |

101 | 0.009 | 0.018 |

*A gain of 1.25 is impossible to obtain with matched capacitors for the low-pass case. |

For a gained filter response, the use of a fixed-gain op amp reduces cost and component count. It also decreases sensitivity, because the internal, factory-trimmed, precision gain-setting resistors provide 0.1% gain accuracy. To design a second-order Butterworth low-pass or high-pass filter using a fixed-gain op amp, follow these steps:

- Determine the corner frequency f
_{C}. - Select a value for C.
- For the desired gain value, locate X under the proper column heading in the table.
- Calculate R
_{1}and R_{2}using the equations.

Choosing C and then solving for R_{1} and R_{2} lets you optimize the filter response by selecting component values as close to the calculated values as possible. C can be lower than 1000pF for most corner frequencies and gains. Fixed-gain op amps come optimally compensated for each gain version and provide exceptional gain-bandwidth products for systems operating at high frequencies and high gain. Suppose, for example, you must design a low-pass filter with a 24kHz corner frequency and a gain of 10. Step 1 is complete (f_{C} = 24kHz). Next, complete Step 2 by selecting a value for C, say, 470pF. In the table, note that X = 0.076 for a low-pass filter with a gain of 10. Substitute these values in the equations:

R_{2} = 1/(2π f_{C} √) = 1/(2π × 24kHz × 470pF × √ ) = 51kΩ, and R_{1} = XR_{2} = 0.076 × 51kΩ = 3.9kΩ.

A similar version of this article appeared in the July 6, 2000 issue of *EDN.*