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Maxim > Design
support > App notes
> Amplifier
and Comparator Circuits > APP 700

Jan 22, 2001

Keywords: low pass filters, high pass filters,
filter designs, op amps as filters, Sallen-Key filter

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APPLICATION NOTE 700

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
RF and RG is greater than unity. What's
more, the cookbook component-value formulas can yield
unrealistic values for the capacitors and the
resistors.

Figure
1. Sallen-Key filters use fixed-gain op amps to realize a
second-order Butterworth response.

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: R2 = 1/(2πfC √) and R1 =
XR2.

Butterworth-Filter-Design
Criteria

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 fC.

Select a value for C.

For the desired gain value, locate X under the proper
column heading in the table.

Calculate R1 and R2 using the
equations.

Choosing C and then solving for
R1 and R2 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 (fC = 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:

R2 = 1/(2π
fC √) = 1/(2π × 24kHz ×
470pF × √ ) = 51kΩ, and R1 = XR2 =
0.076 × 51kΩ = 3.9kΩ.

Figure
2. Using the circuit values in the text, a simulation of the
circuit in Figure 1a produces this Butterworth
response.

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

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Download,
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APP 700: Jan 22, 2001

AN700, AN 700, APP700,
Appnote700, Appnote 700

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