Active Filter Design Techniques
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Op Amps for Everyone
Literature Number: SLOD006A
Active Filter Design Techniques
What is a filter?
A filter is a device that passes electric signals at certain frequencies or frequency ranges while preventing the passage of others. — Webster. Filter circuits are used in a wide variety of applications. In the field of telecommunication, band-pass filters are used in the audio frequency range (0 kHz to 20 kHz) for modems and speech processing. High-frequency band-pass filters (several hundred MHz) are used for channel selection in telephone central offices. Data acquisition systems usually require anti-aliasing low-pass filters as well as low-pass noise filters in their preceding signal conditioning stages. System power supplies often use band-rejection filters to suppress the 60-Hz line frequency and high frequency transients.
In addition, there are filters that do not filter any frequencies of a complex input signal, but just add a linear phase shift to each frequency component, thus contributing to a constant time delay. These are called all-pass filters.
At high frequencies (> 1 MHz), all of these filters usually consist of passive components such as inductors (L), resistors (R), and capacitors (C). They are then called LRC filters. In the lower frequency range (1 Hz to 1 MHz), however, the inductor value becomes very large and the inductor itself gets quite bulky, making economical production difficult. In these cases, active filters become important. Active filters are circuits that use an operational amplifier (op amp) as the active device in combination with some resistors and capacitors to provide an LRC-like filter performance at low frequencies (Figure 16–1). L R
VIN VOUT VIN
Figure 16–1. Second-Order Passive Low-Pass and Second-Order Active Low-Pass Chapter 16
Fundamentals of Low-Pass Filters
This chapter covers active filters. It introduces the three main filter optimizations (Butterworth, Tschebyscheff, and Bessel), followed by five sections describing the most common active filter applications: low-pass, high-pass, band-pass, band-rejection, and all-pass filters. Rather than resembling just another filter book, the individual filter sections are written in a cookbook style, thus avoiding tedious mathematical derivations. Each section starts with the general transfer function of a filter, followed by the design equations to calculate the individual circuit components. The chapter closes with a section on practical design hints for single-supply filter designs.
16.2 Fundamentals of Low-Pass Filters
The most simple low-pass filter is the passive RC low-pass network shown in Figure 16–2. R
Figure 16–2. First-Order Passive RC Low-Pass
Its transfer function is:
where the complex frequency variable, s = jω+σ , allows for any time variable signals. For pure sine waves, the damping constant, σ, becomes zero and s = jω . For a normalized presentation of the transfer function, s is referred to the filter’s corner frequency, or –3 dB frequency, ωC, and has these relationships: s
With the corner frequency of the low-pass in Figure 16–2 being fC = 1/2πRC, s becomes s = sRC and the transfer function A(s) results in:
The magnitude of the gain response is:
For frequencies Ω >> 1, the rolloff is 20 dB/decade. For a steeper rolloff, n filter stages can be connected in series as shown in Figure 16–3. To avoid loading effects, op amps, operating as impedance converters, separate the individual filter stages. Fundamentals of Low-Pass Filters
Active Filter Design Techniques 16-3
Figure 16–3. Fourth-Order Passive RC Low-Pass with Decoupling Amplifiers The resulting transfer function is:
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