"Chapter 1 - The Op Amp's Place in the World" - HTL Wien 10

Fundamentals of Low-Pass Filters
16-2
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.
Figure 16–2. First-Order Passive RC Low-Pass
Its transfer function is:
A(s)
V IN
R
C
1
RC
s 1
RC
1
1 sRC
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, ω C, and has these relationships:
s s
C j
j
C f
j
fC With the corner frequency of the low-pass in Figure 16–2 being f C = 1/2πRC, s becomes
s = sRC and the transfer function A(s) results in:
A(s) 1
1 s
The magnitude of the gain response is:
|A|
1
1 2
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.
V OUT