english filtress, Politechnika Lubelska, Studia, Semestr 7, 7 semestr elektrotechnika, lab układów ...
[ Pobierz całość w formacie PDF ]
Chapter 16
Active Filter Design Techniques
Literature Number SLOA088
Excerpted from
Op Amps for Everyone
Literature Number: SLOD006A
Chapter 16
Active Filter Design Techniques
Thomas Kugelstadt
16.1 Introduction
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 sig-
nal conditioning stages. System power supplies often use band-rejection filters to sup-
press 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 op-
erational 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).
C
2
R
1
R
2
L
R
V
IN
V
OUT
V
IN
V
OUT
C
C
1
Figure 16–1. Second-Order Passive Low-Pass and Second-Order Active Low-Pass
16-1
Fundamentals of Low-Pass Filters
This chapter covers active filters. It introduces the three main filter optimizations (Butter-
worth, 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 fil-
ters. Rather than resembling just another filter book, the individual filter sections are writ-
ten 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 cal-
culate 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
V
IN
V
OUT
C
Figure 16–2. First-Order Passive RC Low-Pass
Its transfer function is:
1
RC
1
A(s)
RC
1
1
sRC
s
where the complex frequency variable,
s = j
+
, allows for any time variable signals. For
pure sine waves, the damping constant,
, becomes zero and
s = j
.
For a normalized presentation of the transfer function,
s
is referred to the filter’s corner
frequency, or –3 dB frequency,
C,
and has these relationships:
C
j
s
f
s
C
j
f
C
j
With the corner frequency of the low-pass in Figure 16–2 being
f
C
= 1/2
RC
,
s
becomes
s = sRC
and the transfer function A(s) results in:
1
A(s)
1
s
The magnitude of the gain response is:
1
|A|
1
2
For frequencies
>> 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.
16-2
 Fundamentals of Low-Pass Filters
R
V
IN
R
R
C
R
C
C
V
OUT
C
Figure 16–3. Fourth-Order Passive RC Low-Pass with Decoupling Amplifiers
The resulting transfer function is:
1
1
1
s
1
2
s
(
1
n
s
)
A(s)
In the case that all filters have the same cut-off frequency, f
C
, the coefficients become
1
2
n
n
2
1
, and f
C
of each partial filter is 1/
times higher than f
C
of the overall filter.
Figure 16–4 shows the results of a fourth-order RC low-pass filter. The rolloff of each par-
tial filter (Curve 1) is –20 dB/decade, increasing the roll-off of the overall filter (Curve 2)
to 80 dB/decade.
Note:
Filter response graphs plot gain versus the normalized frequency axis
(
= f/f
C
).
Active Filter Design Techniques
16-3
 Fundamentals of Low-Pass Filters
0
–10
–20
1st Order Lowpass
–30
–40
4th Order Lowpass
–50
–60
Ideal 4th Order Lowpass
–70
–80
0.01
0.1
1
10
100
Frequency —
Ω
0
1st Order Lowpass
Ideal 4th
Order Lowpass
–90
–180
–270
4th Order Lowpass
–360
0.01
0.1
1
10
100
Frequency —
Curve 1: 1
st
-order partial low-pass filter, Curve 2: 4
th
-order overall low-pass filter, Curve 3: Ideal 4
th
-order low-pass filter
Note:
Figure 16–4.
Frequency and Phase Responses of a Fourth-Order Passive RC Low-Pass Filter
The corner frequency of the overall filter is reduced by a factor of
2.3 times versus the
–3 dB frequency of partial filter stages.
16-4
Â
[ Pobierz całość w formacie PDF ]