21-01-2013, 01:15 PM
A Basic Introduction to FiltersÐActive, Passive, and Switched-Capacitor
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INTRODUCTION
Filters of some sort are essential to the operation of most
electronic circuits. It is therefore in the interest of anyone
involved in electronic circuit design to have the ability to
develop filter circuits capable of meeting a given set of
specifications. Unfortunately, many in the electronics field
are uncomfortable with the subject, whether due to a lack of
familiarity with it, or a reluctance to grapple with the mathematics
involved in a complex filter design.
This Application Note is intended to serve as a very basic
introduction to some of the fundamental concepts and
terms associated with filters. It will not turn a novice into a
filter designer, but it can serve as a starting point for those
wishing to learn more about filter design.
Filters and Signals: What Does a Filter Do?
In circuit theory, a filter is an electrical network that alters
the amplitude and/or phase characteristics of a signal with
respect to frequency. Ideally, a filter will not add new frequencies
to the input signal, nor will it change the component
frequencies of that signal, but it will change the relative
amplitudes of the various frequency components and/or
their phase relationships. Filters are often used in electronic
systems to emphasize signals in certain frequency ranges
and reject signals in other frequency ranges. Such a filter
has a gain which is dependent on signal frequency. As an
example, consider a situation where a useful signal at frequency
f1 has been contaminated with an unwanted signal
at f2. If the contaminated signal is passed through a circuit
(Figure 1) that has very low gain at f2 compared to f1, the
undesired signal can be removed, and the useful signal will
remain. Note that in the case of this simple example, we are
not concerned with the gain of the filter at any frequency
other than f1 and f2. As long as f2 is sufficiently attenuated
relative to f1, the performance of this filter will be satisfactory.
In general, however, a filter's gain may be specified at
several different frequencies, or over a band of frequencies
The Basic Filter Types
Bandpass
There are five basic filter types (bandpass, notch, low-pass,
high-pass, and all-pass). The filter used in the example in
the previous section was a bandpass. The number of possible
bandpass response characteristics is infinite, but they all
share the same basic form. Several examples of bandpass
amplitude response curves are shown in Figure 5 . The
curve in 5(a) is what might be called an ``ideal'' bandpass
response, with absolutely constant gain within the passband,
zero gain outside the passband, and an abrupt boundary
between the two. This response characteristic is impossible
to realize in practice, but it can be approximated to
varying degrees of accuracy by real filters. Curves (b)
through (f) are examples of a few bandpass amplitude response
curves that approximate the ideal curves with varying
degrees of accuracy. Note that while some bandpass
responses are very smooth, other have ripple (gain variations
in their passbands. Other have ripple in their stopbands
as well. The stopband is the range of frequencies
over which unwanted signals are attenuated. Bandpass filters
have two stopbands, one above and one below the
passband.
All-Pass or Phase-Shift
The fifth and final filter response type has no effect on the
amplitude of the signal at different frequencies. Instead, its
function is to change the phase of the signal without affecting
its amplitude. This type of filter is called an all-pass or
phase-shift filter. The effect of a shift in phase is illustrated
in Figure 15 . Two sinusoidal waveforms, one drawn in
dashed lines, the other a solid line, are shown. The curves
are identical except that the peaks and zero crossings of
the dashed curve occur at later times than those of the solid
curve. Thus, we can say that the dashed curve has undergone
a time delay relative to the solid curve.
Elementary Filter Mathematics
In 1.1 and 1.2, a few simple passive filters were described
and their transfer functions were shown. Since the filters
were only 2nd-order networks, the expressions associated
with them weren't very difficult to derive or analyze. When
the filter in question becomes more complicated than a simple
2nd-order network.
Filter Approximations
In Section 1.2 we saw several examples of amplitude response
curves for various filter types. These always included
an ``ideal'' curve with a rectangular shape, indicating that
the boundary between the passband and the stopband was
abrupt and that the rolloff slope was infinitely steep. This
type of response would be ideal because it would allow us
to completely separate signals at different frequencies from
one another. Unfortunately, such an amplitude response
curve is not physically realizable. We will have to settle for
the best approximation that will still meet our requirements
for a given application. Deciding on the best approximation
involves making a compromise between various properties
of the filter's transfer function. The important properties are
listed below.
Butterworth
The first, and probably best-known filter approximation is
the Butterworth or maximally-flat response. It exhibits a
nearly flat passband with no ripple. The rolloff is smooth and
monotonic, with a low-pass or high-pass rolloff rate of
20 dB/decade (6 dB/octave) for every pole. Thus, a 5th-order
Butterworth low-pass filter would have an attenuation
rate of 100 dB for every factor of ten increase in frequency
beyond the cutoff frequency.