TL/H/11221
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National Semiconductor
Application Note 779
Kerry Lacanette
April 1991
A Basic Introduction to
FiltersÐActive, Passive,
and Switched-Capacitor
1.0 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 mathe-
matics 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.
1.1 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 fre-
quencies to the input signal, nor will it change the compo-
nent 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 fre-
quency f
1
has been contaminated with an unwanted signal
at f
2
. If the contaminated signal is passed through a circuit
(Figure 1) that has very low gain at f
2
compared to f
1
, 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 f
1
and f
2
. As long as f
2
is sufficiently attenuated
relative to f
1
, the performance of this filter will be satisfacto-
ry. In general, however, a filter’s gain may be specified at
several different frequencies, or over a band of frequencies.
Since filters are defined by their frequency-domain effects
on signals, it makes sense that the most useful analytical
and graphical descriptions of filters also fall into the fre-
quency domain. Thus, curves of gain vs frequency and
phase vs frequency are commonly used to illustrate filter
characteristics,and the most widely-used mathematical
tools are based in the frequency domain.
The frequency-domain behavior of a filter is described math-
ematically in terms of its transfer function or network
function. This is the ratio of the Laplace transforms of its
output and input signals. The voltage transfer function H(s)
of a filter can therefore be written as:
(1)
H(s)
e
V
OUT
(s)
V
IN
(s)
where V
IN
(s) and V
OUT
(s) are the input and output signal
voltages and s is the complex frequency variable.
The transfer function defines the filter’s response to any
arbitrary input signal, but we are most often concerned with
its effect on continuous sine waves. Especially important is
the magnitude of the transfer function as a function of fre-
quency, which indicates the effect of the filter on the ampli-
tudes of sinusoidal signals at various frequencies. Knowing
the transfer function magnitude (or gain) at each frequency
allows us to determine how well the filter can distinguish
between signals at different frequencies. The transfer func-
tion magnitude versus frequency is called the amplitude
response or sometimes, especially in audio applications,
the frequency response.
Similarly, the phase response of the filter gives the amount
of phase shift introduced in sinusoidal signals as a function
of frequency. Since a change in phase of a signal also rep-
resents a change in time, the phase characteristics of a filter
become especially important when dealing with complex
signals where the time relationships between signal compo-
nents at different frequencies are critical.
By replacing the variable s in (1) with j0, where j is equal to
0
b
1 , and 0 is the radian frequency (2qf), we can find the
filter’s effect on the magnitude and phase of the input sig-
nal. The magnitude is found by taking the absolute value of
(1):
(2)
l
H(j0)
l
e
À
V
OUT
(j0)
V
IN
(j0)
À
and the phase is:
(3)
arg H(j0)
e
arg
V
OUT
(j0)
V
IN
(j0)
TL/H/11221–1
FIGURE 1. Using a Filter to Reduce the Effect of an Undesired Signal at
Frequency f
2
, while Retaining Desired Signal at Frequency f
1
C
1995 National Semiconductor Corporation RRD-B30M75/Printed in U. S. A.

As an example, the network of Figure 2 has the transfer
function:
(4)
H(s)
e
s
s
2 a
s
a
1
TL/H/11221–2
FIGURE 2. Filter Network of Example
This is a 2nd order system. The order of a filter is the high-
est power of the variable s in its transfer function. The order
of a filter is usually equal to the total number of capacitors
and inductors in the circuit. (A capacitor built by combining
two or more individual capacitors is still one capacitor.)
Higher-order filters will obviously be more expensive to
build, since they use more components, and they will also
be more complicated to design. However, higher-order fil-
ters can more effectively discriminate between signals at
different frequencies.
Before actually calculating the amplitude response of the
network, we can see that at very low frequencies (small
values of s), the numerator becomes very small, as do the
first two terms of the denominator. Thus, as s approaches
zero, the numerator approaches zero, the denominator ap-
proaches one, and H(s) approaches zero. Similarly, as the
input frequency approaches infinity, H(s) also becomes pro-
gressively smaller, because the denominator increases with
the square of frequency while the numerator increases lin-
early with frequency. Therefore, H(s) will have its maximum
value at some frequency between zero and infinity, and will
decrease at frequencies above and below the peak.
To find the magnitude of the transfer function, replace s with
j0 to yield:
(5)
A(0)
e
l
H(s)
l
e
À
j0
b
0
2 a
j0
a
1
À
e
0
0
0
2 a
(1
b
0
2
)
2
The phase is:
(6)
i(0)
e
arg H(s)
e
90
§
b
tan
b
1
0
2
(1
b
0
2
)
The above relations are expressed in terms of the radian
frequency 0, in units of radians/second. A sinusoid will
complete one full cycle in 2q radians. Plots of magnitude
and phase versus radian frequency are shown in Figure 3.
When we are more interested in knowing the amplitude and
phase response of a filter in units of Hz (cycles per second),
we convert from radian frequency using 0
e
2qf, where f is
the frequency in Hz. The variables f and 0 are used more or
less interchangeably, depending upon which is more appro-
priate or convenient for a given situation.
Figure 3(a) shows that, as we predicted, the magnitude of
the transfer function has a maximum value at a specific fre-
quency (0
0
) between 0 and infinity, and falls off on either
side of that frequency. A filter with this general shape is
known as a band-pass filter because it passes signals fall-
ing within a relatively narrow band of frequencies and atten-
uates signals outside of that band. The range of frequencies
passed by a filter is known as the filter’s passband. Since
the amplitude response curve of this filter is fairly smooth,
there are no obvious boundaries for the passband. Often,
the passband limits will be defined by system requirements.
A system may require, for example, that the gain variation
between 400 Hz and 1.5 kHz be less than 1 dB. This specifi-
cation would effectively define the passband as 400 Hz to
1.5 kHz. In other cases though, we may be presented with a
transfer function with no passband limits specified. In this
case, and in any other case with no explicit passband limits,
the passband limits are usually assumed to be the frequen-
cies where the gain has dropped by 3 decibels (to
0
2/2 or
0.707 of its maximum voltage gain). These frequencies are
therefore called the
b
3 dB frequencies or the cutoff fre-
quencies. However, if a passband gain variation (i.e., 1 dB)
is specified, the cutoff frequencies will be the frequencies at
which the maximum gain variation specification is exceed-
ed.
TL/H/11221–3
(a)
TL/H/11221–5
(b)
FIGURE 3. Amplitude (a) and phase (b) response curves
for example filter. Linear frequency and gain scales.
The precise shape of a band-pass filter’s amplitude re-
sponse curve will depend on the particular network, but any
2nd order band-pass response will have a peak value at the
filter’s center frequency. The center frequency is equal to
the geometric mean of the
b
3 dB frequencies:
f
c
e
0
f
I
f
h
(8)
where f
c
is the center frequency
f
I
is the lower
b
3 dB frequency
f
h
is the higher
b
3 dB frequency
Another quantity used to describe the performance of a filter
is the filter’s ‘‘Q’’. This is a measure of the ‘‘sharpness’’ of
the amplitude response. The Q of a band-pass filter is the
ratio of the center frequency to the difference between the
2