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By ELMO V. JANSZ
A filter is one of the most common types of circuit used
in electronic equipment. By definition, a filter passes some
frequencies and suppresses or attenuates others.
Filters can be active or passive, depending on their con
struction. Passive filters use passive components such as
resistors, capacitors and inductors, whereas active filters
include an amplifying device, such as a transistor or
operational amplifier, in addition to a number of passive
components. The presence of the amplifier gives the filter
very good isolation between its input and output and a
certain amount of amplification as well.
In this article, we shall learn how to design active filters
using simple calculations.
Let us start by establishing a few basic ideas about active
filters. Fig.1 shows the idealised amplitude response of a
lowpass filter. A lowpass filter is one that passes all frequencies up to a point and heavily attenuates or suppresses
Fig.1: idealised
amplitude
response of a
lowpass filter.
Fig.2: idealised
amplitude
response of a
highpass filter.
frequencies beyond this point. The amplitude response is
a plot of the gain of the filter against frequency. The gain
is calculated by dividing the output voltage by the input
voltage in the equation:
G = 20 log10(Vo/Vi)
where G is the gain expressed in decibels; Vo is the output
voltage; and Vi is the input voltage.
In Fig.1, the frequency fc is called the cutoff frequency
while region AB in which the gain is constant is called
the filter’s passband. Beyond fc, the gain drops rapidly and
this region is called the stopband.
The rate at which the line BD falls is measured in dB/
octave or dB/decade. The is the “slope” of the filter. An
octave is a doubling or halving of frequency; ie, for a
frequency of 2kHz, octaves above are 4kHz, 8kHz and so
on, while octaves below are 1kHz, 500Hz, etc. Decades
are a tenfold increase or decrease in frequency. For a
January 1994 37
Fig.3: response
characteristic of
a practical lowpass filter.
Fig.5: basic circuit for a first order lowpass
Butterworth active filter.
where G is the passband gain in decibels; W is the normalised angular frequency; and n is the order of the filter.
The normalised frequency is given by W/Wc where W is
the frequency in question and Wc is the cut off frequency
Fig.4: a filter
with ripples in
the passband
is called a
Chebyshev filter.
frequency of 2kHz, decades above are 20kHz, 200kHz and
so on, while decades below are 200Hz, 20Hz, 2Hz, etc.
We now come to another important definition, the
“order” of a filter. This is the rate at which the line BD in
Fig.1 falls off, or the filter’s ability to attenuate frequencies
outside its passband.
A “first order” filter has an attenuation outside its passband of 6dB/octave or 20dB/decade. The order of a filter
is also referred to as its rolloff or falloff.
A “second order” filter has a rolloff of 12dB/octave or
40dB/decade; ie, twice that of the first order filter. A third
order filter will have a rolloff of three times that of a first
order filter and so on for higher order filters.
A highpass filter is the complement of a lowpass filter and will have an idealised response characteristic as
shown in Fig.2.
Notice that frequencies below fc are attenuated heavily.
The rolloff has the same values as stated above but in this
case will have the opposite sign.
A practical lowpass filter will have the response charac
teristic shown in Fig.3. The cutoff frequency in this case
is not a sharp transition point as shown in Figs.1 & 2 but
the frequency at which the gain is reduced by 3dB, from
its passband value.
A filter with a response as shown in Fig.3 – ie, one
having a flat response in the passband – is called a Butterworth filter. A filter could also have a response as shown
in Fig.4, with ripples in the passband. This is called a
Chebyshev filter.
The shape of the filter’s response is determined by a
constant (alpha) called the Damping Factor. There are
other filters called Cauer, Bessel and Thompson filters
but in this article we shall confine ourselves to Butterworth filters, as they are the most popular due to their
design simplicity.
The general equation for a Butterworth lowpass filter
of order n is given by:
Gain = 20 log [G/(1 + W2n)½ ]
38 Silicon Chip
Design of a first order filter
Let us now design a first order lowpass Butterworth
active filter. The basic circuit is shown in Fig.5. The portion within the dotted line is a lowpass passive filer. The
operational amplifier is connected in the noninverting
mode.
The cutoff frequency (fc) and passband gain (G) are
given by the following formulas:
fc = 1/(2πRC)
G = 1 + RB/RA
Suppose we wish to construct a lowpass filter with a
cutoff frequency of 2kHz. We start by selecting a value for
C. Let this be .022µF. By using the formula fc = 1/(2πRC),
we arrive at:
R = 1/(2π x 2 x 103 x 0.022 x 106) = 3.617kΩ
This would be selected as 3.6kΩ, using the closest value
in the E24 (5%) range.
Let us set the passband gain required equal to 2. There
fore, using the formula for gain:
RB/RA = G  1 = 2  1 = 1
Therefore, we can make RA equal to RB and set both at
10kΩ. A 741 could be used for the operational amplifier
and then you have your basic first order low pass filter. By
interchanging C and R, you can produce the corresponding
high pass filter.
Second order lowpass filter
The basic circuit of a second order lowpass filter is
shown in Fig.6. Here again a network of passive components is placed around an op amp. Second order active
Fig.6: basic circuit of a second order lowpass filter.
Fig.7: the circuit for a unity gain lowpass active filter.
filters are also often referred to as SallenKey filters. This
circuit has two RC networks, hence it is a second order
filter. The cutoff frequency fc for this filter is given by:
fc = 1/2π(R1.R2.C3.C4)½
and the midband gain is given by:
G = 1 + RB/RA
In practice, two versions of this circuit are possible:
either a filter with a passband gain of unity, or a filter with
equal components; ie, R1 = R2 and C3 = C4.
Unity gain
For this example, it is customary to make R1 = R2 and
then C3 and C4 are fixed in the ratio C3 = 2C4, in order
to satisfy the damping factor (alpha) requirements for a
Butterworth response. The required circuit is shown in
Fig.7. Note that the op amp has been configured for unity
gain, as a voltage follower, by connecting its inverting
input to its output.
Using the formula fc = 1/2π(R1.R2.C3.C4)½
and remembering that R1 = R2 = R and C3 = 2C4 (ie, if C4 = C
then C3 = 2C), the above equation can now be written as:
fc = 1/2π(R x R x 2C x C)½ = 1/2πCR√2
If we select R = 10kΩ and if a cutoff frequency of 1kHz
is desired, C can be calculated from the above equation to
give: C = 1/(2π x 103 x 10 x 103 x √2) = 0.01µF.
Therefore, we can select C3 = 0.02µF and C4 = 0.01µF.
The final design is now R1 = R2 = 10kΩ; C3 = 0.02µF; C4
= 0.01µF.
Fig.9: unity gain second order highpass filter.
The passband gain for a Butterworth filter is defined
by the equation:
G=3α
and since α = √2, G = 1.586. Unfortunately, this is the
only gain that will permit the circuit to operate correctly.
By selecting R = 5kΩ and a cutoff frequency of 1kHz,
the above equation gives C = .032µF. A .033µF polyester
capacitor would be suitable. The gain of G = 1.58 can be
satisfied by making RB = 27kΩ and RA = 47kΩ (using preferred values). The final circuit is shown in Fig.8.
Second order high pass filters
High pass filters can be set up by interchanging the R
and C components of the lowpass circuit. Two versions of
this circuit are possible, as for the lowpass configurations
– ie, a unity gain circuit and an equal component circuit.
These are shown in Figs. 9 & 10.
For Fig.9, if C1 = C2, then R4 = 2R3 in order to satisfy
the damping requirements for a Butterworth response.
Equal component filter
If R1 = R2 = R and C3 = C4 = C, then the equation
fc = 1/2π(R1.R2.C3.C4)½ becomes fc = 1/2πRC
Fig.10: equal component highgain Butterworth filter.
Fig.8: equal component lowpass Butterworth filter.
For the equal components version of Fig.10, if R3 = R4
and C1 = C2, then the gain is fixed by the equation:
G=3α
With alpha = √2, this again fixes the gain at 1.586.
Higher order filters can be obtained by cascading appro
priate filter sections. For example, a fifth order filter can
be produced by cascading two second order and one first
order sections.
Filters can also be set up to pass a band of frequencies
and so are called bandpass filters. A bandpass filter can
be obtained by cascading an appropriate highpass and
SC
lowpass section.
January 1994 39
