Active Filter Design
Introduction
It’s a jungle out
there. Somewhere in the dense wilderness, hidden
from the view of the masses is a small tribe, much
sought after by the head hunters of the surrounding
plains. The tribe knows it is under threat and its
numbers are dwindling at an alarming rate, killed
off by the ever accelerating advancement of modern
technology. This is the tribe of the Analogue
Engineer.
Now every tribe has
its elder, the one shrouded in mystique, who imparts
wisdom while reminiscing of better days. Even in the
tribe of the Analogue Engineer there is the guru 
the rarest of the rare, the one you never get
to see, even with an appointment. On the throne of
his kingdom sits the Analogue Filter Designer. You
call him ‘Sir’.
Many books
have been written on active filter design and they
normally include countless pages of equations that
frighten most small dogs and some children. This
article sets out to unravel the mystery of filter
design and to allow the reader to design continuous
time, analogue filters based on op amps in the
minimum of time and with the minimum of mathematics.
The Theory of Analogue Electronics
There are two distinct sides to analogue
electronics: the theory that the academic
institutions teach (equations of stability, phase
shift calculations) and the practical that most
engineers use (tweak the gain with a capacitor to
avoid oscillation). Unfortunately, active filter design is
based firmly on long established equations and
tables of theoretical values. Since designing
practical circuits from theoretical equations can
prove arduous, this text has derived the response of
a general purpose filter 'building block' and
translating the theoretical tables into practical
component values involves nothing more than minor
maths.
The Fundamentals
A simple RC Low Pass Filter has the transfer
function
Cascading filters similar to the one above will give
rise to quadratic equations in the denominator of
the transfer function and hence further complicate
the response of the filter. In fact, any second
order Low Pass filter has a transfer function with a
denominator equal to
and substituting different values of a, b
and c determine the response of the filter
over frequency. Now, anyone who remembers high
school maths will notice that the above statement
can have a value of zero for certain values of s,
given by the equation
hence our transfer function can present infinite
gain.
It is the points at which the quadratic equation in
the denominator equals zero that the transfer
function has theoretically infinite gain. These
points are known as the poles of the
quadratic equation and it is the values of these
poles that establish the performance of each type of
filter over frequency. These poles normally come in
pairs and take the form of a complex number and
its complex conjugate ,
although sometimes jb is zero.
In practice it is the real part, a, of the
pole that indicates how the filter will respond to
transients and the imaginary part, jb,
that indicates how the filter will respond over
frequency. We only ever apply frequencies
to a filter, hence need not worry about the system
going unstable.
This text will describe how to transfer the tables
of poles found in many text books into component
values suitable for circuit design.
Types of Filter
The three most common filter characteristics, and
the ones discussed in this text, are Butterworth,
Chebyshev and Bessel, each giving a different
response. There are many others, but 90% of all
applications can be solved with one of the above
implementations.
The Butterworth implementation ensures flat response
('maximally flat') in the pass band and an adequate rolloff. This type
of filter is a good ‘all rounder’, simple to
understand and is good for applications such as
audio processing.
The Chebyshev implementation gives a much steeper
rolloff, but has ripple in the pass band, so is no
use in audio systems. It is however, far superior in
applications where there is only one frequency
present in the pass band but several other
frequencies that need removing (e.g. deriving a
sinewave from a square wave by filtering out the
harmonics).
The Bessel filter gives a constant propagation delay
across the input frequency spectrum. Therefore
applying a square wave (consisting of a fundamental
and many harmonics) to the input of a Bessel filter
will yield a ‘square’ wave on the output with no
overshoot (i.e. all the frequencies will be delayed
by the same amount). Any other filter will delay
different frequencies by different amounts and this
will manifest itself as overshoot on the output
waveform.
There is the one more popular implementation, the
Elliptical Filter, that is a much more complicated
beast so won't be discussed in this text. Similar to
the Chebyshev response, it has ripple in the pass
band and severe rolloff at the expense of ripple in
the stop band.
The Standard Filter Blocks
Figure 1a shows the generic filter structure and by
substituting capacitors or resistor in place of
components G1G4, either a High Pass or Low Pass
implementation can be realised. Considering the
effects of these components on the op amp feedback
network, it is quite easy to picture that a Low Pass
filter can be derived from Fig 1a by making
components G2/G4 into capacitors and G1/G3 resistors
and vice versa for a High Pass implementation.
FIG 1
The transfer function of the Low Pass Filter shown
in Fig 1b is
(to understand how this transfer characteristic was
determined, please see the article on Nodal Analysis
of Op Amps on this website).
Here the equation is simpler if conductances
are used, where the capacitors have a conductance of
sC and the resistors have a conductance of G.
If this looks too complicated, life is made much
easier if the above equation is 'normalised'. This
involves selecting either the resistors or the
capacitors to be equal to 1Ω or 1F respectively and
changing the surrounding components to fit the
response. Thus, making all resistor values equal to
1Ω , the Low Pass transfer function is
Equation 1
This is the transfer function describing the
response of our generic second order Low Pass
Filter. We are now faced with taking the theoretical
tables of poles describing the three main filter
responses and translating these into real component
values.
The Design Process
The designer should determine the type of
filter required by selecting the pass band
performance needed from the descriptions above. The
simplest way of determining the desired order
of the filter is to design a second order filter,
then cascade multiple stages of this filter. Check
to see if the implementation gives the desired stop
band rejection, then proceed to design with the
correct pole locations, shown in the tables in the
appendix. Once the pole locations are established,
the component values can soon be calculated.
Firstly, each pole location needs to be transformed
into a quadratic expression similar to that in the
denominator of our generic second order filter. If a
quadratic equation has poles of ,
then it has roots of
and
.
When these are multiplied together, the resulting
quadratic equation is
In the pole tables, a is always negative, so
it is sometimes more convenient to use
and use the magnitude of a regardless of its
sign.
To put this into practice, consider a fourth order
Butterworth filter. Below are the poles of a fourth
order Butterworth filter with the associated
quadratic expression of each pole location. The
poles can be found from the file at the end of this
document.
Poles
(a±jb)
Quadratic
0.9239 ± j0.3827
0.3827 ± j0.9239
From this, a fourth order Butterworth Low Pass
Filter can be designed. Simply substitute the values
from the above two quadratic expressions into the
denominator of Equation 1.
Thus, in the first filter
C2C4 = 1; 2C4 =
1.8478 implying
C4 = 0.9239F; C2 = 1.08F
and for the second filter
C2C4 = 1; 2C4 =
0.7654 implying
C4 = 0.3827F; C2 = 2.61F
And all the resistors in both filters equal 1Ω.
Cascading both second order filters will yield a
fourth order Butterworth response with a roll off
frequency of 1 rads^{1} but with components
impossible to find. For those who do not find the
above frequency or component values suitable, read
on.
It just so happens that as long as the ratio
of the reactances to the resistors is maintained,
the circuit response remains unchanged. Therefore
one may elect to have 1kΩ resistors. Then divide the
capacitor values by 1000 to ensure that the
reactances increase in the same proportion as the
resistances.
Unfortunately, we still have the perfect Butterworth
response, but with a roll off frequency of 1rads^{1}.
To change the frequency response of the
circuit, we must ensure the ratio of the reactances
to the resistors is maintained, but simply at a
different frequency. Keeping the resistors at 1kΩ ,
for a roll off at 1kHz (rather than 1rads^{1}),
the capacitor value has to be further reduced by a
factor of 2π x 1000. Therefore the capacitors
reactance does not reach our original (normalised)
value until a higher frequency.
In conclusion, a fourth order Butterworth Low Pass
Filter with a 1kHz roll off takes the form of FIG2
FIG 2a
FIG 2b shows the LTspice^{®}
plot of the output.
Perfectly flat response in the passband and slow
roll off after the break frequency.
FIG 2b
Cascading any number of second order filters will
obtain any evenorder filter response using the
above technique. At this point it is important to
note that a true fourth order Butterworth filter is
not simply obtained by calculating the components
for a second order filter, then cascading two of
these stages. Two second order filters have to be
designed, each with different pole locations.
If the filter we are trying to design has an odd
order, we can simply cascade second order filters,
then add an RC network in the circuit to gain the
extra pole.
For example, a 5^{th} order, 1dB ripple
Chebyshev filter has the following poles
Poles
Quadratic
0.2265 ± j0.5918 s^{2}
+ 0.453s + 0.402 ≡ 2.488s^{2} + 1.127s + 1
0.08652 ± j0.9575 s^{2}
+ 0.173s + 0.924 ≡ 1.08s^{2} + 0.187s + 1
0.2800
see text
The first two quadratics have been multiplied by a
constant to ensure the last term equals unity, thus
conforming to the generic filter described by
Equation 1.
Thus, in the first filter
C2C4 = 2.488; 2C4 = 1.127
implying C4 = 0.5635F; C2 = 4.41F
and for the second filter
C2C4 = 1.08; 2C4 = 0.187
implying C4 = 0.0935F; C2 = 11.55F
Earlier in the text, it was shown that an RC circuit
has a pole when
i.e.
Therefore to obtain the final pole at s = 0.28, a
capacitor of C = 3.57F is needed if R = 1.
Normalising for a roll off frequency of 1kHz, using
1k resistors gives the circuit shown in FIG 3
FIG 3
Thus the designer can now boldly go and design many
Low Pass filters of any order at any frequency.
Now, all of the above theory can also be applied to
the design of a High Pass Filter.
It has been shown that a simple RC Low Pass filter
has the transfer function.
Similarly, a simple RC High Pass filter has the
transfer function
Normalising these to correspond with the normalised
pole tables gives:
for
a Low Pass
and
for
a High Pass
It can be seen that the High Pass pole positions,
s, can be obtained by inverting the Low Pass
pole positions. Inserting the values into the High
Pass Filter Block will ensure that the correct
frequency response is obtained. To obtain the
transfer function of the High Pass filter block, we
need to go back to the transfer function of the Low
Pass filter block. Thus from
interchanging capacitors and resistors it can easily
be seen that the transfer function of the equivalent
High Pass Filter Block is
Again, life is made much simpler if the
capacitors are normalised instead of the
resistors thus,
and putting
and
gives
Equation 2
This is the transfer function of the High Pass
filter block and this time we calculate the
resistor values instead of capacitor values.
Once the general High Pass filter response has been
obtained, the High Pass pole positions can be
derived by inverting the Low Pass pole positions and
continuing as before.
Now, inverting a complex pole location is easier
said than done. Take the example of the 5^{th}
order, 1dB ripple Chebyshev filter shown earlier. It
has two pole positions at
(0.2265 ±j0.5918)
The easiest way of inverting a complex number is to
multiply then divide by the complex conjugate, to
obtain a real number in the numerator. Finding the
reciprocal is then a matter of inverting the
fraction.
Thus
Gives
Inverting gives
The newly derived pole positions can then be
converted into the corresponding quadratic and the
values calculated as before.
We end up with
Poles
Quadratic
0.564 ± j1.474 s^{2}
+ 1.128s + 2.490 ≡ 0.401s^{2} + 0.453s + 1
Thus from Equation 2 we can calculate the first
filter component values to be
R2R4 = 0.401; 2R2 = 0.453
implying R2 = 0.227Ω ; R4 = 1.77Ω
This process can then be repeated for the other pole
locations.
A simpler way to achieve the above is to design for
a Low Pass filter using the suitable Low Pass
poles, then treat every pole, s, in the
filter as a single CR circuit since it has been
shown that .
Inverting each Low Pass pole to obtain the
corresponding High Pass pole simply involves
inverting the value of CR. Once the High Pass pole
locations have been obtained, we can interpose the
capacitors and resistors to ensure the correct
frequency response.
For clarification consider that so far, for the Low
Pass implementation a normalised capacitor value was
calculated assuming that R = 1Ω. Hence the value of
CR is equal to the value of C. Finding the
reciprocal of the value of C will find the High Pass
pole and treating this as the new value of R will
yield the appropriate High Pass component value.
Considering again the fate of the 5^{th}
order 1dB ripple Chebyshev Low Pass filter, the
capacitor values calculated were:
C4 = 0.5635F; C2 = 4.41F.
To obtain the equivalent High Pass resistor
values, invert the values of C (to get the High Pass
pole location) and treat these as the new normalised
resistor values. i.e. R4 = 1.77 and R2 = 0.227.
It can be seen that the same results are obtained as
the more formal method mentioned earlier.
Thus the circuit in FIG 3 can now be converted into
a High Pass Filter with a roll off at 1kHz by
inverting the normalised capacitor values,
interposing resistors and capacitors and scaling the
values accordingly. Previously, the normalised Low
Pass values were scaled by dividing by (2πfR).
In this case, the scaling factor used is (2πfC),
where C is the capacitor value and f is the
frequency in Hertz.
The resulting circuit is thus
Below we can see the LTspice schematic and transfer
characteristic.
FIG 4
Finally, just for kicks, let's implement a 9th order
Chebyshev low pass filter with 1dB ripple, shown in
FIG 5
FIG 5
and a close up on the
ripple shows it is indeed 1dB
FIG 6
Please
download the LTspice files and try them yourself:
4th
Order Low Pass Butterworth
4th Order
Low Pass Bessel
5th Order
High Pass Chebyshev
9th Order Low
Pass Chebyshev
Conclusions
Using the text above, the designer can now design
Low Pass and High Pass filters with response at any
frequency. Band Pass and Band Stop filters can be
implemented with single op amps using techniques
similar to those shown, but the scope of this
article does not cover this. However, by cascading
Low Pass and High Pass filters, both Band Pass and
Band Stop filters can be implemented.
Butterworth,
Chebyshev, Bessel pole tables
LTspice
is a registered trademark of Linear Technology
Corporation
