# [sdiy] Help with a strange filter

Richie Burnett rburnett at richieburnett.co.uk
Wed Jun 13 14:13:35 CEST 2012

```> Can anyone tell me more about following filter design?
> http://www.electricdruid.net/images/Filter.png

It's a cheap way to make a 4-pole active filter.

> Is it any good?

It saves the cost of one op-amp but compromises on performance compared to
the normal practice of cascading two active biquads to get a 4th order
filter:

http://4.bp.blogspot.com/_vU9dVUMN-SU/TGmxrSVvVjI/AAAAAAAAAMk/Dg5VCSdA6-4/s1600/4th_order_LP_filter_B.jpg

> If it is, why doesn't everyone use 4-pole filters that only use a single
> op-amp? Dead handy for antialiasing filters and such like.

Since it creates a 4th order transfer function in one stage it's likely to
be sensitive to component tolerances, and the design calculations will be
quite complicated compared to the normal 2nd order Sallen-Key filter
equations.  The topology may also inherently limit the placement of the 4
poles in the frequency domain.

(For an anti-aliasing filter the "Elliptic" Cauer filter is hard to beat.
You can get a very sharp transition from passband to stopband, and place the
first notch in the response at the point where aliasing would start to
occur.)

> How would I design a version with different caps?

By deriving the circuit equations from Thevenin theory, nodal voltage
analysis etc and then solving the equations to obtain your required 4th
order transfer function.  As they did in the example you can simplify the
equation-solving by assuming R1, R2, R3, R4 are all equal and some arbitrary
value like 15k.  The remaining combinations of C1, C2, C3, C4 will determine
the natrual frequencies and damping factors of the two pairs of poles.

> Is this some mutant offspring of the multiple-feedback filter topology?

It's a continuation of the normal Sallen-Key filter.  Which is itself
classed as a voltage controlled voltage source (VCVS) filter.

> What type of filter is it? (Butterworth, probably, but I'd like to know)

I plotted the response, (and I see you also have.)  I guess it's close to a
Butterworth but it is not as tight around the transition frequency (droopy
transition region.)  Also on my simulator it barely makes the -4 slope
(-24dB/oct) before it reaches -90dB and the bahaviour of the op-amp distorts
the response.  The majority of the visible roll-off is about -20dB/oct, so
whilst mathematically 4th order the 4th pole doesn't kick in until you are
well into the stop-band region.  Not the best 4 pole filter, and in some
ways similar to the TB-303 filter.  This also has its poles scattered widely
across the frequency axis due to coupling between un-buffered RC stages.

> Is it genuinely a 4-pole active filter, or (as I suspect) in fact a 2-pole
> active filter with 2 extra passive poles stuck on it?

It's a 4-pole active filter.  Just not a very good one.  The question really
should be:  Is it a sloppy 4-pole filter because of bad design decisions, or
is the placement of the poles inherently limited by the topology of the
filter.  I don't know the answer, so I'll give the original designer the
benefit of the doubt and assume that it's not possible to do any better with
that cheap arrangement of components.

In order to make a 4-pole Butterworth filter it requires 2 pairs of complex
poles both with the same natural frequency, but each pair with unique and
specific damping factors.  I'd hazzard that its not possible to achieve
those exact placements with a 4-pole filter implemented in one active stage.
Can any of the mathematicians prove or disprove this?

> I came across this design all those years ago (almost thirty!) and I've
> never, ever seen anything like it since. It doesn't ever appear in
> textbooks or on webpages about filters.

That alone probably speaks volumes! ;-)

Students doing EE degrees are actually taught in linear systems or control
class to construct systems of order higher than 2 by cascading the
appropriate number of 2nd and 1st order filter stages.  Not only does it
make the design easier, but it also makes subsequent analysis easier because
high-order systems are already factorised into 2nd order polynomials.  And
the behaviours and properties of 2nd order systems are drilled into every EE
from year one so they have the tools to analyse a system of any order.

-Richie,

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