[sdiy] Help with a strange filter

[Tom Wiltshire]

Thanks Richie.

I'm slightly suspicious of the usual "use 2nd order active sections for everything" approach which seems to be what people are taught and what you find everywhere on the web. It's useful, but it shouldn't be the only tool in the box.

I recently managed to find a page talking about cascading passive RC sections (which *no-one* recommends) but it's actually possible to build a reasonable 3-pole filter with only 6 components. The trick is scaling the Rs and Cs along the filter:

	http://www.electricdruid.net/images/SimplePWMFilter.png

Obviously there are better 3-pole filters out there, but it scores highly for compactness and simplicity. For some applications (stompboxes?) that's a winning combination. For my purposes (PWM filter for PIC output) it works fine, and is way simpler than the alternative. It's not even like the values are awkward.

Given that the equations for the 4th order Sallen Key are probably a hideous mess, my current technique of simply scaling the R or C values up or down to get a cutoff near to what I need is probably about as good as any technique for that particular filter. Back when I saw this, everything seemed to be based on single 741 op-amps, so probably saving an op-amp made more difference then than it does now.

Thanks,
Tom



On 13 Jun 2012, at 13:13, Richie Burnett wrote:

>> 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,