Zeroes have value
To get the most stopband attenuation for a given passband bandwidth, you need a filter with stopband zeroes. These are the features that look like “notches” in the response at particular frequencies. Such filters are often loosely called “elliptic” filters, but that term only actually applies to a specific type of filter having flat passband and stopband response and the maximum possible number of ripples in both bands. Stopband zeroes help to “pull” the stopband response down quickly. The tradeoff is that the stopband rejection doesn’t become indefinitely larger as frequency rises, but ends up bouncing around, ideally staying below some defined minimum level, in our case 60 dB.

Standard tables exist for some filters like this but in that case, they are not only not “dissipated” (the term to describe passive filters designed for inductors with loss) but are also only given for doubly-terminated networks, which is no use to us here, as discussed in “Lowpass Filters that Don’t”.

So I gave an Excel Solver spreadsheet the job of fitting a 5-pole DC-free filter circuit (or rather, the prototype uses to develop the filter) to a response that tracks the Gaussian amplitude profile out to 9.5 dB at the standard ‘normalized’ cutoff frequency of 0.159 Hz, and that is then down at 60 dB at slightly less than twice that frequency. It worked!

When scaled and transformed into our final filter, we end up with the circuit of Figure 7. The schematic doesn’t show the surge protection diodes or TVS devices that you really should use in any circuit like this, to prevent the enormous voltage switch-on surges from damaging the low voltage-rated op-amp inputs).

Stopband zeroes are introduced into the prototype with a small capacitor across each inductor, which translates into a small resistor in series with the input to our almost-D-element. Despite the approximations inherent in trying to form stopband zeroes with this filter structure, we easily reach the stopband requirements. I tried several values of ‘dissipation factor’ and ended up with one that gave nice round numbers in the final filter. The Excel result and its simulated response are shown in Figures 8 and 9.

The settling behavior, shown in Figure 10, is a bit “nervous” in comparison to the Gaussian filter but still easily beats it, at 58 ms to less than 0.1%. If you’re happy with 1% settling, it is much faster than the Gaussian version (18 ms versus 46 ms).

We haven’t completely conquered that boosting at the amplifier outputs, as Figure 11 shows, though it’s much better controlled, and we’ll be able to cope with our 1 V_PP of 120 Hz ripple with no problem. More intervention in the optimization process could probably improve this and perhaps trim a little more total capacitance off (maybe also compensate for any ESR effects). Nevertheless, the capacitors shown add up to only 137 µF, not far short of a 100x reduction in comparison to the single-pole case. We might actually get it to fit!

Will it actually work? Figure 12 shows the op-amp output voltages and currents when running on a ripple waveform on 1 V_PP created by a full-wave rectified power supply feeding a 1000 µF reservoir cap and a 1.37 kΩ load. The voltage swings are well inside the 5 V supply rail (the non-inverting inputs are biased at half the supply voltage, of course) and the required output current is safely inside the 25 mA rating of the op-amps I used. And the ripple level on the output is too small to show on that scale of graph. So, yes, I think it will!

Are there other ways of doing this? I’m happy to receive homework submissions and will publicize the best one. But I don’t yet know where I’d start if I couldn’t solve the problem this way.

Happy (rapid, low-capacitance) bias filtering!

References
[1] Williams & Taylor, “Electronic Filter Design Handbook”, McGraw-Hill
[2] Zverev, “Synthesis of Filters”, Wiley

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