Designing high order active filters

I've been getting a lot of use out of Simon Bramble's webpage for designing active filters recently - https://www.simonbramble.co.uk/techar...ter_design.htm. Its a great resource.

Right down at the bottom of the page the last filter he shows the schematic of is a 9th order Chebyshev, 1dB ripple, with a corner frequency of 1kHz. A textbook frequency response plot is obtained using LTC6241s. I latched on to this and tried changing the corner frequency to 18kHz, wondering if I could use such a design for an anti-imaging filter for my DACs. So I divided all the capacitor values by 18 and ran the sim. Disaster! The frequency response I obtained is below - a 7dB spike at 17kHz.

The problem seems to be inadequate Q - high order filters are composed of sections which increase in Q (more positive feedback) and the chosen opamps aren't fast enough (18MHz GBW). I went to a faster opamp for the highest Q stage which brought about some improvement but still not as beautiful as the 1kHz corner frequency plot. Since doing those sims I've studied a bit more and it may be that the opamp's CM input capacitance is as much, or more, of an issue here than its GBW. A helpful TI paper (here - https://www.ti.com/lit/ml/sloa088/sloa088.pdf) has a formula (p53) for estimating the required GBW from the Q.

A presentation from TI about component (mostly R and C) sensitivity in opamp-based active filters. They look at the effect of finite GBW on the highest Q stage (close to the end of the text) and conclude for their 8kHz filter that 10MHz GBW ought to be sufficient. However that's not quite what I found in LTSpice with my filter so that may be a pointer that GBW isn't the primary issue, opamp input capacitance could well be.

https://www.ti.com/lit/ml/sprp524/sprp524.pdf

Given there's no way I'd go anywhere near as low as 1k for resistors driven by opamps in my designs (too much PSU noise from the classAB output stage going over to classB) there looks to be a snowball in hell's chance for scaling those 1ks upwards if parasitic capacitances are coming into play already. The capacitors are in expensive NP0 territory (332nF would call for 3 paralleled 100ns) already so higher resistor values would help the BOM cost if they could be accommodated.

After some reflection I decided to try with a discrete component approach, rather than using opamps. I was inspired by John Linsley Hood's 'Liniac' where he uses a current source loaded darlington - I opted for a current source loaded darlington wired as a follower - i.e. close to unity voltage gain. I chose a darlington because they're cheaper than the more traditional JFET and wanted to find out if I could get aurally pleasing results for less money.

The darlington needs to have a good chance of high current gain at low emitter currents - I'd like the approach not to use more current than say a TL084 (1.4mA typical per stage). Scanning a few DSs I settled on the BCV27 - its shown (in NXP data) to have a typical current gain around 60k, minimum 4k at 1mA. Its also attractively cheap - below a fifth the cost of a BF862.

Update - since beginning this post my curiosity has gotten the better of me and I've gone back to LTSpice and found out what it takes to get a textbook response out of this filter. It turns out not to be anything to do with common-mode input capacitances. Rather its due to loading effects on the higher Q stages from their following stage. What gave the game away was seeing a 'kink' in the output of the last-but-one stage, just at the point where the peak appeared in the highest Q stage. So I stuck in a perfect buffer between the two highest Q stages - and the kink vanished. The irony here is that it does indeed take higher resistor values to get a textbook FR plot - including in the last RC stage - the 1k loading is too low. Kudos to LTSpice for having opamp models which reflect the loading they're subject to - presumably there's an Rout in the model? In which case if the loading effect is a function of the NFB (Rout is reduced by loop gain) then higher GBW opamps should fix up the problem. When I experimented with a higher GBW opamp before, I think I only substituted the highest Q stage with the faster opamp - arguing that it would only be needed for that stage. Faulty reasoning - so let's go back and substitute the 2nd-highest Q opamp with a faster one and remove the buffer....

Turns out yes - with 50MHz GBW opamps (LTC6244) in the two highest Q stages the response can be pretty well perfect. So this implies the 18MHz opamps will work well enough if the resistor values are tweaked upwards, the capacitors proportionately downwards. But how low can the GBW go before the Q gets negatively affected? And how low can the capacitors be before parasitics start to kick in? This could be where building gives a faster answer than simulating. It is however an interesting discovery that the TI papers have missed an important influence - that of opamp loading.

After some more tweaking in LTSpice I got a textbook response with the LTC6244 in the highest Q stage but could not achieve it with the LTC6241 there even when increasing the impedances. With the slower part there was always a nagging 0.3dB peak which wouldn't go away. Using 10k resistors the response is perfect with a buffer and the faster opamp - removing the buffer though and there's about a quarter dB shift. The faster LTC6244 comes with lower input parasitic capacitance so when exchanging them the cap nearest the input needs a couple of pF tweak. Plot now attached of the response I get with two LTC6244s in the two most demanding stages, 10k resistors used.

Update - I've now included a schematic of a second-order filter section which is designed to do NOS droop correction. Its accurate to within 0.1dB up to 18kHz just has a little roll off at the very top. Note its not using the BCV27 rather two discretes as LTSpice doesn't have the darlington - with the BCV27 the 3e5 resistor isn't required. The voltage source (1.75V) and series 33R is in reality a red LED.

Update2 - the first filter that I built with the BCV27 buffers was an 8th order Chebyshev - four cascaded sections. When I first listened to it on my headphones I discovered it suffered from a LOT of HF noise. Going back to LTSpice and gaining understanding of how to use the noise analysis function quickly revealed why this is so. The final stage of the filter is the highest Q one, for reasons of dynamic range - we don't want to clip this stage so it must appear in the signal chain after the HF has been sufficiently attenuated by the earlier stages. Having high Q though means high gain and from high gain follows high noise. The predominant source of noise being the shot current flowing through the input resistors. Reducing the resistor values (scaling up the Cs in tandem) reduces the noise so I did this and got the noise acceptable on my 'phones. However last night I tried this filter out on my speaker system, auditioning it against a straight wire. It clearly is still a bit too noisy but other than that I didn't notice any other audible artifacts being introduced.

So I'd say that 8th order is probably the practical limit in audio with this topology for reasons of noise, if not Q stability against component variations. Fortunately with discrete transistors we can increase the supply voltages to get more dynamic range if noise is a problem. 6th order might well be the sweet spot though on battery power so that's what I'm including in my latest DAC design, together with the NOS droop correction section.

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