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Chebyshev LC filter design with real-world inductors.

Nowadays there are lots of resources available to help with LC filter design, the one I use most is at https://rf-tools.com/lc-filter/. This site is invaluable but it only gets a design so far, as soon as you use real-world inductors (i.e. with losses) then the very nice flat, even equi-ripple response the math gives goes out of the window. Over a number of years I've developed a way to try to restore flatness and evenness to a filter incorporating real-world inductors but haven't documented it. I did mention it to Elvee here though. I figured it was probably worth fleshing out the steps in a relatively short blog post.

RF-tools allows you to specify your passband ripple - meaning the variation in frequency response in the passband. In general, the higher the passband ripple, the higher the Q in the filter. Introducing losses (inductor DCR) lowers the Q hence we need to aim for higher Q when designing than we actually need. Which means design for higher ripple and it'll be reduced when real inductors are substituted. The amount of additional ripple is going to depend on how lossy your inductors are - I find typically its in the region of tenths of a dB with relatively high Q ferrite-cored inductors.

Next up comes the corner frequency - we need to aim for a higher corner frequency than we need in practice as the process of straightening out the frequency response inevitably lowers it. Lastly, the terminating impedance needs to be specified higher as we'll lower that on the way to optimizing the design.

Another aspect to be considered is - are you winding inductors yourself or going to buy them off-the-shelf? In both cases inductor values are quantized but off-the-shelf inductors are much more coarsely quantized in values than self-wound ones. Some ranges are only E6, E12 is fairly common but E24 is rather rare. If you specify the corner frequency and the terminating impedance tightly then you'll have no flexibility over what inductor values fall out of the design process. But normally those two parameters are at least a little bit flexible, so it makes sense to aim to hit an inductor value which is available. This isn't at all difficult when there's only one inductor in the design, but what about higher order filters?

Hitting off-the-shelf inductor values with a 5th order filter (two inductors, three caps) isn't too hard but does take patience and a certain amount of flexibilty over the end result. You can get the two inductors both to fall on off-the-shelf values if you give up control over the pass-band ripple as the ratio of the two inductor values directly relates to the ripple parameter. With more ripple, the inductor values move closer together - by 10dB ripple (altogether quite useless for audio) they're practically co-incident. Moving the other way, having no ripple at all makes for an effective Butterworth filter with a 1.9 ratio. So you can see with E12 values you'll have a choice of ~20% or ~45% or ~75% ratio between inductors and this gives a choice of three ripples - ~0.3dB, ~0.01dB and effectively flat. With lower ripple you lose stopband rejection so the 0.3dB option is the most attractive one to me giving inductors with next-door values in the E12 series.

(to be continued)