Class D vs speaker impedance

https://dk.rs-online.com/web/p/spoler-med-tilledninger/1857421

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I see.There exist gapped ferrite toroids as well. For lowest distortion gapped ferrite outperforms any metal powder imho. As a DIY solution I would prefer some gapped ferrite pot core.
For lowest THD with TPA3251 amps I use the biggest ferrite cores with saturation beyond 20amps. This makes - at least measurable - a difference.
 
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The speaker impedance changes the output filter Q, not the frequency, but the speaker is not a resistor either so maybe not much. The amplifier voltage feedback attempts a zero output impedance so the FET on resistance affects efficiency but not the output impedance. It depends somewhat whether the feedback is taken pre or post filter. Whatever current your coil and FETs are capable of, there is a rail voltage that matches that current to the load impedance.
A gapped toroid may be a mechanical issue. Cores come with a wide range of permeability ferrites and if the inductance you want comes with only a few turns then you need a different core, maybe a different style core that is a better choice for gapping. High permeability is bad news for core saturation and not suited for power applications.
 
I have now tried with a thinner copper wire, about half the thickness, but the value is still far too high.
I think it is the core itself that is too big / powerful.
Will it make any sense to wind a coil like the ones in the picture, but on plastic, wood or something non-magnetic?
That kind of air coil?

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I do not see any relationship between wire thickness and inductance. It is true aircoils do not saturate, but that is only one aspect. Without a core, you get a maximum of turns required. Maybe something in the ballpark of 100 turns. Thus a coreless toroid requires a lot of copper, otherwise DC resistance would be inacceptable. And without a core magnetic leaking flux would be considerably bigger.

There is a reason that air coil output inductors are rare birds in reality, sometimes appearing in boutique amps.