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*Does the air core inductor ...behave the same way as a ferrite core inductor,*
Air-core has a

**constant** inductance at any current (until it melts, which is a huge current).

Iron/ferrite/powder cores allow a smaller (and sometimes cheaper) part, but the inductance declines at high current. You MUST be sure it is rated for more current than you have, and you must accept that the actual inductance could vary 2:1 from low current to rated current. (Cored chokes sold for speaker use "should" have a reasonably constant inductance up to rated current.)

In this case: iron/etc-core may be fine. The exact inductance is not critical, and the inductor current is known.

>

*what is the math*
Pick C1 as a reasonable compromise between ripple and cost/size. Generally you aim for 5% ripple, though you may have to re-consider this.

If that is not clean enough, add a R-C or L-C filter. For best ripple reduction, the impedance of the C should be much less than the load R, the impedance of the R or L should be more than the load R.

Speaker amp power supply impedance is around 50 ohms.

Take 100Hz as the ripple frequency (double the line frequency). In 60Hz lands it will be 120Hz ripple, but we don't need to be precise and I'm lazy.

32uFd is 50 ohms at 100Hz. We will want much-much more than that.

High-current chokes are hard to find. 1mH is readily available (you can use a pound of fat magnet wire on a plastic spool). 1mH at 100Hz is 0.6 ohms. This is "less" than our 50 ohm load. However, as air-core, we will probably find ~1 ohm DC resistance. In fact air-core is not a lot of good for reducing 100Hz ripple, unless you use a LOT of copper (big, costly). Iron/etc-core will give higher AC impedance at lower DC resistance. (However, even if the air-core choke acts like a resistor at 100Hz, it can have high impedance at 1,000Hz, which will reduce the most annoying part of the ripple.)

Using the above values, 32mFd is 50 ohms at 100Hz and 1mH is 0.6 ohms at 100Hz. The ripple reduction is about 0.6/(0.6||50), or about 1:0.99, not much at all. We need more!

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*desired output voltage (V) / total current draw (mA) => inductor size (H)*
An OK starting point. However this calls for 50V/2,000mA= 25mH. It may be hard to find 25mH at 2 Amps.

Note that C1 "has" to be >1,000uFd for tolerable ripple in a speaker-amp. A good first-guess is that C2 should also be >1,000uFd. Both act as reservoirs, the choke prevents the amp from getting full use of C1, so C2 should be large too. And in the real world, a bag of same-size caps is cheaper than an assortment of different values. And another 1,000uFd costs little.

We would like the choke's AC (100Hz) impedance to be very high (with low DC resistance), but that may not be easy.

We would like the capacitor's AC impedance to be very low. Since the choke can't be as good as we like, we pick a cap with very-very-very low impedance.

This works really well when caps are cheap, as they have been since 1940. On today's market, caps of any size are a commodity product, chokes are all "specials".

Taking 1,000uFd and sticking with 1mH, 1000uFd at 100Hz is 1.6 ohms, choke is 0.6 ohms, ripple is reduced about as 1.6/(0.6+1.6) or 1:0.7, still not much.

So the answer seems to be: find the biggest practical choke. At these currents, 2mH may be as good as it gets without custom-winding a heavy hunk of iron. Then super-size the capacitors, spending about equally on C1 and C2.

In this case the designer got to 2x22,000uFd before he either met his ripple spec or ran out of space/money.

44,000uFd at 100Hz is 0.036 ohms. 2mH at 100Hz is 1.2 ohms. C1 ripple is about 0.5%. Ripple is then reduced about 0.036/(0.036+1.2)= 1:34 by L1 C2. Output ripple is around 0.015%. If the amplifier has zero ripple reduction, output buzz is very roughly -75dB from full output.