Design of a choke-input rectifier requires the minimum load current desired for continuous output current. For welding, that will be the least current for which you want smooth DC. Which should maybe be lower than usual, in case you want to run at lower currents, or have more leeway swinging around the stinger (more inductance means more time at higher distance as you adjust the arc length, or shift position or whatever, before it extinguishes). For say 50A, at 120Hz, that would be around 265uH.
You can make a "swinging choke" by air-gapping the choke in a distributed manner: a section with low airgap to give high inductance up to some margin, then the remainder of the gap of normal length. The close-spaced section saturates first, turning it into effective airgap*, and inductance decreases to a 2nd nominal value, until current goes high enough that the whole core becomes saturated.
*Albeit still loaded with metal, and the metal doesn't stop being a conductor; so the fringing field in this area will increase losses somewhat. This is really just for point of interest; it's irrelevant for a welder -- you have tons of losses everywhere, and a couple, or maybe dozens, of watts of core loss around the airgap isn't going to affect your operating duty cycle.
Alternately, you could make the choke with a variable airgap, so you can shrink it to increase inductance for low-current operation, or vice versa.
A microwave oven transformer might make a good start here. A typical old/large one is around 2500 mm^2 cross section, and ~1500 mm^2 winding area.
To handle 100A at 265uH, you need to store 1.3J. 1.2T is 0.57 mJ/mm^3 so you need 2270 mm^3; at 2500 mm^2 area that's a spacing of 0.91mm, not bad. This gives AL = 3.2 uH/t^2, so N >= 9.
Let's see if the current rating is reasonable. At reduced duty cycle, current density of 10 A/mm^2 is a reasonable starting point. 9t * 100A = 900At is required, and fits into 1500 mm^2 maximum or 0.6 A/mm^2. I mean, that'll be lower due to winding factor (if you're using welding cable looped around there, you'll lose a lot to the bulky insulation), but that sounds very chill. Maybe an MOT is an overestimate -- or you can get even more inductance (more turns and air gap) from one, or use a smaller one probably more typical of compact/modern-ish MOTs.
Or I've made a gross mistake, which, is entirely possible.
L1 is used to charge a cap that usually sits across the load.
That limits the available current to recharge the cap and this arrangent is not for smoothing per se.
L1 might be rated at 25mH @50A with a 0.1F capacitor bank.
Assuming the cap is charged, before the arc is struck there is current flowing to charge L1 so no energy is stored in it.
Its the capacitor energy that provides the energy for the strike.
Stick welding died out with the dinosaurs.
A large enough capacitor makes a relaxation oscillator (spark discharge, recharges thru inductor, repeat). Strike energy isn't needed, at least if your scratch technique isn't awful. Stick welding remains a useful, and highly affordable, process.
HF start can also be added; it's basically a spark gap Tesla coil without the long resonant secondary.
Such a power supply could be used for TIG as well, if one is so inclined. The lack of ready adjustment will be, probably rather annoying to use though.
But with DC output, there's potential to add a switching power stage to make it adjustable, implement pulse modes, etc. A lot easier than making a whole-ass offline supply (if still a challenge to build from scratch!).
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As for the rectifier, if you don't have one yet -- you can get diodes that big, and that's the preferred solution, but you may find it attractive to build from smaller parts instead.
You can't simply wire these in parallel, but you can parallel them with ballast resistors. Put one resistor in series with each AC terminal of each rectifier, and size them to give, say, at least 0.2V drop at rated current. Also, put them all on the same heatsink, preferably a very thick one (or an even chunkier aluminum plate, with heatsinks bolted to that, all joints greased with thermal paste of course). So that would be, say 0.4V at 35A is pretty much 0.01Ω, (35A)^2 * (0.01Ω) = 16W so choose 20-50W rated resistors*, and, for say a 200A rating, six GBPC3502 (or similar) in parallel should suffice.
*You may find aluminum-case resistors attractive here, but mind they must be heatsinked. The bother of putting everything on such an even massiver heatsink may prove less worthwhile; I would prefer metal-link or wirewound style resistors, personally. You can also make your own from thick nichrome wire, if you have the means to join it (i.e. crimping). Or ribbon, for that matter... (Stainless steel is also acceptable, having not much more tempco, not much different resistance, from nichrome. Neither one you want to operate anywhere near glowing temperatures, for safety, anyway, so the stainless-ness of it, or the maximum operating temp limit, doesn't matter here, both are plenty good enough. Stainless can be joined by brazing (use appropriate flux) or bolting.)
Tim