Google Honda GX 390 wiring diagram. If you wish to increase the charging output get the Honda Charging coils, they are much bigger. They will require a tad longer mounting bolts and a regulator. Pretty simple.
Not certain, but probably 4-5 amps. Probably plenty to keep the battery charged. I think my regulator/rectifier took a dump on mine though. I intent to put a John Deere type 20 amp permanent magnet unit on mine. I already built a pulley that sits behind the driver unit of my 40 series TAV.How much can one expect from the honda charging coils (two installed)

Not certain, but probably 4-5 amps. Probably plenty to keep the battery charged. I think my regulator/rectifier took a dump on mine though. I intent to put a John Deere type 20 amp permanent magnet unit on mine. I already built a pulley that sits behind the driver unit of my 40 series TAV.
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Tractor Supply sells weld on pulleys. You buy the pulley you want then a hub for your shaft. I bought a 4” pulley for a 1/2” belt. Instead of a hub I got a shaft collar which was real close to fitting in a pulley. I spun the collar in my drill press and used a file to take it down the tiny amount needed. Welded it up. It runs straight and true. Can’t really get a key in it that close to the engine. And although I thing the TAV tightened down will hold it I still filed down a set screw that fits in the keyway. It’s going nowhere. I have the power sports TAV and it requires quite a few flat washers to set up correctly. I only used a couple thin flats and the pulley is the rest.I am considering the same thing... but I believe I'm going to go with a mini denso 30A. I'd be interested in hearing what you did for a pulley behind the driver.