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What kind of applications are transformers with a single turn secondary used for? I believe a high current is generated, but couldn't really figure out a good practical example where this might be used.

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A Weller gun-style soldering iron is one good example.

soldering gun http://static.emedco.com/media/catalog/product/Cooper-Hand-Tools-Wellerreg--Professional-Soldering-Guns-DD678-ba.jpg

Also, pretty much any spot-welder for sheet metal.

spot welder
(source: harborfreight.com)

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I've used a single turn secondary with the wires passing through a current transformer for testing it at hundreds of amps. The primary was fed from a variac so I could control current fed through the CT. Checking linearity was the objective.

Rf transformers are quite commonly made with one or a few turns to make them work effectively at VHF and UHF.

I've also made data coupling transformers where one winding was on a rotating spindle. Single turns had to be used because the data rate was so high ie tens of Mbps.

Same sort of idea for transmitting power across a small gap on rotating apparatus. Single turns were used and resonated with tuning capacitors.

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I saw one in a 12V 800W server power supply (Ablecom PWS-801-1R).

In case you were wondering - the PSU provides 12V and 5VSB, the other voltages are generated by the backplane.

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