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So the author is wrong in his conclusion? I am happy to know the correct answer.

The author's solution may still help somewhat, but yes, they are almost certainly wrong. This transistor really just isn't being driven properly, because it really shouldn't generate heat in this application.

It could a whole slew of problems and it's impossible to tell without measuring. But, I would guess that gate voltage is not far enough above the threshold to drop the Rds(on) to it's low loss on-state.

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