Does the autotransformer balance the output and the input 120 lines, or just the output, or just the input? I'm confused!
Auto-transformers are a single ended device, it is one transformer winding, typically with a single center tap. More advance models have multiple taps on the 220V end, giving 220,23,240 to 120 at the center. Common terminal is Neutral. There is no balancing.
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