The simplest way to do this is to chop the DC to where you essentially have a square wave (basically a switch turning on and off at the desired AC frequency). Then run that through capacitors to “smooth” the square into something resembling a sine wave. Then “level” the 0V to the middle of that wave and amplify that voltage (easily done with a center tap transformer, for example) and voila “AC” voltage at the output.
Nowadays, there are lots of electronic inverters that do this in a more efficient fashion but the principle is just about the same.
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