I am building a mobile Sound System powered by a car battery. I have a small mixing table that runs on 12V but it needs AC Power (50Hz) and won't work with DC (i tried that, with very weird results).

Currently I am using a 12V to 230V Inverter and the provided power supply to convert it back to 12V alternating current. This seems wasteful and I do not want to introduce another voltage level into my design just to power the mixing board.

How could one build a simple circuit that does this? I have basic electronics skills (soldering, know about resistance, voltage, current) but never build a circuit board.