You can't just rectify 110V AC power and feed it into a battery, you'll destroy the battery. Rectified 110V AC power will actually vary from 0-162V! The normal black line is regular 110V AC power. The top of the graph (black/red/black/red) is what you get when you rectify 110V AC power. In order to get the 110V RMS, you would need to create some additional circuitry to smooth the power output out. Unfortunately, I can't help you with that because I don't know how.
