Two things.
1- You still haven't answered the question about which converted you're using. If you want us to help you figure out what's going on, you need to give us some basic information here. As an example, if you're using a 30 amp converter, it will take six or more hours to get from 10 volts to 14 volts.
2- For the sake of your battery life, you really shouldn't let it get much below 12.5 volts. I would never let mine get to 10 volts. Realistically, there's not very much battery capacity between 10 volts and 12.5 volts with a lithium ion phosphate battery anyway. There is a big difference in how many years the battery will last though.
100 Proof