You need an INVERTER, not a converter as some have told you. An inverter transforms DC battery power to AC. For watching TV an inverter rated at 300 watts or more should work.
Do not try to run an inverter that size off of a cigarette lighter as some have suggested. A lighter socket is not designed to supply that much power. You will need to wire the inverter directly to the batteries.
We typically watch television for three or four hours an evening on battery power with no problem, but we have 220 amp-hour battery capacity.
12V to the antenna from the booster is just the way they have the antenna wired so it will work in all configurations. I would not count on a big inverter plugged into a cigarette lighter. Larger inverters should be hard wired.
I too will say this: You expect a 12 volt battery to run a 120 volt TV? I don't
At least not without an inverter.
I have a xantrex Prosine 2.0.. it runs the TVs and related electronics, The Microwave and the GFCI "loop" (Recepticals) which includes the baths, kitchen, dining room and outside outlet. The remaining outlets including the bedroom. and the 20 amp outlet I put in for an electric heater... DEAD w/o shore or generator power.
Nothin adds excitment like something that is none of your business
Kenwood TS-2000 housed in a 2005 Damon Intruder 377
Lots of folks have hinted, but not come out and said- The 12V system is pretty limited- a little lighting, water pump, a little bit of TV (if you have a 12V TV or inverter), and one or two days of running the furnace.
Now, some folks (like me) go for that kind of camping from time to time- use some commmon sense, limit your electricity and the battery can last for several days.(OR you can do the generator thing)
If the 110V outlets were working, and you plugged in much of anything, you will suck the battery down very quickly.