Hi,
You probably only have one circuit that is powered by the inverter. Try the receptacles closest to the TV plug in. Also the inverter sucks the battery dry fairly quickly, so normally it is left off until you need to run it, so you will have to manually turn on the inverter someplace. You might have a remote control, or might need to get into one of the storage compartments to turn it on there. The inverter when left on, and you are not using any power will use about 1 amp hour per hour. Even when powering a 20 watt laptop, the inverter is not 100% efficient, and likely will use the same 12 watts as when not powering anything and an additional 24 watts to provide the 20 watts to the computer. So effectively, you will be using 3 amps per hour from your battery to power a 20 watt load, via the inverter. Your battery should hande the load OK, because it is only a 3 amp load for about 3-4 hours, or less than 15 amp hours. Your battery total is probably 200 AH for a pair of batteries, and 400 amp hours if you have 4 batteries.
The lights will consume another 1.1 amps per hour they are on, while the furnace will be around 6-8 amp per hour.
When the power is restored, the converter will recharge the batteries in a couple of hours. If not then driving will top them off in about 5-7 hours.
So remember that most of your RV receptacles are not inverter powered, only a couple of them. Best to mark them now. Use a radio or small wattage portable lamp to quickly find the working lights.
Fred.
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