The battery is an integral part of the 120-volt AC system. It supposedly helps to smooth things out by absorbing voltage spikes. But having said that, if your converter is operating correctly you should not need the battery when you are on shore power.
Here are a few steps to help diagnose the power failures reported above...
In general, any electrical circuit is a circle: Power source (12V battery or 120V AC), a load (lights, pump, jack, etc), the wiring running throughout the trailer and the controls (switches, fuses and circuit breakers). The failure is in one or more of these components.
Check if there is a battery selector switch and make sure its on. Check the kill switch as found in popups.
Check battery terminals for corrosion and clean with baking soda. Check the electrolyte level and water as needed.
Make sure all connections are tight. Start at the battery and torque the terminal connections as needed. Follow the wiring from the battery to an inline fuse and make sure its not blown. Replace fuse or reset breaker (depends on what you have). Look for bare wires or metal touching wires that can cause a short (sometimes trailers are wired poorly in such a way that they rub against the frame and cause a short over time).
Check the converter that the fuses are not blown. Check the AC breakers under shore power, as well.
Check the batteries for voltage. Are you sure its charged? Above 12.6v is fully charged. 10.5 is full discharged. You may want to use a hydrometer to check individual cells as that will also tell you if there is a bad cell.
If you have a multimeter and are familiar with it, you can then go on to checking for an open circuit somewhere.
DW, 3 Kids and our Goldens
2012 Shamrock 233S
2008 Toyota Sequoia 5.7L 4WD