How many 200 Watt solar panels do I need to power a 3.5kVa 48DCV inverter for 24 hours if the load on it is 1500 Watts and the batteries are 200Ah, 12DCV each in a country with about 6hrs of sunlight?
OK, let's assume that the charger and the inverter are each about 80% efficient, and the batteries themselves are about 70% efficient. That means that the overall power-in to power-out efficiency is 0.8 × 0.7 × 0.8 = 0.45.
You need 1500 W × 24 h = 36 kWh per day at the load, so your panels need to supply 36 kWh / 0.45 = 80 kWh per day.
But if they can only do that for 6 hours each day, then they need to supply 80 kWh / 6 hours = 13.3 kW when they're operating.
13.3 kW / 200 W per panel = 67 panels.
Now, what about the batteries? Each battery holds nominally 200 Ah × 12 V = 2.4 kWh. But you really don't want to cycle them by more than about 50% of their nominal capacity, so call it 1.2 kWh.
80 kWh / 1.2 kWh per battery = 67 batteries