Re: Panels and Battery Bank Draw and Refill..
I think you're getting lost in the numbers.
The 77% efficiency we talk about is panel & controller average.
The 52% efficiency is for over-all system operation (name plate rating of the panels to AC Watt hours out the door).
So 870 Watts of panel would produce about 670 Watts DC; important for determining charge rate with an MPPT controller.
In 6 hours of equivalent good sun it would produce 2.7 kW hours AC, providing you can make use of it.
Your battery bank at 330 Amp hours @ 12 VDC can supply up to 1980 Watt hours DC, about 1600-1700 Watt hours AC.
Your A/C unit uses 550 Watts. From the battery alone that would be up to 3 hours total run time (at full power consumption).
From the panels only it would be perhaps as much as 4 hours; a lot depends on how depleted the batteries are when charging begins and thus how much panel power has to go to recharging before the A/C can be used "for free". This is not too far off from what you expect.
The difference between the A/C's 550 Watts when converted to DC consumption and the panels' output ability is not very much. At best it would be 612 Watts (550 @ 90% conversion efficiency) from 670 Watts (870 Watts @ 77% efficiency) or a meager 58 Watts to spare. But don't panic:
A couple of things: first, the A/C is not going to use exactly 550 Watts all the time it is running. The start surge will be much higher, and some times the compressor will not be running. Second, if the batteries aren't drawn down over night they may not need much charging in the morning.
Usually, as a guide to keep it straight for discussion purposes, it helps if you think of Amps only when dealing with the DC side, and Watts on the AC side. Although both exist in either realm, it's easier to consider it as "Amp hours" (as in the battery rating) DC and "Watt hours" (as in what the utility bills you for) AC.
In between the available DC Amp hours @ Voltage and the AC Watt hours is the inverter's own consumption and its conversion efficiency.
Four 175 Watt panels, OB MX60, 232 Amp hrs, OB 3524, Honda eu2000.
Ohm's Law: Amps = Volts / Ohms
Power Formula: Watts = Volts * Amps