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I'm inclined to agree with DavidEM in theory regarding A/C use. What's the point of having a 3000 W inverter if you can't generate 3000 W.
I do have a soft start as recommended above, but that doesn't reduce the overall power requirements of the A/C, just smooths out the curve.
I've measured the voltage drop under load across various cables, fuses and connectors and it seems like each cable or connector contributes on the order of 0.05 V of drop or about 7 W of waste heat each. Maybe 70-100 W of heating is enough to make the cables heat up?
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