Inverters, depends a lot on the efficiency of the inverter as to the amp-draw, cheap ones are ineffient and have lousy output waveforms, your power supply will run HOT and might be damaged eventually.

Try staging them so that you don't get all of the inrush current at once: plug in the inverter, turn it on, plug in the power-supply & laptop, then turn it on.

I don't know what a separate fuse would be protecting, the cigarette lighter is already properly fused.

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Gone to the dark side

- Jeff