I thought they were extinct
Looks like an idea from about 1980.
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That's 29A at 13.8V (nominal open terminal voltage on a standard car battery). I think cigar lighter sockets, last I looked, were rated for 8A max. I balk at connecting the notebook's 70W LV power adapter; even though it doesn't exceed the rated current it still gets bloody warm due to the comparatively poor contacts one finds in these things. I shudder to think what a 400W inverter (I hope that's input and not output) will do. Meltdown, probably. One look at the cable the manufacturers use to connect the cigar lighter to the fusebox should convince you that this is a bad idea.
Don't you just love these "plug in and bugger the consequences" gizmos?
Flames, for obvious reasons.
Indeed. I have an inverter on the boat, but that is a direct connection to a leisure battery. That is beside the point. I was talking about the connection method, the cigar lighter socket and associated cabling. Believe me, I've seen what a faulty cigar lighter connection does to a car's wiring loom first hand. If anyone remembers the old Rover 200 series Mk1, their cigar lighter went straight to the main 60A link fuse via what looked like 16/0.2 hookup wire. It blew said link fuse, but not before the white wire with a tracer melted its way through the dash wiring. Moral is that you cannot know what setup a vehicle has until you look. Yes, I did manage to rewire that poor unfortunate car. The cause? A faulty mobe charger.
This pulls far more current from that connection than is safe. If you connect a current consumer of this magnitude to something like a cigar lighter with a comparatively high R contact set and wiring that will probably show resistance of several ohms at this current, it WILL "deep six" something. It's almost guaranteed. If you're lucky, the fuse will go first. If not, enjoy the smell of melting PVC. Next time you look at your inverter, pay close attention to the gauge of cable that connects the inverter to power. Then, if you can be bothered, compare it to the wiring leading to your cigar lighter socket. You will soon see why any inverter over a certain power rating is supplied not with a cigar lighter plug, but croc clips for direct connection to a lead-acid style battery.
"Wattage is output." OK, so let's assume it's, oh, I'll be generous, 90% efficient. (400*(1/0.9))/13.8=? It's 32A - from a cigar lighter socket. Unwise, my friends, very unwise. It'll also drain the average car battery in a little over an hour. You alternator can probably manage to offset this current draw with nothing else consuming power, unless you're an audio buff with an uprated alternator, in which case you'll have a 45A or 60A alternator fitted which can easily accommodate these currents.
Contrary to what you imply, I am fully versed in DC theory and the engineering applied to a car's LV wiring. This sort of thing is the whole rationale behind increasing a car's LV from 12V to 48V posited a couple of years ago: Higher voltage equates to less current for a given power, thus the manufacturers can get away with thinner gauges/longer runs of wiring for the same power draw.