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Originally Posted by mora
.. So if that component is missing circuit isn't complete.
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Good. So we know it is in series with the negative input lead.
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There was no writing on the other side but PCB had silkscreen printing "L2".
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Ah. I'd say it's 99% sure that this component is an inductor.
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Should I try bridging that component with some wire?
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You could do that, but I'd run it from a current limited source like a power supply that you can set the current limit on. I'd start with perhaps a 1 amp limit, even though it might not pull in at that current. There is a good chance that whatever caused L2 to overheat will still be causing way too much current to flow. If the power supply limits itself to a low voltage with 1A flowing continuously, you can say with reasonable certainty that there is still a fault there.
[ Edit: one amp at 12 V is still plenty of power to let smoke out of components. So be very watchful still; don't imagine that the 1 A limit will prevent all possible adverse events, like components burning up. ]
If it all works, [ edit 2: and you don't get a better clue about the value of L2, e.g. by measuring the inductance from a good economizer ] you could do worse than replacing L2 with a surface mount inductor that is capable of handling 3.8 A peak. It looks like a 1515 (4mm x 4mm) component to me, but it's hard to tell from the photo. Any inductance in the range 5 to 50 microhenries would be appropriate, I'd guess. Perhaps less, perhaps down to one microhenry, because of the rather large currents involved.
It will probably work fine with L2 shorted permanently; it's just that a little more of a spike may find its way back into the 12 V circuits that way. But consider that L2 acted as a kind of fuse last time; replacing it with another inductor may prevent worse burning up of components if this sort of thing happens again. It doesn't have to be an accident with a cable; voltage spikes in cars are a fact of life. The inductor will also help protect the economizer circuit from moderate spikes on the 12 V line from elsewhere. (The accidental cable connecting would be considered a jumbo spike

.)