The alternator has nothing to do with this problem, at least up to this point. It's job is to recharge the battery after it has cranked the engine, and to supply all the current for the entire electrical system, but only once the engine is running. This problem is related to a defective battery or one that has been severely discharged. When a good battery becomes this badly discharged to reach 11.9 volts, it will take a good ten to 15 minutes on a charger before it starts to take a charge. It takes that long for the acid to fill with electrons and become conductive. As that happens, charging current will gradually increase from near 0 amps to no more than 20 amps to prevent overheating the battery. Once it is fully-charged, the current from the charger will slowly drop to around five amps. It can be considered fully-charged at that point as current will not drop much lower.
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Thursday, January 3rd, 2019 AT 4:29 PM