The "Battery" light means the battery isn't being recharged while you're driving. It's the charging system you need to have tested, not the battery. If the generator is totally dead, you'll be able to drive no more than an hour after charging the battery fully, and even less if the heater fan, radio, head lights, or other things are turned on.
When you're able to drive for a day or two, or even a couple of weeks before the battery runs down, it's likely one of the six diodes in the generator is bad. That will reduce its maximum current capacity to exactly one third of its rating. 30 amps from the common 90 amp generator is not enough to run the entire electrical system under all conditions. The battery has to make up the difference until it runs down.
You need a professional load tester to test for full-load output current and "ripple" voltage. If rippIe voltage is high and maximum output current is low, a diode has failed. It really isn't practical to replace a set of diodes. The generator is replaced instead.
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Sunday, September 6th, 2015 AT 10:32 PM