Sounds like you may have a failed diode inside the generator. That will cause it to develop a maximum of only one third of its design output current which is not enough to meet the car's demand, so it will slowly drain the battery to make up the shortfall. The average system voltage can often be maintained under most conditions, and that's what the voltage regulator looks at to determine when to turn on the warning light.
Start by having the generator load-tested. The most common ones are designed to produce around 90 amps. If yours has a bad diode, 30 amps is all it will produce and "ripple" will be very high. It is not practical to repair these generators. They are normally replaced. The bad news is it is common to go through four to six of them in the life of the car. The good news you already did what it takes to reduce the number of repeat failures. That is to replace the battery unless it is less than about two years old. Due to their design, these generators develop huge voltage spikes that can destroy the diodes, internal voltage regulator, and can interfere with Engine Computer sensor signals. As the battery ages, it loses its ability to dampen and absorb those spikes.
Friday, December 28th, 2012 AT 6:36 AM