First thing is NEVER unhook a battery while the engine is running. 2 things could happen and both are something no one would want. First the battery could make an arc and ignite the gases around the battery and cause it to blow up. Second, if there is a problem with the regulator and it is OVER charging you could actually TOTAL the car by distroying all the electronics. SO NEVER unhook the battery while the engine is running.

Now on to your problem.
The voltage at the battery with the engine running and everything else off and fully charged battery should be 13.0 to 13.8 Volts. Domestic cars will be 14 to 14.8 volts.
With the engine at idle, check for AC voltage. It should be less than 1V. If it is more, than the rectifiers are bad in the alternator. Now back to DC volts at the battery, turn on the lights and the AC fan to high. The voltage will drop but it should stay above 12.5V. Rev the engine to 2000 RPM and the voltage should go back up to 13.0 to 13.8 V. If not, check the alt belt first. It should be tight enought that you can't turn it by hand.(Be careful, it will be hot). If the belt will slip when turning it by hand, it is toooo loose. Tighten it and recheck.
Let us know what you find.

Paul's Auto Electric

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84 500 SEL (307,xxx miles)