NEVER REMOVE THE BATTERY CABLE WHILE THE ENGINE IS RUNNING!
That was a trick done many years ago to tell if the generator was working and was done by mechanics who didn't understand how these simple systems work. If you do that at the dealership I worked at, you got a stern lecture in the manager's office the first time, and you were offered the chance to find a job somewhere else the second time. The battery is the key component is helping the voltage regulator maintain system voltage to a safe level. Every year I did a demonstration on the alternator test bench to show my students what could happen. It was real easy for the output voltage to exceed 30 volts. That will destroy every computer on the car and every light bulb that is turned on. You are very lucky your generator wasn't working.
To tell if the system is working, measure the battery voltage with the engine running. It must be between 13.75 and 14.75 volts. If it doesn't come up to that, measure the voltages on the two smaller wires that are plugged into the generator and tell me what those are. I think one will be yellow and one is green with a red stripe. If the yellow one does not have 12 volts on it, check the fuses. Also observe if the battery warning light on the dash goes off with the engine running and if it is on with the ignition switch on but the engine not running. If the generator is working, you should also find around 6 volts on the white or white / black wire.
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Friday, October 26th, 2012 AT 7:14 PM