You do not want to disconnect the battery while the engine is running. This is because the battery is essentially a big capacitor that acts as a filter for your car's electrical system, keeping voltage spikes down. Running a vehicle with the battery disconnected can cause damage to sensitive electronics due to voltage spikes.

The correct way to check an alternator is to observe the voltage level that is output. The voltage at the battery should be somewhere around 14V with engine running. With engine off, battery should read 12.6V. If the voltage level does not change from 12.6V after you start the car, the alternator is not charging. This could be due to a faulty alternator or voltage regulator.

Another test to perform, if the alternator is still suspect, is to hook up an ammeter in series with the big alternator wire, and turn on all electrical accessories with engine running, and observe that the alternator is outputting a good amount of current (approx. 40-80 amps depending on alt.) Note that you'll need a decent ammeter/multimeter to do this.
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85 245 DL - 91 Mazda MX6 Turbo