At least not on a modern vehicle with computers.
Positive battery cable to alternator.
Without it full alternator output voltage can spike to 150 volts more than enough to fry every computer in your vehicle along with your stereo fuses remote door locks etc.
Never test an alternator by removing battery cable.
Have the alternator tested and replace the battery.
As others have stated by contrast with a generator the alternator is designed exclusively to recharge the battery.
Some may recommend running the engine with the negative battery cable disconnected to test the alternator.
Reconnecting battery cables generates a second spike.
The alternator charge wire routes direct to the battery and not through any switch connection the alternator will not operate correctly if not connected direct to battery or directly through the ammeter.
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Acdelco 20943125 gm original equipment alternator battery jumper cable.
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A bad battery can do it.
A scraped up wire can do it.
Connect b battery cable to alternator.
Fixing your battery or alternator can be an expensive repair.
Install a jumper from pin f in harness plug to b terminal on alternator.
All you can do is replace the cable to the battery and possibly the negative battery cable terminal since that one is giving you problems removing the terminal.
In modern cars the alternator is actually disabled when the battery is full and the electronics will run from the battery for a while reducing load on the engine and improving fuel economy.
If faced with electrical issues it is always a good idea to clean up your battery cables and check your grounds first before buying at replacement components.
However this is not a good idea because it could damage the electrical system of your vehicle and cause greater problems.
Doing this would allow you to check whether the alternator is putting out enough current to run the vehicle.
To do a quick check on an alternator i was told that you could start the car and then remove the negative battery cable.