What is the process and difficulty of removing and replacing the alternator on a 1995 Hyundai Elantra GLS with the 1.8L engine?
My nephew's car stopped running. We changed out the battery to get it started and drive it home. When home I disconnected the negative cable from the battery (with the car running), and as suspected, it stopped running. This is how I quickly concluded there was a bad alternator. Any help would be appreciated as I am trying to save him money without getting ourselves in trouble.
Disconnecting the battery cable while the engine is running in order to verify charging system operation is a backyard method of testing the alternator. It may work, or it may not, and you run the risk of an explosion from battery fumes while making or breaking the connection. Today's cars and trucks are much more electronically complex than years past, and certain sub-systems may be sensitive to losing a connection to the battery.
You can more easily determine basic alternator output with a simple voltmeter attached to the battery. This is by no means a complete test, but a reading of about 14 volts is good, while 12 volts and falling is bad.
If your diagnosis was correct, and the charging system was not producing any current, that doesn't necessarily condemn the alternator. It could be as simple as a blown fuse or a poor connection in one of the charging system circuits. The proper way to test the system is with modern battery/charging system/starter diagnostic equipment, followed by a confirmation of all necessary power and ground sources to the alternator.
Many auto parts stores which carry a tester will gladly check the system and help you correctly diagnose the problem. As far as replacement of the alternator is concerned, I show about $350 for the genuine Hyundai part, and 1.1 hours labor for a qualified technician to install. It's not the toughest job in the world, but your level of skill is going to determine if it's worth the effort.