The brief
The 520i was hard to start from cold, gave an intermittent squeal when the air-con and the headlights came on together, and the battery warning had come up on the dash. The owner brought it in braced for a dead alternator.
Those symptoms point at the charging side, but not always at the alternator itself. The alternator is spun by the accessory belt off the crankshaft, through an overrunning pulley that smooths out the engine's pulses. If that pulley fails, or the belt slips on it, the alternator does not get turned hard enough at low revs to keep the battery topped up, which is the warning light, the squeal, and the slow cranking the next cold morning.
The diagnosis
A multimeter on the output told the story: spun fast, the alternator put out fine, but at idle the belt was slipping on the alternator pulley and costing the system its charge. The overrunning pulley had sunk on its clutch and was no longer holding the belt in line, and the belt itself had taken so much abuse from the slip that it had started to shred.
So it was a charging-drive problem rather than a worn-out alternator winding, but with the pulley failed and the belt coming apart the sensible scope was a new alternator with a fresh overrunning pulley, plus a new belt and tensioner.
The work
Belt tension was released and the failed alternator removed, then a new BMW-spec alternator with a fresh overrunning pulley fitted in its place. A new belt and a new tensioner went on, the belt routed correctly through the accessory drive and re-tensioned to spec.
Then the charging output was checked across a load test, at idle and at speed, to confirm the system was holding voltage properly under load.
The outcome
No squeal, charging output back to spec at idle and at speed, normal cold cranking, and no warning light.
The 520i went home with the charging system right again. For the owner the practical win is a car that fires every cold morning and a battery that stays charged, plus, having caught the slipping belt before it shredded entirely, no breakdown with a dead alternator at the worst possible moment.