IMPORTANT: always disconnect the battery before working on the alternator; the battery should be disconnected -ve terminal first; reconnected -ve terminal last - if you work on the +ve battery terminal while the -ve terminal is still hooked up you risk damage, fire and injury. Always disarm your alarm just before removing the -ve terminal (you have <60s from disarming alarm to disconnecting terminal; otherwise alarm goes ape).
For the S1 I would lay money on the little wire (and possibly the big wire too) having broken off right at the back of the alternator. You'll need to remove the undertray and reach up and pull on the wires to see if any of them come loose. A little mirror on a telescopic stick (like the sort of thing a dentist uses) might help too. If you ignored my advice, you will just have discovered why you needed to disconnect the battery
For the S2 the failure is more likely to be inside the plastic cover on the back of the alternator itself, though still just a fractured wire that could be repaired by removing the alternator, removing the back cover, spotting broken wire and the rewiring it.
In all cases, finding something durable to rewire it with is the hard part. The crimp terminals are prone to vibration fracture and the wires are exposed to a lot of radiated heat from the exhaust manifold - you need to make sure all your heatshields are in order. Where the heatshields are broken (or for early S1s) not fitted, replace or fit them - otherwise you'll have repeat failures.
If the crimp terminals/associated wiring fails more than once every couple of years you have a heatshielding or vibration issue, IMHO.
In either case you could still have a broken regulator pack; this can be replaced without replacing the alternator if the coils and brushes are still in good working order - but I don't have a source for the regulator pack, so a recon unit might be best bet. The packs sometimes break when the control or load wires go make/break/make/break repeatedly while the alternator is under load - the broken wire is the root cause of the problem, but the reg pack takes it in the shorts at the same time unfortunately.
Cheers,
Robin

