My 2002 s-type 4.0 randomly had a battery die, replaced it, tested the voltage with multimeter at the battery with car off and turned on. Car off read 12.6V and while on read only 12.2V so my battery isn’t being recharged. Looking at the alternator and the belt is fine(tight, moves while running, doesn’t make noise) and I can see the internals of the alternator spinning, doesn’t smell like it’s burning and no visible damage. I’m pretty sure the pulley is fine because the inside of the alternator spins while running. Do you guys think the issue actually is the alternator or could it be another component in the recharging system? If it is the alternator, is there a way to test and see if it can be repaired?