PA 28-181 Flight Instructor reported an alternator failure during cruise; while on a training flight. The Instructor took control of the aircraft; diverted and landed safely.

Narrative

I; Instructor; was flying a PA-28-181 with my student; on a cross-country flight under IFR with the following route: ZZZ1 ZZZ2 ZZZ. We terminated a practice instrument approach into ZZZ2 with a touch-and-go; followed by a departure to the north to resume on-course to ZZZ. We received direct routing to ZZZZZ (the IAF for the RNAV RWY XX approach at ZZZ) with a climb up to an altitude of 4000 ft. Upon reaching this altitude; we leveled-off and put the airplane in the cruise configuration; following the checklist. About five minutes after; the 'LOW BUS VOLTAGE' and 'ALTERNATOR INOP' annunciators both illuminated. At this point; the ammeter read '0.' My student and I promptly ran the associated checklist. This resulted in the lights being extinguished and the ammeter returning to an indication of +17 to +20 for about 30 seconds. After this; both lights illuminated again and the ammeter returned to 0. We complied with the rest of the checklist; which included shedding electrical load (exterior lights; GNS430 #2; dimming our PFD and MFD; turning off the fan; etc.). At this point; we were about 36 NM WNW of ZZZ3; and I took full control of the aircraft and elected to follow-through with a diversion to that airport. I informed ZZZ3 Approach that we were experiencing a possible alternator failure and could lose electrical power within half an hour. I stated my intentions to divert to ZZZ3 and requested direct routing. I maintained the highest forward speed that was safe at the time; so that I could attempt to make it to the airfield prior to losing battery power (and with it; radio communications). The entire diversion from that point to ZZZ3 took about 25 minutes. When I was about 11 miles away from the field; with the airport in sight; I elected to cancel our IFR flight plan with Approach; and received radar vectors for traffic; followed by clearance for a VFR descent towards the airport. I never [requested priority handling]; but suspect that ATC did on our behalf. Approach inquired about the number of souls on-board and the fuel remaining (X souls and about X hours of fuel remaining). We received priority handling all the way to ZZZ3 and got cleared for a straight-in visual approach to RWY X; followed by a taxi to the ramp via TWY 1. The landing was safe; smooth; and uneventful. Upon securing the aircraft at the ramp; a gentleman from Airport Operations approached me and asked if we were flying the [priority handling] aircraft. I told him that we never [requested priority handling]; but suspected that ATC did on our behalf. He asked if we were okay and if the landing was safe. I answered in the affirmative to both inquiries. He copied my name; the tail number of the aircraft we were flying; and my phone number. The total flight time per the OUT/IN Hobbs was X.X hours (X.X hours OFF/ON). VFR conditions prevailed throughout the entire duration of our flight.