Whether it's an "emergency" or not within the FAA definition of the term is both debatable and situational, but in most planes that have one, the failure of the only alternator renders the aircraft unairworthy, and it's written that way in the manuals for those aircraft. For example, if you look in a C-172 POH, you'll see the alternator is an R-item, meaning it is required to be operable for even Day VFR operations, and the aircraft is unairworthy if it's not working.
That is not the definition of airworthiness, The wording in 91,3 and the FAA's definition of airworthiness both give the pilot the desecration to make the decision.
Quoted from the FAA training guide
What is the meaning of airworthy?
2 conditions :
1 CONFORMS WITH TYPE DESIGN.
Attained when the proper components are installed, and they are consistent with drawings, specifications, and other data that are a part of the type certificate, and supplemental type certificates and field approval alterations.
2 THE AIRCRAFT IS IN A CONDITION FOR SAFE OPERATION.
This refers to the condition of the aircraft with relations to wear and deterioration.
that's directly from the FAA and says nothing about the equipment being in a working order. It does say the aircraft must be in a condition for safe operation that is why we have 91.3
Loosing an alternator does not render the C-182 unsafe to operate, because we have redundant systems, and there is no requirement in "Airworthy" to have all equipment in working order, or we could not have deferred maintenance.
If you would like a copy of the Power point presentation I will send it to you. give me an e-mail (nc19143(at)comcast.net)