Since you are unaware that most cars now have alternators, not generators, I respectfully suggest you take you car to a mechanic that does "Automotive-Electrical" work (specializing in Starters and Alternators, most commonly).
A gauge can be put on the Alternator to easily measure the output amperage (assuming you do not have or can't trust the gauge on the dash) and you could watch and have them show you the electrical output. For anyone who has the gauge and basic knowledge on how to connect it, this is a very simple and quick procedure and many would check it free and tell you if its good or bad, and then try to get you to let them replace or fix it if its bad.
You should know that you have choices besides buying a new ALTERnator if it is not working. There are places that do almost nothing else but repair or "rebuild" alternators. They have "brushes" in them and sometimes that is all they need .. new brushes. Thats a relatively quick and easy job and should cost around $30, under $50 most plance, most alternators (even less sometimes if you take it off and take it in to them so they can just put it on bench and do the work).
You can also buy a "REBUILT alternator" at many places, including Auto Zone and other Auto Parts stores. Means one that was used but it was taken apart and new main components were put in and any problems fix and tested and working fine when you get it. Check on short warranty on it, but should be some warranty, less than new one and it will cost a LOT less than getting a new alternator and they can work fine for long time usually.
Taking an alternator off is a dirty job and can bang knuckles, and have hard to reach bolts .. but other than that its a simple job for anybody who does any basic work on cars (less than a "mechanic" job) .. if you know anyone who works on their car and can help you out. But do NOT take it out until it is TESTED in car in operation!
Personally, I have changed them on my boat motors when needed (as getting a boat mechanic to come to an in-water bigger boats is not so easy, convenient, fast or cheap and out on water (which I can jump in and with breeze) I don't mind so much. But under hood of a car in heat and all the grease and knuckle banging and all ... is just NOT for me. I will take it in and let them test and replace with a REBUILT with some guarantee and I don't think I've ever had any problems with REBUILT alternators or starters; but I do perfer it come from a known parts place so I can easily go see them if it is defective and I feel better about the rebuiding they do, as compared to the mechanic saying ... "I just rebuilt this one, I can put it in" .. no thanks; too may mechanics are LIARS. Sorry to the honest ones, but its a FACT.
BTW, an experienced mechanic should be able to test yours, pull it out and put in another one in an hour or less, in most cases. That's if he has one on hand like you need ... add wait time if he has to order it or go pick one up,
How this helps you Chris.
You asked for generator so I'll give you generator....
Disconnect the generator from the regulator (probably easiest to simply take the wires off the generator).Connect a jumper from DF on the generator to the generator frame. Now run the engine and measure the voltage from ground to D+ on the generator.
Some generators have different types of terminals;verify which is DF and which is D+.
As you increase the engine speed, the voltage should jump up to +35 volts or so (@3000 RPM).If it passes this test, the generator is good.
(Don't run this test longer than necessary as it will overheat the generator.)
Unless your car is from the early 60's or older you don't have a generator. You probably have an alternator. You can test it as follows, using a digital volt meter connect it to the battery and take a reading before starting the vehicle. The meter should read 12.68 volts if the battery is fully charged. You need the battery fully charged to get an acturate reading of the alternator output. Next, start the vehicle and take a reading at an idle with no accessories on, you should have a reading of 14.5 volts (plus or minus .2) if it is healthy. Next, turn on some accessories such as the heater motor on high and the headlights. The volts should not drop below 13.8 if everything is healthy. If the vehicle doesn't reach those readings your alternator is probably bad.
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