Good observation. Battery voltage must be between 13.75 and 14.75 volts with the engine running. 13.78 is not impressive but technically it's okay.
First check the battery voltage with the engine off. It will be 12.6 volts if it's good and fully-charged. If it's closer to 12.2 volts, it's good but discharged. If you find it's around 11 volts or less, it has a shorted cell and must be replaced.
The generator will usually not cause a slow drop in charging voltage. That is due to a failing voltage regulator or a wiring problem in the circuit where it senses system voltage. It can be caused too by a failed diode inside the generator. That will limit the current output to exactly one third of its maximum rating, and that is often not enough to meet the demands of the electrical system under all conditions. The battery will have to make up the difference until it slowly runs down.
The voltage tests are just the first part of testing the charging system. You need a professional load tester to measure full-load output current and "ripple" voltage. If you find the maximum current is one third of what it should be, replace the generator.
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Monday, June 29th, 2015 AT 8:53 PM
