When you get the low voltage alarm it would be interesting to measure the voltage at the battery terminal connected to the starter solenoid on the hydraulic jack motor, both with shore power connected and no shore power connected to see if there is a difference. Just curious if the jack controller monitors this voltage point as opposed to the 14.4v chassis battery voltage you found at the controller terminal. If your coach happens to use the coach batteries to run the hydraulic motor (as mine does).....maybe that is the voltage that is low with no shore or gen running. If that is true, then the other issue is when the engine is running why is the alternator not also charging the coach batteries. Under some circumstances the BCC does not connect the coach and chassis batteries together via the interconnect relay. I hope you can resolve your issue.
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