Since it "can" be charged with a solar array, it can be charged while simultaneously providing power but that's not perpetual motion, because the sun does go down and batteries are depleted. The catch is the solar panel(s) need to be sufficiently large in wattage to keep whatever storage it had from being depleted under load (at a correct angle to the sun) while the sun is up. The catch in the case of the little power stations with ~100-200 watts of panels won't help keep the battery up too much.
What a decent solar system does (4Kw in my case) the solar panels will provide charge current AND load power while the sun is up, but large appliances like pool pumps, clothes dryers, whole house home AC units and electric cook ranges aren't practical without a very large system and the power stations are not that.
What I was referring to was trying to keep an inverter's 12V-12.5V DC input current within an acceptable range by plugging a battery charger (which is itself, just another transformer) into the same inverter's 120V AC side and attaching it's output leads to the same battery that is presently powering the inverter.....Problem there being that every battery charger I've tried would drain the battery's power faster than it could be restored by the same charger. Maybe that wouldn't be the case if one was using a bank of solar panels instead, I dunno. My gut says that even if it could, it would take an awful lot lot of panels to accomplish that....and what if it rained for two or three days in a row, what then?
In other words, be it a fuel powered generator, electrical inverter (transformer),
thermoelectric generator (
TEG), windmill driven, water fall driven, steam driven, or what have you...they all require some outside form of energy to operate, they all have their own drawbacks, and none that I know of are capable of putting out enough power to run themselves while doing the job they are intended to do.
All in all, I've grown fond of the 12V-DC to 120V-AC inverter concept. Like I said in an earlier post, I can rig one of those up for about $700 and it'll do most of everything I need done during a power outage, including washing clothes, heat and/or A/C. Of course, that is because when I embarked on that adventure, I already had a way to keep the dual batteries which run it recharged. Still though, a little 4amp-6amp trickle charger don't cost much.
Sorry for being so long winded.
DGW