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"As previously stated, if you are connecting a charger to your battery put a voltmeter on it and monitor what it is doing. Do not leave a charger on a battery unattended."
In a perfect world that's good advice. If a person IS going to do it though - for instance because it's minus 10 degrees in the hangar for weeks at a time or for whatever reason people think putting a maintainer on the battery is necessary, make sure it's a good one. Sears/Wally-World stuff is not anything to leave on a battery for longer than it takes to bring it to a full charge - around 13 volts including the surface charge.
The little "$6.99 on sale" battery maintainers they sell at Harbor Freight are actually better for long term use than anything with the name Schumacher, et. al. on them. It's foolish perhaps, since it's possible they could short out or do something weird, but at least they will bring the battery up to 13.1 and keep it there. They won't overcharge - which is what the other ones do that I've mentioned. It's the overcharging that overheats the batteries, boils out the fluid, warps the plates, and turns them into junk.
I'm guessing the Battery Tender brand is okay. I know for sure the Pro-Logic charger/maintainer I use is safe. Safer still though is to not leave anything connected to the battery but in really cold climates you pretty much have to. Freezing temperatures for extended periods is not good for batteries, as I understand things. Probably better to take the battery out in that situation and take it home. Keep it warm and charge it once a month.
I have an 55 Ah AGM in my plane. They like a very slow charge. Flooded cell, AGM, Gel-cell and Li-Ion all have different charge needs . . . which is why I like the Pro-Logix intelligent charger. It handles each type differently. Mine is the small one, PL-2140 and works beautifully. I keep an eye on it with a voltmeter and have learned to trust it so far. Pro-Logix makes larger ones but the small one has been fine for the plane, motorhome, bike and cars. If left on for really extended periods - over a week - they'll shut down and wait until the battery voltage drops to a certain point before coming back on. They will also do a de-sulfating mode if left on for weeks at a time - in other words, after about a month or more of being left connected to a battery, they will draw current down on the battery then take the charge up to a high number to knock the sulphate off the plates, then bring the voltage down to a maintenance level then shut off again.
That little cheapo from Harbor Freight has also been good over the years. It never charges higher than 13.1. It doesn't do any of the fancy stuff like I just mentioned but it will keep a battery at 13.1 and never take it higher. I used one for years and it never did anything weird. The battery I used it on lasted almost 8 years. $6.99 on sale. They are not for use with a badly discharged battery. The battery needs to have at least 11 volts in it before connecting one of these HF maintainers. If you're careful about that, these little inexpensive things are fine and will keep your battery healthy for many years . . . that's been my experience.
If you have something with the name Schumacher on it, or one with a different name that is probably made by the same company, don't trust them as I've found they overcharge, overheat, and destroy batteries in short order, no matter what they say in the book that came with it. Use a voltmeter on them and you'll see.
I agree it's not good to play with your electronics with a charger hooked up to your battery. I've read cautions against doing that. I don't know enough about how it all fits together to know why it's not a good idea, but I've read it's not, so I don't do it. I'll just use the ships power if the battery is at a full charge. Once the electronics are turned off, I'll put the maintainer or charger back on the battery to bring it back up to full.
Here's the one I use. Pro-Logix have higher amperage maintainer/chargers for about the same price but this one has been fine for everything I've needed it for.
http://www.amazon.com/SOLAR-PL2140-P.../dp/B00U3ZBU2E
It was recommended to me by a battery expert so I took the advice and it's been good.