I think that's the part that bibilit had in mind. This is a very common problem with these (and other similar) chargers, apparently.

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Yes, if you are comfortable with a soldering iron, this is pretty easy to fix, you can find several links to do so.

I rebuilt one of mine, and it worked well for awhile. Until I left it unused for a month or two. Back to square one...

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As far as i know, the battery will only be charged if the battery is showing a minimum amount of voltage.