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This might be a dumb question but:

If you have a Li-ion with a built-in protection circuit, and you discharge it until it cuts out (3V?), can you still recharge it? Is the protection circuit to protect the battery to improve it's life, or just to protect it from leaking/exploding?

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The protection circuitry disables the cell output, to protect it from deep discharge, which is damaging to Li-Ion chemistry. If you start charging it, and if the cell is not completely dead, it should be fine.

BTW, if you discharge the cell below protection threshold, and then disconnect any load, a normal healthy cell will re-bound by itself to some degree, slightly above the cut-off threshold.

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Yes, you can still recharge it. The point of the circuit is to protect from excessive discharge which shortens battery life at no benefits (since it does not provide enough current under 3V for most things to still function). Unprotected cells will not leak, but can get very hot (up to starting a fire) if a larger than supported current is drained from them. As for the overheat protection, most circuits do not have such protection (no way to limit current), but there is one by design for the high power cells used in power tools to prevent accidents due to short-circuits (those type of cells have a fuse which will burn-out in case of over-current).

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