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I scavenged a li-ion battery from a broken laptop. It is made of 6 Samsung ICR18650 cells (3.75V 2800mAh), parallels two by two so that the total pack is in fact the sum of 3 cells at 3.75V, so 11.25V (it's written 11.1V on the pack).

I have a raspberry pi which can run out of only one power boosted cell to 5V.
But I also have a TFT screen which runs on 7-12V (consumes less battery near 12V than near 7V).

I would like to know if I can actually plug the screen to the full battery, and the raspi to only one third of it.
I know cells have some kind of "balance" so I am wondering if that could make the battery unstable as one part of it would discharge at a faster rate.

As another solution to my problem, would it be possible to powerboost 5V to 7V or 12V so that everything runs from the same number of cells ?

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    \$\begingroup\$ It would be much easier to buck 11V-ish to 5V. Usually easier to get more efficient that way around too. If you want a long life time you need to have per cell protection, like the laptop battery had/has. \$\endgroup\$ – Asmyldof Jul 4 '15 at 17:47
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No: you have to treat all cells of a LiIon battery alike, or you will overdischarge or overcharge some cells. Yes, there are "balance" chargers which handle small discrepancies in the capacity or charge level of the different cells, but that's meant to deal with naturally-occurring differences, not deliberate differences in charge or discharge rate.

@Asmyldof is right: you should best use either a buck-boost converter to convert the full voltage of the battery pack to the 5V needed for your Raspberry Pi.

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You could use the battery to power the screen directly and use a regulator to power the Pi

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