I was wondering whether it would be possible to use a rechargeable battery from an ordinary cell phone to power some electronics such as a motor shield for Arduino? Can that be done and if yes then how?
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It is very unlikely that a cellphone battery can power a motor shield, assuming motors will be actually operated off it. Motors typically draw large currents, and while modern Lithium-Ion batteries have fairly high current rating, they are designed to supply just enough for a small portable device.
There are however very low-current motors available, such as pager motors, which can be run from a cell phone battery.
If the objective is only to run an Arduino board and conceivably a shield, but not operate a motor, things begin to look more positive.
Success would depend on the specifics of the cell phone battery, but in general:
How this can be done:
If the devices to be operated do not support 3.x Volt supplies, a boost regulator circuit can be used to boost the battery voltage to 5 Volts, with 75 to 90% efficiency.
This will work, but the current delivery capability of the boost regulator will be lower than that of the battery alone.