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I would like to know if a 3.7 V external Battery would be able to charge a phone battery of the same voltage. From my understanding, the batteries would charge each other until the charge in both the batteries is equal. Is this True?

Let's say the external battery has a capacity of 500 mAh, while the phone's battery has a capacity of 1000 mAh. Contingent on the prior statement being true, would it be right to say that in an ideal case, the external battery will be incapable of charging the phone once both the batteries reach a capacity of 250 mAh?

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Current will flow between the batteries based on the difference in terminal voltages, not based on the difference in State Of Charge (SOC), and certainly not based on absolute capacity (equal mAh remaining). The terminal voltage vs. SOC curve is highly dependent on the chemistry and manufacturer of the cell, so don't assume that two cells will behave the same even if they have the same nominal voltage. A typical voltage vs. SOC curve for a Lithium ion Polymer cell looks like this:

open circuit voltage vs. SOC

In short, while you might succeed in moving a little charge from a full cell to an empty cell, current will most likely stop flowing between the cells well before they have equalized SOC, and they'll never end up a the same amount of stored energy remaining given that they're different sizes.

If you want to use one battery to charge another battery, consider a boost regulator in conjunction with a charge management IC. In your application, you would use a boost regulator to get to 5V from your external battery, then use your normal USB charger to leverage the existing charge control circuitry for you phone. You won't even void your warranty!

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There has to be voltage difference for current to flow, once the phone battery reaches the same voltage as external battery, the charge will stop.

One the side note, from the fact, that you are using 3.7V batteries i imply, those are using Lithium - based batteries, be aware that you have to limit current, otherwise you can damage phone battery by pushing too much current into it.

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