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The charging curve for Li-Ion batteries (explained here) has a final constant voltage stage where the charge voltage plateaus and the charge controller slowly shuts off the current until it reaches some threshold (10% in the video's example) and charger considers the battery full.

I can take a battery and measure the voltage. But here the voltage is constant, so how does the charge controller know when to cut off the current?

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Current flows into the battery from the charger regardless of the battery voltage measured since this voltage is artificially depressed due to the internal resistance of the battery. The battery voltage therefore is not an indicator of when the battery is fully charged.

The current flowing into the battery is measured via a current shunt resistor; when the current flowing into the battery falls below a programmed limit the charger cuts it off completely and only then is the battery considered fully charged.

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