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Lithium batteries start off at about 4.2v and drop to 3.0v when empty with 3.6-3.7v for most of the time in the middle. AA batteries start at 1.6v and drop to about 1.0v, so three of them is vaguely equivalent to a lithium.
White LEDs need about 2.9-3.0 volts, and there'll be a resistor (or sometimes a proper driver circuit) between the batteries and the LED to cover the difference in voltage. Some torches don't have an actual resistor and rely on the resistance of the connections and the voltage of the battery sagging under load.
I suspect your small torch has a resistor and if you ran in on a lithium it would work but be rather dim. If you can take it apart and change the resistor to a lower value it should work fine. I suspect they decided to use four button cells rather than three because the voltage sags a lot at high current.
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grams
I've got a work torch which has a little 2nd torch at the end you pull out. It uses four LR44 AAA button cells in a sheath. I believe they are 1.5V each. It's a bit dim and is running out, so they need replacing. But have not been used much, so I assume will just run out quickly again.
Can I swap these out for something like a 10180?
It seems to be close enough in dimensions and should last longer as well as being rechargeable, right?
I have no real knowledge of batteries, but doesn't 4 (stacked) x 1.5v = 6v? So if a 10180 is 3.6V then it might not be powerful enough.
The thing that prompted me was that my car torch can take either AA or (I think) 18650, which makes the light more powerful.