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I'm guessing these batteries are the same thing.

The shake-to-charge flashlights appear to be a long tube with a coil of wire wrapped around it at the middle, with a permanent magnet with soft bumpers able to slide up and down the tube. You can feel the bump when the magnetic field hits the coil; I'm guessing that the rest of the circuitry consists of a bridge rectifier and a capacitor. (And an LED, of course.)

Presumably in the "battery", the LED is replaced with a power supply that can boost 0.8V or whatever up to 1.5V, or reduce 2V down to 1.5V when needed.

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