The basic idea is shown in the image above. Power from a USB breakout board is supplied to a supercapacitor, which in turn supplies an LED. Note that you'll need the appropriate resistors here to prevent too much current from entering or exiting the 'cap.
See below for the build story, including a second version toward the end where I use a constant voltage driver to keep the LED bright until the supercapacitor discharges. This does work, but severely limits the lighting time. Also, you can find a bit more discussion about this project and supercapacitors in general here. Note that the linked article was written before I tried the constant voltage supply.




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