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How do LED flashlights work with just one battery?

They use boost converters. There are many ways to do this.

LED flashlight "bulbs" with boost converters built in are available - check out Craig Johnson"s Night Pearl Bulb Page.

UPDATE 8/10/2003 - A higher power LED flashlight bulb that supposedly accepts an amazing range of voltages is now available. Craig Johnson has a web page on this one here.

One old traditional way of powering LEDs from a single 1.5 volt cell was to use National Semiconductor"s LM3909 LED flasher IC. One can pulse the LED fast enough to appear continuously on, or one can add a diode and a filter capacitor. However, the LM3909 is generally not the best way and will not boost 1.5 volts to a voltage that will power blue, white, or non-yellowish green LEDs. Adding a diode and a filter capacitor is recommended if you use this to boost 3 volts for blue, white or non-yellowish green LEDs, since those LEDs are generally more efficient with steady current than with pulsed current.
UPDATE 3/24/2008 - The LM3909 is discontinued, but there is now a published LM3909 emulator circuit at Red Circuits.

Another IC that looks better for this is the Texas Instruments TPS61010DGS.

UPDATE 3/26/2009: Boost converter Ics for powering white LEDs from 1.5 volts and similar low voltages are now increasingly available. In addition, there are boost converter Ics designed with regulation circuitry, but that can be used to power white LEDs from 1.5 volts. Some are known as "synchronous" converters, for having switching transistors passing current that would otherwise flow through lossier diodes.