High  efficiency white LEDs have advanced to the point where they can replace  glow bulbs and other light sources not only as indicators, but also for  illumination. While many of the claims made about the LEDs efficiency,  light quality, lifetime and economy are mostly exaggeration, the truth  is that for very low light levels they are now competitive. They have  equal or slightly higher efficiency than a flashlight bulb, a longer  lifetime, and are very much tougher. On the other hand, they are still  far more expensive than a bulb, for a given light output.

It follows that LEDs are almost ideal for very tiny, low power  flashlights, in the less-than-one-watt category. But such a low power  flashlight makes sense only if the whole flashlight is small and  lightweight, and has a reasonable battery lifetime. But white LEDs  require about 3.3 volts each, and typically some extra voltage is needed  to provide room for current regulation! Thats why most commercial LED  flashlights use at least three alkaline or NiMH cells, or a lithium  cell. And often they cant use their batteries all the way down to the  true end of their charges!
Using three AA cells isnt really practical for a small flashlight,  simply because it will no longer be small! Lithium cells are expensive.  So some manufacturers use three button cells, but these last only for  minutes and are also expensive compared to their tiny energy contents!  So I set out to build a circuit that lights a string of white LEDs,  using a single alkaline or NiMH cell. That allows using the widely  available and inexpensive AA cell, obtaining a small size, low cost and  good runtime. 
A typical white LED has its best power-efficiency combination at about  20mA, and needs about 3.3V. This makes for a power of about 66mW per  LED. I decided to use seven LEDs, because they can be arranged in a nice  and compact way with one in the middle and the other six around, and  the whole array runs at close to one half watt, which is a reasonable  power for a tiny pocket flashlight. To avoid having to control the  current separately for each LED, the LEDs were arranged in series. So, I  needed a driver circuit that will provide about 23V at 20mA, when fed  from a 1.2V NiMH rechargeable cell  or from a 1.5V alkaline cell. It  should be ultra simple, low cost, efficient and reliable. And here it  is! 
The circuit is a self-oscillating boost converter, and I certainly  cannot claim having invented it. It is ages old! I only did the detail  design of this one, and optimized it in the course of one afternoon. It  runs with a beautifully clean waveform, with all components except the  LEDs staying completely cold to the touch. At this low power level, even  that doesnt guarantee a good efficiency, but I measured it at about  72%, which is quite good for a circuit operating from such a low  voltage!
How it works: When switching it on, R1 and D1 bias the transistor into the linear  range, through the feedback winding on T1. That causes a current through  the 18 turn winding, and thanks to the positive feedback the transistor  is driven into saturation. At this moment there will be a base current  defined like this: The 1.2V of the cell, plus the 0.2V induced in the  feedback winding, minus the 0.7V base-emitter drop of the transistor,  make a total of 0.7V, which applied to the 22 ohm resistor gives about  32mA base current. D1 is not conducting a significant current at this  time, because the transistor clamps the base voltage to 0.7V and the 3  turn winding subtracts 0.2V from this, so that we end up with only 0.5V  across the diode.
This base current keeps the transistor in saturation until its collector  current reaches approximately 1A, while the transformer loads up. At  this point the transistor will start getting out of saturation, which  makes the feedback voltage drop. This very quickly puts the transistor  into blockage. The collector voltage will soar as T1 forces current to  keep flowing, until D2 starts conducting and discharges the transformer  into C2, by means of a quite narrow pulse. During operation this pulse  is about 24V high, so that the feedback winding develops -4V, which  results in applying about -3.3V to Q1s base, enough to switch it off  very fast, but not enough to make the base reverse-conduct.
As soon as the transformer has fully discharged into C2, the voltage on  it breaks down, and the transistor enters conduction to start a new  cycle. The oscillating frequency is 30kHz, and the transformer operates  at a peak flux density of 0.1 tesla, far away from saturation, and low  enough to have very low loss. C2 has to eat the load pulses that start  at about 1A, and has to keep the voltage constant enough to feed the  LEDs an almost smooth DC. The value given works well. If anyone wants to  build this circuit to run 24 hours a day for 30 years, it would be good  to pick a capacitor rated for low ESR and a relatively high ripple  current, but for flashlight use a plain standard 47µF, 35V electrolytic  capacitor works great.
C1 is not strictly necessary. With a good NiMH cell, the circuit works  the same without it, so you can save a few cents here. But with the  capacitor in place, the circuit keeps working better when the cell is  almost fully discharged and its internal resistance gets higher, so its  better to include it.
Components:Of course, the part over which most builders will stumble is the  transformer. I used an Amidon EA-77-188 core, because I had it at hand,  and it was the smallest core I had. I should say that this core is still  at least five times larger than required! So feel free to use the  smallest ferrite double-E core you can find, or any other ferrite core  that offers a closed loop and the possibility of assembling it with an  air gap. But then you will have to redo the math!
The main winding has 18 turns, and I wound it with 7 strands of #30  enameled wire twisted together, simply because there is room enough to  do so. But this thick wire bundle is huge overkill, like the whole  transformer is! The feedback winding  was wound with a single strand of  that same #30 wire, and it has just three turns. The phasing is like  shown in the diagram, of course. If you get the phasing wrong, the  circuit wont work and the transistor will get warm.
I used masking tape to hold the windings in place on the bobbin. No  special insulation is required, because the voltages are so low that the  enamel on the wire is insulation enough.
Now comes a very important step: This transformer is airgapped. The two  core halves need to be separated by a distance of 0.1mm. I simply stuck  little pieces of masking tape on the three legs of one core half, taking  advantage of the fact that my masking tape is just the right thickness!  Then I assembled the core, wrapping masking tape around it to hold it  together.
If you have to use a different ferrite core, you can use my transformers  and coils article to learn how to design your transformer. The turns  ratio will of course remain 6:1, but the absolute number of turns will  change in inverse proportion  to the cores cross section. You can look  up the data of my core on Amidons or Bytemarks websites, compare that  to the data for your core, and go from there. After calculating the  turns numbers, you have to calculate the required air gap to obtain an  inductance of the main winding of about 40µH.
The transistor I used, the 2SC1226A, is a pretty old part and may no  longer be available. I have a bunch of them, so I used it. It has a  soft, thin copper tab which can easily be cut off, which is an advantage  in this circuit, because it allows saving some space! The transistor  works cold, so it doesnt really need the tab! If you have to use  another transistor instead, feel free, but look for one which has the  proper characteristics: It should have a breakdown voltage of about 40V,  a maximum continuous current of about 3A, be reasonably fast (mine is  very fast, having an Ft of 150MHz!), it should have good saturation  characteristics, and it should have a reasonably high hfe (at least 30,  ideally about 100) at a current of 1A.
Any different transistor will most likely require a change in the value  of R1, to set the proper power level for the LEDs. You can  experimentally determine that resistor value, by placing a  milliamperemeter in series with the LED string, and selecting the  resistor for 20mA in the LEDs. By the way, if you want to build this  circuit for an alkaline cell instead of a NiMH cell, the resistor should  be a bit higher. D2 is a Schottky rectifier. A non-Schottky ultrafast  diode could be used too, but the Schottky is better. D1 instead is any  plain simple silicon diode.
If your power switch doesnt have very low resistance, it might cause a  significant loss in this low voltage circuit! If that happens, you could  instead place the power switch in series with R1, leaving the rest of  the circuit permanently energized. That will cost almost no lost battery  power, because the only current drain when off will be the leakage  through the parts, which should be in the microampere range. But if you  place the switch at R1, you should also place a 1 megaohm resistor (or  almost any other high value) in parallel with D1, to make sure that the  transistor really does stay fully off when it should! 
 
 
Source: Humo Ludens