I too have a burning thirst for knowledge for Flashlight technology I was given some cheap 2cell flash llights and some of them didn't work. so One day I took it apart to see why it didn't work and if I could fix it. Cause that's what I do.
I made a schimatic of the circuit and its really simple, though it took me several weeks and many days on the compute to find a device that would do
what this schimatic shows. keep in mind all I had was the PCB, I could measure the inductor and capacitor. a series resistor told me the forward voltage drop of the diode, but nothing on the device
so after several weeks and being boared on weekends I tried to find a jfet that might work. I did the best I could w limited knowledge and selected this fet. MMBF4391L not being able to determine what would work I ordered and installed into the circut.
it really works but its dim. the fet only allows about 6ma to the diode, but it did work. I was elated, for awhile. soon I wanted to get the current up to make it as bright as the flashlights that did work.

so, My question to the group, how would find and select a fet that would allow about 20ma to the diode w the 2.5v source to gate voltage?

for grins and becasue I bought 100 of these jfet, I was able to stack 4 high and solder the pins togather to try and increase the current by paralling fets.
I think it should work, will know in next couple of weeks when I solder them into the smd board


Please let me know what device might work for this application. I have one more broken flash light

 

link1 I am certain the bottom picture does not have a FET in it. There are several companies that make SOT-23-3 LED drivers. Here is an example of one. I used a part PAM2400.... no I can't remember the part number. Diodes corp.


----edited-----
Sorry I did not see this was a very old post.

 

that really looks like it would fit the circuit.
I will see if I can buy some if cheap enough
thanks for the help

 

You could build a low voltage oscillator with a transformer ratio to raise voltage for the high overhead.
The coil winding like on a T50 toroid 1 : 5 ratio to a storage capacitor and big mosfet drives the CC.
Simulation using an arbitrary load Irms 58mA increase the resistor to get 20 mA, decreased frequency to 20 Hz
Then replace the temporary load with the led
7mW on each jfet not bad at all.

TL431 low battery cut off

 

Although I agree that @DickCappels conjecture is likely, that circuit could be a Jewel Thief if the inductor has four pins. It has exactly the right components.

 

Bobs on a roll. Thats true the variants of a blocking generator.
If your browser has the translator as the website is in Romanian, it does touch on sub 1 volt blocking generators.
I am not sure ZXLD probably does uses the term PWM correctly. PWM has grown. The novelty of using jewel thief
was a popular trend when a radio shack sold battery operated toys.

Before the blocking oscillator was used in televisions, Later a prior study done at MIT for Navy was released to public.
I am not aware of anything before that which has sufficient documentation. The solid state versions are numerous and scattered.
Romanian Some Simple LED Power Schemes
Blocking oscillators
Blocking Oscillators: An Introduction Into its Working, Types, and Uses

 

You could build a low voltage oscillator with a transformer ratio to raise voltage for the high overhead.
The coil winding like on a T50 toroid 1 : 5 ratio to a storage capacitor and big mosfet drives the CC.
Simulation using an arbitrary load Irms 58mA increase the resistor to get 20 mA, decreased frequency to 20 Hz
Then replace the temporary load with the led
7mW on each jfet not bad at all.

TL431 low battery cut off

View attachment 335479

keep in mind this is a 2 cell flashlight, which I may not have said, and the photo of the circuit, the inductor is just that. its not a transformer.
but I like the idea.
infact in one of the photos I submitted, I soldered 4 fets in parallel to increase the current . one fet gave me about 8 ma, 4 gave me about 25ma. not sure why it wasn't a staright 4x8 ma, could be I damaged one of the fets.
Thanks

 

Although I agree that @DickCappels conjecture is likely, that circuit could be a Jewel Thief if the inductor has four pins. It has exactly the right components.

Nope its only 2 leads, I really think its similar to the PAM2400 that ronsimpson posted. I have looked up similar parts to the PAM2400, but can't find anyone that really sells them unless you buy 30,000 pieces.