It seems that many "Joule Thief" circuits depend on a clunky (bulky and heavy) toroid or "donut" that has to be carefully wound with copper wire. But now there are several very small 4 legged ICs available that do the job using only a simple inductor, single cell battery and a LED. In effect, the 4 legged IC replaces the clunky toroid.
I came across these ICs when I disassembled some solar powered yard lights. I looked for a toroid but only found a four legged IC and a part that looked like a resistor but actually was a very physically small inductor (coil). Both of these parts along with wire attachment points were soldered to a small circuit board. I was able to remove parts, attach wires to them and assemble them on a Radio Shack type of "Breadboard" to test and better understand this circuit.
But then I created a very crude and minimal circuit to better understand some of the key parts of a "Joule Thief."
I used a Yellow LED that requires 2 volts (or a little more). My 1.5 volt "Rocket Battery" has been worn down to 1.4 volts. As a result, the LED is off and is not even close to conducting any current. Points A and B on the inductor coil L are at pretty much the same voltage, 1.4 volts compared to "ground" or the minus of the battery.
When the switch is pushed and and held ON, briefly, current flows through the coil and creates a magnetic field around the coil. Points A and B are still positive with point A being slightly more positive than point B.
But when the switch is released and turned OFF, the magnetic field suddenly collapses and creates a 1.4 volt voltage with a reverse polarity. This means that point B is now 1.4 volts higher (more positive) than point A. It is as if the coil has become like a temporary battery connected in series with the actual battery, presenting 2.8 volts to the LED. The LED reacts to this by flashing on for a very short moment. Pushing the switch again repeats this cycle. If I could push the switch rapidly enough, the LED would appear to be solidly ON.
The pictures that follow will reveal how simple it would be to recreate this. The coil or inductor is 12 feet of 24 gage wire wrapped (200 turns) around a 1/4 inch diameter soft iron nail.