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."
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.
And it turns out that the IC I mentioned at the beginning of all of this does just that. This IC turns the LED on and off somewhere between 50,000 and over 100,000 times a second, making the LED appear solidly ON.
With the QX5252F IC connected as shown, Pin 4 on the IC very rapidly connects and disconnects to Pin 3, the negative on the battery, causing the LED to repeatedly flash similar to the crude "Joule Thief" already mentioned. But here the LED flashes so rapidly that the LED appears completely ON, more so than some florescent lights that have a flicker to them.
The bulky, handwound coil-on-a-nail can be replaced with the tiny inductor shown between the IC and an LED.
Note: If the battery is replaced by a rechargeable battery, a solar cell can be connected to "Unused" Pin 1 and Pin 3 on the IC. During the day, the IC will turn off the LED while also trying to charge the battery, depending on sunlight. During the night, the IC will turn on the LED. But this is another story since this operation takes us away from our "Joule Thief" discussion...