Transistor and LED Tester Circuit

The circuit looks to be very simple but it uses an air-cored transformer to produce the voltage needed to illuminate the LED indicators and the circuit only works when the transistor is connected correctly. There are two separate circuits, one for NPN transistors and one for PNP transistors. We will cover the NPN section:
The circuit turns ON when the NPN transistor is fitted and the current through the 30 turn coil and 1k5 resistor turns ON the transistor and produces expanding flux in the 40 turn coil. This flux cuts the turns of the 30 turn coil and produces a voltage in the coil that adds to the supply voltage and increases the current into the base. This turns the NPN transistor ON more. This action continues until the transistor is fully turned ON. At this point the current in the 40 turn coil is a maximum but it is not expanding flux and the 30 turn coil ceases to see the extra voltage. Thus the current into the base reduces and this turns the transistor OFF slightly. The flux produced by the 40 turn coil now becomes collapsing (or reducing ) flux and it produces a voltage in the opposite direction to greatly reduce the current into the base. In a very short period of time the transistor becomes TURNED OFF and it is effectively removed from the circuit. The flux in the 40 turn coil collapses quickly and it produces a voltage in the 40 turn coil that is higher than the supply voltage and is in the opposite direction. This means the voltage produced by the 40 turns ADDS to the supply voltage and is delivered to the LEDs to illuminate them.

FLYBACK
The circuit is called a FLYBACK OSCILLATOR and is characterised by the fact that the transistor turns off abruptly while the transformer is passing a high current. This current is producing a high flux density and the abrupt ceasing of the current causes the flux to collapse and cut the turns of the wire (all the turns of all the windings) to produce a voltage in the opposite direction. Because the flux collapses very quickly, the voltage produced is higher than the supply voltage and this voltage is added to the supply voltage to produce a result as high as 10v or more. However, as soon as the voltage reaches 1.7v + 2.3v = 4v, the red and green LED illuminate and all the energy from the collapsing magnetic field goes into illuminating the two LEDs.
For the FLYBACK Oscillator to work, the feedback winding (connected to the base) must be connected so that the voltage (and current) it produces (when the transistor is turning ON), will ADD to the voltage supplied by the 1k5 resistor, to turn the transistor ON harder. It keeps doing this until the transistor is fully saturated.
During this "turn-on" period, the primary winding (connected to the collector) is producing EXPANDING FLUX and this flux is cutting the turns of the feedback winding and the voltage across the primary winding (about 1v) is reflected in (passed to) the turns of the feedback winding according to the ratio of the turns. If 1v is across 40 turns, the feedback winding will see 1/40 x 30 = 0.75v. There is no "flyback" or "high-voltage" produced during this part of the cycle. The voltage impressed (delivered to) the feedback winding is called "transformer action" and is determined by the ratio of the turns and the voltage on the primary winding. The 0.75v is added to the 1.5v delivered by the 1k5 resistor to produce a higher current into the base of the transistor and this causes it to turn-ON more. The base of the transistor never rises above 0.7v and this extra voltage supplied by the feedback winding is converted to current to produce this part of the cycle.
Understanding how the transformer works is a very important concept. Normally a flyback transformer is wound on a ferrite core, however air has a permeability of 1 and will transfer energy via magnetic flux from one coil to another at low flux densities. If the density is too high, air becomes saturated and most of the flux will not be transferred, but if the density is low (as in our case) the transfer will take place and a high voltage will be produced.
You have to "sit down" and study "transformer action" for hours, until you understand how the transformer works, because we are talking about energy being converted from low-voltage, high-current to high-voltage, low-current. We also talk about a collapsing magnetic field producing a voltage in the opposite direction. This is one of the magic secrets of how a transformer works when it is in fly-back mode.
We also talk about the primary producing a high voltage and illuminating one or more LEDs. When this happens, the primary voltage is "dampened" or "restricted" or "limited" and this will have an effect on the peak voltage delivered by the feedback winding. However the feedback winding is also restricted by the fact that is has a load connected to it in the form of the base connection of a transistor. This means the feedback winding will not produce a high voltage during the fly-back portion of the cycle but the voltage will be high enough to completely turn the transistor OFF.
Now we come to the theory behind the number of turns on the primary and feedback windings.
The actual resistance of the primary winding is only a few ohms and if the wire is simply a straight piece of wire, the circuit will take a very high current and not work AT ALL.
The only reason why it works is due to the fact that the wire is wound in a coil.
When it is wound in a coil, the current passing through each turn cuts the other turns and creates an opposing voltage. This voltage makes it difficult for the applied voltage to cause a hgi current to flow and the current is considerably reduced. It's only when the transistor is fully turned ON that the current is no longer increasing and the magnetic flux is no longer expanding. At this point the current is a maximum by the flux is no longer cutting the turns of the feedback winding and the transistor starts to turn OFF.
This action happens very fast and the transistor effectively "comes out of circuit" and the energy generated by the collapsing flux creates a high voltage. This voltage is sufficient to illuminate 1, 2 or more LEDs. 

As you can see, the circuit is much more complex than first meets the eye and only after about 100 hours of visually conceiving how the circuit works, will you be able to see the circuit in action "in your mind."