The
Wikipedia article (almost cited by
@KJ6EAD) provides a "beginners" explanation of bipolar junction transistors, or BJTs. The starting equation for BJT design is the Shockley Diode Equation which led to the Ebers-Moll model. As
@Merlin3189 implies in his post #9 above, this is only an approximation to reality. A better approximation is the Gummel-Poon model mentioned by
@LvW.
As computers become ever more powerful, we will eventually reach the point where an almost exact quantum-mechanical model of any semiconductor device can be produced. We aren't quite there yet, but any model (however accurate it may be) is only a model, not reality. The "map is not the territory" is an old saying that is appropriate here.
I think most of the "controversy" of current-control
versus voltage-control revolves around cause-and-effect arguments, which may be non-productive to getting things accomplished at the beginner level.
It is certainly true that NO collector current (except reverse leakage current) flows through the reverse-biased base-collector junction if the base-emitter junction of a BJT is shorted. That is, zero base-emitter voltage and zero base-emitter current results in no collector current.
And it is certainly true that if there is an
electrical field applied to forward-bias the base-emitter junction, with a reverse-bias field applied to the base-collector junction, then current will flow in the collector that is proportional to the base-emitter junction voltage. This is cause and effect: the electrical field in the base-emitter junction
causes the collector current
effect in the base-collector junction.
Perhaps it is not so obvious that it is base-emitter voltage that produces an electrical field across the junction between the base area and emitter area. The junction between base and emitter is very thin, so a small voltage can produce a very large electrical field. It is this electrical field that cause carriers to migrate from the heavily doped emitter region into the base region. Without the field, some do wander into the base region by thermal motion, where they neutralize oppositely charged carriers, creating a depletion region whose thickness is then diminished by a forward biasing electric field.
What happens as a result of this migration of carriers from the emitter is complicated, but the end result is collector current that now flows through the reverse-biased base-collector junction. The fact that there is a base current is almost incidental. It is the base-emitter voltage causing an electrical field to be produced across the base-emitter junction that is responsible for the collector current.
Externally, the base-emitter voltage is almost always a consequence of a voltage applied to a resistor in series with the forward-biased base-emitter junction, which voltage then causes a base current, Ib, to flow as approximated by the Shockley Diode Equation. Experimentally, this base current is related to the collector current, Ic, by a "gain" factor, beta, so Ic is approximately equal to Ib times beta. This is a "good enuf" explanation for simple circuit design, but it doesn't accurately describe everything that is going on. No model, so far, accurately describes
everything that goes on in a BJT.
Consider for example a Darlington-connected pair of transistors, or a Sziklai Darlington-connected pair of complementary NPN and PNP transistors. It appears that current in the base of the first transistor is multiplied by its beta to cause a much larger base current in the second transistor, which then multiplies that base current by its own beta to create an overall collector current of Ib x beta1 x beta2. It is circuits like these that perpetuate the "myth" that transistors are current-controlled devices.
My advice? Go buy a few hundred 2N3904 (NPN) and 2N3906 (PNP) small-signal transistors. These are really inexpensive, a few cents each, so you can afford to "blow up" or let the "magic smoke" out of few while learning how they work. Also buy a bag of 1/4-watt resistors in assorted values, a solderless prototyping board, two 9 V "transistor radio" batteries, and snap-on leads for the batteries. Buy a decent 3-1/2 digit multimeter so you can measure electrical things: voltage, current, resistance. Now sit down and build stuff from simple circuits you can find on-line.
Oh, most people also throw in a few cheap LEDs (Light Emitting Diodes) to the basic experimenter's kit since these give immediate visual feedback when you do something right. You can add stuff, like mechanical switches or relays, to the kit as you progress in your learning. A few low-power LM555-type timers are fun to play with. Eventually you will want your very own line-operated bench power supply. Come back here and ask questions, but do some web surfing yourself to answer most questions. Google is your friend.