MrAl,
I hope what you are saying is just a point for clarification, and not a reason for saying that causality cannot be determined. Yes, charge flow requires a voltage, and energy is exchanged and transferred when current exists. Voltage can be generated by the separation of charges like the Wimshurt machine does, or by magnetic means the way a dam generator works. That is not important.
What we are talking about is not about the cause and effect of current and voltage in a conductor. We are talking about what is the most basic control element in a BJT. I maintain that it is Vbe because both Ic,Ie,and Ib are intimately tied to it. It does not matter whether Vbe or Ib happen before or after eventwise, or whether the source supplying Vbe has to be able to supply the energy. It does not matter what occurs in Quantumland. The causality is what is really controlling the BJT, and that is Vbe. I will reiterate again that controlling Ib is the functional control of the transtor, and should always be used for design and calculation. But "under the hood" it is really Vbe that is calling the shots.
I illustrated my point earlier with a helmsman and rudder. The helmsman has functional control of the boat, but it is really the rudder that causes the boat to turn. That is what I mean by causality.
Ratch
But how does Vbe relate to being "the rudder". Vbe exists because p-n jcns are not ideal. An ideal revtifying jcn would be open in reverse, short in forward direction. If a semiconductor w/ zero bandgap energy existed, charges entering the emitter would transit through the base onward to the collector. The no. of e- collected is controlled by the no. of e- emitted. After Ib/Ie are changed by Sue, Ic increases as a result, but a little later, Vbe increases. But Vbe increasing is not the control mechanisn. In order to supply more electrons for the emitter to emit, work must be done. When Sue cranks it up, it is the increased current/voltage from the mic resposible for Ie/Ib increasing. As a consequence, Vbe increases after the fact, but Ic has already begun to increase w/o Vbe increasing yet.
The intermediate variable at the jcn is Ie. Although Ib & Vbe change as well, only Ie contributes to Ic. Ib & Vbe are necessary because perfect semiconductors cannot be produced. The Ie is not controlled by the Vbe. The eqns you keep referencing are functional relations, not causal.
Start w/ the n-p-n as a pair of back to back diodes. The upper diode is the b-c jcn, described by Shockley's diode eqn
Ic = Ics*exp((Vbc/Vt)-1). Since Vbc is negative, a very small Ic exists, leakage of the reverse biased p-n b-c jcn.
The lower p-n jcn, b-e is forward biased. So we have
Ie = Ies*exp(Vbe/Vt)-1), a large current since Vbe is positive.
If the base region was very wide, say 1.0 mm, the above relations are what we get. Ie = Ib, a large value, w/ Ic being a small leakage.
But if the base region is made ultra thin, say 1.0 um, we find that Ic measures almost as large as Ie, w/ Ib being quite small, Ib = Ie-Ic. Instead of 2 mere series diodes, we have
transistor action. The b-c jcn is reverse biased, but Ic is almost Ie. The Shockley eqn needs another term. The additional current is Ic = alpha*Ie. So we get
Ic = alpha*Ies*exp((Vbe/Vt)-1) + Ics*exp((Vbc/Vt)-1).
But sonce the 2nd term is much smaller than the 1st, we round off & use Ic = alpha*Ies*exp((Vbe/Vt)-1). Also, alpha is very near to unity, so some texts omit alpha, w/ the resulting eror being 1 to 2%.
The Vbe eqn, which I've called eqn 2), is derived from Shockley's diode eqn, plus the eqn 3), the transistor action eqn. Ie & Vbe do have a functional relationship per eqn 2). But Vbe is directly related to Ib & Ie, but indirectly related to Ic. Ie has a direct relation w/ Ic per eqn 3).
In order for Ic to change, all that is needed is a change in Ie. In the process, Ib & Vbe change as well, but that is not what changes Ic. Ic starts increasing prior to Vbe increasing, & Vbe continues to increase after Ic has settled.
No brainer at all. Transistor action is Ic = alpha*Ie. Any test w/ instruments will affirm. A good software simulator might show this, but the models have to be really accurate. Low end packages won't work.