"Many errors, of a truth, consist merely in the application of the wrong names of things." - Spinoza
I think, our discussion has reached a point where two terms need some more clarification - otherwise, misunderstandings cannot be avoided.
1.) What means "control"? I think, we have two alternatives:
Imagine that the BJT is within a black box and I only have access to the three terminals (C, B, E).
We will observe that a current "injection" into the B node (read point 2 for "injection") will result in an Ic increase. I think, in this case and without knowing what happens within the box, it is correct to say that the input current controls the output current (from the practical point of view).
However, with some knowledge about transistor physics we can be more precise and say "in fact, it is the voltage across the B-E path that controls the current Ic.".
Now - the question arises where this controlling voltage comes from (see next point).
2.) Does a current IB really "produce" a voltage drop VBE? I don`t think so.
I am aware that we all speak about "voltage drops" and "current injection" - but these terms are closely related to the question "do current sources exist ?". My answer is : NO.
Each technical device we call "current source" is in reality a voltage source with an internal source resistance that is much larger than the load resistance.
Basic law: No current without a driving voltage!
From this it follows that a so called "current source" feeding a current into a load is nothing else than a simple voltage driven resistive voltage divider.
Hence, it is not the current which "produces" a voltage at the load resistor. Instead, we have a current - determined by the sum of both resistances - and a voltage division according to the resistor ratios.
As a result - physically spoken, it is not correct to say that an "injected" current IB would "control" the voltage VBE.
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I hope that some of these theoretical considerations can help to avoid misunderstandings while using the terms "control", "current injection", "current source".