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In npn Ic is related to Vbe in a exponential way, Vbe positive In pnp Ic is related to Vbe in a exponential way, Vbe negative I BPJ transistor symbols the arrow on the emitter indicated direction of positive convention current flow, which is also pointed to the n-type material.
In both NPN and PNP transistors collector to emitter current is controlled by the current flowing into (for NPN) or out of (for PNP) the BASE-EMITTER junction. The current IS NOT controlled by the voltage across the base-emitter junction. Because voltage is flowing between the base and the emitter when the transistor is turned on, a voltage develops across the base-emitter junction. But that voltage is a result of the current flowing in the base-emitter junction, not the other way around.
A question in semantics Base current is related to the voltage between the Base and Emitter in a natural exponential way. And the Collector current is hFE, Beta, times the Base current as long as the Base current is much greater than the leakage, saturation, current. Generally, as in complimentary transistors silicon transistors 2N3904; 2N3906, VBE is between .3 and .9 volts giving a appropriate Base current but hFE varies within these ranges. When designing on uses an appropriate VBE, IB, hFE
With all due respect, and I certainly mean no disrespect, I think it is more than a question of semantics. There is, however, one particular practical application exception to this, which I will discuss below. But for a clear understanding of the operation of bipolar transistors it seems important to realize that a bipolar transistor is a current operated device. Certainly some voltage must be applied to Vbe to cause base current to flow. But it is that base current that controls the operation of the bipolar transistor, as illustrated by the key spec hFe and by the actual surface effect physics of the semiconductor itself. Vbe is more of an effect that a cause. So my concern is that your explanation may lead the student to erroneously analyze a bipolar transistor as a voltage mode device rather than the current mode device that it is. A MOSFET, on the other hand, IS a voltage controlled device. An easy test of this is to take multiple samples of the same transistor, say your 2N3904. Choose ones that are carefully selected to have the same hFe. Then apply exactly the same voltage to Vbe to each of the devices. You will find a very wide range of collector currents result. The device is controlled by the base current, not Vbe. The one particular application exception I mentioned is a brilliant and very famous one: the current mirror. Bob Widler (when at National Semiconductor) realized that two identical bipolar transistors fabricated right with a common base (located right next to each other on an integrated circuit) would have identical characteristics. Looking for ways in which this might be useful, he brilliantly invented the first current mirror. By causing the same Vbe across both devices (by applying a known current to the base of one of them and connecting that collector to the base) he discovered that the same collector current would flow in both devices. Thus was born the current mirror which changed the world of integrated analog circuitry dramatically and gave rise (among many other things) to the first low leakage, low input bias, and high input impedance integrated op amps.
I've studied the Widler and the Wilson current sources, mirrors, quit extensively. The fact that you say the Widler can be analyzed using VBE on a IC insinuated it really can be done discretely. When you say 2 2N3904 with the same hFE could have different IB, IC, with the same VBE insinuates to me other possible problems. Maybe its the difference in Base spreading resistance, rbb' or emitter resistance, re'e, or the procedure in measuring hFE wasn't that accurate. The question may come down to do you measure hFE at a specific IB or a specific VBE, ideally they should be the same, in the discrete world I would choose VBE because this would be used to set bias.
I didn't mean to be misleading with my comments about matched discrete transistors. I don't actually think you can make practical high-precision Widlar style current mirrors using discrete transistors. Matching discrete transistors well enough would be far too expensive and they could never be mounted in such a way as to prevent some temperature gradient between them. Don't get me wrong, I assume you can make a useful current mirror with discrete parts but you could never (IMHO) make one good enough to make a discrete amplifier with (untrimmed) input characteristics that can hold a candle to (even cheap or old) integrated op amps - even old ones like the LM101. I have vivid memories of Bob arguing with me about all the reasons the two transistors must be fabricated as one device. I admit to being highly influenced by that and I have not actually tried to make a good current mirror with discrete parts. So I could be wrong. I assume you can make a great one using an op amp and feedback. But, like I say, I haven't tried it myself. Fun chatting.
I've designed a jig to measure IC, VBE, and hFE, with IC equal to 10ma, peak value for hFE in 2N3904, to measure hFE within +-2%. The typical stock has a hFE = 200 +- 50%. The reason VBE must be measured is to account for variances in saturation current, IS. Initially I developed it to sell select transistors but I decided not to, but during the process I gain a great understanding of the practical measurements of a transistor. The jig actually uses a Wilson current mirror with emitter resistors to measure Base current and determine hFE. I choose +-2% to simulate IC's values. I've also generated equation for hFE at different junction temperatures and variations in VCE which requires that I must also measure the early voltage.