operational amplifier - How can there even be an input voltage on a long-tailed pair?

Assuming the following:

1. The voltage across the base-emitter junction of a BJT is constant in forward-active and saturation mode


2. The BJTs in a long-tailed pair are normally forward-active or saturated

(If any of these assumptions are incorrect, read not further and correct me)

then how is the difference between V_in⁺ and V_in^- not fixed to the difference between the base-emitter voltage drops of the two transistors? (Zero, if the transistors are identical)

Answer

This is a classic misconception: "V_BE is constant, BJT transistors are current-controlled devices." Nope, wrong.

So, your #1 above is wrong. Long-tailed pairs are based on changes to V_BE. The V_BE becomes the transistor's voltage-input.

In BJT transistors, the collector current is determined by the potential-barrier of the EB junction. (This is the same way that diodes work, with an exponential V-I curve where the V_F voltage-drop determines diode current.) The base current of a BJT actually has no direct effect on the collector current. Transistors are voltage-input devices, so the I_B and the beta are mostly irrelevant to explaining their internal workings.

However, much technician-level training material (especially in the military) teaches that BJTs are current-input devices. They assume that V_BE is fixed at 0.7V. This oversimplification is fine at the black-box level, and works for the tech students who never will become engineers or scientists. (It's a "lie to children," like teaching kids that electrons orbit inside atoms just like little planets.) The full engineering version of BJT explanation is too complicated. It involves the built-in voltage-fields of the PN junction, depletion zones and the diode exponential V-I curve, and the Ebers-Moll equation I_C = I_S*(e^V_BE/V_T -1).

The classic circuits which rely on direct V_BE voltage-input are: DC amplifiers (op amps, long-tailed pair,) current-mirrors, cascode circuits.

In other words, suppose a student had been taught that "hfe is primary, V_BE is irrelevant," and has never even encountered the idea that Ic is actually controlled by V_BE ...then that student will have no hope of understanding the inside of a modern op-amp. Those ICs are composed of voltage-input transistor circuits: current-mirrors, long tailed pairs, cascode stages!

For a good textbook which directly attacks this "hfe mistake," see Art of Electronics by Horowitz and Hill. Their lab-manual especially delves into all the problems caused by beta-based thinking in transistor design. Win Hill on C4 even discusses how he first encountered this issue.

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[Note that hfe and base current can be a helpful concept. After all, base current can be used to determine V_BE, and then the EB junction potential-barrier will control the collector current ...so if we oversimplify and remove the middle part, we can pretend that base current can affect collector current. Because both currents are linked to V_BE, the two currents are roughly proportional, and this is a useful concept. But it can become a mental stumbling-block for students encountering BJT circuits which harness V_BE directly, where the base current plays no role except as unwanted leakage. ]