A few days ago I decided to see what happens in the forum... and I came across this question. It seemed simple... but actually, the whole philosophy of the differential pair was hidden behind its answer.
Obviously, the HelpMee's “naive” question (persistently repeated several times) is about the nature of the “tail” - he/she wants to know WHAT, HOW and WHY is done there. I perfectly know the answer… but how do I explain it in the possibly simplest way?
OK, let’s start with a short answer: The differential pair consists of two emitter followers connected in parallel to a common load Re so that each of them tries to make the common emitter voltage equal to its input voltage.
At differential mode, they oppose each other by changing their currents in different directions so both voltage across and current through Re stay constant. In this case, the humble resistor does a good job. It should be resistor (not an inductor or something else) to set the common emitter current according to Ohm's law. This current stays constant; only the partial collector currents change (crossfade, redistribute or are steered). We take one of them, convert to voltage by the corresponding collector resistor and use as a single-ended output.
At common mode, the two emitter followers contribute each other by changing their currents in the same direction. As a result, the common voltage across Re and current through Re, partial collector current and output voltage change as well… which is undesirable. To suppress these current variations, we can make Re dynamic. Then when the emitter voltage increases, the Re resistance increases with the same extent and v.v. By this trick we artificially keep constant the ratio Ve/Re in Ohm's law, which is the common emitter current Ie.
See also this RG discussion - Can we reveal the basic idea behind the long-tailed pair and explain its operation in an intuitive way?
(Another clever trick would be to leave the constant ohmic resistor and make Vee variable. When the emitter voltage increases, the Vee magnitude decreases with the same extent and v.v. Thus we artificially keep constant both the voltage across Re - the numerator in Ohm's law, and common current Ie. See the RG discussion Are there situations in which the common emitter current source of the long-tailed pair can be replaced with a humble resistor?)
Now I will answer the question in detail as HelpMee's wants (and I like) - using “an analytic step by step explanation”. I will do it as a fictional story about the invention of the “long-tailed pair”. I will refrain from using special terms that discourage understanding.
1. Single common-emitter amplifier. In this configuration, we apply the input voltage to the transistor base and take the collector current (converted to voltage by the collector resistor) as an output signal. We can change the gain by inserting various “things” with different (differential) resistance dRe between the emitter and ground. Thus the stage acts as an emitter follower loaded with some “load”... but surprisingly, we do not use the voltage drop across this load as an output… instead we use the collector current that creates this voltage drop. Here are three typical cases:
dRe = 0. If we insert a voltage source, voltage stabilizing element (e.g. Zener diode)... or simply ground the emitter, the emitter voltage will be fixed. Figuratively speaking, the emitter is “stiff”, “immovable”. The whole input voltage is applied to the base-emitter junction and the gain is maximum.
dRe = Re. If we insert an ordinary ohmic resistor with static resistance Re, the emitter voltage will begin “moving” in the same direction as the input voltage. Now the emitter is “soft”, “movable”. A part of the input voltage is compensated and the gain is decreased (as they say, there is a negative feedback or emitter degeneration).
dRe -> infinity. If we insert a current source, or more likely, current stabilizing element (usually, a collector-emitter junction of a transistor), the emitter will become extremely “soft” and will exactly follow the input voltage at the base. As a result, the input voltage is fully compensated and the gain is zero.
The conclusion is that we can increase the gain by “hardening” the common emitter voltage and decrease it by “loosening” this voltage. This trick will lead us to the differential pair...
2. Paired common-emitter amplifier. To make a differential amplifier, it is not enough just to take two single common-emitter amplifiers for at least two reasons. First, we want to have a single-ended output but here we have a differential one. Second, they will amplify both input signals - differential and common-mode. Somehow we have to make them amplify as much as possible the differential signal but do not amplify (and even attenuate) the common-mode signal. According to the trick above, this means to “harden” the emitter voltages at differential mode and “soften” them at common mode.
We can do this magic by connecting their emitters to a common current stabilizing resistor (“current source” or a resistor in the simplest case). Now, at differential mode, each of them will fix the other emitter voltage thus providing maximum gain (the other transistor will have the “feeling” that a voltage source is connected in its emitter). At common-mode, both will work on a common current-stabilizing load that provides a minimum gain (both transistors will have the “feeling” that a current source is connected in their emitters).
See also my answer to the question Why the common-mode gain of the differential pair is almost zero?