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The first approach :Assuming 0v at both bases, when the base input of Q1 rises, its collector current rises and its collector voltage goes down. When the base input voltage of Q2 rises by the same amount, the same happens and its collector voltage also goes down,so the voltage difference between the two collector voltages will be zero.

The second approach : the same thing happens as the first approach but each emitter of both transistors acts like an input to the other and each transistor acts like a common base amplifier and the collector voltage will be in phase with emitter of the same transistor, and out of phase with the amplified signal from its base so they will cancel and the voltage of any collector to ground will be 0v.

I think i confused you a little but here's a photo from "electronic devices" by floyd to make the second approach clear.

enter image description here

I know i got something wrong -as usual- but i want to know where.

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First approach: In principle, correct - however, the voltage decrease at both collector nodes is very small because the resistor RE is selected with a rather large value (sometimes with a 3rd transistor working as a "constant" current source). Hence, we have a very strong feedback effect for such common-mode signals.

Second approach: ....and out of phase with the amplified signal from its base".

I don`t know if you speak about symmetrical or unsymmetrical operation (base of Q2 at ground). In the latter case, it is correct that - as seen from the left - the two transistors work in common-collector-common-base configuration. Hence, the collector of Q1 is in anti-phase and the collector of Q2 is in phase with the input at the base of Q1.

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Your first approach is correct.

As both bases rise together and you have correctly said that the collector differntial voltage remains at zero, you could regard this as paralleling the transistors. As you lift the combined base voltage, the emitter voltage rises, drawing more current through Re, which passes more current through the Rcs, lowering the collector voltages.

In your second approach, you have neglected that while the emitter output of each transistor acts as an input to the other as a common base amplifier, the increase of emitter voltage increases the emitter current, which drops the collector volts. This conteracts the collector volts rising due to the common base amplification. Of course both approaches are equivalent.

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