# Why are the $2R_{EE}$ and $r_o$ in parallel?

Here is the BJT common mode and its half circuit in Microectronic Circuits.

The book says $$\2R_{icm}=(1+\beta)(2R_{EE}||r_o)\$$.

Why are the $$\2R_{EE} \$$ and $$\r_o\$$ in parallel?

I drew the T model of that half circuit:

simulate this circuit – Schematic created using CircuitLab

I still don't understand why are they parallel and ignore the $$\r_e\$$.

• Here you will find the full expression for the input resistance of CE amplifier electronics.stackexchange.com/questions/662191/…
– G36
Commented Jun 20, 2023 at 14:29
– LvW
Commented Jun 20, 2023 at 14:49
• @LvW I think that the answer will be the same. The only difference is that now, for the common-mode operation of a diff amp we have $2R_{EE}$ instead of $R_E$
– G36
Commented Jun 20, 2023 at 15:03
• @G36 No - I do not think so. The input resistance at each input (base) does not only depend on the common emitter resistor but also on the input resistance at the emitter of the second transistor: Remember - the long-tailed pair is nothing else than a series connection of two stages: Common collector and common base.
– LvW
Commented Jun 20, 2023 at 18:32

If you read the book carefully, I hope that all the used neglections/simplifications are mentioned in the book. Simplifications are OK - however, they must/should always be mentioned.

Certainly, the shown expression for r_in is, indeed, a simplified formula. This becomes clear because even the base-emitter resistance (hie=rbe) is not included in the expression.

Regarding your question: The series connection (ro+Rc) is in parallel to 2R_EE because both are connected between a common node (between re and R_EE) and ground. In the given formula, Rc is neglected in comparison to the value of ro (ro>>Rc).

More than that, the input resistances are small-signal parameters. That means that they are differential (dynamic) terms and should therefore - in contrast to static quantities - always be identified by small symbols (r instead of R).

Example: I have used a symbolic simulation program for computing the common-mode input resistance for 2 cases:

• Case 1 (ro infinite): 2ri_cm=hie+(beta+1)2R_EE (with hie=rbe).

• Case 2 (finite ro): The resulting expression is too long for reproducing it here: A quotient with 9 separate expressions in the numerator and 4 expressions in the denominator.

• thanks for your explanation,so if we have to consider the $r_o$,we can ignore the $R_C$ or $R_L$ directly if the output is at the common node ? Commented Jun 21, 2023 at 8:16
• BTW, your "Rc is neglected in comparison to the value of ro (ro>>Rc)." is really a key to my question Commented Jun 21, 2023 at 8:17
• Attention: We can neglect Rc when it appears in a sum together with ro. However, in the expression for voltage gain both appear as a parallel connection! In this case, the opposite is true: We often can neglect ro.
– LvW
Commented Jun 21, 2023 at 8:22