Small-signal transfer function from this differential input

[CorHawk]

Homework Statement

2. Homework Equations [/B]
Small Signal Equivalent Circuit, Kirchhoff Current Law, and BJT equations mentioned in the question.

The Attempt at a Solution

It looks like I did something wrong. Can someone help me with fixing this?

 

[rude man]

Homework Statement

View attachment 81419
2. Homework Equations [/B]
Small Signal Equivalent Circuit, Kirchhoff Current Law, and BJT equations mentioned in the question.

The Attempt at a Solution

View attachment 81420
It looks like I did something wrong. Can someone help me with fixing this?

I would not use equivalent circuits for this. You're given all the equations you need to solve the problem without resorting to equiv. ckts.
Hint: at some point use e^x ≅ 1 + x for 0 ≤ x << 1.
OR you can do it with just g_m.

 

[CorHawk]

I would not use equivalent circuits for this. You're given all the equations you need to solve the problem without resorting to equiv. ckts.
Hint: at some point use e^x ≅ 1 + x for 0 ≤ x << 1.
OR you can do it with just g_m.

I don't understand how you want to get to the answer 1/2*g_m by only using g_m and nothing else. And I don't see how e^0=1 would be related to this.

 

[rude man]

I don't understand how you want to get to the answer 1/2*g_m by only using g_m.

I'll get you started:
g_m = dI₁/d(V_b1 - V_e) = -dI₂/dV_e
where V_b2 = 0 is assumed wlog ("without loss of generality"). V_b is base voltage, V_e is emitter voltage.

And I don't see how e^0=1 would be related to this.

That would come in handy if you choose to use I = I_sexp(qV_be/kT) instead of g_m directly.
I recommend using g_m directly, it's shorter.