How to find common-mode voltage gain of a Differential Amp

[DODGEVIPER13]

Homework Statement

Beta=100,Thermal Voltage=0.026 Volts, Va=infinity, and Vbe=0.75 Volts for the differential amplifier. Please look at the upload for a clearer picture.

Homework Equations

Ad=betaRc/2(rpi+Rb)
Acm=-gmRc/(1+(2(1+beta)Ro/(rpi+Rb)))

The Attempt at a Solution

I have uploaded my attempt but I am wondering if Ro is really zero I mean Va is infinity so I would think so but this would cause Acm=0 and CMRR= infinity this seems incorrect what am I doing wrong?

 

[rude man]

I think what you did was correct.
The 'tail' in this diff amp is a pure current source (Ic3) and so not changed by a common mode input voltage.
Thus, the common mode gain, absent any transistor asymmetries of which none are specified, is indeed zero and the CMRR is infinity.

For Ad you seem to have two answers: "Ad = 73.169 = 28.846". What gives with that? Anyway, I got close to your the second number by approximating as
i = is exp(V/VT), ∂i/∂V = is (1/VT) exp(V/VT) = (1/VT) i = (1/.026)0.75e-3 = 34.7
so your number of 28.846 looks about right since my approx. will come in high.

 

[DODGEVIPER13]

Well darn the 28.846 is actually my Gm the 73.169 is my gain I need to check that

 

[rude man]

I myself goofed in computing g_m. It should have been 0.029 A/V.
That times the 8K resistor gives A_d = 232 but that ignores beta, r_pi and the 2K base resistor so the actual gain will be somewhat less.

 

[DODGEVIPER13]

Sweet thanks again man!

 

[DODGEVIPER13]

Hey I had one last question I had a friend bring up the point that the circuit is not ideal due to current biasing. So I resolved for R0 by running a kvl from 12 to Vec3 soI got 12-Ib(2)-0.7=Vec=IcRo and Ro=7.54 k ohms

 

[rude man]

Hey I had one last question I had a friend bring up the point that the circuit is not ideal due to current biasing. So I resolved for R0 by running a kvl from 12 to Vec3 soI got 12-Ib(2)-0.7=Vec=IcRo and Ro=7.54 k ohms

I don't quite understand this. R_o is due to the Early voltage V_A which is given to you as infinite, so R_o should also be infinite.

" the circuit is not ideal due to current biasing "?? Every transistor circuit is current biased, or maybe there is an interpretation here I don't know about. I realize the "tail" is a constant-current source = 1.5 mA.

Maybe you can elaborate on your equation.

 

[DODGEVIPER13]

Alright man appreciate it well that kind of sums it up Ro=infinite

 

[rude man]

OK viper. One last suggestion - watch your units, like mixing mA and Ohms etc. It's probably best if you change everything to ohms, amps and volts before computing numbers.