# Differential Amplifier Common-mode Output Resistance

1. Jun 4, 2013

### hisotaso

1. The problem statement, all variables and given/known data

I am learning about differential amplifiers, and am having trouble determining the common-mode output resistance. In the text, we have the two sections "Differential-mode gain and input and output resistances" and "Common-mode gain and input resistance." Notice the lack of common mode output resistance.

The only place I can find reference to common-mode output resistance is in the section "Two-port model for differential pairs", and it is given by Roc =~ 2*μf*Ree (in this section, no derivations are shown). At the end of this section is an "Exercise", and it asks for Rod and Roc. Rod = 3.2MΩ and Roc = 4.8GΩ. In other words, Roc is much bigger.

Now, in the homework problem I am working on, VA is given as infinite, which means ro =~ VA/Ic is infinite, which means μf = gm*ro is infinite. However, in the solutions (authors solution manual) given my prof., Roc is given as Rc/2 which is much less than Rod (in this problem Rod =~ 2*Rc = 94kΩ, and and the given solution to Roc = Rc/2 = 13.5kΩ).

So my question is: what is Roc? I have googled around a bit and have not been able to find any reference to it. I would be happy with a link to a derivation (preferable actually) as opposed to just an expression, though at this point any help would be appreciated.

2. Relevant equations

3. The attempt at a solution

2. Jun 4, 2013

### hisotaso

Ok well I have a line or reasoning that I think works out, please correct me if this is invalid.

Roc=$\frac{voc}{ic1+ic2}$

ic1=ic2=ic

=$\frac{\frac{vc1+vc2}{2}}{2ic}$

=$\frac{\frac{ic*Rc1+ic*Rc2}{2}}{2ic}$

Rc1=Rc2=Rc

=$\frac{ic*Rc+ic*Rc}{2*2ic}$

=$\frac{2Rc}{4}$

=$\frac{Rc}{2}$

Last edited: Jun 4, 2013
3. Jun 4, 2013

### hisotaso

Don't know why itex isn't working, at least on my end...

4. Jun 4, 2013

### The Electrician

Don't use itex; it's just tex. And, don't use the SUB /SUB construct within tex (but you can use it in the non-tex ordinary part of your post); use the underline prefix instead. For example your first tex expression should be like this:

Roc=$$\frac{v_{oc}}{i_{c1}+i_{c2}}$$

That didn't work out as intended. It looks like you get an extra carriage return when following the non-tex part with some tex. Maybe you should put everything inside tex:

$$R_{oc}=\frac{v_{oc}}{i_{c1}+i_{c2}}$$

5. Jun 4, 2013

### The Electrician

Maybe you should ask your instructor what the definition of common mode output impedance is.

If you put an AC short (a big capacitor) between the differential outputs and calculate the impedance from that point to ground, I think you get the result you quote for Roc.