Differential pair output voltage formula derivation

Question

I am trying to get to the simplified form of the equation of the differential output voltage of a differential pair:

I know the correct equation is:

*Image above from Franco, S. Analog Circuit Design. Please let me know if it is not proper to show this excerpt here.

I am stuck with simple algebra, really. I want to get the equation 4.73. Here is what I did:

Notation: In the book, VoD. In my text, Vodif In the book, ViD. ViD is the complete differential input voltage, vid =vi1 - vi2. In my text, I named it Vdif.

Also I've just considered αf = 1, so it is not explicit in my equations. Considering Ree too large and just ignored.

I am stuck at this point for a while now. This is not homework or alike. Just reviewing some basic concepts.

Answer 1

You can follow these steps: $$i_{C1} = \frac{I_{EE}}{1+exp(\frac{-v_{id}}{Vt})}$$ $$i_{C2} = \frac{I_{EE}}{1+exp(\frac{v_{id}}{Vt})} = \frac{I_{EE}exp(\frac{-v_{id}}{Vt})}{1+exp(\frac{-v_{id}}{Vt})}$$ $$v_{od} = R_c(i_{C2} - i_{C1}) = \frac{I_{EE}R_c(exp(\frac{-v_{id}}{Vt})-1)}{1+exp(-\frac{v_{id}}{Vt})} = I_{EE}R_ctanh(\frac{-v_{id}}{2Vt})$$

EDIT
There is nothing wrong with your steps, just proceed as follows:

$$v_{od} = IeeR_c\frac{exp(-v_{id}/Vt) - exp(v_{id}/Vt)}{2+exp(-v_{id}/Vt) + exp(v_{id}/Vt)}$$ Multiply numerator and dinominator by \$exp(-v_{id}/Vt)\$: $$v_{od} = IeeR_c\frac{exp(-2v_{id}/Vt) - 1}{2exp(-v_{id}/Vt) +exp(-2v_{id}/Vt) + 1}$$

$$v_{od} = IeeR_c\frac{(exp(-v_{id}/Vt) - 1)(exp(-v_{id}/Vt) + 1)}{(exp(-v_{id}/Vt) + 1)^2}$$ $$v_{od} = IeeR_c\frac{exp(-v_{id}/Vt) - 1}{exp(-v_{id}/Vt) + 1}$$

Answer 2

HW_SW_engineer - I think, the simplest way is to start with the currents:

In general, we have: Ic=Ico[(exp(Vbe/Vt)-1]

Ic1/Ic2=exp[Vbe1-Vbe2)/Vt]=exp[Vd/Vt]

with Iee=Ic1+Ic2 we arrive at

Ic1=Iee/[1+exp(-Vd/Vt)] and Ic2=Iee/[1+exp(+Vd/Vt)]

From mathematics: 2/[1+exp(-x)]=....[1+th(x/2)]

Ic1=0.5*Iee[1+th(Vd/2Vt)] and Ic2=0.5*Iee[1-th(Vd/2Vt)]

Ic1-Ic2=Iee*th(Vd/2Vt)

Vo1=Vdc - RcIc1 and Vo2=Vdc - RcIc2 (Vdc: DC quiescent voltage)

Vd=Vo1-Vo2=-IeeRc*th(Vd/2Vt)

Comment: I have assumed Iee=Ic1+Ic2 (for each transistor: coll=emitter current); hence the factor "alpha" does not appear in the equations)