Deriving mosfet differential current expressions

[promach]

Could anyone help to derive mosfet differential current expression (9.4) ?

Note: Screenshot taken from book "Low-power-wireless-communication-circuits-and-systems-60GHz-and-beyond"

 

[Dominik Przyborowski]

Could be a bit hard, because Lambert W function is not an elementary function.
If you would like to deriv similar formula using square law approximation, check Gray, Hurst, Meyer book "Analysis and Design...", chapter related to diff pairs.

 

[vivekroy]

See the attached image for the proof of 9.4.

Edit: If you want the proof for MOSFETs, it can be found in Razavi's 'Fundamental of Microelectronics'. I have extracted the most important parts of the expressions..

 

[promach]

In the proof for BJT , why do you add Io (DC bias point current) ?

I thought gmc*V1 is the total current flowing through the bottom mosfet (or in your proof, a current source) ?

 

[FvM]

Io + gmc*V1 is just copied from post #1...

 

[promach]

@FvM

Io + gmc*V1 is just copied from post #1...

Where ? The book screenshot itself does not even indicate what exactly "Io" refers to.

I have checked the whole section of the chapter, there is no mention about Io.

Why you call "Io" as DC bias point current ?

 

[vivekroy]

I thought gmc*V1 is the total current flowing through the bottom mosfet (or in your proof, a current source) ?

But that is just the small signal current. All of these expressions were derived NOT assuming small signal models (and thus all the complicated expressions)... And thus, I assumed the DC component of the tail current is Io.

EDIT 1: (Some more comments)


I cannot see it being anything other than a DC component anyway.

 

[FvM]

The nature of Io as bias current is suggested by using a capital letter, also by symmetry considerations, you have i0a=Io+gmc*v1 and i0b=Io-gmc*v1.