# Deriving mosfet differential current expressions

1. ## Deriving mosfet differential current expressions

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"

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2. ## Re: Deriving mosfet differential current expressions

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.

3. ## Re: Deriving mosfet differential current expressions

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..

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4. ## Re: Deriving mosfet differential current expressions

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) ?

5. ## Re: Deriving mosfet differential current expressions

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

6. ## Re: Deriving mosfet differential current expressions

@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 ?

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7. ## Re: Deriving mosfet differential current expressions

Originally Posted by promach
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.

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8. ## Re: Deriving mosfet differential current expressions

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.

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