# Obtain the expression of the noise figure for this circuit

Here is the circuit:

Here is the solution (taken from the book RF microelectronics by Behzad Razavi):

My attempt:

First i tried to find the gain expression for this circuit, i noticed that M2 is in diode configuration so in the small signal model M2 is equivalent to a resistor (= 1/gm2).

I would say everything is ok here as i got the same gain as the solution.

Now for the total noise at the output:

The noise i got at the output by the contribuition of IdM1 is not the same as the solution.

In the solution it's Id^2 * gm2 while i got Id^2 * 1/gm2 . Also in the solution there's a term somewhat related to the Vm1,Rs noise i computed but instead it has a id^2/gm1 factor. I dont know where that came from.

What am i doing wrong here?

• Your equations do not seem to include the noise of M2. I bet that this is the term with the $$\\cdot g_{m2}\$$ that you don't seem to be able to match.
• The solution avoids dealing with $$\r_{ds1}\$$ as it is likely negligible compared to $$\g_{m2}^{-1}\$$ (i.e. $$\r_{ds1}//g_{m2}^{-1}\approx g_{m2}^{-1}\$$). You probably already saw this however.
• The first line of the solution does not include the noise term for the resistor $$\R_s\$$. They probably consider that noise source as part of the input, i.e. for computing the input SNR for the noise figure.