I thought I'd post this n the homework section as it's more likely to get an answer here, although I haven't been in school for quite a while!

I want to find the output noise current of a cascode, using the attached equivalent circuit. The problem is to find i in terms of i2, i4 and gm. But I end up going in a circle! Where am I going wrong?

My solution:
i = i1 + i2 + gm.V

But:
V = -i3.R2 = -(i-i4)R2 = (i4-i).R2

Substituting this into the previous expression:
i = i1 + i2 + gm.(i4-i).R2

But:
i1 = i - i2 - gm.V = i - i2 - gm.(i4-i).R2

Substituting this into the previous expression:
i = i - i2 - gm.(i4-i).R2 + i2 + gm.(i4-i).R2

Hence:
i = i

 

You need another independent equation that relates i, i1, R1 & R3 to the control voltage V.

 

AhA! So using:

V = (i4-i).R2 = i.R3 + i1.R1

But: i1 = i - i2 - gm.V = i - i2 - gm.(i4-i).R2..........................1

Hence: V = i.R3 + (i - i2 - gm.(i4-i).R2).R1............................2


Solution:
i = i1 + i2 + gm.V

Sub in 2:
i = i1 + i2 + gm.(i.R3 + (i - i2 - gm.(i4-i).R2).R1)

Sub in 1:
i = (i - i2 - gm.(i4-i).R2) + i2 + gm.(i.R3 + (i - i2 - gm.(i4-i).R2).R1)

Hence:
i = [R2.i4 + R1.i2 + gm(R1.R2.i4)] / [R1 + R2 + R3 + gm(R1.R2)]

 

Yep - looks correct.