# MOSFET biasing with PMOS load

Not a homework problem, I'm refreshing before semester starts. Problem is from chapter 7 of Razavi Fundamentals. Given are $$\V_{th} = 0.4\text{ V}\$$, $$\\mu_nC_{ox} = 200\text{ µA/V}^2\$$, $$\\mu_pC_{ox} = 100\text{ µA/V}^2\$$.

Part (a) I figured out, $$\A_v = g_{m1}\left(r_{o1} \parallel r_{o2}\right)\$$ and $$\r_{o1}\$$ and $$\r_{o2}\$$ are $$\1/(\lambda I_d)\$$. That gets me $$\g_{m1}\$$, and I set $$g_{m1} = \sqrt{2 \mu_nC_{ox} \times (W/L)_1 \times I_d}$$ and solve for $$\(W/L)_1\$$. I got 7.8125.

I'm now struggling on part (b). The equation for bias $$\I_d\$$ of each transistor is $$\frac{1}{2} \mu C_{ox} \frac{W}{L} \times (V_{GS}-V_{TH})^2 \times (1 + \lambda V_{DS})$$ both equal to each other (obviously using the PMOS version for the PFET).

Where I'm having problems is how I go about finding $$\V_{GS}\$$ for M1 so that I can find $$\V_{DS}\$$ for M1[attachimg=1], and where $$\V_{\text{out}}\$$ is biased at (i.e. what is $$\V_{DS}\$$ or $$\V_{SD}\$$?). The drain-source voltage for M1 and M2 add up to $$\V_{DD}\$$ but that's all I know. I was going to assume $$\V_{\text{out}}\$$ sits at $$\V_{DD}/2\$$ but that seems wrong. Any pointers?

• You do not need to know the Vgs1. In part B you need to solve for Vb at Id = 0.5mA.
– G36
Sep 10, 2021 at 16:48

from Razavi (1st ed. eq 2.28). $$\ g_m \$$ (with $$\\lambda\$$ included) is equal to $$\sqrt{\frac{(2\times \mu_n\times C_{ox} \times \frac {W}{L}\times I_d)}{(1+\lambda\times V_{DS})}}$$ Since you know $$\g_m,\mu_n ,C_{ox},\lambda,\frac{W}{L},\$$ and $$\I_d\$$ (from your comments and problem statement(s) parts a and b) you can get $$\V_{DS1}\$$.
If you know $$\V_{DS1}\$$, then you can get $$\V_{DS2}\$$ (given fixed supply). If you know $$\V_{DS2}\$$, then you have enough information to get $$\V_{GS2}\$$ from the $$\I_d\$$ equation (including $$\V_{DS}\$$ and $$\\lambda\$$ effects).