# DC analysis of common-drain NMOS amplifier

The DC analysis is regarding the amplifier calculations, but that is not relevant to the topic. The equivalent DC circuit of the amplifier is:

The known values are: $$\R_{g1}=300\text{ k}\Omega\$$, $$\R_{g2}=200\text{ k}\Omega\$$, $$\R_s=100\text{ k}\Omega\$$, $$\k_n=25\ \mu\text{A/V}^2\$$, $$\\lambda=0.02\text{ V}^{-1}\$$, $$\V_{tn}=1\text{ V}\$$, $$\V_{dd}=10\text{ V}\$$.

Now the problem is to find the bias point (drain current - $$\I_D\$$, voltage $$\V_{GS}\$$ and voltage $$\V_{DS}\$$).

First, I calculated the gate voltage as: $$V_G=\frac{R_{g2}}{R_{g1}+R_{g2}}V_{dd}=4\text{ V}$$ Then, I assumed that the transistor is operating in saturation mode, and set up these equations: $$I_D=k_n(V_{GS}-V_{tn})^2(1+\lambda V_{DS})$$ $$V_G=V_{GS}+R_s I_D$$ $$V_{dd}-V_{DS}-R_s I_D=0$$

The problem is, I cannot solve those equations, as there always seems to be one element missing. Any ideas on how to solve this?

• 4 eqn and 4 unknowns...hmm Jan 31, 2018 at 21:22
• @TonyStewart.EEsince'75 Actually, there are three unknowns and three equations, the first equation is just calculation of $V_G$. But my guess is, there is something else we could assume in this case, so the equations get a bit less complex to solve.
– A6EE
Jan 31, 2018 at 21:23
• Yes so what's the problem? Jan 31, 2018 at 21:24
• Note: Vgs = Vg - Vs, the only unknown in your circuit is Vs Jan 31, 2018 at 21:26
• I can't seem to solve those three equations and there is probably something else to consider here for the equations to get more simple, but I'm not sure what.
– A6EE
Jan 31, 2018 at 21:30

For quick hand analysis, I would personally not include the impact channel length modulation.

Knowing, $$V_{GS} = V_G - V_S\;\;\;\&\;\;\; V_{DS} = V_{dd} - V_S$$

and that the drain current equals,

$$I_D=k_n(V_{GS}-Vtn)^2(1+\lambda V_{DS})$$

Since gate current of Q1 is zero the drain current is also,

$$I_D = \dfrac{V_S}{R_S}$$

Put it all together as,

$$\dfrac{V_S}{R_S} = k_n(V_G - V_S-Vtn)^2(1+\lambda (V_{dd} - V_S))$$ and solve for $V_S$.