# MOSFET sub-threshold operation

I have a question related to the sub-threshold operation of MOSFETS.

Let's say I have the following setup:

How can I calculate the sub-threshold current ( Id when Vgs < Vgs(th) ) at Vgs = 0.5V using the data from the datasheet and data from the circuit?

Also what value will Rds have and what will be the value of Vout (Vds) at Vgs=0.5V?

I am very new to the domain of electronics so if I understand it correctly when Vgs < Vgs(th) the MOSFET will not be turned off completely but instead the "resistance" of the MOSFET (Rds) will be very high (in the range of MOhms) and there will be a very low drain leakage current (in the range of uAmps). I would like to know if there is a way to get a more precise calculation of this values (Id, Rds, Vds when Vgs < Vgs(th)) using only the available data from the datasheet and the circuit.

-
 As soon as I read the question I knew it was you, Buzai. :-) – stevenvh May 27 '12 at 6:48 Does this mean I am on the wrong track for learning electronics? :) – Buzai Andras May 27 '12 at 9:41

For $V_{GS}<V_{th}$, there is weak-inversion current, which varies exponentially with $V_{GS}$, as given by

$I_{D}\approx I_{D0}·e^\dfrac{V_{GS}-V_{th}}{n\frac{kT}{q}}$

with

$I_{D0}= I_{D}$ when $V_{GS}=V_{th}$

$k=$ Boltzmann constant=$1.3806488(13)·10^{−23} J·K^{-1}$

$T=$ temperature in kelvins

$q=$ charge of a proton=$1.602176565(35)·10^{−19}$ C

$n=$ slope factor$=1+\dfrac{C_D}{C_{ox}}$

$C_D=$ capacitance of the depletion layer

$C_{ox}=$ capacitance of the oxide layer

You can use either experimental data, or a few points from graphs in the datasheet (like the one that Armandas suggests), to estimate $I_{D0}$ and $n$, and then use them to estimate $I_{D}$ for any $V_{GS}$ and $T$.

Reference: Modes of operation of a MOSFET.

Added: with my paragraph "You can use either..." I meant that you can do curve fitting to find the values for $n$ and $I_{D0}$ that best fit the data you have available, either from experiments (if you can do them), or from graphs from the datasheet (if there is any that is useful). In your case, Figure 2 (above), together with the equation above, might allow you extrapolate $I_D$ for lower $V_{GS}$ values. I'm not saying that you will end up with a high-quality estimate. I'm saying this is the best I could think of.

-
I'm removing my answer, as yours seems to be exactly what the OP wants. – Armandas May 27 '12 at 5:55
12 significant digits, Telaclavo? To a physicist the charge may be 1.602176565(35)·10$^{−19}$ C, to an engineer it's 1.60·10$^{−19}$ C. – stevenvh May 27 '12 at 6:46
Thank you for the answer. I read the reference article before I posted the question, but I didn't know (and still don't know) how to calculate 'n' from the given formula using only the data from the datasheet. I am not sure where can I find values for Cd and Cox in the given datasheet. Also I am not sure how could I further calculate Rds and Vds when Vgs < Vgs(th). Unfortunately I see that Armandas removed his answer :(. – Buzai Andras May 27 '12 at 9:55
@Buzai - I still can see Armandas's deleted answer (users with >10k rep can). He didn't provide an equation. – stevenvh May 27 '12 at 9:57
@stevenvh Come one. Let's not spend times with these things. It takes me the same effort, to copy and paste those constants with all known digits. The OP will take as many as he wants. – Telaclavo May 27 '12 at 12:55