Find the input resistance of the following MOSFET circuit

[Jalo]

Homework Statement

Given the following circuit find the input resistance as seen through vo.

Homework Equations

The Attempt at a Solution

To solve this problem I started by drawing the signal equivalent circuit:

As we can see the voltage v_gs = v_i - v_o. To find the input resistance I used Thenevin's theorem, placing a DC test voltage, V_T, between the output terminals. By finding the relationship between V_T and the current that passes through it I hoped to find the value of the input resistance.

This, however, didn't take me to the correct result. I was told the result was equal to 1/g_m || r_o || R_s. I do not understand how to get to this result tho.. If someone could point me in the right direction I'd appreciate!

 

[CWatters]

Looking at your small signal model you can see that the output voltage V_T = V_gs.

So the current source g_mV_gs is actually g_mV_T.

Then compare ohms law I=V/R with I = g_mV_T.

What's the equivalent resistance of the current source?

Edit: Sorry I'm not explaining that very well. V_T = -V_gs when V_i = 0. See page 5..

http://web.mit.edu/6.012/www/SP07-L20.pdf

 

[rude man]

You can't apply a hard voltage to the output and determine the resulting current from your hard voltage source. Why? Because that rides roughshod over the entire circuit. When you change vs you also change vg-vs which changes id etc.

So, question: what is the definition of output impedance really? It obvously has something to do with the following: I put a signal vg on the gate, I get a signal vs on the source. Now I parallel Rs with a test resistor R and the voltage drops to some extent. If the output impedance is zero there is no drop in vs. If the output impedance is 1K then if R = 1K the output voltage drops to one-half. Etc.

So your first (and main!) task is to determine an equation which relates the change in vs to a change in Rs. (I am lumping Ro in with Rs, which is justified by your equivalent circuit. Your equiv. ckt. is OK, use it.

Hint #1: change your Rs and Ro to conductances and work exclusively with conductances, not resistances.