MOSFET Common-Source Amplifier Small-Signal

[CoolDude420]

Homework Statement

Hi,

so I was trying to derive the small-signal model of a common-source mosfet amplifier that I learned from the notes. Here is what I did,

Now the issue is the i_d branch in the final small-signal circuit. In my lecture notes, that i_d branch is not there at all but when I derived it I do have it there.
The lecture notes goes on to say that the output resistance for this circuit is Rd ∥ Rl ∥ ro. However if you include i_d like in my case, that no longer holds.

How is it possible to remove that id branch?

Homework Equations

The Attempt at a Solution

 

[donpacino]

It all depends on what you use for a model. If you use a current source (id), RD will go away. If you use a resistance and voltage model, then RD will be present, but the voltage will go to zero when finding the transfer function vo/vi.

 

[bartman_bartman]

In your first schematic, you label id as the "drain current" on your small-signal MOSFET model. Later, you use id to represent a completely different current involving two external resistors, Rd and RL. I'd choose another name, because id in your 2nd pic is definitely not the same as the current flowing into the drain of the MOSFET model.

For the problem statement... I'm not sure what you're trying to do... presumably find the output impedance or solving for id. Not sure from your problem statement, so I'll run through both. You're already 90% there.

For problems like this, you can find the small-signal output impedance by inspection. In your 2nd pic, it looks right and the output impedance is the equivalent Z you see BETWEEN the nodes defined by output voltage, Vo. Small-signal conditions means large capacitors are approx. a short circuit & VDD is approx. connected to ground. Therefore, as you draw in 2nd pic, small-signal assumptions connect ro, Rd and RL in parallel... and that parallel combination attaches to + & - nodes of Vo. Therefore, Zout = ro || Rd || RL. The current in each resistor is simply i_rd = Vo/Rd and load current i_rl = Vo/RL and i_ro = Vo/ro... just pay attention to the sign of the current. Going back to solving for id in your original MOSFET model (bottom of first pic), id is the sum of currents in dependent source & ro. Therefore, id = gm*Vgs + Vo/ro