# Output Resistance of Small-Signal MOSFET Circuit

I have some doubts concerning the calculation of the small-signal output resistance of a common-source stage with source-degeneration. This is related to another question I posted but I have refined my question in detail here as to exactly what is troubling me

## The Problem

Calculate the small-signal output resistance of the following circuit including the effect of channel-length modulation and ignoring the body effect.

## My Solution

I have two approaches to find the output small-signal resistance, they both involve,

• Drawing the small-signal model of the circuit
• Zero all independent sources (voltage sources = short, current sources = opens)
• Applying a voltage Vx at the output and measure the resulting current Ix flowing.
• Output resistance will then be Rout = Vx/Ix

This gives us the following circuit

Now from here on, I have two approaches - the first approach gives me the correct answer, however the second approach gives me an incorrect answer.

### Approach 2 (Resistors)

Why does my second approach give me an incorrect answer? I suspect it is something to do with the fact that the ro now only gets a current Ix flowing through it.? Something that didn't sit well with me in the second approach was that after decomposing the current source into a resistor, I had to get rid of the negative sign of the 1/gm. So current flow direction is strange to me too.

• I have already told you why: you can't make transform gm*Vs into a fixed resistor because Vs varies with I(Rs). Your approach implies a fixed Vs, but that's one possibility out of an infinity. If Vs is fixed then yes, you can equivalate gm*Vs with a fixed resistor, but that will only be true for that particular value of Vs. As long as there is current flowing through Rs, the voltage level at the source will change, with or without Vgs. Commented Mar 14, 2021 at 18:40
• I agree with @aconcernedcitizen. When I got to the part in your derivation where you equate the a current source with a resistor, my mind went "huh?" Commented Mar 14, 2021 at 18:43
• Not only, but even if the calculations with the first approach are OK, in the 3rd picture the current Ix circulating inside ro is wrong. Commented Mar 14, 2021 at 21:15
• Thank you for the responses. I think I understand it now. Apologies for so many questions. @barrow The current circulating in the loop is between the VCCS and ro is gmvgs before the sign of vgs has been taken into account. Commented Mar 14, 2021 at 21:31
• @AlfroJang80, please, look attentively to your third picture. It shows a current Ix circulating inside ro. Well, this is completely wrong and confusing. Commented Mar 14, 2021 at 21:50

However, the voltage across that current source is $$\V_{ds}\$$, while its current is proportional to $$\V_{gs}\$$. Thus, the current is not proportional to the voltage across the current source, and replacing it with a resistor is invalid.