Electronic – Output Resistance of Small-Signal MOSFET Circuit

cmosmosfetoperational-amplifiersmall-signal

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.

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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
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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 1 (KVL)

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Approach 2 (Resistors)

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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.

EDIT:
Extra Info for Comments:
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Best Answer

You've written "Current source whose current depends on voltage across it = resistor".

This is true, as long as the current is linearly proportional to the voltage across the current source.

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.

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