# Degenerated common source output resistance by inspection

In Razavi's Design of Analog CMOS Integrated Circuits (2nd edition), on page 65 he describes a way to derive the output resistance of a source-degenerated CS stage by inspection:

I understand how to derive this result from the small signal model but I don't understand the reasoning/intuition behind the inspection analysis.

He starts off by saying that if you apply a $$\\Delta V\$$ at the drain and measure the resulting $$\ \Delta I\$$, then all the $$\ \Delta I \$$ must pass through the degeneration resistor $$\R_S\$$. Sounds good, since the current has nowhere else to go.

He then derives a voltage divider equivalent circuit as shown in the linked image. However, he uses the result that the resistance looking into the source of a MOSFET is $$\ \frac{1}{g_m+g_{mb}} \$$ to insert a resistor in parallel with $$\ R_S \$$, which I don't understand. My confusion is that the resistance looking into the source of a MOSFET is derived when you have an ideal independent source applied at that terminal and calculate the voltage/current ratio (typical Thevenin resistance). How is that scenario applicable here? This analysis started off by saying that all of the $$\ \Delta I\$$ must pass through the resistor $$\ R_S\$$ but the model derived in figure 3.30(c) seems to imply that some of it is going through this other impedance.

(A side-doubt that I have is that here $$\ \lambda \$$ is not assumed to be zero since $$\r_o\$$ is finite but in then in this case resistance looking into the source shouldn't be exactly $$\ \frac{1}{g_m+g_{mb}} \$$, not sure if that makes a difference here.)

• The $\frac{1}{g_m+g_{mb}}$ is the impedance that you would see when looking into the source of a MOSFET in a common gate circuit (or grounded gate, which is what we have here!). So have a look at the small signal model of the common gate and determine the impedance at its source then it should become more clear. Saying that the voltage change (resulting from $\Delta I$) assumes that the MOSFET "isn't doing anything", but that's not true. The MOSFET does do something, it puts an impedance $\frac{1}{g_m+g_{mb}}$ in parallel with $R_s$ Commented Sep 20, 2020 at 20:45
• @Bimpelrekkie Thanks for the clarification. I suppose the core of my confusion is, why is the action of MOSFET to put that impedance in parallel with $R_S$? I can see that it will generate a current in response to sensing a change in voltage at the source, but I don't see why that would manifest here as an impedance between source and ground, since we're applying a voltage to the drain. Commented Sep 20, 2020 at 21:28
• @Bimpelrekkie Okay, I think I got it, thanks for the guidance. The key point for me was to realize that when you consider that the mosfet in this inspection diagram is just the two current sources, the mosfet senses the voltage change at its source and carries away from the source a positive current that is proportional only to the source voltage, it behaves exactly like a resistor, which is a component that carries positive current away from the source and the current depends only on that source voltage. Commented Sep 20, 2020 at 22:09
• Indeed the MOSFET's $V_{GS}$ changes when the voltage at its source changes since its gate is grounded. Then the MOSFET draws a current in response to that. I am happy to see that I pointed you in the right direction which enabled you to find the answer yourself! Excellent. Commented Sep 21, 2020 at 5:44

The short answer is that when you use visual shortcuts like drawing the $$\r_o\$$ in parallel with the MOSFET symbol, you're separating out the MOSFET into two parts: one part which holds the dependence on $$\v_{ds}\$$ only ($$\r_o\$$) and one part that holds the dependence on $$\v_{gs}\$$ only (the traditional MOSFET symbol). At the source, the resistor $$\ R_S \$$ senses $$\ \Delta V_{RS} \$$ and carries away a current proportional to it (and no other voltages). The MOSFET symbol (since it "holds" only the VCCSs $$\g_m v_{gs} \$$ and $$\ g_m v_{bs}\$$) will also sense the voltage $$\ \Delta V_{RS}\$$ and carry away from the source a current proportional to it (and no other voltages). This means the MOSFET symbol has the exact same function as the resistor $$\R_S\$$, so you can model the MOSFET symbol as a resistor and put it in parallel as shown.