How can this circuit remove $R_{D2}$ in small signal analysis for this nmos?

The following is differential pair small signal analysis from the book Design of Analog CMOS Integrated Circuits.

Let us set $$\V_{in2}\$$ to zero and find the effect of Vin1 at X and Y [Fig. 4.17(a)]. To obtain $$\V_X\$$ , we note that M1 forms a common-source stage with a degeneration resistance equal to the impedance seen looking into the source of M2 [Fig. 4.17(b)]. Neglecting channel-length modulation and body effect, we have $$\R_S = 1/g_{m2}\$$ The following is analysis of differential pairs. How can this circuit remove $$\R_{D2}\$$ in (c)?

This is what I thought about

• Can you explain to us why do you think that you should include the Rd2 resistor? Notice that in this case, M2 is working as a common-gate amplifier. What is Rin of a CG stage?
– G36
Commented Nov 30, 2023 at 13:23
• @G36 M2 is seen as a resistor $R_S$. Which transistor of $R_{in}$ you are talking about?
– kile
Commented Nov 30, 2023 at 14:30
• Yes, M2 is seen as a resistor Rs. This Rs resistance is equal to the input resistance of a CG stage. Because in this case M2 is working as a common-gate amplifier.
– G36
Commented Nov 30, 2023 at 14:54
• @G36 M2 is in series with $R_{D2}$. Why should we remove $R_{D2}$?
– kile
Commented Nov 30, 2023 at 14:56
• Find Rin of a CG state (also do not forget to neglect channel-length modulation) and you will see why. Or try to find Rin in a circuit that contains an ideal current source in series with a resistor. And you will see that a series resistor does not influence the current.
– G36
Commented Nov 30, 2023 at 15:03

The following is Thevenin analysis.

$$g_m V_T = I_T$$

$$\frac{V_T} {I_T} = 1 / g_m$$

For Thevenin voltage

$$V_{eq} = V_T$$ $$g_m (V_T- V_{eq}) = 0$$ So current source is zero.

• So, do you see now why we do not include Rd2 in Rin equation?
– G36
Commented Nov 30, 2023 at 15:33
• @G36 Thank you. Is Thevenin $V_{eq}$ same as $V_{in}$?
– kile
Commented Nov 30, 2023 at 15:35
• To be honest I do not understand what you are trying to achieve involving thevenin's voltage here. For what purpose.
– G36
Commented Nov 30, 2023 at 15:38
• @G36 In last example we have to calculate Veq electronics.stackexchange.com/questions/691011/…
– kile
Commented Nov 30, 2023 at 15:40
• But we do not have to do it here because Vin2 = 0V (M2 gate is at AC GND).
– G36
Commented Nov 30, 2023 at 15:43