How can I find the equivalent resistance, Rxy, of the two NMOS transistors in Figure 5.25 in Razavi's "Design of Analog CMOS Integrated Circuits"?

Question

While calculating the gain (\$-G_mR_{out}\$) of the differential pair (with active current mirror), in order to find \$R_{out}\$, how is Razavi able to substitute M1 and M2 with \$R_{xy}=2r_{01,2}\$?

Answer 1

To find \$R_{xy}\$, we redraw the portion of Figure 5.25 of interest below, substituting small signal models for M1 and M2 (with \$\gamma=0\$, \$g_{m1}=g_{m2}=g_m,r_{o1}=r_{o2}=r_{o}\$), and apply a test voltage \$v_{test}\$ across the terminals:

From the above network we can use KVL to write,

\$v_{test}=(i_{test}+g_mv_p)r_o + (i_{test}-g_mv_p)r_o\$

\$v_{test}=2i_{test}r_o + (g_mv_p-g_mv_p)r_o\$

\$R_{xy}=\frac{v_{test}}{i_{test}}=2r_o=2r_{o1,2}\$.