Although G_m [cm] of a differential pair with a current-mirror load can be calculated exactly from a small-signal diagram where mismatch is allowed, the calculation is complicated because the mismatch terms interact, and the results are difficult to inter-pret. In practice, the mismatch terms are often a small fraction of the corresponding average values, and the interactions between mismatch terms are often negligible. Using the following steps as a guide, calculate an approximation to G_m[cm] including the effects of mismatch.

(a) Derive the ratio i₂/υ_ic included in (4.165) and show that this ratio is approximately 1/2r_tail as shown in (4.185) if ∈_d ≪ 2, g_m2 r_o2 ≫ 1, and 2g_m2 r_tail ≫ 1.

(b) Use (4.173) to calculate ∈_d with perfect matching, where ∈_d represents the gain error of the differential pair with a pure common-mode input and is defined in (4.161).

(c) Calculate ∈_d if 1/g_m3 = 0 and if the only mismatch is g_m1 ≠ g_m2.

(d) Calculate ∈_d if 1/g_m3 = 0 and if the only mismatch is r_o1 / = r_o2.

(e) Now estimate the total ∈_d including mismatch by adding the values calculated in parts (b), (c), and (d). Show that the result agrees with (4.186) if g_m3 r_o(dp) ≫ 1.

(f) Calculate ∈_m, which represents the gain error of the current mirror and is defined in (4.133). Show that the result agrees with (4.187).

(g) Calculate the value of G_m[cm] using (4.185) and the CMRR for the data given in Problem 4.18. Compare your answer with a SPICE simulation. Also, compare your answer to the result that would apply without mismatch.