As with the telescopic cascode of the previous example, we begin with the power and swing specifications. Allocating 1.5 mA to the input pair, 1.5 mA to the two cascode branches, and the remaining $330 \mu \mathrm{~A}$ to the three current mirrors, we first consider the devices in each cascode branch. Since $M_5$ and $M_6$ must each carry 1.5 mA , we allow an overdrive of 500 mV for these transistors so as to keep their width to a reasonable value. To $M_3-M_4$, we allocate 400 mV and to $M_7-M_{10}, 300 \mathrm{mV}$. Thus, $(W / L)_{5,6}=400,(W / L)_{3,4}=313$, and $(W / L)_{7-10}=278$. Since the minimum and maximum output levels are equal to 0.6 V and 2.1 V , respectively, the optimum output common-mode level is 1.35 V .

The minimum dimensions of $M_1-M_2$ are dictated by the minimum input common-mode level, $V_{G S 1}+V_{O D 11}$. For example, if the input and the output CM levels are equal (Fig. 9.20), then $V_{G S 2}+V_{O D 11}=1.35 \mathrm{~V}$. With $V_{O D 11}=0.4 \mathrm{~V}$ as an initial guess, we have $V_{G S 1}=0.95 \mathrm{~V}$, obtaining $V_{O D 1,2}=0.95-0.7=0.25 \mathrm{~V}$, and hence $(W / L)_{1,2}=400$. The maximum dimensions of $M_1$ and $M_2$ are determined by the tolerable input capacitance and the capacitance at nodes $X$ and $Y$ in Fig. 9.18.

We now calculate the small-signal gain. Using $g_m=2 I_D /\left(V_{G S}-V_{T H}\right)$, we have $g_{m 1,2}=0.006 \mathrm{~A} / \mathrm{V}, g_{m 3,4} 0.0038 \mathrm{~A} / \mathrm{V}$, and $g_{m 7,8}=0.05 \mathrm{~A} / \mathrm{V}$. For $L=0.5 \mu \mathrm{~m}, r_{O 1,2}=r_{O 7-10}=13.3 \mathrm{k} \Omega$, and $r_{O 3,4}=2 r_{O 5,6}=6.67 \mathrm{k} \Omega$. It follows that the impedance seen looking into the drain of $M_7$ (or $M_8$ ) is equal to $8.8 \mathrm{M} \Omega$ whereas, owing to the limited intrinsic gain of $M_3$ (or $M_4$ ), that seen looking into the drain of $M_3$ is equal to $66.5 \mathrm{k} \Omega$. The overall gain is therefore limited to about 400.

In order to increase the gain, we first observe that $r_{O 5,6}$ is quite lower than $r_{O 1,2}$. Thus, the length of $M_5- M_6$ must be increased. Also, the transconductance of $M_1-M_2$ is relatively low and can be increased by widening these transistors. Finally, we may decide to double the intrinsic gain of $M_3$ and $M_4$ by doubling both their length and their width, but at the cost of increasing the capacitance at nodes $X$ and $Y$. We leave the exact choice of the device dimensions as an exercise for the reader. Note that the op amp must incorporate common-mode feedback