We begin with the power budget, allocating 3 mA to $M_9$ and the remaining $330 \mu \mathrm{~A}$ to $M_{b 1}$ and $M_{b 2}$. Thus, each cascode branch of the op amp carries a current of 1.5 mA . Next, we consider the required output swings. Each of nodes $X$ and $Y$ must be able to swing by $1.5 \mathrm{~V}_{p p}$ without driving $M_3-M_6$ into the triode region. With a 3-V supply, therefore, the total voltage available for $M_9$ and each cascode branch is equal to 1.5 V , i.e., $\left|V_{O D 7}\right|+\left|V_{O D 5}\right|+V_{O D 3}+V_{O D 1}+V_{O D 9}=1.5 \mathrm{~V}$.

Since $M_9$ carries the largest current, we choose $V_{O D 9} \approx 0.5 \mathrm{~V}$, leaving 1 V for the four transistors in the cascode. Moreover, since $M_5-M_8$ suffer from low mobility, we allocate an overdrive of approximately 300 mV to each, obtaining 400 mV for $V_{O D 1}+V_{O D 3}$. As an initial guess, $V_{O D 1}=V_{O D 3}=200 \mathrm{mV}$.

With the bias current and overdrive voltage of each transistor known, we can easily determine the aspect ratios from $I_D=(1 / 2) \mu C_{o x}(W / L)\left(V_{G S}-V_{T H}\right)^2$ or simulated I/V characteristics. To minimize the device capacitances, we choose the minimum length for each transistor, obtaining a corresponding width. We then have $(W / L)_{1-4}=1250$, and $(W / L)_{5-8}=1111$, and $(W / L)_9=400$.

The reader may think that the above choice of overdrives is arbitrary and leads to a wide design space. However, we must emphasize that each of the overdrives has but a small range. For example, we can change the allocated values by only a few tens of millivolts before the device dimensions become disproportionately large.

The design has thus far satisfied the swing, power dissipation, and supply voltage specifications. But, how about the gain? Using $A_v \approx g_{m 1}\left[\left(g_{m 3} r_{O 3} r_{O 1}\right) \|\left(g_{m 5} r_{O 5} r_{O 7}\right)\right]$ and assuming minimum channel length for all of the transistors, we have $A_v=1416$, quite a lot lower than the required value.

In order to increase the gain, we recognize that $g_m r_O=\sqrt{2 \mu C_{o x}(W / L) I_D} /\left(\lambda I_D\right)$. Now, recall that $\lambda \propto 1 / L$, and hence $g_m r_O \propto \sqrt{W L / I_D}$. We can therefore increase the width or length or decrease the bias current of the transistors. In practice, speed or noise requirements may dictate the bias current, leaving only the dimensions as the variables. Of course, the width of each transistor must at least scale with its length so as to maintain a constant overdrive voltage.

Which transistors in the circuit of Fig. 9.11 should be made longer? Since $M_1-M_4$ appear in the signal path, it is desirable to keep their capacitances to a minimum. The PMOS devices, $M_5-M_8$, on the other hand, affect the signal to a much lesser extent and can therefore have larger dimensions. ${ }^3$ Doubling the (effective) length and width of each of these transistors in fact doubles their $g_m r_O$ because $g_m$ remains constant while $r_O$ increases by a factor of 2. Choosing $(W / L)_{5-8}=2222 \mu \mathrm{~m} / 1.0 \mu \mathrm{~m}$ and hence $\lambda_p=0.1 \mathrm{~V}^{-1}$, we obtain $A_v \approx 4000$. Thus, the PMOS dimensions can be somewhat smaller. Note that with such large dimensions for PMOS transistors, we may revisit our earlier distribution of the overdrive voltages, possibly reducing that of $M_9$ by 100 to 200 mV and allocating more to the PMOS devices.

In the op amp, the input CM level and the bias voltages $V_{b 1}$ and $V_{b 2}$ must be chosen so as to allow maximum output swings. The minimum allowable input CM level equals $V_{G S 1}+V_{O D 9}=V_{T H 1}+V_{O D 1}+V_{O D 9}=$ 1.4 V . The minimum value of $V_{b 1}$ is given by $V_{G S 3}+V_{O D 1}+V_{O D 9}=1.6 \mathrm{~V}$, placing $M_1-M_2$ at the edge of the triode region. Similarly, $V_{b 2, \max }=V_{D D}-\left(\left|V_{G S 5}\right|+\left|V_{O D 7}\right|\right)=1.7 \mathrm{~V}$. In practice, some margin must be included in the value of $V_{b 1}$ and $V_{b 2}$ to allow for process variations. Also, the increase in the threshold voltages due to body effect must be taken into account. Finally, we should remark that this op amp requires common-mode feedback.