The problem involves two matched MOS transistors with the following parameters:

• $\frac{W}{L} = 12$
• $k_n' = 390 \mu A/V^2$
• $V_m = 0.5 V$
• Desired output current: $I_O = 90 \mu A$
• $R = 9 \, \Omega$

We are tasked with:

1. Designing the circuit to provide $I_O = 90 \mu A$.
2. Calculating $V_{DD}$.
3. Finding the minimum value of $V_O$ while transistor $Q_2$ is still in saturation.

1. Design the Circuit to Provide $I_O = 90 \mu A$

For a MOS transistor in saturation, the drain current $I_D$ is given by: $I_D = \frac{k_n' (W/L)}{2} (V_{GS} - V_T)^2$ Where:

• $I_D = I_O = 90 \mu A$
• $k_n' = 390 \mu A/V^2$
• $\frac{W}{L} = 12$
• $V_m$ is often treated as $V_T = 0.5V$.

Substituting the known values, we have: $90 \mu A = \frac{390 \mu A/V^2 \times 12}{2} (V_{GS} - 0.5)^2$

Simplifying: $90 \mu A = 2340 \mu A/V^2 (V_{GS} - 0.5)^2$ $\frac{90 \mu A}{2340 \mu A/V^2} = (V_{GS} - 0.5)^2$ $0.03846 = (V_{GS} - 0.5)^2$ $V_{GS} - 0.5 = \sqrt{0.03846}$ $V_{GS} - 0.5 = 0.1961$ $V_{GS} = 0.6961 V$

Thus, $V_{GS} = 0.6961 V$.

2. Calculate $V_{DD}$

The voltage across $R$ (denoted as $V_R$) is given by Ohm's Law: $V_R = I_O \times R = 90 \mu A \times 9 \Omega = 0.81 V$

The total supply voltage $V_{DD}$ is the sum of $V_{GS}$ and $V_R$: $V_{DD} = V_R + V_{GS} = 0.81 V + 0.6961 V = 1.5061 V$

3. Minimum Value of $V_O$ for $Q_2$ in Saturation

For transistor $Q_2$ to remain in saturation, the condition is: $V_{DS2} \geq V_{GS2} - V_T$ Since $V_{GS2} = V_{GS1}$ (because the transistors are matched and in the same configuration): $V_{DS2} = V_O \geq V_{GS2} - V_T = 0.6961 V - 0.5 V = 0.1961 V$

Thus, the minimum value of $V_O$ for $Q_2$ to remain in saturation is $V_O = 0.1961 V$.

Summary of Answers:

1. Circuit design provides $I_O = 90 \mu A$.
2. $V_{DD} = 1.5061 V$.
3. The minimum value of $V_O$ for $Q_2$ to remain in saturation is $0.1961 V$.

Do you want more details or clarifications?

Related Questions:

1. How does the $W/L$ ratio affect the operation of MOS transistors in this circuit?
2. Can you derive the expression for $I_D$ in the linear region of operation?
3. What happens if $V_{DS}$ is less than $V_{GS} - V_T$ in transistor $Q_2$?
4. How would increasing $R$ affect the required $V_{DD}$?
5. How can you ensure that both transistors remain in saturation under different load conditions?

Tip:

When designing MOSFET circuits, always verify the operating region by comparing $V_{DS}$ with $V_{GS} - V_T$.