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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 \( V_{GS} - V_T \).

For two matched MOS transistors with W/L=12, k'n=390µA/V^2, and V_m=0.5V, design the circuit to provide Io=90µA. Given that R=9Ω, find V_DD and the minimum value Vo can have while Q2 still operates in saturation.

Learn how to design a circuit using two matched MOS transistors with W/L=12, k'n=390µA/V^2, and Vm=0.5V. This problem involves calculating Io, V_DD, and the minimum Vo value required for Q2 to operate in saturation. Suitable for electrical engineering students.

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