Let's solve the current mirror problem step by step: --- ## **Given Data** - \( I_{in} = 80\, \mu A \) - \( Q_1: \frac{W}{L} = \frac{100}{1.6} = 62.5 \) - \( Q_2: \frac{W}{L} = \frac{25}{1.6} = 15.625 \) - Use 0.8-μm NMOS process from Table 1.5: - \( \mu C_{ox} = 92\, \mu A/V^2 \) - \( V_{T0} = 0.80\, V \) - \( \lambda \cdot L = 0.12\, (1/V) \cdot \mu m \implies \lambda = 0.12 / 1.6 = 0.075\, V^{-1} \) --- ## **Step 1: Nominal Output Current (\( I_{out} \))** For a current mirror: \[ I_{out} = I_{in} \times \frac{(W/L)_2}{(W/L)_1} \] Plug in the values: \[ I_{out} = 80\, \mu A \times \frac{15.625}{62.5} = 80\, \mu A \times 0.25 = \boxed{20\, \mu A} \] --- ## **Step 2: Output Resistance (\( r_{out} \))** The output resistance is given by: \[ r_{out} = \frac{1}{\lambda I_{out}} \] Recall: \[ \lambda = \frac{0.12}{1.6} = 0.075\, V^{-1} \] So, \[ r_{out} = \frac{1}{0.075 \times 20 \times 10^{-6}} = \frac{1}{1.5 \times 10^{-6}} = \boxed{666.7\, k\Omega} \] --- ## **Step 3: Minimum Output Voltage (\( V_{out, min} \))** For both transistors to remain in saturation (active region): \[ V_{DS} \geq V_{GS} - V_{T} \] Find \( V_{GS} \) for \( Q_2 \): The saturation current equation: \[ I_{out} = \frac{1}{2} \mu C_{ox} \frac{W}{L} (V_{GS} - V_T)^2 \] \[ 20 \times 10^{-6} = \frac{1}{2} \times 92 \times 10^{-6} \times 15.625 \times (V_{GS} - 0.8)^2 \] Calculate constant: \[ \frac{1}{2} \times 92 \times 10^{-6} \times 15.625 = 0.00071875 \approx 7.19 \times 10^{-4} \] So, \[ 20 \times 10^{-6} = 7.19 \times 10^{-4} (V_{GS} - 0.8)^2 \] \[ (V_{GS} - 0.8)^2 = \frac{20 \times 10^{-6}}{7.19 \times 10^{-4}} = \frac{20}{719} \times 10^{-2} = 0.0278 \] \[ V_{GS} - 0.8 = \sqrt{0.0278} = 0.167 \] \[ V_{GS} = 0.8 + 0.167 = 0.967\, V \] So, \[ V_{out, min} = V_{GS} - V_T = 0.967 - 0.8 = \boxed{0.167\, V} \] --- ## **Summary of Answers** 1. **Nominal Output Current (\( I_{out} \))**: \(\boxed{20\, \mu A}\) 2. **Output Resistance (\( r_{out} \))**: \(\boxed{666.7\, k\Omega}\) 3. **Minimum Output Voltage (\( V_{out, min} \))**: \(\boxed{0.167\, V}\) Let me know if you'd like more details or derivations!

Certainly! Here's a concise, step-by-step explanation of each part with theoretical reasoning: --- **Step 1: Calculating the Nominal Output Current** In a current mirror, the output current is scaled from the reference current based on the ratio of the transistor widths and lengths. Assuming both transistors are identical in process parameters, the current ratio simplifies to the W/L ratio. Therefore, the output current equals the input current multiplied by the ratio of W/L of the output transistor to that of the reference transistor. This proportionality stems from the quadratic current equation in saturation, where the drain current depends on device geometry, enabling predictable current copying. --- **Step 2: Determining the Output Resistance** The output resistance of a MOSFET in saturation is primarily influenced by channel-length modulation, modeled by the parameter λ. It quantifies how the drain current varies with changes in drain-to-source voltage, effectively creating a finite output impedance. The resistance is inversely proportional to the product of λ and the drain current. This resistance impacts the current mirror's output impedance, affecting its ability to maintain a constant current under varying load voltages. The calculation reflects the device's physical channel-length modulation behavior, which depends on process parameters. --- **Step 3: Finding the Minimum Output Voltage for Saturation** A MOSFET remains in saturation when the drain-to-source voltage exceeds \((V_{GS} - V_T)\). To ensure both transistors stay active, the output voltage must be at least \(V_{DS} \geq V_{GS} - V_T\). Using the quadratic current equation, we solve for \(V_{GS}\) that produces the desired current. This \(V_{GS}\) then determines the minimum \(V_{out}\) to keep the output transistor in saturation. The calculation ensures the device operates in its active region, preventing cutoff or triode operation, which would compromise the current mirror's accuracy. --- Let me know if you'd like further elaboration!