Question

Consider the CMOS circuit shown in the figure (substrates are connected to their respective sources). The gate width $(W)$ to gate length $(L)$ ratios of the transistors are as shown. Both the transistors have the same gate oxide capacitance per unit area. For the pMOSFET, the threshold voltage is $-1\ \text{V}$ and the mobility of holes is $40\ \text{cm}^2/(\text{V·s})$. For the nMOSFET, the threshold voltage is $1\ \text{V}$ and the mobility of electrons is $300\ \text{cm}^2/(\text{V·s})$. The steady state output voltage $V_O$ is ________________.

• A. equal to 0 V
• B. more than 2 V
• C. less than 2 V
• D. equal to 2 V

Hint:

In CMOS steady state, equate pMOS and nMOS saturation currents. Always consider threshold voltages and mobility ratios carefully.

Answer 1

The Correct Option is C

At steady state, both MOSFETs are in saturation and the drain currents must match (KCL at the output node). For the pMOS, the current magnitude is \[ I_p = \frac{1}{2}\mu_p C_{ox} \left( \frac{W}{L} \right)_p (V_{SG} - |V_{TP}|)^2. \] For the nMOS, the drain current is \[ I_n = \frac{1}{2}\mu_n C_{ox} \left( \frac{W}{L} \right)_n (V_{GS} - V_{TN})^2. \] Step 1: Substitute given parameters.
pMOS: \( \mu_p = 40,\ W/L = 5,\ V_{TP} = -1,\ V_{SG} = 4 - V_O \). nMOS: \( \mu_n = 300,\ W/L = 1,\ V_{TN} = 1,\ V_{GS} = V_O \). \[ I_p = \frac{1}{2} (40) C_{ox} (5) (4 - V_O - 1)^2 = 100 C_{ox} (3 - V_O)^2. \] \[ I_n = \frac{1}{2} (300) C_{ox} (1) (V_O - 1)^2 = 150 C_{ox} (V_O - 1)^2. \] Step 2: Apply current equality (magnitude). \[ I_p = I_n. \] \[ 100 (3 - V_O)^2 = 150 (V_O - 1)^2. \] Divide by 50: \[ 2 (3 - V_O)^2 = 3 (V_O - 1)^2. \] Step 3: Solve for $V_O$. Expand: \[ 2(9 - 6V_O + V_O^2) = 3(V_O^2 - 2V_O + 1). \] \[ 18 - 12V_O + 2V_O^2 = 3V_O^2 - 6V_O + 3. \] Rearrange: \[ 0 = V_O^2 + 6V_O - 15. \] Quadratic formula: \[ V_O = \frac{-6 \pm \sqrt{36 + 60}}{2} = \frac{-6 \pm \sqrt{96}}{2} = -3 \pm 2\sqrt{6}. \] Only the positive root is valid: \[ V_O = -3 + 2\sqrt{6}. \] Since \(2\sqrt{6} \approx 4.90\): \[ V_O \approx 1.90\ \text{V}. \] Step 4: Conclusion.
\[ V_O \approx 1.9\ \text{V}<2\ \text{V}. \] Final Answer: less than 2 V