GATE EC 2016 Set 2 — Question 22

The question asks to identify the incorrect statement regarding the resistance of a long-channel NMOS transistor biased in the linear region.The drain current ($I_D$) for an NMOS transistor in the linear (or triode) region is given by the equation:$$I_D = \mu_n C_{ox} \frac{W}{L} \left[ (V_{GS} - V_{th})V_{DS} - \frac{V_{DS}^2}{2} \right]$$where:

• $\mu_n$ is the electron mobility
• $C_{ox}$ is the gate oxide capacitance per unit area
• $W$ is the channel width
• $L$ is the channel length
• $V_{GS}$ is the gate-to-source voltage
• $V_{th}$ is the threshold voltage
• $V_{DS}$ is the drain-to-source voltage

The transistor is used as a resistance, which is defined as $R_{DS} = \frac{V_{DS}}{I_D}$.
Given that $V_{DS} = 50$ mV, which is a small value, the condition $V_{DS} \ll 2(V_{GS} - V_{th})$ for linear region operation is satisfied, and we can approximate the current by neglecting the $V_{DS}^2$ term.$$I_D \approx \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})V_{DS}$$The resistance $R_{DS}$ is then:$$R_{DS} = \frac{V_{DS}}{I_D} \approx \frac{1}{\mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})}$$Now, let's analyze each statement based on this expression for $R_{DS}$:(A) If the device width $W$ is increased, the denominator increases, so the resistance $R_{DS}$ decreases. This statement is correct.(B) If the threshold voltage $V_{th}$ is reduced, the term $(V_{GS} - V_{th})$ increases. This increases the denominator, so the resistance $R_{DS}$ decreases. This statement is correct.(C) If the device length $L$ is increased, the denominator decreases (since $L$ is in the denominator of the $\frac{W}{L}$ term), so the resistance $R_{DS}$ increases. This statement is correct.(D) If $V_{GS}$ is increased, the term $(V_{GS} - V_{th})$ increases. This increases the denominator, so the resistance $R_{DS}$ decreases. The statement says the resistance increases, which is contrary to our finding. This statement is NOT correct.The question asks for the statement that is NOT correct. Therefore, option (D) is the correct answer.