# GATE EC 2018

 Question 1
Two identical nMOS transistors $M_{1}$ and $M_{2}$ are connected as shown below. The circuit is used as an amplifier with the input connected between G and S terminals and the output taken between D and S terminals. $V_{bias}$ and $V_{D}$ are so adjusted that both transistors are in saturation. The transconductance of this combination is defined as $g_{m}=\frac{\partial i_{D}}{\partial V_{GS}}$ while the output resistance is $r_{0}=\frac{\partial V_{GS}}{\partial i_{D}}$ , where $i_{D}$ is the current flowing into the drain of $M_{2}$. Let $g_{m1}$ , $g_{m2}$ be the transconductances and $r_{01}$ , $r_{02}$ be the output resistances of transistors $M_{1}$ and $M_{2}$ , respectively. Which of the following statements about estimates for $g_{m}$ and $r_{0}$ is correct?
 A $g_{m}\approx g_{m1}\cdot g_{m2}\cdot r_{02} \;and \; r_0 \approx r_{01}+r_{02}.$ B $g_{m}\approx g_{m1}\ + g_{m2} \; and \; r_{0} \approx r_{01}+r_{02}.$ C $g_{m}\approx g_{m1} \; and \; r_{0}\approx r_{01} \cdot g_{m2}\cdot r_{02}.$ D $g_{m}\approx g_{m1} \; and \; r_{0}\approx r_{02}$.
Analog Circuits   FET and MOSFET Analysis
Question 1 Explanation: $g_{m}=\frac{\Delta I_{D}}{\Delta V_{\text {in }}}=\frac{i_{D}}{v_{g s}}=\frac{i_{D 1}}{v_{g s}}=g_{m 1}$
To calculate $r_{o}$ : \begin{aligned} v_{\pi 2} &=-I_{x} r_{01} \\ I_{x} &=g_{m 2} v_{\pi 2}+\frac{\left(V_{x}-I_{x} r_{01}\right)}{r_{02}} \\ I_{x} &=-g_{m 2} r_{01} I_{x}+\frac{V_{x}}{r_{02}}-I_{x} \frac{r_{01}}{r_{02}} \\ V_{x} &=r_{02}\left[1+r_{01} g_{m 2}+\frac{r_{01}}{r_{02}}\right] I_{x} \\ r_{0} &=\frac{V_{x}}{I_{x}}=r_{01}+r_{02}+r_{01} r_{02} g_{m 2} \\ & \approx r_{01} r_{02} g_{m 2} \end{aligned}
 Question 2
In the circuit shown below, the op-amp is ideal and Zener voltage of the diode is 2.5 volts. At the input, unit step voltage is applied, i.e. $v_{IN}(t)= u(t)$ volts. Also, at t= 0, the voltage across each of the capacitors is zero. The time t, in milliseconds, at which the output voltage $v_{OUT}$ crosses -10 V is
 A 2.5 B 5 C 7.5 D 10
Analog Circuits   Operational Amplifiers
Question 2 Explanation:
$\text{For} \quad t \gt 0,$ $I=\frac{1 V}{1 \mathrm{k} \Omega}=1 \mathrm{mA}$
Till $t=2.5 \mathrm{msec}$, both $V_{1}$ and $V_{2}$ will increase and after $t=2.5 \mathrm{msec}$, $V_{2}=2.5 \mathrm{V}$ and $V_{1}$ increases with time.
\begin{aligned} \text { when } v_{\text {out }}(t) &=-10 \mathrm{V} \\ & V_{1}=7.5 \mathrm{V}\\ \text{So,}\\ \frac{1}{1 \mu F} \int_{0}^{t}(1 \mathrm{m} \mathrm{A}) d t &=7.5 \mathrm{V} \\ 10^{3} t &=7.5 \\ t &=7.5 \mathrm{msec} \end{aligned}

 Question 3
A good transimpedance amplifier has
 A low input impedance and high output impedance. B high input impedance and high output impedance. C high input impedance and low output impedance. D low input impedance and low output impedance.
Analog Circuits   Feedback Amplifiers
Question 3 Explanation:
A good transimpedance amplifier should have low input impedance and low output impedance
 Question 4
Let the input be u and the output be y of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:
 A $\frac{d^{3}y}{dt^{3}} + a_{1} \frac{d^{2}y}{dt^{2}} + a_{2}\frac{dy}{dt} + a_{3}y =$ $b_{3}u+b_{2}\frac{du}{dt}+b_{1}\frac{d^{2}u}{dt^{2}}$ (with initial rest conditions) B $y(t)=\int_{0}^{t}e^{a(t-r)}\beta u(\tau)d \tau$ C $y= au +b$, $b \neq 0$ D $y=au$
Signals and Systems   Basics of Signals and Systems
Question 4 Explanation:
$y=a u+b, b \neq 0$ is a non-linear system.
 Question 5
The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE:
 A Both the criteria provide information relative to the stable gain range of the system. B The general shape of the Nyquist plot is readily obtained from the Bode magnitude plot for all minimum-phase systems. C The Routh criterion is not applicable in the condition of transport lag, which can be readily handled by the Nyquist criterion. D The closed-loop frequency response for a unity feedback system cannot be obtained from the Nyquist plot.
Control Systems   Frequency Response Analysis

There are 5 questions to complete.