# 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
 Question 6
Consider p(s) = $s^{3}+a_{2}s^{2}+a_{1}s+a_{0}$ with all real coefficients. It is known that its derivative ${p}'(s)$ has no real roots. The number of real roots of ${p}(s)$ is
 A 0 B 1 C 2 D 3
Engineering Mathematics   Numerical Methods
Question 6 Explanation:
If p(s) has "r" real roots, then $p^{\prime}(s)$ will have atleast $"r-1^{\prime \prime}$ real roots.
 Question 7
In a p-n junction diode at equilibrium, which one of the following statements is NOT TRUE?
 A The hole and electron diffusion current components are in the same direction. B The hole and electron drift current components are in the same direction. C On an average, holes and electrons drift in opposite direction. D On an average, electrons drift and diffuse in the same direction.
Electronic Devices   PN-Junction Diodes and Special Diodes
Question 7 Explanation:

$\longrightarrow$ Hole diffusion
$\longleftarrow$ Electron diffusion
$\longleftarrow$ Hole drift
$\longrightarrow$ Electron drift
$\longrightarrow$ Hole diffusion current
$\longrightarrow$ Electron diffusion current
$\longleftarrow$ Hole drift current
$\longleftarrow$ Electron drift current
 Question 8
The logic function f(X,Y) realized by the given circuit is
 A NOR B AND C NAND D XOR
Digital Circuits   Logic Families
Question 8 Explanation:
From pull-down network,
\begin{aligned} \overline{f(X, Y)}&=\bar{X} \bar{Y}+X Y=X \odot Y \\ f(X, Y)&=\overline{X \odot Y}=X \oplus Y \end{aligned}
 Question 9
A function F(A,B,C) defined by three Boolean variables A, B and C when expressed as sum of products is given by

$F=\bar{A}\cdot \bar{B} \cdot \bar{C} + \bar{A}\cdot B \cdot \bar{C} + A\cdot \bar{B} \cdot \bar{C}$

where,$\bar{A},\bar{B} \;and \; \bar{C}$ are complements of the respective variable. The product of sums (POS) form of the function F is
 A $F=(A+B+C)\cdot (A+\tilde{B}+C)\cdot (\bar{A}+B+C)$ B $F=(\bar{A}+\bar{B}+\bar{C})\cdot (\bar{A}+B+\bar{C})\cdot (A+\bar{B}+\bar{C})$ C $F=(A + B + \bar{C}) \cdot (A + \bar{B} + \bar{C} ) \cdot (\bar{A} + B + \bar{C}) \cdot$ $(\bar{A}+\bar{B}+C) \cdot (\bar{A}+\bar{B}+\bar{C})$ D $F=(\bar{A} + \bar{B} + C) \cdot (\bar{A} + B + C) \cdot$ $(A + B + \bar{C}) \cdot (A+B+C)$
Digital Circuits   Boolean Algebra
Question 9 Explanation:
\begin{aligned} F(A, B, C, D) &=\bar{A} \bar{B} \bar{C}+\bar{A} B \bar{C}+A \bar{B} \bar{C} \\ &=\Sigma m(0,2,4)=\Pi M(1,3,5,6,7) \\ =&(A+B+\bar{C})(A+\bar{B}+\bar{C})(\bar{A}+B+\bar{C}) \\ &(\bar{A}+\bar{B}+C)(\bar{A}+\bar{B}+\bar{C}) & \end{aligned}
 Question 10
The points P, Q, and R shown on the Smith chart (normalized impedance chart) in the following figure represent:
 A P: Open Circuit, Q: Short Circuit, R: Matched Load B P: Open Circuit, Q: Matched Load, R: Short Circuit C P: Short Circuit, Q: Matched Load, R: Open Circuit D P: Short Circuit, Q: Open Circuit, R: Matched Load
Electromagnetics   Transmission Lines
Question 10 Explanation:
For Short circuit,
$r=x=0 \quad \Rightarrow \text { Point } " P^{\prime \prime}$
For Open circuit,
$r=x=\infty \quad \Rightarrow \text { Point }^{\prime \prime} R^{\prime \prime}$
For Matched load,
$r=1 \text { and } x=0 \Rightarrow \text { Point " } Q^{\prime \prime}$
P: Short Circuit, Q: Matched Load R: Open circuit
There are 10 questions to complete.