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.
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