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?
g_{m}\approx g_{m1}\cdot g_{m2}\cdot r_{02} \;and \; r_0 \approx r_{01}+r_{02}.
g_{m}\approx g_{m1}\ + g_{m2} \; and \; r_{0} \approx r_{01}+r_{02}.
g_{m}\approx g_{m1} \; and \; r_{0}\approx r_{01} \cdot g_{m2}\cdot r_{02}.
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
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
low input impedance and high output impedance.
high input impedance and high output impedance.
high input impedance and low output impedance.
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:
\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)
y(t)=\int_{0}^{t}e^{a(t-r)}\beta u(\tau)d \tau
y= au +b, b \neq 0
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:
Both the criteria provide information relative to the stable gain range of the system.
The general shape of the Nyquist plot is readily obtained from the Bode magnitude plot for all minimum-phase systems.
The Routh criterion is not applicable in the condition of transport lag, which can be readily handled by the Nyquist criterion.
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.