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?

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}. |

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

The time t, in milliseconds, at which the output voltage v_{OUT} crosses -10 V is

2.5 | |

5 | |

7.5 | |

10 |

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}

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. |

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 | |

y=au |

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. |

There are 5 questions to complete.