GATE EC 2003 Set 1 Q1 — Circuit Analysis

The Laplace Transform of the signal $x(t)=u(t-2) *(t u(t))$ is given by which of the following expressions? ["*" represents convolution operator]

The relation between the input current ( $I$ ) and the output voltage ( $V$ ) of a circuit is governed by the equation: $C \dfrac{d V}{d t}=I(t)-m(t)$. The circuit is excited by $I(t)=q \delta(t)$, where $q$ is a real valued constant. $V$ at $t=0^{-}$is $V_{0}$. Which of the following is an equivalent representation of the above case?

The output voltage $V_{o}$ (in Volt) for the network given in the Figure is $\_\_\_\_$ . (rounded off to two decimal places)

Consider the circuit shown in the Figure, where the input $v_{i}(t)$ is in Volt. The average power (in $\text{mW}$ ) dissipated in the load resistance of $1 ~\mathrm{k} \Omega$ at the resonant frequency is $\_\_\_\_$ . (rounded off to two decimal places)

For a lossless passive two-port network, $\left|S_{11}\right|$ and $\left|S_{21}\right|$ intersect at $-3$ dB . For a lossy passive two-port network, $\left|S_{11}\right|$ and $\left|S_{21}\right|$ intersect at $-4$ dB . The percentage of power dissipated in the lossy network at the intersection frequency is $\_\_\_\_$. (rounded off to two …

In the given circuit, $L=1 \mu \mathrm{H}$ and $C=1 \mu \mathrm{~F}$. The phasor diagram for $I_{C}$ and $I_{L}$ is also shown. Assume that the phase $\left(\theta_{1}+\theta_{2}\right)$ is $90^{\circ}$ at a frequency of $159.15 ~\text{kHz}.$ Among the following options, what is the nearest integer value of $R_{C} \times R_{L}?$