GATE Electrical Engineering Solved Papers

Let $a_R$ be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of $n$, the divergence of the radial vector field $f(R) = a_R \frac{1}{R^n}$ is independent of $R$?

Consider two coupled circuits, having self-inductances $L_1$ and $L_2$, that carry non-zero currents $I_1$ and $I_2$, respectively. The mutual inductance between the circuits is $M$ with unity coupling coefficient. The stored magnetic energy of the coupled circuits is minimum at which of the following value(s) of $\frac{I_1}{I_2}$?

Let $(x, y) \in \mathfrak{R}^2$. The rate of change of the real valued function

$$V(x, y) = x^2 + x + y^2 + 1$$

at the origin in the direction of the point $(1, 2)$ is

Consider ordinary differential equations given by

$$\dot{x}_1(t) = 2x_2(t)$$

$$\dot{x}_2(t) = r(t)$$

with initial conditions $x_1(0) = 1$ and $x_2(0) = 0\text{ (zero)}$. If $r(t) = \begin{cases} 1, & t \ge 0\text{ (zero)} \\ 0\text{ (zero)}, & t < 0\text{ (zero)} \end{cases}$, then at $t = 1$, $x_1(t)$ is

Let $C$ be a clockwise oriented closed curve in the complex plane defined by $|z| = 1$. Further, let $f(z) = jz$ be a complex function, where $j = \sqrt{-1}$. Then,

$$\oint_C f(z) dz =$$

The op-amps in the following circuit are ideal. The voltage gain of the circuit is

The switch (S) closes at $t = 0\text{ sec}$. The time, in sec, the capacitor takes to charge to $50\text{ V}$ is

In an experiment to measure the active power drawn by a single-phase RL Load connected to an AC source through a $2\ \Omega$ resistor, three voltmeters are connected as shown in the figure below. The voltmeter readings are as follows:

$$V_{\text{source}} = 200\text{ V}, \quad V_R = 9\text{ V}, \quad V_{\text{Load}} = 199\text{ V}$$

Assuming perfect resistors and ideal voltmeters, the Load-active power measured in this experiment, in W, is

In the Wheatstone bridge shown below, the sensitivity of the bridge in terms of change in balancing voltage $E$ for unit change in the resistance $R$, in $\text{mV}/\Omega$, is

The steady state capacitor current of a conventional DC-DC buck converter, working in CCM, is shown in one switching cycle. If the input voltage is $30\text{ V}$, the value of the inductor used, in mH, is

An ideal low pass filter has frequency response given by

$$H(j\omega) = \begin{cases} 1, & |\omega| \le 200\pi \\ 0\text{ (zero)}, & \text{otherwise} \end{cases}$$

Let $h(t)$ be its time domain representation. Then the value of $h(0)$ is

Consider the state-space model

$$\dot{x}(t) = Ax(t) + Br(t)$$

$$y(t) = Cx(t)$$

where $x(t)$, $r(t)$, and $y(t)$ are the state, input, and output, respectively. The matrices $A$, $B$, $C$ are given below:

$$A = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \end{bmatrix}$$

The sum of the magnitudes of the poles of the system is

Using shunt capacitors, the power factor of a 3-phase, $4\text{ kV}$ induction motor (drawing $390\text{ kVA}$ at $0.77\text{ pf}$ lag) is to be corrected to $0.85\text{ pf}$ lag. The line current of the capacitor bank, in A, is

Two units, rated at $100\text{ MW}$ and $150\text{ MW}$, are enabled for economic load dispatch. When the overall incremental cost is $10,000\text{ Rs./MWh}$, the units are dispatched to $50\text{ MW}$ and $80\text{ MW}$ respectively. At an overall incremental cost of $10,600\text{ Rs./MWh}$, the power output of the units are $80\text{ MW}$ and $92\text{ MW}$, respectively. The total plant MW-output (without overloading any unit) at an overall incremental cost of $11,800\text{ Rs./MWh}$ is

A controller $D(s)$ of the form $(1 + K_D s)$ is to be designed for the plant $G(s) = \frac{1000\sqrt{2}}{s(s+10)^2}$ as shown in the figure. The value of $K_D$ that yields a phase margin of $45^{\circ}$ at the gain cross-over frequency of $10\text{ rad/sec}$ is