Master previous years' GATE questions with verified analytical derivations, mathematical phasor solutions, and fundamental theorems across Electric Circuits, Electromagnetic Fields, Signals & Systems, Electrical Machines, Power Systems, Control Systems, Electrical Measurements, Analog & Digital Electronics, and Power Electronics.
Marking pattern: +1.00 / +2.00 positive, -0.33 / -0.66 negative for MCQs (Standard GATE Scheme).
The operating region of the developed torque ($T_{em}$) and speed ($\omega$) of an induction motor drive is given by the shaded region OQRE in the figure. The load torque ($T_L$) characteristic is also shown. The motor drive moves from the initial operating point O to the final operating point S. Which one of the following trajectories will take the shortest time?
The input voltage $v(t)$ and current $i(t)$ of a converter are given by,
$$v(t) = 300 \sin(\omega t) \text{ V}$$
$$i(t) = 10 \sin\left(\omega t - \frac{\pi}{6}\right) + 2 \sin\left(3\omega t + \frac{\pi}{6}\right) + \sin\left(5\omega t + \frac{\pi}{2}\right) \text{ A}$$
where, $\omega = 2\pi \times 50 \text{ rad/s}$. The input power factor of the converter is closest to
Instrument(s) required to synchronize an alternator to the grid is/are
The open-loop transfer function of the system shown in the figure, is
$$G(s) = \frac{K s (s + 2)}{(s + 5)(s + 7)}$$
For $K \ge 0\text{ (zero)}$ which of the following real axis point(s) is/are on the root locus?
A continuous time periodic signal $x(t)$ is
$$x(t) = 1 + 2 \cos(2\pi t) + 2 \cos(4\pi t) + 2 \cos(6\pi t)$$
If $T$ is the period of $x(t)$, then the value of $\frac{1}{T}\int_{0}^{T} |x(t)|^2 dt$ is
The maximum percentage error in the equivalent resistance of two parallel-connected resistors of $100\ \Omega$ and $900\ \Omega$ with each having a maximum $5\%$ error is
Consider a distribution feeder, with $R/X$ ratio of 5. At the receiving end, a $350\text{ kVA}$ load is connected. The maximum voltage drop will occur from the sending end to the receiving end, when the power factor of the load is
The bus impedance matrix of a 3-bus system (in pu) is
$$Z_{\text{bus}} = \begin{bmatrix} j0.059 & j0.061 & j0.038 \\ j0.061 & j0.093 & j0.066 \\ j0.038 & j0.066 & j0.110 \end{bmatrix}$$
A symmetrical fault (through a fault impedance of $j0.007\text{ pu}$) occurs at bus 2. Neglecting pre-fault loading conditions, the voltage at bus 1, during the fault in pu is
In the circuit with ideal devices, the power MOSFET is operated with a duty cycle of 0.4 in a switching cycle with $I = 10\text{ A}$ and $V = 15\text{ V}$. The power delivered by the current source, in W, is
The induced emf in a $3.3\text{ kV}$, 4 pole, 3-phase star connected synchronous motor is considered to be equal and in phase with the terminal voltage under no load condition. On application of a mechanical load, the induced emf phasor is deflected by an angle of $2^{\circ}$ mechanical with respect to the terminal voltage phasor. If the synchronous reactance is $2\ \Omega$, and stator resistance is negligible, then the motor armature current magnitude, in ampere, during loaded condition is closest to
Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y = X^2$ and
$$P_X(x) = \begin{cases} 1, & x \in (0, 1] \\ 0, & \text{otherwise} \end{cases}$$
Which one of the following options is correct?
A Boolean function is given as
$$f = (\overline{u} + \overline{v} + \overline{w} + \overline{x}) \cdot (\overline{u} + \overline{v} + \overline{w} + x) \cdot (\overline{u} + v + \overline{w} + \overline{x}) \cdot (\overline{u} + v + \overline{w} + x)$$
The simplified form of this function is represented by
In the circuit, $I_{\text{DC}}$ is an ideal current source. The transistors $M_1$ and $M_2$ are assumed to be biased in saturation, wherein $V_{\text{in}}$ is the input signal and $V_{\text{DC}}$ is a fixed DC voltage. Both transistors have a small signal resistance of $r_{ds}$ and trans-conductance of $g_m$. The small signal output impedance of this circuit is
In the circuit shown below, if the values of $R$ and $C$ are very large, the form of the output voltage for a very high frequency square wave input, is best represented by
Let continuous-time signals $x_1(t)$ and $x_2(t)$ be
$$x_1(t) = \begin{cases} 1, & t \in [0, 1] \\ 2 - t, & t \in [1, 2] \\ 0, & \text{otherwise} \end{cases}$$
and
$$x_2(t) = \begin{cases} t, & t \in [0, 1] \\ 2 - t, & t \in [1, 2] \\ 0, & \text{otherwise} \end{cases}$$
Consider the convolution $y(t) = x_1(t) * x_2(t)$. Then $\int_{-\infty}^{\infty} y(t) dt$ is
Let $G(s) = \frac{1}{(s+1)(s+2)}$. Then the closed-loop system shown in the figure below, is
The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system S. The output is observed to be the continuous-time unit step signal $u(t)$. Which one of the following statements is true?
The transformer connection given in the figure is part of a balanced 3-phase circuit where the phase sequence is "abc". The primary to secondary turns ratio is 2:1. If $(I_a + I_b + I_c = 0)$, then the relationship between $I_A$ and $I_{ad}$ will be
A DC series motor with negligible series resistance is running at a certain speed driving a load, where the load torque varies as cube of the speed. The motor is fed from a $400\text{ V}$ DC source and draws $40\text{ A}$ armature current. Assume linear magnetic circuit. The external resistance, in $\Omega$, that must be connected in series with the armature to reduce the speed of the motor by half, is closest to
A 3-phase, $400\text{ V}$, 4 pole, $50\text{ Hz}$ star connected induction motor has the following parameters referred to the stator:
$$R_r' = 1\ \Omega, \quad X_s = X_r' = 2\ \Omega$$
Stator resistance, magnetizing reactance and core loss of the motor are neglected. The motor is run with constant $V/f$ control from a drive. For maximum starting torque, the voltage and frequency output, respectively, from the drive, is closest to,
The 3-phase modulating waveforms ($v_a(t)$, $v_b(t)$ and $v_c(t)$), used in sinusoidal PWM in a Voltage Source Inverter (VSI) are
$$v_a(t) = 0.8 \sin(\omega t) \text{ V}$$
$$v_b(t) = 0.8 \sin\left(\omega t - \frac{2\pi}{3}\right) \text{ V}$$
$$v_c(t) = 0.8 \sin\left(\omega t + \frac{2\pi}{3}\right) \text{ V}$$
where $\omega = 2\pi \times 40\text{ rad/s}$ is the fundamental frequency. The modulating waveforms are compared with a $10\text{ kHz}$ triangular carrier whose magnitude varies between $+1$ and $-1$. The VSI has a DC link voltage of $600\text{ V}$ and feeds a star connected motor. The per phase fundamental RMS motor voltage, in volts, is closest to
An ideal sinusoidal voltage source $v(t) = 230\sqrt{2}\sin(2\pi \times 50t)\text{ V}$ feeds an ideal inductor $L$ through an ideal SCR with firing angle $\alpha = 0^{\circ}$. If $L = 100\text{ mH}$, then the peak of the inductor current, in ampere, is closest to
In the following circuit, the average voltage
$$V_o = 400\left(1 + \frac{\cos\alpha}{3}\right)\text{ V}$$
where $\alpha$ is the firing angle. If the power dissipated in the resistor is $64\text{ W}$, then the closest value of $\alpha$ in degrees is
In the system shown below, the generator was initially supplying power to the grid. A temporary LLLG bolted fault occurs at F very close to circuit breaker 1. The circuit breakers open to isolate the line. The fault self-clears. The circuit breakers reclose and restore the line. Which one of the following diagrams best indicates the rotor accelerating and decelerating areas?
An air filled cylindrical capacitor (capacitance $C_0$) of length $L$, with $a$ and $b$ as its inner and outer radii, respectively, consists of two coaxial conducting surfaces. Its cross-sectional view is shown in Fig. (i). In order to increase the capacitance, a dielectric material of relative permittivity $\epsilon_r$ is inserted inside 50% of the annular region as shown in Fig. (ii). The value of $\epsilon_r$ for which the capacitance of the capacitor in Fig. (ii), becomes $5C_0$ is