In the following circuit, the input voltage $${V_{in}}$$ is $$100\sin \left( {100\pi t} \right).$$ For $$100\pi RC = 50,$$ the average voltage across $$R$$ (in volts) under steady-state is nearest to

When a bipolar junction transistor is operating in the saturation mode, which one of the following statements is TRUE about the state of its collector-base (CB) and the base-emitter (BE) junctions?

In the following circuit, the transistor is in active mode and $${V_C} = 2V.$$ To get $${V_C} = 4V,$$ we replace $${R_C}$$ with $${R_C}.$$ Then the ratio $$R{'_C}/{R_C}$$ is __________.

The operational amplifier shown in the figure is ideal. The input voltage (in Volt) is $${V_i} = 2\sin \left( {2\pi \times 2000t} \right).$$ The amplitude of the output voltage $${V_0}$$ (in Volt) is __________.

The filters $${F_1}$$ and $${F_2}$$ having characteristics as shown in figures $$(a)$$ and $$(b)$$ are connected as shown in Figure $$(c).$$

The cut-off frequencies of $$F1$$ and $$F2$$ are $${f_1}$$ and $${f_2}$$ respectively. If $${f_1}\, < \,{f_2},$$ the resultant circuit exhibits the characteristic of a

The saturation voltage of the ideal op-amp shown below is $$ \pm 10\,V.$$ The output voltage $${V_O}$$ of the following circuit in the steady-state is

The unit step response of a system with the transfer function $$G\left( s \right) = {{1 - 2s} \over {1 + s}}$$ is given by which one of the following waveforms?

An open loop control system results in a response of $${e^{ - 2t}}\left( {\sin 5t + \cos 5t} \right)$$ for a unit impulse input. The DC gain of the control system is __________.

The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables $$𝑥$$ and $$𝑦.$$ The integration time step is $$h.$$ $$${{{x_{k + 1}} - {x_k}} \over h} = {y_k},\,\,\,\,\,{{{y_{k + 1}} - {y_k}} \over h} = {x_k}$$$

For this discrete-time system, which one of the following statements is TRUE?

An open loop transfer function $$G(s)$$ of system is $$G\left( s \right) = {k \over {s\left( {s + 1} \right)\left( {s + 2} \right)}}$$

For a unity feedback system, the breakaway point of the root loci on the real axis occurs at,

Nyquist plots of two functions $${G_1}\left( s \right)$$ and $${G_2}\left( s \right)$$ are shown in figure.

Nyquist plot of the product of $${G_1}\left( s \right)$$ and $${G_2}\left( s \right)$$ is

For the system governed by the set of equations: $$$\eqalign{ & d{x_1}/dt = 2{x_1} + {x_2} + u \cr & d{x_2}/dt = - 2{x_1} + u \cr & \,\,\,\,\,\,y = 3{x_1} \cr} $$$
the transfer function $$Y(s)/U(s)$$ is given by

Consider the following Sum of products expression, $$F.$$
$$F = ABC + \overline A \overline B C + A\overline B C + \overline A BC + \overline A \overline B \overline C $$

The equivalent Product of Sums expression is

A Boolean function $$f\left( {A,B,C,D} \right) = \Pi \left( {1,5,12,15} \right)$$ is to be implemented using an $$8 \times 1$$ multiplexer ($$A$$ is $$MSB$$). The inputs $$ABC$$ are connected to the select inputs $${S_2}{S_1}{S_0}$$ of the multiplexer respectively.

Which one of the following options gives the correct inputs to pins $$0,1,2,3,4,5,6,7$$ in order?

In the following sequential circuit, the initial state (before the first clock pulse) of the circuit is $${Q_1}{Q_0}$$ $$=00.$$ The state $$\left( {{Q_1}{Q_0}} \right),$$ immediately after the $${333^{rd}}$$ clock pulse is

A series RL circuit is excited at t = 0 by closing a switch as shown in the figure. Assuming zero initial conditions, the value of $$\frac{\mathrm d^2\mathrm I}{\mathrm{dt}^2}$$ at t=0⁺ is

Find the transformer ratios a and b that the impedance (Z_in) is resistive and equal 2.5 Ω when the network is excited with a sine wave voltage of angular frequency of 5000 rad/s.

In the given network $$V_1=100\angle0^\circ V$$, $$V_2=100\angle-120^\circ V$$, $$V_3=100\angle+120^\circ V$$. The phasor current i (in Ampere) is

Two identical coils each having inductance L are placed together on the same core. If an overall inductance of $$\alpha$$L is obtained by interconnecting these two coils, the minimum value of $$\alpha$$ is ________.

A symmetrical square wave of 50% duty cycle has amplitude of ±15V and time period of 0.4$$\mathrm\pi$$ ms. This square wave is applied across a series RLC circuit with R = 5Ω, L = 10 mH, and C = 4μF. The amplitude of the 5000 rad/s component of the capacitor voltage (in Volt) is __________.

The current i (in Ampere) in the 2 Ω resistor of the given network is _________.

A capacitive voltage divider is used to measure the bus voltage V_bus in a high-voltage 50Hz AC system as shown in the figure. The measurement capacitors C₁ and C₂ have tolerances of $$ \pm $$10% on their nominal capacitance values. If the bus voltage V_bus is 100 kV rms, the maximum rms output voltage V_out (in kV), considering the capacitor tolerances, is __________.

Match the following.

A 3-phase balanced load which has a power factor of 0.707 is connected to a balanced supply. The power consumed by the load is 5 kW. The power is measured by the two-wattmeter method. The readings of the two wattmeters are

The coils of a wattmeter have resistances $$0.01$$ $$\Omega $$ and $$1000$$ $$\Omega $$; their inductances may be neglected. The wattmeter is connected as shown in the figure, to measure the power consumed by a load, which draws $$25$$ $$A$$ at power factor $$0.8$$. The voltage across the load terminals is $$30$$ $$V$$. The percentage error on the wattmeter reading is______________.

A three-phase, $$11$$ $$kV,$$ $$50$$ $$Hz,$$ $$2$$ pole, star connected, cylindrical rotor synchronous motor is connected to an $$11$$ $$kV,$$ $$50$$ $$Hz$$ source. Its synchronous reactance is $$50\Omega $$ per phase, and its stator resistance is negligible. The motor has a constant field excitation. At a particular load torque, its stator current is $$100$$ $$A$$ at unity power factor. If the load torque is increased so that the stator current is $$120$$ $$A,$$ then the load angle (in degrees) at this load is ____________.

A 4-pole, separately excited, wave wound DC machine with negligible armature resistance is rated for 230 V and 5 kW at a speed of 1200 rpm. If the same armature coils are reconnected to form a lap winding, what is the rated voltage (in volts) and power (in kW) respectively at 1200 rpm of the reconnected machine if the field circuit is left unchanged?

A shunt-connected DC motor operates at its rated terminal voltage. Its no-load speed is 200 radian/second. At its rated torque of 500 Nm, its speed is 180 radian/second. The motor is used to directly drive a load whose load torque T_L depends on its rotational speed $$\omega_r$$ (in radian/second), such that $$T_L=2.78\times\omega_r$$. Neglecting rotational losses, the steady-state speed (in radian/second) of the motor, when it drives this load, is _______.

With an armature voltage of 100 V and rated field winding voltage, the speed of a separately excited DC motor driving a fan is 1000 rpm, and its armature current is 10 A. The armature resistance is 1 Ω. The load torque of the fan load is proportional to the square of the rotor speed. Neglecting rotational losses, the value of the armature voltage (in Volt) which will reduce the rotor speed to 500 rpm is ______.

Two three-phase transformers are realized using single-phase transformers as shown in the figure. The phase difference (in degree) between voltages V₁ and V₂ is _______.

A balanced (positive sequence) three-phase AC voltage source is connected to a balanced, star connected load through a star-delta transformer as shown in the figure. The line-to-line voltage rating is 230 V on the star side, and 115 V on the delta side. If the magnetizing current is neglected and $${\overline{\mathrm I}}_\mathrm s=100\angle0^\circ$$, then what is the value of $${\overline{\mathrm I}}_\mathrm p$$ in Ampere?

A three-winding transformer is connected to an AC voltage source as shown in the figure. The number of turns are as follows: N₁=100, N₂=50, N₃=50. If the magnetizing current is neglected, and the currents in two windings are $${\overline I}_2=2\angle30^\circ$$ A and $${\overline I}_3=2\angle30^\circ$$ A, then what is the value of the current $${\overline I}_1$$ in Ampere?

The figure shows the per-phase equivalent circuit of a two-pole three-phase induction motor operating at 50 Hz. The 'air-gap' voltage, V_g across the magnetizing inductance, is 210V_rms, and the slip, s, is 0.05. The torque (in Nm) produced by the motor is ________.

A 220 V, 3-phase, 4-pole, 50 Hz inductor motor of wound rotor type is supplied at rated voltage and frequency. The stator resistance, magnetizing reactance, and core loss are negligible. The maximum torque produced by the rotor is 225 % of full load torque and it occurs at 15% slip. The actual rotor resistance is 0.03 Ω/phase. The value of external resistance (in Ohm) which must be inserted in a rotor phase if the maximum torque is to occur at start is ______.

Match the following.

Two semi-infinite conducting sheets are placed at right angles to each other as shown in the figure. A point charge of +𝑄 is placed at a distance of 𝑑 from both sheets. The net force on the charge is $$\frac{Q^2}{4{\mathrm{πε}}_0}\frac{\overrightarrow K}{d^2}$$ , where $$\overrightarrow K$$ is given by

Two semi-infinite dielectric regions are separated by a plane boundary at y = 0. The dielectric constants of region 1 (y< 0) and region 2 (y>0) are 2 and 5, respectively. Region 1 has uniform electric field $$\overrightarrow E=3{\widehat a}_x\;+\;4{\widehat a}_y\;+\;2{\widehat a}_z$$, where $${\widehat a}_x$$, $${\widehat a}_y$$, and $${\widehat a}_Z$$ are unit vectors along the x, y and z axes, respectively. The electric field in region 2 is

A circular turn of radius 1 m revolves at 60 rpm about its diameter aligned with the x-axis as shown in the figure. The value of μ0 is $$4\mathrm\pi\times10^{-7}$$ in SI unit. If a uniform magnetic field intensity $$\overrightarrow H=10^7\;\widehat z\;A/m$$ is applied, then the peak value of the induced voltage, V_turn ( in Volts), is _________.

Given $$f\left( z \right) = g\left( z \right) + h\left( z \right),$$ where $$f,g,h$$ are complex valued functions of a complex variable $$z.$$ Which ONE of the following statements is TRUE?

The Laplace transform of $$f\left( t \right) = 2\sqrt {t/\pi } $$$$\,\,\,\,\,$$ is$$\,\,\,\,\,$$ $${s^{ - 3/2}}.$$ The Laplace transform of $$g\left( t \right) = \sqrt {1/\pi t} $$ is

We have a set of $$3$$ linear equations in $$3$$ unknown. $$'X \equiv Y'$$ means $$X$$ and $$Y$$ are equivalent statements and $$'X \ne Y'$$ means $$X$$ and $$y$$ are not equivalent statements.

$$P:$$ There is a unique solution.
$$Q:$$ The equations are linearly independent .
$$R:$$ All eigen values of the coefficient matrix are non zero .
$$S:$$ The determinant of the coefficient matrix is non-zero .

Which one of the following is TRUE?

The volume enclosed by the surface $$f\left( {x,y} \right) = {e^x}$$ over the triangle bounded by the lines $$x=y;$$ $$x=0;$$ $$y=1$$ in the $$xy$$ plane is ________.

Match the following.

List-$${\rm I}$$
$$P.$$ Stoke's Theorem
$$Q.$$ Gauss's Theorem
$$R.$$ Divergence Theorem
$$S.$$ Cauchy's Integral Theorem

List-$${\rm I}{\rm I}$$
$$1.$$
$$2.$$
$$3.$$
$$4.$$

Two coins $$R$$ and $$S$$ are tossed. The $$4$$ joint events $$\,\,\,\,\,\,{H_R}{H_S},\,\,\,\,{T_R}{T_S},\,\,\,\,{H_R}{T_S},\,\,\,\,{T_R}{H_S}\,\,\,\,\,\,\,$$ have probabilities $$0.28,$$ $$0.18,$$ $$0.30,$$ $$0.24$$ respectively, where $$H$$ represents head and $$T$$ represents tail. Which one of the following is TRUE?

A differential equation $$\,\,{{di} \over {dt}} - 0.21 = 0\,\,$$ is applicable over $$\,\, - 10 < t < 10.\,\,$$ If $$i(4)=10,$$ then $$i(-5)$$ is

In the given rectifier, the delay angle of the thyristor $${T_1}$$ measured from the positive going zero crossing of $${V_s}$$ is $${30^ \circ }$$. If the input voltage $${V_s}$$ is $$100$$ $$sin$$ $$\left( {100\,\pi t} \right)$$ $$V,$$ the average voltage across $$R$$ (in Volt) under steady-state is ________.

For the switching converter shown in the following figure, assume steady-state operation. Also assume that the components are ideal, the inductor current is always positive and continuous and switching period is $${T_s}.$$ If the voltage $${V_L}$$ is as shown, the duty cycle of the switch $$S$$ is __________.

The incremental costs (in rupees/$$MWh$$) of operating two generating units are functions of their respective powers $${P_1}$$ and $${P_2}$$ in $$MW,$$ and are given by $$${{d{C_1}} \over {d{P_1}}} = 0.2{P_1} + 50$$$ $$${{d{C_2}} \over {d{P_2}}} = 0.24{P_2} + 40$$$
Where, $$$\eqalign{ & 20\,MW \le {P_1} \le 150\,MW \cr & 20\,MW \le {P_2} \le 150MW. \cr} $$$
For a certain load demand, $${P_1}$$ and $${P_2}$$ have been chosen such that $$\,\,d{C_1}/d{P_1} = 76\,Rs/MWh\,\,$$ and $$\,d{C_2}/d{P_2} = 68.8\,Rs/MWh.\,\,$$ If the generations are rescheduled to minimize the total cost, then $${P_2}$$ is ____________.

A $$3$$-phase transformer rated for $$33 kV/11kV$$ is connected in delta/star as shown in figure. The current transformers $$\left( {C{T_s}} \right)$$ on low and high voltage sides have a ratio of $$500/5.$$ Find the currents $${i_1}$$ and $${i_2}$$, if the fault current is $$300$$ $$A$$ as shown in figure.

A $$3$$-bus power system network consists of $$3$$ transmission lines. The bus admittance matrix of the uncompensated system is
$$\left[ {\matrix{ { - j6} & {j3} & {j4} \cr {j3} & { - j7} & {j5} \cr {j4} & {j5} & { - j8} \cr } } \right]\,pu$$
If the shunt capacitance of all transmission lines is $$50$$% compensated, the imaginary part of the $$3$$^rd row $$3$$^rd column element (in $$pu$$) of the bus admittance matrix after compensation is

The synchronous generator shown in the figure is supplying active power to an infinite bus via two short, lossless transmission lines, and is initially in steady state. The mechanical power input to the generator and the voltage magnitude E are constant. If one line is tripped at time t₁ by opening the circuit breakers at the two ends (although there is no fault), then it is seen that the generator undergoes a stable transient. Which one of the following waveforms of the rotor angle $$\delta $$ shows the transient correctly?

A composite conductor consists of three conductors of radius $$R$$ each. The conductors are arranged as shown below. The geometric mean radius (GMR) (in cm) of the composite conductor is kR. The value of k is ______.

For linear time invariant systems, that are Bounded Input Bounded stable, which one of the following statement is TRUE?

Consider a signal defined by $$$x\left(t\right)=\left\{\begin{array}{l}e^{j10t}\;\;\;for\;\left|t\right|\leq1\\0\;\;\;\;\;\;\;for\;\;\left|t\right|>1\end{array}\right.$$$ Its Fourier Transform is

The Laplace transform of f(t)=$$2\sqrt{t/\mathrm\pi}$$ is $$s^{-3/2}$$. The Laplace transform of g(t)=$$\sqrt{1/\mathrm{πt}}$$ is

The z-Transform of a sequence x[n] is given as X(z) = 2z+4−4/z+3/z². If y[n] is the first difference of x[n], then Y(Z) is given by