Consider the set of points which minimize the real valued function
Which of the following statements is true about the set S?
Thus,
Calculating stationary points:
Let
Let
where is an arbitrary constant.
Thus, these are infinitely many stationary points.
Option (b) is correct.
Let and be the two eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix. Which one of the following statements is true?
The eigen vectors of a real symmetric matrix corresponding to different eigen values are always pair-wise orthogonal.
So, for the given two eigen vectors and ,
option (B) is correct.
Let , and . Then, the system of linear equations has
will have an infinite no. of solutions
Option (b) is correct.
Let and let I can be identity matrix. Then is equal to
Option (a) is correct.
Consider discrete random variables and with probabilities as follows:
Given , the expected value of is
Given: and
Option (b) is correct.
Which one of the following statements is true about the small signal voltage gain of a MOSFET based single stage amplifier?
For common source Amplifier,
∴ Inverting Amplifier
For common gate Amplifier,
Non-Inverting Amplifier
For common drain Amplifier,
∴ Non-Inverting Amplifier
Hence, Option (c) is correct.
Assuming ideal op-amps, the circuit represents a
Using virtual short analysis,
&
Option (d) is correct.
The I-V characteristics of the element between the nodes X and Y is best depicted by
Hence, Option (b) is correct
A nullator is defined as a circuit element where the voltage across the device and the current through the device are both zero. A series combination of a nullator and a resistor of value, R, will behave as a
V=0 and I=0
Overall circuit will also behave as a Nullator.
Hence, option (b) is correct.
Consider a discrete-time linear time-invariant (LTI) system , where
Let
where is the discrete-time unit impulse function. For an input signal , the output is
Given a discrete-time LTI system ,
In other way,
Hence, option (a) is correct.
Consider a continuous-time signal
where is the continuous-time unit step function. Let be the continuous time unit impulse function. The value of is
Hence, option (a) is correct.
Selected data points of the step response of a stable first-order linear time-invariant (LTI) system are given below. The closest value of the time-constant, in sec, of the system is
For standard order system,
Output,
where and Time Constant
For
Hence, option (b) is correct.
The Nyquist plot of a strictly stable having the numerator polynomial as encircles the critical point -1 once in the anti-clockwise direction. Which one of the following statements on the closed-loop system shown in figure, is correct?
Given that open loop system is stable.
No. of open -loop poles in RH of s-plane
Also, No. of encirclements about critical point is once,
Assume Nyquist Contour is clockwise
N = 1
N = P-Z
P= Open Loop Poles in RHP = 0
Z = Closed Loop Poles in RHP
Hence Z = -1 which is invalid
Now assume Nyquist Contour is Anti-Clockwise
N = -1 = P – Z
Hence Z = 1, which means one closed loop pole
lies in RHP and hence closed loop system is unstable
Option (d) is correct.
During a power failure, a domestic household uninterruptible power supply (UPS) supplies AC power to a limited number of lights and fans in various rooms. As per a Newton-Raphson load-flow formulation, the UPS would be represented as a
During a power failure, the only power source is UPS. Since for convergence of load flow solution we need to declare one bus as Slack Bus and that bus needs to have power source. In our case only one bus has power source i.e. UPS and hence that bus needs to be treated as Slack Bus.
Which one of the following figures represents the radial electric field distribution caused by a spherical cloud of electrons with a volume charge density, for (both are positive and is the radial distance) and for ?
Given:
For ,
Option (c) is correct
The operating region of the developed torque and speed () of an induction motor drive is given by the shaded region OQRE in the figure. The load torque 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 shortest time to reach the final operating point S happens when the motor accelerates with the maximum possible torque.
In the given characteristics, the maximum developed torque corresponds to the uppermost boundary of the shaded region, the line .
So, the fastest trajectory will follow: .
Hence, correct answer is option (a).
The input voltage and current of a converter are given by,
where, . The input power factor of the converter is closest to
Option (a) is correct.
Instrument(s) required to synchronize an alternator to the grid is/are
The correct answer is option (a) Voltmeter and (c) synchroscope.
Voltmeter: It is used to measure the voltage of the alternator and ensure it matches the grid voltage before synchronization.
Synchroscope: It helps to match the phase angle and speed of the alternator with the grid's frequency before connecting it.
The open-loop transfer function of the system shown in the figure, is
For , which of the following real axis point(s) is/are on the root locus?
( Root locus lies on real axis when the sum of poles and zeros on its right side is odd).
Root locus lies from to and to .
from the given options, -1 and -6 lies on the Root locus.
Hence, option (a) & (c) is correct.
A continuous time periodic signal x(t) is
If is the period of , then __________ (round off to the nearest integer).
This expression represents Power contained in the signal as per Parseval's Energy Theorem Since, the signal is in terms of a Fourier Series all the signals are orthogonal and hence their powers can be directly added.
The maximum percentage error in the equivalent resistance of two parallel-connected resistors of and with each having a maximum error is _________ % (round off to nearest integer value).
Correct answer is 5.
Consider a distribution feeder, with ratio of 5. At the receiving end, a 350 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 ________ (round off to three decimal places).
For maximum voltage drop,
Load power factor angle
The bus impedance matrix of a 3-bus system (in pu) is
A symmetrical fault (through a fault impedance of ) occurs at bus 2. Neglecting pre-fault loading conditions, the voltage at bus 1 , during the fault is __________ pu (round off to three decimal places).
when the fault occurs at bus-2,
Voltage at bus -1, (during fault)
In the circuit with ideal devices, the power MOSFET is operated with a duty cycle of 0.4 in a switching cycle with and . The power delivered by the current source, in W, is __________ (round off to the nearest integer).
Duty Cycle, and
when MOSFET is ON,
when MOSFET is OFF current source makes diodes ON, and
Power delivered by current source,
The induced emf in a 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 mechanical with respect to the terminal voltage phasor. If the synchronous reactance is , and stator resistance is negligible, then the motor armature current magnitude, in ampere, during loaded condition is closest to, _________ (round off to two decimal places).
poles P=4, connected synchronous motor
Under No-load- (equal in magnitude & in-phase)
At load: mechanical (given),
From phasor diagram,
And,
In
Let and be continuous random variables with probability density functions and , respectively. Further, let and
Which one of the following options is correct?
Given: X and Y be two continuous random variables with Probability density function (PDF) and .
And,
For continuous Probability density function,
Option (a) is correct.
A Boolean function is given as
The simplified form of this function is represented by
The is given in POS form
Option (a) is correct.
In the circuit, is an ideal current source. The transistors and are assumed to be biased in saturation, wherein is the input signal and is fixed DC voltage. Both transistors have a small signal resistance of and trans-conductance of . The small signal output impedance of this circuit is
infinity
Transistors and are biased in saturation.
In AC analysis: (DC Supply) = Short Circuited
Current Source = Open circuited
Concept:
where
Option (c) is correct.
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
Step 1: Understanding the Square Wave Input
The input voltage is a square wave defined as:
This pattern repeats every T seconds.
Step 2: Charging Phase ( )
At t=0, the source switches to +1V, and the capacitor begins charging from its initial voltage (where ).
The charging equation for the capacitor voltage is:
Here:
- = +1V (target voltage).
- (starting voltage).
Substituting these values:
At , the capacitor voltage reaches +V (due to steady-state symmetry):
Step 3: Solving for V
Rearranging the equation:
For high RC( ) , the exponential term can be approximated using the Taylor series:
Substituting this approximation:
Simplifying:
Neglecting the small term :
The exact solution is:
For , so .
Step 4: Discharging Phase
At , the source switches to , and the capacitor begins discharging toward from .
The discharging equation is:
At , the capacitor voltage reaches , completing the cycle.
Final Expression for Capacitor Voltage
For :
For :
Conclusion
The capacitor voltage oscillates between and , where:
This is because the capacitor cannot fully charge/discharge due to the high RC time constant. The waveform resembles a clipped exponential curve, oscillating below the input square wave amplitude.
Let continuous-time signals and be
Consider the convolution . Then is
from area property,
Option (a) is correct.
Let . Then the closed-loop system shown in the figure below, is
stable for all .
unstable for all .
unstable for all .
stable for all .
Open Loop Transfer function,
for unity feedback system, characteristic equation is:
for stable system, and
From all options, unstable for all is correct.
The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system . The output is observed to be the continuous-time unit step signal . Which one of the following statements is true?
Every bounded input signal applied to results in a bounded output signal.
It is possible to find a bounded input signal which when applied to results in an unbounded output signal.
On applying any input signal to , the output signal is always bounded.
On applying any input signal to the output signal is always unbounded.
Given
Since the unit impulse input produces output as u(t). Hence u(t) is the impulse response of the system.
For input (bounded signal),
Any signal convolved with , gives its running integral,
Thus, every bounded signal does not produces every bounded o/p signal
h(t) = u(t) is not an absolutely integrable signal
Hence, the system is not BIBO stable so it can produce unbounded output for bounded input
Hence, option (b) is correct.
The transformer connection given in the figure is part of a balanced 3-phase circuit where the phase sequence is "". The primary to secondary turns ratio is . If , then the relationship between and will be
and lags by .
and leads by .
and lags by .
and leads by .
Given : .
Primary and secondary phase currents of same phase are drawn. parallel to each other in phasor.
Thus, lags by .
Option (a) is correct.
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 V DC source and draws 40 A armature current. Assume linear magnetic circuit. The external resistance, in , that must be connected in series with the armature to reduce the speed of the motor by half, is closest to
Given: ,
Series resistance,
Load Torque,
In series motor,
….(i)
…(ii)
…(iii)
Under balance condition,
From eq (iii),
On Solving,
A 3-phase, pole, 50 Hz star connected induction motor has the following parameters referred to the stator:
Stator resistance, magnetizing reactance and core loss of the motor are neglected. The motor is run with constant control from a drive. For maximum starting torque, the voltage and frequency output, respectively, from the drive, is closest to,
Given:
- Rotor resistance referred to stator:
- Stator + rotor leakage reactance at
- Constant V/f control:
Key Insight:
For maximum torque at start, the starting slip (s=1) must equal the slip for maximum torque:
Where is the total reactance at the new frequency .
Step 1: Set for Maximum Starting Torque
Step 2: Find New Frequency
Since reactance scales with frequency:
Set :
Step 3: Compute New Voltage
Using constant :
Line-to-line voltage:
The 3-phase modulating waveforms and , used in sinusoidal PWM in a Voltage Source Inverter (VSI) are
where is the fundamental frequency. The modulating waveforms are compared with a 10 kHz triangular carrier whose magnitude varies between +1 and -1. The VSI has a DC link voltage of 600 V and feeds a star connected motor. The per phase fundamental RMS motor voltage, in volts, is closest to
Peak of each phase modulating waveforms,
Peak of triangular carrier wave,
For Sinusoidal PWM Inverter,
Peak of Fundamental phase voltage,
feeds an ideal inductor through an ideal SCR with firing angle . If , then the peak of the inductor current, in ampere, is closest to
for , SCR will behave as diode.
at
Peak of inductor current occurs at
In the following circuit, the average voltage
where is the firing angle. If the power dissipated in the resistor is 64 W, then the closest value of in degrees is
Average output Voltage,
Load: and
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?
Given: A temporary LLLG bolted fault occurs at point F near to bus-bar .
Let : Reactance of Transmission lines
After fault occurs (when CB open the faulty line),
Initial operating point: on Curve 1 (under Normal conditions).
when 3-phase symmetrical fault occurs near bus-bar,
(electrical power transfer during fault)
Operating pt. moves to .
to : Duration of fault or Time taken by to clear the fault
At pt. opens and isolate the faulty line.
Due to which operating pt. moves to pt. c on curve-2.
Then, due to accelerating power, operating pt. accelerate upto pt. on curve 2 .
At some pt. D, CB again closes after rectification of fault & thus curve again moves to curve 1, operating pt. E. and can swing upto pt. F maintaining stable conditions.
for stable conditions, Accelerating Area decelerating Area
Hence, option (b) is correct.
An air filled cylindrical capacitor (capacitance ) of length , with and 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 is inserted inside of the annular region as shown in Fig. (ii). The value of for which the capacitance of the capacitor in Fig. (ii), becomes is
It is given that
Since half of the region is filled with air and half with a dielectric the two parts can be considered to be connected in parallel and the capacitance is equally divided as
Option (c) is contest.
Let be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of , the divergence of the radial vector field is independent of ?
1
2
Step 1: Compute Divergence in Spherical Coordinates
For a purely radial field , the divergence is:
Given , substitute:
Step 2: Simplify the Expression
.
Take the derivative:
Thus, the divergence becomes:
.
Step 3: Identify When Divergence is Independent of R
For the divergence to be independent of R, the exponent of R must vanish:
But wait! There's another case where the divergence is independent of R:
If the prefactor , the divergence becomes zero (trivially independent of R).
This occurs when:
Verification for Both Cases:
1. For :
2. For :
Consider two coupled circuits, having self-inductances and , that carry nonzero currents and , respectively. The mutual inductance between the circuits is with unity coupling coefficient. The stored magnetic energy of the coupled circuits is minimum at which of the following value(s) of ?
for minimizing the stored Magnetic energy,
Thus, for minimum stored energy,
Let . The rate of change of the real valued function,
at the origin in the direction of the point is __________ (round off to the nearest integer).
Unity vector directed from to ,
Rate of change of at in direction of point
=0 (round off to Nearest Integer).
Consider ordinary differential equations given by
with initial conditions and .
If , then at __________ (round off to the nearest integer).
Given:
Initial conditions: &
Input,
Let be a clockwise oriented closed curve in the complex plane defined by . Further, let be a complex function, where . Then, __________ (round off to the nearest integer).
Method-1
Given:
(-ve sign as contour is in clockwise direction, which is negative direction of angle)
Method-2
No poles of exist inside the contour
is an analytic function inside the contour .
By Cauchy Integral Theorem
The op-amps in the following circuit are ideal. The voltage gain of the circuit is ________ (round off to the nearest integer).
For ideal op-amps,
Using at Node A
Using Virtual short concept,
Using at Node
Using virtual short concept,
The switch (S) closes at . The time, in sec, the capacitor takes to charge to 50 V is ________ (round off to one decimal place).
For t<0 sec,
Under steady-state,
For ,
= 25V
Under steady state (when capacitor is fully charged and current becomes zero),
Let time takes to charge the capacitor to 50 V.
In an experiment to measure the active power drawn by a single-phase RL Load connected to an AC source through a resistor, three voltmeters are connected as shown in the figure below. The voltmeter readings are as follows: . Assuming perfect resistors and ideal voltmeters, the Load-active power measured in this experiment, in W, is __________ (round off to one decimal place).
Let be the Load pf angle.
In the Wheatstone bridge shown below, the sensitivity of the bridge in terms of change in balancing voltage for unit change in the resistance , in , is __________ (round off to two decimal places).
Initially,
, wheat stone bridge is balanced
For unit change in resistance R,
Change in balancing voltage
As No sign of is given in question, take only magnitude
Sensitivity,
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 V, the value of the inductor used, in mH, is ________ (round off to one decimal place).
Input voltage,
In DC-DC buck converters, during conduction (ON) of switch, i.e,
from 0 to , (capacitor current) increases.
when switch is off, i.e from to decreases.
from wave form, μsec
Slope of slope of
Also,
(as Load current is constant)
When switch is ON :
An ideal low pass filter has frequency response given by
Let be its time domain representation. Then __________ (round off to the nearest integer).
Consider the state-space model
where are the state, input and output, respectively. The matrices are given below
The sum of the magnitudes of the poles is __________ (round off to nearest integer).
Given:
Characteristic equation,
Poles of the system is calculated as:
Sum of Magnitudes of poles
Using shunt capacitors, the power factor of a 3 -phase, 4 kV induction motor (drawing 390 kVA at 0.77 pf lag) is to be corrected to 0.85 pf lag. The line current of the capacitor bank, in A, is _________ (round off to one decimal place).
Induction Motor: ,
After correcting capacitor bank, .
From above phasor, complex power drawn by motor
complex power drawn by total Load (motor + capacitor bank)
Reactive power supplies by capacitor,
Line current of capacitor bank =9.05 A
Two units, rated at 100 MW and 150 MW, are enabled for economic load dispatch. When the overall incremental cost is 10,000 Rs./MWh, the units are dispatched to 50 MW and 80 MW respectively. At an overall incremental cost of , the power output of the units are 80 MW and 92 MW, respectively. The total plant MW-output (without overloading any unit) at an overall incremental cost of is __________ (round off to the nearest integer).
Given: For &
For
Let
For lossless system,
&
On solving,
for &
On solving,
Now, Rs/ MWh,
Since,
Thus,
Total plant Mw -output without overloading any
A controller of the form is to be designed for the plant as shown in the figure. The value of that yields a phase margin of at the gain cross-over frequency of is ___________ (round off to one decimal place).
Given:
Gain cross-over frequency, .
Phase Margin at
Open-loop Transfer function of system, OLTF
…(i)
From equation (i),
