TANGEDCO AE EE Full Test - 3 - Electrical Engineering (EE) MCQ

Given complex no.

Rationalize the above expression

Concept:

Cauchy-Riemann Equation (Condition for a function to be analytic):

If z = x + iy and w = f(z) = u(x,y) + iv(x,y) then

i.e. u_x = v_y and u_y = -v_x

Calculation:

Given:

w = z² ⇒ (x + iy)²

∴ w = (x² – y²) + i(2xy)

Comparing with standard form i.e. w = f(z) = u(x,y) + iv(x,y)

u(x, y) = x² – y² and v(x, y) = 2xy

u_x = 2x, -v_x = -2y

u_y = -2y, v_y = 2x

C-R equations are satisfied every where in the finite part of complex plane and all first order partial derivatives u_x, u_y, v_x, v_y are continuous function of x and y

∴ f(z) = z² is analytic everywhere i.e., f(z) = z² is entire function.

Other Related Points

Entire function:

A function f(z) which is analytic at every point of the finite complex plane is known as entire functions.

Eg. polynomial and exponential functions are entire functions.

Concept:
System of linear equations are defined by
a₁₁x₁ + a₁₂x₂ + ⋯ + a_1nx_n = b₁
a₂₁x₁ + a₂₂x₂ + ⋯ + a_2nx_n = b₂
⋮ ⋮ ⋮
a_m1x₁ + a_m2x₂ + ⋯ + a_mnx_n = b_n
‘m’ number of equations and ‘n’ number of unknowns.
Augmented matrix is defined as:

Calculation:
2x - 3y + 6z - 5t =3 
y - 4z + t = 1
4x - 5y + 8z - 9t = k
Number of unknowns is 4 here.
Matrix ‘A’ is defined as

Performing Row operations and making Matrix into Echelon Form and it is defined below.

In this matrix, ‘k’ is a non zero value.
R₃ → R₃ – 2R₁

R₃ → R₃ - R₂

The number of non zero rows is 2. So, rank = 2
Now augmented matrix is (A|B)

For k = 7, R(A|B) = 2 and R < n
So we get an infinite number of solutions.

Concept:

If any surface is given by f = f(x, y, z), To find the directional derivative at any point P(x₁, y₁, z₁) on the surface in the direction of the vector. Follow the following steps.

Find,

Then, find the value of ∇f at point P(x₁, y₁, z₁).

Calculation:

Given:

f(x, y, z) = 2x² + 3y2 + z² - 11

⇒ î(4x) + ĵ(6y) + k̂(2z)

At point P(1, 0, 3), ∇f = î (4 × 1) + ĵ (6 × 0) + k̂ (2 × 3) = 4î + 6k̂

Direction vector () = î + 2ĵ + k̂

Newton's forward difference formula is given as:

Where

Newton's backward difference formula is given as:

Where

Newton's divided difference formula is given as:

Gauss forward difference formula is given as:

Where and

∴ The correct option is 1.

Intrinsic Impedance of free space is given by

μ = 4 π x 10 -7 H/m

ϵ = 8.85 x 10 -12 F/m

• In a synchronous generator, the magnetic field rotates at synchronous speed and the rotating magnetic field is created in the stator. These two fields are not fully aligned. The stator field lags the rotating field. This lagging angle is called a load angle or torque angle or power angle. It is denoted by ‘δ’.
• For stable operation, the maximum angle of torque angle is 90° i.e. 0 < δ < 90°. But in practical stable systems, the normal value of ‘δ’ lies between 0 to 30°.
• For stability and economic reasons, we operate the transmission line with a power angle in the range of 30° to 45°
• In an unstable system, δ increases indefinitely with time, and the machine loses synchronism.

Important Points:
Equal Area Criteria:

• The equal-area criterion is a simple graphical method for concluding the transient stability.
• This principle does not require the swing equation for the determination of stability conditions.
• The stability conditions are recognized by equating the areas of segments on the power angle diagram.

​Starting with the swing equation:
Where,
M is the moment of inertia or angular momentum
δ is the power angle between rotor rotating field and stator rotating field
P_s is a mechanical input power
P_e is the electrical output
Multiply the above equation with dδ / dt
we get,
Rearranging, multiplying by dt, and integrating, we have
At steady-state condition, the torque angle was not changing i.e. before the disturbance.
dδ / dt = 0
Also, if the system has transient stability the machine will again operate at synchronous speed after the disturbances, i.e.,
dδ / dt = 0
Hence the condition for stability is dδ / dt = 0

Corona loss (P_L):

Where,

f = Supply frequency in Hz

r = Radius of conductor

d = distance between two adjacent conductors

V_o = Operating voltage in kV

V_C = Critical disrputive voltage in kV

δ = Air density correction factor

• Frequency of supply: Corona loss increases as the supply frequency increases
• Air Pressure: In hilly areas, the corona effect is more dominant due to reduced pressure
• By increasing conductor size, the voltage at which corona occurs is raised and hence corona effects are considerably reduced.
• By increasing the spacing between conductors, the voltage at which corona occurs is raised and hence corona effects can be eliminated.

Concept:

• Friction: Friction is an opposing force that comes into play when the body is actually moving (slide or roll) or even tries to move over the surface of another body.
  • Static friction(F_s): the opposing force that is active when the body tends to move over the surface, but the actual motion is yet not started.
  • Kinetic friction(F_k): kinetic or dynamic force is an opposing force that is active when the body is actually moving on another surface.
• Kinetic friction is always slightly less than static friction.
  • F_k < F_s​

• The unbalance force (P – F_k) produces acceleration.
  • F_s ≤ μ_sN
  • where μ_s is called the coefficient of static friction.
• The magnitude of the kinetic friction force:
  • F_k = μ_kN
  • where μ_k is called the coefficient of kinetic friction.

The coefficients μ_s and μ_k do not depend on the surface area in contact but depend on the nature of the surfaces in contact.
The coefficient of kinetic friction is generally less than the coefficient of static friction.

Concept:
Le Chatelier's principle states that - "If an equilibrium is subjected to a change of concentration, pressure or temperature, the equilibrium shifts in the direction that tends to undo the effect of the change".
Explanation:
In the reaction, , the most favourable conditions of temperature and pressure for greater yield of SO₂ are high pressure and low temperature.

• High pressure is maintained because the moles of the product are less than the moles of reactants.

• When pressure is increased, the equilibrium will shift to product side (which contains less number of moles of gaseous species).
• This nullifies the effect of increase in pressure. Hence, more and more product will be formed.
• Low temperature favours exothermic reaction. Heat is evolved during the reaction (positive value of enthalpy change).
• When an inert gas is added to an equilibrium system at constant volume, there will be an increase in the total pressure of the system. The concentrations of the products and reactants will not change.
• Therefore, there will be no effect on the equilibrium.

So, the reaction is favoured by only the increase in pressure.

The correct answer is option 2, i.e. The principle involved in paper chromatography is Partition.

• Chromatography is a separation technique which is used to separate the components of a mixture.
• There are two phases: a stationary phase and a mobile phase
  • Stationary phase provides a differential resistance to the components of the mixture so that they move with a different pace.
  • Mobile phase carries the components of the mixture with it and moves over the stationary phase.
• In paper chromatography, the water trapped in the paper acts as a stationary phase and a suitable solvent acts as a mobile phase.
• The components get separated due to the difference in the way they get attracted to water or the mobile phase.

Based on the nature of the operation, instruments can be classified into three types:
Indicating Instruments: These indicate the quantity being measured by means of a pointer that moves on a scale. These instruments indicate the instantaneous value of the electrical quantity being measured at the time at which it is being measured.

• Example: Ammeter, Voltmeter, Wattmeter

Recording Instruments: These instruments record continuously the variation of any electrical quantity with respect to time. In principle, these are indicating instruments but so arranged that a permanent continuous record of the indication is made on a chart or dial.
Any electrical quantity like current, voltage can be recorded by a suitable recording mechanism.

• Example: A potentiometric type of recorder used for monitoring temperature records the instantaneous temperatures on a strip chart recorder, CRO, DSO.

Integrating Instruments: These instruments record the consumption of the total quantity of electricity, energy, etc. during a particular period of time. These instruments give reading for a specific period of time but no indication of reading for a particular instant of time.

• Example: Ampere-hour meter, Household energy meter, kilovolt ampere-hour meter.

Concept:
Nuclear reactor:

• It is a device in which a nuclear reaction is initiated, maintained, and controlled.
• It works on the principle of controlled chain reaction and provides energy at a constant rate.

Explanation:

• Nuclear Fuel: It is a fissionable material to be used for the fission process to take place.
  • Commonly used fuels in a nuclear reactor are U²³³, U²³⁵, PU²³⁹, etc.
  • Generally, uranium oxide pellets are inserted end to end into long hollow metal tubes constituting the fuel rods. When slow neutrons interact with the fuel, the fission starts, and the energy is released.
• Ceramics hold a unique position in nuclear fission reactors since they are used for fuel, the coating for fuel particles and pressure or reactor vessels, and as the materials of moderator and reflector, control and shielding. Therefore option 2 is correct.

Concept:
Linearity: Necessary and sufficient condition to prove the linearity of the system is that linear system follows the laws of superposition i.e. the response of the system is the sum of the responses obtained from each input considered separately.
y{ax₁[n] + bx₂[t]} = a y{x₁[n]} + b y{x₂[n]}
Conditions to check whether the system is linear or not.

• The output should be zero for zero input
• There should not be any non-linear operator present in the system

Causality: A system is causal, if the output of the system does not depend on future inputs, but only on past input.
Time-Invariance: If the input to a time-invariant system is shifted in time, its output remains the same signal, but is shifted equally in time.
If the output for an input x(t) is y(t), then for a time shift of t₀ in the input gives the t₀ shift in the output.
x(t) → y(t), then x(t – t₀) → y(t – t₀)
Application:
(i) y(n) = n x(n)
Here in the system, there is a nonlinear operator and hence it is linear.
(ii) y(n) = e^x(n)
Here there is a non-linear operator (exponential) and hence it is nonlinear.

Concept:
The Q factor is 2π times the ratio of the stored energy to the energy dissipated per oscillation cycle, or equivalently the ratio of the stored energy to the energy dissipated per radian of the oscillation.
Mathematically, the Q-factor is given by:

It is also defined in terms of Bandwidth, i.e.

Where,
fr = Center frequency of the filter or the resonant frequency.
BW = Bandwidth of the tuned amplifier.
Calculation:
Given:
Center frequency (f_r) = 3 MHz
Bandwidth (BW) = 5 KHz


Important Point
The sharpness of the resonance in the RLC series resonant circuit is measured by the quality factor and is explained in the figure shown below:

Observations:

• Less the Bandwidth, more the Quality factor.
• More the Bandwidth, less is the Quality factor.
• For the above figure, BW₂ > BW₁, so Q₂ < Q₁

Concept:
The z-transform of a function x(n) = αn u(n) is given as:

Time shifting property of z-transform:

Calculation:
Given,
This can be written as,

Now, if , its IZT = αⁿ u(n)

On comparing, we get k = 10.

• Layers of OSI model: Physical Layer, Data Link Layer, Network Layer, Transport Layer, Session Layer, Presentation Layer, Application Layer
• Packet segmentation is the process of dividing a data packet into smaller units for transmission over the network; Packet segmentation happens at the transport layer
• Data compression (bit-rate reduction) involves encoding information using fewer bits than the original representation; Data compression is done at Presentation Layer

Note:

• When an unloaded transformer is switched on, it draws a large initial magnetizing current. It is called as magnetizing inrush current.
• As the inrush current flows only in the primary winding, the differential protection will see this inrush current as an internal fault.
• The harmonic contents in the inrush currents are different than those in the usual fault current.
• The dc component varies from 40 to 60%, the second harmonic 30 to 70% and the third harmonic 10 to 30%.
• The third harmonic and its multiples do not appear in CT leads connected as these harmonics circulate in the delta winding of the transformer.
• As the second harmonic is more in the inrush current than in the fault current, this feature can be utilized to distinguish between a fault and magnetizing inrush current.

The figure shows a high-speed biased differential scheme incorporating a harmonic restraint feature.

Operating principle:

• The operating principle is o filter out the harmonics from the differential current, rectify them and add them to percentage restraint.
• The tuned circuit X_CX_L allows only the current of the fundamental frequency to flow through the operating coil.
• The dc and harmonics, mostly second harmonics in case of magnetic inrush current, are diverted into the restraining coil.
• The relay is adjusted so as not to operate when the second harmonic (restraint) exceeds 15% of fundamental current (operating).

Concept:

Ripple frequency at the output = m × supply frequency

fo = m × fs

Where m = types of the pulse converter

Calculation:

A three-phase full-wave AC to DC converter is a 6-pulse converter

Number of pulses (m) = 6

fo = 6 × supply voltage frequency

∴ f₀ = 6 x 50

f₀ = 300 Hz

Concept:
Time constant τ = C_eqR_eq
Where,
τ is the time period of the circuit
R_eq is the equivalent resistance in the circuit
C_eq is the equivalent capacitance is the circuit
Calculation:
For calculating equivalent Resistance:
All the voltage sources should be short-circuited and all the current sources must be open-circuited.
Both the resisters are in series so,

= 3 + 3 = 6Ω
For calculating equivalent Capacitance:
Two capacitors are parallel
So,
C_eq1 = C₁ + C₂
= 1 + 1 = 2F
Now these parallel capacitors are in series with another capacitor of capacitance 1F

Time constant τ = R_eqC_eq
= 6 × 2/3
= 4s
The time constant of the above circuit is 4s

• Usually, unexcited single-phase synchronous motor runs at a constant speed equal to the synchronous speed of revolving flux
• They do not need a dc excitation for their rotors
• That’s why they are called as unexcited single-phase synchronous motors
• These motors are divided into two types, one is reluctance motor and another one is hysteresis motor

Concept:
The plug setting multiplier of a relay is defined as the ratio of secondary fault current to the pick-up current.
PSM = Secondary fault current / Relay current setting
Pick up current or operating current = (Rated secondary current in CT) x (Current setting)
Calculation:
Given that,
Relay current setting = 125% = 1.25, Rated secondary current = 5 A
Therefore, operating current of relay = 5 x 1.25 = 6.25 A

A media access control address (MAC address) is a unique identifier assigned to a network adaptor for use as a network address in communications within a network segment. This use is common in most IEEE 802 networking technologies, including Ethernet, Wi-Fi, and Bluetooth.

Important Points:
MAC broadcast address (MAC destination) consists of 6 bytes i.e. 48 bits and all are 1’s
MAC broadcast address → FF:FF:FF:FF:FF:FF
Limited broadcast address (IP-32 bit) → 255.255.255.255

Hysteresis losses: These are due to the reversal of magnetization in the transformer core whenever it is subjected to alternating nature of magnetizing force.

Where

x is the Steinmetz constant = 1.6

Bm = maximum flux density

f = frequency of magnetization or supply frequency

v = volume of the core

At a constant V/f ratio, hysteresis losses are directly proportional to the frequency.

Wh ∝ f

Eddy current losses: Eddy current loss in the transformer is I2R loss present in the core due to the production of eddy current.

Where,

K - coefficient of eddy current. Its value depends upon the nature of magnetic material

Bm - Maximum value of flux density in Wb/m2

t - Thickness of lamination in meters

f - Frequency of reversal of the magnetic field in Hz

V - Volume of magnetic material in m3

At a constant V/f ratio, eddy current losses are directly proportional to the square of the frequency.

We ∝ f2

Concept:
Stateless Protocols are the type of network protocols in which the client sends the request to the server and server response back according to the current state.
It does not require the server to retain session information or status about each communicating partner for multiple requests.

• Application Layer protocol: HTTP and DNS
• Transport Layer protocol: TCP and UDP
• Stateless protocol: HTTP, UDP, DNS

Explanation:
UDP is an unreliable connectionless-transport layer Protocol used for its simplicity and efficiency in applications where error can be provided by the application layer. UDP provided process-to-process communication
Important Point:

• UDP (User Datagram Protocol) → transport layer protocol → connectionless
• TCP (Transmission Control Protocol) → transport layer protocol → connection-oriented
• FTP (File Transfer Protocol) → Application layer protocol → connection-oriented
• NVT → Network Virtual Terminal → Application Layer Protocol → connection-oriented

Concept:

For an AM signal, the transmitted power is given by:

P_c = Carrier Power

μ = Modulation Index

Calculation:

For μ = 0

Pt = P_c (1 + 0)

P_t = P_c

For μ = 1

P_t = 1.5 P_c

We observe that the transmitted power is increased by 50%.

Concept:

Linear Differential Equation: Linear differential equations are those in which the dependent variable, its derivatives occur only in the first degree and they are not multiplied together. It is of the form:

Where, k1, k2, …kn are the constants and X is the function of x only.

The solution of the equation is given as:

y = C.F + P.I

where C.F is the complementary function and P.I is the particular integral.

The above linear differential equation in the symbolic form is represented as

(Dⁿ + k₁ D^n-1 + k₂ D^n-2 +…+ k_n) y = X

X

When X = e^ax:

provided f(a) ≠ 0

Calculation:

Given:

Auxiliary equation is D₂ + 2D - 1

PI is

put D = a

Concept:

The conditional distribution of X given Y is given by

Calculation:

Given X₁ and X₂ are two independent random variables having geometric distribution q^k p = 0;

Let Z = X₁ + X₂;

Now

∴The conditional distribution of X₁ given X₁ + X₂ is uniform.

Concept:

Here, the circular lamina means disc who moment of inertia is given by

Calculation:

Given:

m = 100 kg, r = 10 cm

I = 5000 kg cm²

• The Electromagnetic spectrum: The electromagnetic spectrum is the distribution of electromagnetic radiation according to energy (or equivalently, by virtue of the relations in the previous section, according to frequency or wavelength).

So, the regions of electromagnetic spectrum given in terms of wavelength for Microwave, Visible and Mid. IR spectroscopy are given respectively by

• Microwave: 1 – 100 mm
• Visible wave: 380 – 780 nm
• Infrared wave: 2.5 – 50 μm