FCI JE Electrical Official Paper 2015 (Previous Year Paper)

The correct answer is option 4.

The ratio of voltage (V) and current (I) gives the impedance (Z) of an AC circuit.

Power factor:

The power factor for an AC circuit is given by:

$cosϕ={True \space power\over Apparant\space power}$

$cosϕ={P\over VI}$

$cosϕ ={R\over Z}$​

where, cos ϕ = Power factor

P = Active Power

R = Resistance

Z = Impedance

Concept:

Power = VI

V = voltage

I = current

Calculation:

given that

The maximum power that the lamp can consume = 100W

The maximum voltage that the lamp can tolerate = 100V

Therefore,

By the formula

P=V2/R

The resistance of lamp = (100×100)/100 = 100 Ohm

Now, the main power supply is 250V

But the lamp cannot tolerate this voltage.

Thus, resistance has to be required in series with a lamp which will help in reducing the voltage taken by the lamp from the source and limiting it up to a max of 100 Volt.

Now, let suppose the resistance in series be 'Radd' Ohm

Voltage drop on 100 Ohm (lamp) should be 100V

P = 100 W, V = 100 V

Current = 100/100 = 1 A

Resistance of lamp R = 100/1 = 100 Ω

250 /(100 + Radd) = 1

∴ The value of additional resistance to be connected in series will be Radd = 150 Ω

Circuit Diagram of DC motor:

Applying KVL across armature:

E_b = V - I_aR_a

where, E_b = Back EMF

V = Terminal voltage

I_a = Armature currentt

R_a = Armature resistance

When the motor is at rest, the back EMF is zero.

It is zero at the standstill condition because the back emf opposes the supply voltage and limits the armature current to a safe value. The supply voltage induces the current in the coil which rotates the armature. The electrical work required by the motor for causing the current against the back emf is converted into mechanical energy.

Concept:

The equivalent resistance when 'n' resistances are connected in series:

$R_{eq}=R_1+R_2......R_n$

The equivalent resistance when 'n' resistances are connected in parallel:

${1\over R_{eq}}={1\over R_{1}}+{1\over R_{2}}.........{1\over R_{n}}$

Calculation:

Given, R₁ = 30 Ω 

R₂ = 15 Ω 

R₃ = 5 Ω 

${1\over R_{eq}}={1\over 30}+{1\over 15}+{1\over 5}$

R_eq = 3.33 Ω

Hence, the combine resistance will be less than 5Ω.

Kaplan turbine:

• A Kaplan turbine is a low-head axial flow turbine.
• These micro-level turbines can work for the range of 3 to 0.3 m head with sufficient water stream.
• Kaplan Turbine works on the principle of axial flow reaction. In axial flow turbines, the water flows through the runner along the direction parallel to the axis of rotation of the runner.
• The water at the inlet of the turbine possesses both kinetic energy as well as pressure energy for the effective rotation of the blades in a hydro-power station.
• It is also called as propeller turbine and evolved from the Francis Turbine. It is capable of working at low head and high flow rates very efficiently which is impossible with the Francis turbine.

The function of the commutator is:

1.) Generator:

• ​All generators produce a sine wave, or AC current when the rotor turns in the magnetic field. The commutator on the DC generator converts the AC into pulsating DC. Thus, it is a rectifier in a DC motor.
• The commutator assures that the current from the generator always flows in one direction.
• The brushes ride on the commutator and make good electrical connections between the generator and the load.
• The brushes on the commutator in the DC generator are used to collect the current from the rotating armature.

2.) Motor:

• The commutator on the DC motor converts the DC into AC. Thus, it is an inverter in a DC motor.
• The brushes on the commutator in the DC motor are used to deliver the current back to the rotating armature.

The excessive neutral current is caused by an electronic ballast.

Electric ballast is widely used to turn on the fluorescent. Electric ballast measures the occurrence of harmonic through the actual measurement and it is simulated by modeling an equivalent circuit for harmonic analysis. 

Causes of neutral current:

• An incorrect wiring connection or 3-phase load unbalance issue may cause resultant current flowing on the neutral conductor.
• A high 3rd harmonic current present in each phase may also cause a high neutral current.
• Even though the loads may have been balanced at one time, the normal changing of the system may have caused a large unbalance, leading to a large neutral current.

Synchronous motor:

• The motor which runs at a fixed speed irrespective of load conditions is a synchronous motor.
• A salient pole synchronous motor is running at no load. Its excitation is reduced to zero, the motor will continue running at synchronous speed under no load due to the presence of reluctance torque.
• The torque equations of the salient pole synchronous motor consist of electromagnetic and reluctance torque.
• Since the motor was already running at no load at synchronous speed, and supplies to the stator (and hence rotating MMF) are still present, the rotor will keep on running at the synchronous speed at no load due to reluctance torque.
• The motor now becomes a synchronous reluctance motor.
• If damper windings are present or the field circuit is short-circuited, the motor will run as an induction motor, at a speed slightly less than synchronous.

1ϕ diode rectifier:

The supply power factor is given by:

$cosϕ={V_oI_o\over V_{in(RMS)}I_{s(RMS)}}$

where, cosϕ = Power factor

V_o = Average output voltage

I_o = Average output current

V_in(RMS) = RMS value of supply voltage

I_in(RMS) = RMS value of supply current

Calculation:

Let the input voltage be: V_in = V_m sin(ωt)

Vin(RMS) = $V_m\over \sqrt{2}$

Vo = $2V_m\over \pi$

Iin(RMS) = Load current = I_o

$cosϕ={2V_m\times I_o \times \sqrt{2}\over \pi \times V_m \times I_o}$

cos ϕ = 0.9

2-wattmeter method:

In two wattmeter method, the wattmeter readings are given by:

$W_1=V_LI_Lcos(30-\phi)$

$W_2=V_LI_Lcos(30+\phi)$

The active power is given by:

$P=W_1+W_2=\sqrt{3}V_LI_Lcos\phi$

The reactive power is given by:

$Q=\sqrt{3}(W_1-W_2)$

The power factor is given by:

$cos\phi=cos\space [tan^{-1}{(\sqrt{3}{(W_1-W_2)}\over W_1+W_2}]$

where, V_L = Line Voltage

I_L = Line Current

Pulse width modulation (PWM):

• This is the most efficient method of inverter output voltage control.
• The pulse width modulation control technique in inverters is used for voltage control and harmonic reduction.
• The constant DC input voltage is applied at the input of the inverter and the output voltage is controlled by switching the semiconductor device of the inverter in this method.
• As the low order harmonics ( 3rd, 5th ) reduce whereas higher order harmonics ( 7th, 9th, and 11th ) are filtered out, less requirement for the filter.

Working of PWM inverter:

In this method, the carrier signal is compared with a reference signal.

The time duration in which the value of the carrier signal is less than the reference signal, the inverter gives an output pulse.

The output of the PWM inverter is:

$V_o(t)=\sum_{n=1,3,5}^{}{4V_s\over nπ}sin({nπ\over 2})sin(nd)sin(n\omega t)$

To reduce n^th harmonics:

$sin(nd)=sin \space π$

nd = π

$d={\pi \over n}$

$2d={2\pi \over n}$

where, 2d = Pulse width in each half cycle

Proportional controller:

The transfer function of the P controller is:

${C(s)\over R(s)}={K_pG(s)\over 1+K_pG(s)}$

The steady-state error in the proportional controller is decreased by the factor K_p but cannot be made to zero.

This problem is overcome by the Proportional integral (PI) controller.

Proportional integral controller:

The transfer function of the PI controller is:

${C(s)\over R(s)}={{K_IG(s)\over s}\over 1+{K_IG(s)\over s}}$

${C(s)\over R(s)}={K_IG(s)\over s+K_IG(s)}$

With the introduction of a pole in the PI controller, the steady-state error can be made zero.

Transmission of electrical power

The power is given by:

$P=\sqrt{3}VI\space cos\phi$

${P\over V}=\sqrt{3}I\space cos\phi$

From the above expression, we observe that voltage is inversely proportional to the current.

The transmission of electrical power is done at high voltage because:

• With increased voltage, the value of line current in the transmission line is low. Hence, the losses decreases, and efficiency increases.
• With a low value of the line current, the size of the conductor becomes small.
• With less current, the voltage drop is less and hence the voltage regulation is less.

In a resistor, with silver color band tolerance, the tolerance is ± 10%

Color coding of resistor:

Maximum Power Transfer Theorem:

Consider a linear circuit connected with variable resistance R_L

The maximum power will flow when R_th = R_L

Case 1: Power delivered from source:

$P=V_{th}I$

$P=V_{th}({V_{th}\over R_{th}+R_L})$

At maximum power condition: Rth = RL

$P_{max}={V_{th}^2\over 2R_{th}}$ ............(i)

Case 2: Power received to the load:

$P=I^2R_L$

$P=({V_{th}\over R_{th}+R_{L}})^2R_L$

At maximum power condition: Rth = R_L

$P_{max}={V_{th}^2\over 4R_{th}}$

Transferred power = Delivered power - Received power

Transferred power = ${V_{th}^2\over 2R_{th}}-{V_{th}^2\over 4R_{th}}$

Transferred power = ${V_{th}^2\over 4R_{th}}$

Transferred power = ${1\over 2}\times {V_{th}^2\over 2R_{th}}$ ..........(ii)

Putting the value of equation (i) in (ii), we get:

Transferred power = ${P_m\over 2}$

• The most preferred motor used in the food mixer is the universal motor. Because of the relatively high maintenance commutator brushes, universal motors are best-suited for devices such as food mixers and power tools which are used only intermittently, and often have high starting-torque demands.
• Squirrel cage induction motors are commonly used in many industrial applications. They are particularly suited for applications where the motor must maintain a constant speed, be self-starting, or there is a desire for low maintenance. These motors are commonly used in centrifugal pumps.
• Switched reluctance motors are especially suitable for a variety of coal mining equipment, such as cutters, conveyors, and coal plows. This type of motor provides a low starting current and high torque (30% of rated current gives starting torque up to 150%).
• Series DC motors are generally used where high starting torque is required, and speed variations are possible. These types of direct connection motors are, for instance, used in the traction system, cranes, air compressors, vacuum cleaners, sewing machines, etc.

The speed of the induction motor can be controlled by any of the following methods:

1.) V/f control or frequency control

• The basic idea of a V/f control is to maintain the stator flux constant. In order to operate the machine under nominal conditions, the stator flux must be nominal.
• This control method is commonly applied to blower fans and centrifugal pumps.

2.) Pole changing method

• Pole Changing Method is one of the main methods of speed control of an induction motor.
• This method of controlling the speed by pole changing is used mainly for the cage motor only because the cage rotor automatically develops a number of poles, which is equal to the poles of the stator winding.
• This method is used in AC locomotives.

3.) Supply voltage method

• The speed control of a three-phase induction motor is obtained by changing the supply voltage until the torque required by the load is developed at the desired speed.
• The torque developed by the induction motor is directly proportional to the square of the supply voltage and the current is proportional to the voltage.
• Therefore, the stator voltage control method is suitable for applications where the load torque decreases with the speed, as in the case of a fan load.

4.) Cascade control method

• ​In this method of speed control, two motors are used. Both are mounted on the same shaft so that both run at the same speed.
• One motor is fed from a 3-phase supply and the other motor is fed from the induced emf in the first motor via slip-rings.

Voltage regulation:

The voltage regulation of the transformer is the percentage change in the output voltage from no-load to full-load with respect to the rated voltage.

$VR={V_{nl}-V_{fl}\over V_{rated}}\times 100$

The voltage is given by:

$VR=x{(R_{pu}cos\phi \space \pm \space X_{pu}sin\phi)}$

where, x = Fraction of loading

• is used for lagging power factor

• is used for leading power factor

Voltage regulation curve:

The voltage regulation is negative for the leading power factor.

The voltage regulation is positive and maximum at 0.8 lagging power factor.

Speed control of DC motor

1.) Armature voltage control

• In this method, a speed below normal speed is obtained.
• When the variable resistor reaches its minimum value, the armature resistance is at a normal one. Therefore, the armature voltage drops. When the resistance value gradually increases, the voltage across the armature decreases. This in turn leads to a decrease in the speed of the motor.
• This method is also known as the constant torque variable power method.

2.) Flux control

• In this method, a speed above normal speed is obtained.
• Initially, when the variable resistor keeps at its minimum position, the rated current flows through the field winding due to a rated supply voltage, and as a result, the speed is kept normal. When the resistance increases gradually, the current through the field winding decreases. This in turn decreases the flux produced. Thus, the speed of the motor increases beyond its normal value.
• This method is also known as the variable torque constant power method.

Starting of 3ϕ squirrel cage induction motor

• Star/Delta starters are probably the most common reduced voltage starters.
• They are used in an attempt to reduce the start current applied to the motor during start as a means of reducing the disturbances and interference on the electrical supply.

Induction generator:

When the operating slip of the induction motor becomes negative or less than zero, it starts working as an induction generator.

Under such conditions, the speed of the induction generator is more than the synchronous speed.

Synchronous Generator:

• The synchronous generator is the source of both active and reactive power.
• Synchronous generators are the primary sources of active power generation in a power system, however, they can also provide reactive power to fulfill transmission requirements.
• When a synchronous generator works at a lagging power factor, it delivers reactive power to the load. At this condition, the generator works in an overexcited condition.
• When a synchronous generator works at a leading power factor, it absorbs reactive power from the load. At this condition, the generator works in an under excited condition.
• When a synchronous generator works at the unity power factor, it neither delivers nor absorbs reactive power from the load. At this condition, the generator works in normally exciting conditions.

Additional Information The induction generator is a sink of reactive power.

Losses in the rotating electrical machine (DC, Induction, and synchronous):

1.) Winding loss:

This loss occurs due to the flow of current in the stator and rotor winding.

2.) Core loss:

These losses are due to the heat generated within the core as a result of the alternating current.

3.) Friction loss:

This loss is due to the bearing friction produced in the rotating parts of the machine.

5.) Windage loss:

This loss is due to air friction between the rotor and surrounding air.

Losses in the static electrical machine (Transformer):

There are different kinds of losses that will be occurred in the transformer such as iron, copper, hysteresis, eddy, stray & dielectric.

1.) Iron losses

Iron losses mainly occur through the alternating flux within the transformer’s core. Once this loss occurs within the core then it is called core loss.

This type of loss can be categorized into two types hysteresis as well as eddy current.

a.) Hysteresis loss

This kind of loss mainly occurs when the alternating current is applied to the core of the transformer then the magnetic field will be reversed. 

b.) Eddy's current loss

The flow of current in the emf can be supplied within the body of the material. This flow of current is known as eddy current. This current will occur once the conductor experiences an altering magnetic field.

When these currents are not accountable for doing any functional task, then it generates a loss within the magnetic material. So it is called an Eddy Current Loss.

2.) Copper loss:

Copper losses occur because of the Ohmic resistance in the windings of the transformer.

These losses are also called variable or ohmic losses because these losses will change based on the load.

3.) Stray loss:

These types of losses in a transformer can be occurred because of the occurrence of the leakage field.

4.) Dielectric loss:

This loss mainly occurs within the oil of the transformer. Here oil is an insulating material.

Power gain of amplifier:

The power gain of an amplifier is defined as the product of voltage gain and current gain.

$A_p=A_v\times A_i$

where, A_p = Power gain

A_v = Voltage gain

A_i = Current gain

Comparison between CE, CC, and CB configuration:

Since the voltage and current gain in the CE amplifier are high, therefore the power gain is maximum.

Concept: 

Let the three-phase supply voltages be:

$V_1=V_m\space sin(\omega t)$

$V_2=V_m\space sin(\omega t-120)$

$V_3=V_m\space sin(\omega t+120)$

Now, the three-phase supply currents are:

$i_1=I_m\space sin(\omega t-\phi)$

$i_2=I_m\space sin(\omega t-120-\phi)$

$i_3=I_m\space sin(\omega t+120-\phi)$

The instantaneous power is:

$P=V_mI_m[sin(\omega t) sin(\omega t-\phi)+sin(\omega t-120) sin(\omega t-120-\phi)+sin(\omega t+120) sin(\omega t+120-\phi)]$

$P=3V_mI_m\space cos\phi$ = constant

Additional InformationThe instantaneous power in a single-phase system has a sinusoidal variation with double the supply frequency.

Concept:

The slip of the induction motor is given by:

$s={N_s-N_r\over N_s}$

where, s = Slip

N_s = Speed of stator field

N_r = Rotor speed

Calculation:

The stator field is N_s and the rotor is rotating at the speed 'N_r' in the direction of the stator field.

Then stator field cuts the rotor conductors at the speed of 'N_s - N_r'.

Due to this a rotor field is induced which rotates at the speed of 'sN_s'

with respect to the rotor in its direction

Then, the speed of the rotor field with respect to the stator body is given by

= sN_s + N_r

= N_s - N_r + N_r

= Ns

Power measurement by 1 wattmeter method:

According to the Blondel Theorem:

For the 'n' phase balanced system, 'n-1' wattmeters are required for power measurement.

Explanation:

In two wattmeter method, the wattmeter readings are given by:

$W_1=V_LI_Lcos(30-\phi)$

$W_2=V_LI_Lcos(30+\phi)$

The active power is given by:

$P=W_1+W_2=\sqrt{3}V_LI_Lcos\phi$

The reactive power is given by:

$Q=\sqrt{3}(W_1-W_2)$

The power factor is given by:

$cos\phi=cos\space [tan^{-1}{(\sqrt{3}{(W_1-W_2)}\over W_1+W_2}]$

where, VL = Line Voltage

IL = Line Current

Alloys:

An alloy is a substance composed of two or more metals or of a metal and a nonmetal intimately united usually by being fused together and dissolving in each other when molten.

Brass is an alloy of copper and zinc.

Spelter is an alloy of lead and zinc.

Bronze is an alloy of copper, zinc, and tin.

Solder is an alloy of lead and tin.

Additional Information A metalloid is a type of chemical element which has properties in between, or that are a mixture of, those of metals and nonmetals.

Zener diode:

• A Zener diode is a heavily doped semiconductor device that is designed to operate in the reverse direction.
• When the voltage across the terminals of a Zener diode is reversed and the potential reaches the Zener Voltage (knee voltage), the junction breaks down and the current flows in the reverse direction. This effect is known as the Zener Effect.
• The Zener diode acts as a voltage regulator.

Interconnection of ammeter and voltmeter:

Case 1: When the voltmeter is connected near to load

• A voltmeter has high resistance (infinite in the ideal case) device.
• If connected in series with the load the overall resistance of the circuit will become very high (infinite in the ideal voltmeter case).
• Due to this high resistance of the voltmeter, the current flowing in the circuit will tend to zero i.e. become very low.
• An ammeter consists of a wire of low resistance (ideally zero). and When connected in parallel, a large amount of current passes through it due to the low resistance of the ammeter.
• Hence gets burned i.e. short circuited.

​Case 2: When an ammeter is connected near to load

• The voltmeter has a high resistance value and is linked in parallel so that it draws no current since it needs to draw zero current to measure the voltage across a load in a circuit.
• In a series connection, the current remains the same and the resistance of an ammeter is also very small which it does not affect the current to be measured. Therefore, to measure the current, the ammeter is connected in series.
• So, neither the ammeter nor the voltmeter will burn.

Wattmeter:

• The wattmeter is an instrument for measuring the electric active or average power in watts of any given circuit.
• For the measurement of n-phase power, 'n-1' wattmeters are required.

Working Principle of Wattmeter:

• Generally, a wattmeter consists of a current coil and a potential coil.
• It consists of a fixed which is divided into two halves that are parallel to each other and is connected in series with the load while the moving is connected across the load through a series multiplier resistance.
• The fixed coil is called the current coil while the moving coil is called the potential coil.
• The pressure coil of a wattmeter consists of more turns of fine wire whereas the current coil consists of less number of turns of thick wire.
• The current coil carries the load current and the potential coil carries a current proportional to the load voltage.
• Due to the currents in the coils, a mechanical force acts between them. The result of this mechanical force deflects the moving coil and the pointer over the scale.
• The pointer comes to the rest when the deflecting torque and controlling torque become equal.
• As the reversal of the circuit current reverses the currents in both the current coil and potential coil so that the direction of deflecting torque remains unchanged.
• Therefore, an electrodynamometer wattmeter can measure power in both the AC as well DC circuits.

Absorbing and delivering power:

The power is given by the product of the voltage and current.

P = V × I

When current enters through the +ve terminal, then the device is said to absorb power.

When current leaves through the +ve terminal, then the device is said to deliver power.

$P_A=10\times 10=100 W$

The current is leaving from the +ve terminal, hence device A is delivering 100 W.

$P_B=10\times10=100W$

The current is entering from the +ve terminal, hence device B is absorbing 100 W.

Full wave rectifier:

Case 1: During +ve half cycle

D₁ and D₃ conducts

V_o = V_s

Case 2: During -ve half cycle

D₂ and D₄ conducts

Vo = -V_s

The output waveform is given above:

The average output voltage is:

$V_{o(avg)}={2V_m\over \pi}$

The correct answer is option 3.

Let's assume two inputs be A and B.

and the output is Y.

$Y=\overline{\overline{A}\space .{\overline{B}}}$

Using De-Morgan's Law:

$Y=\overline{\overline{A}}\space +\overline{{{\overline{B}}}}$

$Y=A+B$

Symbols of various logic gates:

1.) OR Gate:

2.) AND Gate:

3.) NOT Gate:

4.) NOR Gate:

5.) NAND Gate:

6.) XOR Gate:

7.) XNOR Gate:

Current mirror circuit:

In the current mirror circuit:

$I_{out}=I_{ref}$

Calculation:

Applying KVL as per loop:

$0+1(I)+0.7-5=0$

I = 4.3 mA

Types of winding in AC machines:

1.) Concentrated winding:

According to concentrated winding, the coil turns are concentrated in one place. So no production of the pitch factor or distribution factor for these winding.

In Concentrated winding, the output voltage has more ripples than the Distributed winding.

Hence, terminals of the distributed winding may deliver pure DC.

2.) Distributed winding:

A winding that is spread throughout the periphery of the rotor/ stator having the least air gap is known as Distributed winding.

This winding is always better to use as it gives the best output waveform much closer to the sinusoidal waveform.

Distributed winding reduces the magnitude of induced emf and reduces the harmonic content.

Concept:

The filter that passes low frequency (s = 0) is a low pass filter.

The filter that passes high frequency (s = ∞ ) is a high pass filter.

The filter that blocks low frequency (s = 0) and high frequency (s = ∞ ) is a Bandpass filter.

The filter that passes low frequency (s = 0) and high frequency (s = ∞ ) is a Band reject filter.

Calculation:

Given, $\rm \frac{V_1(s)}{V_2(s)}=\frac{s^2+c}{s^2+as+b}$

At s = 0

$\rm \frac{V_1(0)}{V_2(0)}=\frac{(0)^2+c}{(0)^2+a(0)+b}={c\over b}$

At s = ∞ 

$\rm \frac{V_1(\infty)}{V_2(\infty)}={s^2(1+{c\over s^2})\over s^2(1+{a\over s}+{b\over s^2})}=1$

It passes both high and low frequencies.

So, it is a Band reject filter.

Symbols of various diodes:

1.) PN junction diode

2.) Zener diode

3.) Tunnel diode

4.) Schottky diode

5.) Varactor diode

6.) Photodiode

AC bridges:

The AC bridges are used for the measurement of inductance and capacitance.

Measurement of inductance:

The inductance of different values of quality factor is measured by Maxwell, Hay, Owen, and Anderson bridge.

Measurement of capacitance:

The dielectric loss of a capacitor can be measured by the Schering bridge.

Counterpoise:

• A counterpoise is simply a long, insulated wire that attaches to the ground connection on your antenna tuner.
• The best counterpoise is 1/4-wavelength at the lowest used frequency.
• Counterpoise is used for reducing transmission tower footing resistance.
• Tower footing resistance is the resistance offered by the metal parts of the tower and the ground resistance and It is important for protection against Surge Voltages mainly back flashover voltage.
• The counterpoise is an effective means of reducing the impedance to ground presented to a lightning strike in areas where high soil resistivity and rocky ground prevent conventional grounding.
• Fundamentally, counterpoise, which should be considered an alternative to other methods of grounding, is a leaky transmission line that is intentionally connected to the earth with large amounts of conductance.
• At the instant of a lightning strike, the counterpoise acts as surge impedance mutually coupled with both the ground wires and phase conductors of the transmission line.
• The energy from the lightning strike travels down the counterpoise and is reflected at the terminal end.
• The counterpoise will act as a series resistance with a distributed leakage to the ground.

Faults in the power system:

Among the given faults, line-to-ground fault (LG) is the most common fault that occurs in the power system.

The order of frequency of occurrence is given below:

LG > LL > LLG > 3 phase fault

The sequence of the severity of fault is:

3 phase fault > LLG > LL > LG

3 phase fault is the most dangerous fault while the LG fault is the least dangerous fault.

Given:

• Insulation resistance of a 20 km cable = 1 MΩ

Concept Used:

Insulation resistance of a cable is inversely proportional to its length:

R ∝ 1 / L

Step 1: Insulation resistance of 10 km cable

Since 20 km cable has resistance 1 MΩ, a 10 km cable (half the length) will have double the resistance.

R_{10 km} = 2 MΩ

Step 2: Equivalent insulation resistance of parallel combination

For parallel resistances:

1 / R_eq = 1 / R₁ + 1 / R₂

1 / R_eq = 1 / 1 + 1 / 2

1 / R_eq = 3 / 2

R_eq = 2 / 3 MΩ

Final Answer:

Equivalent insulation resistance = 0.667 MΩ

Skin effect:

• Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor.
• Skin depth depends on the frequency of the alternating current; as frequency increases, current flow moves to the surface, resulting in less skin depth.
• Skin effect reduces the effective cross-section of the conductor and thus increases its effective resistance.
• The skin effect factors become independent of temperature at high frequencies.

Energy meter:

Working of  the braking system:

• The permanent magnet is used for reducing the rotation of the aluminum disc.
• The aluminum disc induces the eddy current because of its rotation. The eddy current cuts the magnetic flux of the permanent magnet and hence produces the braking torque.
• This braking torque opposes the movement of the disc, thus reducing its speed.
• If the energy meter runs fast, it can be slowed down by adjusting the position of the braking magnet and making it move away from the disc and decreasing the speed, moving it away from the center of the disc.

​The braking torque is given by:

T_d α N ϕ_m d²

N α 1/d

Phantom Loading:

• Phantom loading is used to test the wattmeter and energy meter.
• The pressure coil is supplied from a voltage source( power is negligible) and the current coil is supplied from another circuit operating at low voltage but at the required current.
• So watt meter/energy meter see rated voltage and rated current but the power used for testing is very low.
• This is an indirect way of the load test.
• Direct loading requires rated voltage and current and so power is large.
• Testing at various power factors is also easy in phantom loading.
• It requires only a phase-shifting transformer to shift the phase of voltage with respect to current.
• The power required is less, depending on the voltage required to send rated current( even 5 to 10 percent,) under phantom loading.

Solution:

Phantom loading for testing of energy meters is used to reduce power loss during loading.

Power measurement in 3ϕ circuit

According to Blondel theorem:

For the 'n' phase 'n' wire balanced and unbalanced system, 'n-1' wattmeters are required for power measurement.

For the 'n' phase 'n+1' wire balanced and unbalanced system, 'n' wattmeters are required for power measurement.

Therefore, The power in an unbalanced 3-phase 4-wire circuit can be measured by using 3-wattmeters.

Rectifier type instruments:

• The rectifier instrument has the highest frequency range with accuracy within reasonable limits.
• These instruments are nothing but permanent magnet-moving coil instruments used in conjunction with rectifying devices for AC measurements (current and voltage) from about 20 Hz to 20 kHz.

Electrodynamometer instruments:

• An electrodynamometer or simply a Dynamometer wattmeter is an instrument that is universally used for the measurement of DC as well as AC electric power. It works on the principle of a dynamometer i.e. a mechanical force acts between two current-carrying conductors.

Moving iron instruments:

• The instrument in which the moving iron is used for measuring the flow of current or voltage is known as the moving iron instrument. It works on the principle that the iron placed near the magnet attracts it.
• The force of attraction depends on the strength of the magnet field.

Thermocouple:

• It is a temperature-measuring device consisting of two wires of different metals joined at each end. One junction is placed where the temperature is to be measured, and the other is kept at a constant lower temperature.
• When two wires composed of dissimilar metals are joined at both ends and one of the ends is heated, there is a continuous current that flows in the thermoelectric circuit.

Concept:

The pick-up current of a relay is given by:

Pick-up current = Rated secondary current of CT × Current setting

Calculation:

Given, the CT ratio = 400/5

Secondary current = 5 A

Relay setting = 12.5%

Pick-up current = 5 × 0.125

Pick-up current = 0.625 A

Fuse Law:

• Fuse law determines the current carrying capacity of a fuse wire.
• The current carrying capacity is directly proportional to (diameter)^1.5
• At steady state conditions, the fuse carries an average current without increasing its temperature to the melting limit.
• That means in this steady state condition, heat generated due to the current through the fuse wire is equal to heat dissipated from it.
• The fusing factor is always more than 1.

Concept:

For a series circuit:

1.) Quality factor $(QF)={f_o\over BW}$

2.) Quality factor $(QF)={1\over \omega RC}$

where, f_o = Resonance frequency

BW = Bandwidth

R = Resistance

C = Capacitance

Calculation:

Given, C = 150 pF, BW = 10 kHz and f_o = 1.5 MHz

$QF={1.5\times 10^6\over 10\times 10^3}$

QF = 150

$150={1\over 2\pi \times 1.5\times 10^6\times R\times 150\times 10^{-12}}$

R = 4.7 Ω

Types of Alternators:

1.) Cylindrical rotor:

• Cylindrical (non-salient) pole or round-rotor or drum rotor usually can be found in higher speed higher-power applications, such as turbogenerators.
• A solid rotor is distributed over the edge of the rotor in the rotor with cylindrical poles. However, each slot has different turns, so that sinusoidal flux is created when excited.
• The cylindrical pole rotates at 3600 RPM or 1800 RPM with 2 or 4 poles.

2.) Salient pole rotor:

• Salient pole rotors are large in diameter and shorter in length. They are typically used in lower-speed electrical machines (100 RPM to 1500 RPM).
• Salient pole rotors normally require damper windings to avoid oscillations of rotors during operation.
• They are mostly used in hydropower plants.

V curve of synchronous motor:

• V curve is plotted between the field current and armature current.
• From the V curve, as we increase the field current the value of the armature current decreases at a lagging power factor.
• It becomes the minimum at the unity power factor.
• At the leading power factor, the armature current increases with an increase in field current.

Concept:

When the current is passed through a coil, flux linkage is given by:

$L\times I=N\times ϕ$

$L={N\times ϕ\over I}$

where, L = Mutual inductance

N = No. of turns

ϕ = FLux

I = Current

Calculation:

Given, N = 25

I = 2 A

ϕ =  8 × 10-5 weber

$L={25\times 8\times 10^{-5}\over 2}$

L = 1 mH

Short-pitched coil

• When the coil span is less than the pole pitch, the coil is known as the shorted-pitched coil.
• If the short-pitched angle is α, then the coil pitch factor is equal to $cos\space ({\alpha \over 2})$
• Short pitch winding improves the waveform of generated emf i.e. generated emf can be made to approximate to a sine wave more easily and the distorting harmonics can be reduced or totally eliminated.

Types of torques in measuring instrument

1.) Deflecting Torque

• The deflecting torque causes the moving system to move from its zero position.
• The deflecting torque is produced by the Magnetic effect, Thermal effect, Electrodynamic effect, or Electrostatic effect.

2.) Controlling Torque

• The controlling torque brings the pointer back to zero when the deflecting torque is removed.
• The pointer comes to rest at a position where the deflecting and controlling become equal.
• The controlling torque in indicating instruments may be provided by spring control or gravity control.

3.) Damping Torque

• The damping torque dampens out the oscillations and brings the pointer to the final position quickly.
• The damping torque in indicating instruments can be provided by Air friction damping, Fluid friction damping, or Eddy current damping.

Depletion layer in P-N diode

• The depletion region or depletion layer is a region in a P-N junction diode where no mobile charge carriers are present.
• The depletion layer acts like a barrier that opposes the flow of electrons from the n-side and holes from the p-side.
• In forward bias condition, the positive terminal of the battery injects +ve charge towards p-type.
• These +ve charges recombine with -ve ions present in the depletion region and neutralize them
• Thus, the depletion layer decrease in forward bias.
• During reverse bias, the depletion layer is increased in the P-N diode.

Corona Effect in Overhead Transmission Line

• The ionization of air surrounding the high voltage transmission lines causing the conductors to glow, producing a hissing noise and radio interference, is called Corona Discharge or Corona Effect.
• Air is not a perfect insulator, and even under normal conditions, the air contains many free electrons and ions.
• When an electric field intensity establishes between the conductors, these ions and free electrons experience forced upon them.
• Due to this effect, the ions and free electrons get accelerated and moved in the opposite direction.
• The charged particles during their motion collide with one another and also with the very slow moving uncharged molecules.
• Thus, the number of charged particles goes on increasing rapidly.
• This increase the conduction of air between the conductors and a breakdown occurs.
• Thus, the arc establishes between the conductors.

In an overhead railway system, the pantograph (the device that collects electricity from the contact wire) can lose contact with the contact wire due to several factors, such as vibrations, irregularities in the wire, and changes in the tension of the wire.

At the support (near the poles/towers), the tension in the contact wire is generally higher because it is fixed at the supports. The wire tends to sag more in between supports, leading to potential fluctuations in the distance between the pantograph and the contact wire. The pantograph is more likely to lose contact at the support where there are more dynamic forces acting on the wire.

At mid-span (the point between supports), the tension in the contact wire is usually more uniform, and the distance between the pantograph and the contact wire is less likely to vary as much compared to the support points.

Thus, the loss of contact is typically more at the support than at mid-span.​

Over-voltage protection

• The resistors that are used for over-voltage protection are known as varistors.
• A varistor is a 2 terminal semiconductor device that protects electrical and electronic devices from overvoltage transients.
• The word varistor is formed by the combining variable and resistor. It is also known as a voltage-dependent resistor
• A varistor is an electronic component that varies its electrical resistance according to the applied voltage.
• It is always placed in a shunt with the device being protected. This is basically done to protect the circuit from voltage surges.

The figure below shows the symbolic representation of a varistor:

Concept

The fuse rating of a squirrel cage induction motor is 1.5 times of full load current.

The full load current is given by:

$P=\sqrt{3}V_LI_{L(fl)}$

Also, 1 HP = 745.5 Watt

Calculation

Given, P = 10 HP = 7455 Watt

V_L = 415 V

$I_L={7455\over \sqrt{3}× 415}$

I_L = 10.37 A

Fuse rating = 1.5 × 10.37

Fuse rating = 15 A

Types of distance relay

Mho relay

• Mho relay is referred to as a voltage-controlled directional relay or admittance relay.
• It's characteristic when plotted on an impedance diagram is a circle passing through the origin.
• It is inherently a directional relay as it detects the fault only in the forward direction.
• The relay which is selected for long transmission lines should be less affected due by power swings.
• Hence Mho relay is preferred for the protection of long transmission lines.

Impedance relay

• This relay is a voltage-restrained overcurrent relay.
• This relay operates when the impedance seen from the fault point is less than the relay setting (Z).
• It is used in the protection of medium transmission lines.

Reactance relay

• A reactance relay is an overcurrent relay with directional restraint.
• A reactance relay is suitable for the protection of a short transmission line because its operation is independent of arc resistance.

Solder

• Solder is a fusible metal alloy used to create a permanent bond between metal workpieces.
• Solder is a metal alloy usually made of tin and lead which is melted using a hot iron.
• The iron is heated to temperatures above 600 degrees Fahrenheit which then cools to create a strong electrical bond.
• The most basic soldering flux that has been used is the natural rosin derived from pine tar resin.
• Pine tar resin is dissolved in a solvent and then distilled to yield the clear, water-white rosin used in soldering flux.

The unit of force in the M.K.S system is Newton.

Different standard systems

• The full form of the CGS system is the Centimeter Gram Second system. In the CGS system, fundamental units are Centimeter, Gram, and second
• The full form of the FPS system is the Foot Pound Second system. In the FPS system, fundamental units are Foot, Pound, and second.
• The full form of the MKS system is the Meter Kilogram Second system. In the MKS system, fundamental units are meter, kilogram, and second.
• The full form of SI is System International(International System of Units). In the SI system, the fundamental units are Meter, Kilogram, and Second.

Resistance

The ratio of voltage and current is known as the resistance.

$R={V\over I}$

where, R = Resistance

V = Voltage

I = Current

In circuit theory, the resistance is assumed to be independent of temperature.

Thus in a closed circuit, the ratio of voltage and current remains constant.

Layers in underground cable

1.) Core or conductor

• A cable may have one or more than one core (conductor) depending upon the type of service for which it is intended.
• The conductors are made of tinned copper or aluminum and are usually stranded in order to provide flexibility to the cable.

2.) Insulation

• Each core or conductor is provided with a suitable thickness of insulation, the thickness of the layer depending upon the voltage to be withstood by the cable.
• The commonly used materials for insulation are impregnated paper, varnished cambric, or rubber mineral compound.

3.) Metallic sheath

• In order to protect the cable from moisture, Conductor gases, or other damaging liquids (acids or alkalies) in the soil and atmosphere, a metallic sheath of lead or aluminum is provided over the insulation.

4.) Bedding

• Over the metallic sheath is applied a layer of bedding which consists of fibrous material like jute or hessian tape.
• The purpose of bedding is to protect the metallic sheath against corrosion and mechanical injury due to armoring.

5.) Armouring 

• Over the bedding, armoring is provided which consists of one or two layers of galvanized steel wire or steel tape.
• Its purpose is to protect the cable from mechanical injury while laying it and during the course of handling.

6.) Serving

• In order to protect armoring from atmospheric conditions, a layer of fibrous material (like jute) similar to bedding is provided over the armoring. This is known as serving.

Resistance switching

• Resistance Switching in Circuit Breaker refers to a method adopted for dampening the over-voltage transients due to current chopping, capacitive current breaking, etc.
• In this method, a shunt resistance is connected across the contacts of the circuit breaker.
• It reduces the rate of rising of re-striking voltage and the peak value of re-striking voltage.
• The value of resistance must be such that the circuit is critically damped, then re-striking voltage rises exponentially till recovery voltage is reached.

MOSFET

• Metal Oxide Silicon Field Effect Transistors commonly known as MOSFETs are electronic devices used to switch or amplify voltages in circuits.
• It is a voltage-controlled device and is constructed by three terminals i.e. gate, source, and drain.
• In a MOSFET the gate and channel are separated by a thin layer of SiO₂, they form a capacitance that varies along with gate voltage.
• MOSFET acts like a MOS Capacitor and it is controlled by the input gate to source voltage.
• Hence, MOSFET can be used as a voltage-controlled capacitor.

Electrodynamic type wattmeter

• A wattmeter consists of a current and pressure coil.
• The current coil is connected in series with the load while the pressure coil is connected in parallel with the load.
• Ideally, the current in the pressure coil is in phase but practically the current lags the applied voltage.
• This is due to the presence of inductance in the pressure coil.
• If there was no inductance, the current in the pressure coil will be in phase with the applied voltage, the wattmeter will be free from the effect of power factor and frequency variations.

Series RL circuit:

The total opposition offered by an RL series circuit is called impedance.

The impedance of a series RL circuit is:

$Z=\sqrt{(R)^2+(ω L)^2}$

where, Z = Impedance

R = Resistance

ω = Frequency in radian

L = Inductance

Concept

Case 1: In star-connection

$P_1={3V_p^2\over R}$

$P_1={V_L^2\over R}$

Case 2: When one resistor is removed

$P_2={V_L^2\over 2R}$

% reduction = ${{V_L^2\over 2R}-{V_L^2\over R}\over {V_L^2\over R}}$

% reduction = 50%

Crawling in an Induction machine

• Crawling is the tendency of particularly squirrel cage rotors to run at speeds as low as 1/7^th of their synchronous speed.
• The crawling in the induction motor is caused by harmonics developed in the motor.

Cogging in an Induction machine

• Sometimes, the rotor of a squirrel cage induction motor refuses to start at all, particularly if the supply voltage is low.
• This happens especially when the number of rotor teeth is equal to the number of stator teeth, because of magnetic locking between the stator teeth and the rotor teeth.
• When the rotor teeth and stator teeth face each other, the reluctance of the magnetic path is minimum, which is why the rotor tends to remain fixed. This phenomenon is called cogging or magnetic locking of the induction motor.

Hunting in synchronous machine

• Hunting is the phenomenon of oscillation of the rotor about its steady state position or equilibrium state in a synchronous motor.
• Hence, hunting means a momentary fluctuation in the rotor speed of a synchronous motor.

Resonance:

In a series RLC circuit, when the value of inductive reactance (X_L) is equal to (X_C), it is the condition of series resonance.

Under such conditions, the impedance of the circuit is minimum and equal to the value of resistance (R).

Since the resistance is minimum. So, the value of the current at resonance is maximum.

Faraday's Law of electromagnetic induction

Whenever there is a rate of change in flux in an electrical machine, an AC e.m.f is induced.

E = 4.44 × f × ϕ × N

where, f = Frequency

ϕ = Flux linkage

N = No. of turns

The frequency of the AC e.m.f. induced is given by:

$f={1\over T}$

where, T = Time period

Principle of operation of transformer:

The transformer works on Faraday's Law of electromagnetic induction.

Faraday's law of electromagnetic induction states that, when a change takes place in the magnetic flux which is linked with a circuit, an electromotive force current will induce in the circuit.

$\rm e=\frac{-Nd\Phi}{dt}$

Here, the minus sign is due to Lenz's law.

Lenz’s law tells us the direction of this induced current, which opposes the initial changing magnetic field which produced it. This is signified in the formula for Faraday’s law by the negative sign (‘–’).

Transformer Coupled Amplifier

• The amplifier circuit in which, the previous stage is connected to the next stage using a coupling transformer, is called a Transformer coupled amplifier.
• The reason behind preferring the transformers in amplifiers is, they provide equal impedance through the primary, and secondary windings of the two transformers.

Advantages of transformer coupled amplifier:

• The biggest advantage is it has the feature of impedance matching that can be done by the turn ratio of the transformer. So, one stage's lower impedance can be adjusted with a high impedance of the next stage amplifier.
• The collector resistor and base resistor don’t have any power loss.
• It Provides a higher gain than the RC-coupled amplifier. It offers a 10 to 20 times higher gain value than the RC coupled amplifier.

Disadvantages of transformer coupled amplifier:

• It offers poor frequency responses than the RC coupled amplifier, so gain varies according to the frequencies.
• In this technique, the coupling can be done by using transformers. So looks bulky and expensive for audio frequencies.
• There will be frequency distortions in the speech signal, audio signal, music, etc.

• Short circuit study is used to determine the available fault current or short circuit current at each point in the system.
• Based on these results, short circuit studies are conducted on a given power system in order to design a protection scheme for the system.
• It is used to determine the magnitude of the short circuit current, the system is capable of producing, and compares that magnitude with the interrupting rating of the overcurrent protective devices.
• Short-Circuit Currents are currents that introduce large amounts of destructive energy in the forms of heat and magnetic force into a power system. A short circuit is sometimes called a fault.
• The reliability and safety of electric power distribution systems depend on an accurate and thorough knowledge of short-circuit fault currents that can be present, and on the ability of protective devices to satisfactorily interrupt these currents.

Concept:

The inductive reactance is:

$X_L=2\pi f L$

At low frequency (ω = 0): X_L = 0 means short circuit

At high frequency (ω = ∞): XL = ∞  means open circuit

The capacitive reactance is:

$X_C={1\over 2\pi f C}$

At low frequency (ω = 0): X_C = ∞  means open circuit

At high frequency (ω = ∞) : X_C = 0 means short circuit

Calculation:

At low frequency:

The inductor is short-circuited and the capacitor is open-circuited.

V_o = V_i

At high frequency:

The capacitor is short-circuited and the inductor is open-circuited.

V_o = 0 V

The circuit passes low-frequency and blocks high-frequency input signals.

Hence, it is a low-pass filter.

Comparison between CB, CE, and CC configuration:

The common base has the lowest input resistance while the common collector has the lowest output resistance.

Thevenin's Theorem:

Thevenin’s Theorem states that “Any linear circuit containing several voltages and resistances can be replaced by just one single voltage in series with a single resistance connected across the load“.

Double-cage induction motor:

​​

• A Double Cage Induction motor is a type of motor in which a double cage or two rotor windings or cages are used.
• This arrangement is used for obtaining high starting torque at a low value of starting current.
• In the double cage rotor of an induction motor, there are two layers of bars.
• Each layer is short-circuited by the end rings.
• The resistance of the outer cage is greater than the resistance of the inner cage while the reactance of the inner cage is greater than the outer cage.
• At starting, the voltage induced in the rotor is the same as the supply frequency that is (f₂ = f₁).
• The outer cage winding carries most of the starting current which offers low impedance to the flow of current. The high resistance outer cage winding, therefore, develops a high starting torque.
• As the rotor speed increases, the frequency of the rotor EMF decreases. At normal operating speed, the leakage reactance of both the windings becomes negligibly small.
• The current in the rotor divides between the two cages and is governed by their resistances.
• The resistance of the outer cage is about 5 to 6 times that of the inner cage. Hence, the torque of the motor is developed mainly by the low-resistance inner cage and is developed under normal operating speed.

Reactive Power:

• Reactive power is the part of complex power that corresponds to the storage and retrieval of energy by the inductor and the capacitor.
• The reactive power in a circuit signifies the energy consumed by the magnetic/electric field.
• The inductor stores and dissipate energy in the form of magnetic energy.
• The capacitor stores and dissipate energy in the form of electric energy.

Concept:

The resistance in parallel connection decreases where as it increases in series connection.

R_A < R_B

This means, for the same voltage rating, machine A will have more parallel paths than machine B.

The size of the machine is directly proportional to the no. of parallel paths.

Hence, machine A is bigger than machine B.

Concept:

The power supplied by the source is given by:

$P_s={V_s}{I_s}cosϕ_s$

where, V_s = Source voltage

I_s = Source Current

cos ϕ_s = Phase angle between the supply voltage and supply current

Calculation:

$I_s={100\over 40-j30}$

$I_s={100\over 50∠-36.86}$

I_s = 2∠36.86° 

$P_s={100}\times {2}\times cos(36.86)$

P_s = 160 watts

The correct answer is option 3.

Concept:

Inter-poles:

Inter-poles in DC machines have basically two functions: Automatic neutralization of cross magnetization due to armature reaction. To counter and cancel reactance voltage in the coil undergoing commutation.

Armature resistance and field current:

For the DC shunt generator, the terminal voltage is given by:

$V_t=E_g-I_aR_a$

and $I_a=I_L+I_f$

By Increasing armature resistance (R_a), the voltage drop increases, hence terminal voltage decreases.

By decreasing the field current, the armature current increases, therefore armature drop increases and the terminal voltage decreases.

Armature Reaction:

In a DC machine, two kinds of magnetic fluxes are present; 'armature flux' and 'main field flux'. The effect of armature flux on the main field flux is called an armature reaction.

Armature reaction weakens the main flux. In the case of a dc generator, the weakening of the main flux reduces the generated voltage.

Concept:

The general expression for Magnetic Field Intensity due to an infinite wire is:

$B={μ_o\over 4π}{2I\over d}$

where μ_o = 4π × 10^-7 H/m

I = Current

d = Distance

Calculation:

Given, I = 1 A

d = 10 cm = 0.1 m

$B={4π\times 10^{-7}\over 4π}\times {2\times1\over 0.1}$

B = 2 μ wb/m2 

From the right-hand thumb rule, the direction of the magnetic field is from east to west.

Losses in transformer:

There are two types of losses in transformers: Iron losses and Copper losses.

The iron losses are further two types: Eddy current and hysteresis losses.

In order to separate the iron losses of a transformer, the V/f ratio is kept constant.

The eddy current losses are given by:

$P_e=K_ef^2({V\over f})^2$

If V/f ratio = constant

$P_e\space \alpha\space f^2$

The hysteresis losses are given by:

$P_h=K_hf({V\over f})^{1.6}$

If V/f ratio = constant

$P_h\space \alpha \space f$

Fuse Wire:

It is a metal wire or strip that melts when too much current flows through it, thereby stopping or interrupting the current.

Characteristics of a fuse wire:

A fuse wire should have the following properties:

• Low melting point → It melts easily under excess current, breaking the circuit to prevent damage.
• Low resistivity → Ensures that the fuse wire heats up quickly when current flows through it, leading to melting.
• Moderate conductivity → While conductivity is needed to allow current to pass under normal conditions, the wire should also offer enough resistance to generate heat when excessive current flows.
• Low ohmic loss → To ensure that the fuse wire does not waste energy under normal operating conditions.

Concept:

The MMF in a circular ring is given by:

$MMF = N× I$

where, N = Number of turns

I = Current

Calculation:

Given, N = 500 

I = 3 A

MMF = 500 × 3

MMF = 1500 AT

Additional Information The flux in a magnetic circuit is given by:

$\phi ={MMF\over Reluctance }$

$\phi={NI\over l/μ A}$

where, N = Number of turns

I = Current

μ = Permeability of free space

A = Cross-sectional area

Concept:

Capacitance is defined as that a capacitor having the capacitance of One Farad when a charge of One Coulomb is stored on the plates by a voltage of one volt.

$C={Q\over V}$

where, C = Capacitance

Q = Charge stored

V = Voltage

So the capacitor can be specified by Voltage, Capacitance, and Charge.

The analogy between electric and magnetic circuits:

Transposition of unbalanced three-phase lines:

Transposition of Conductors refers to the exchanging of the position of conductors of a three-phase system along the transmission distance in such a manner that each conductor occupies the original position of every other conductor over an equal distance as shown in the figure below.

 Need of Transposition of Transmission Line:

• To maintain constant voltage drop across each conductor
• To maintain equal voltage at the receiving end terminals
• To reduce communication interference with communication lines.

Kirchoff's current law:

1.) In DC circuits:

The algebraic sum of currents entering the node is equal to the algebraic sum of currents leaving the node.

i5 + i2 = i1 + i3 + i4

In algebraic sum, only the magnitude of the current is added.

2.) In AC circuits:

The phasor sum of currents entering the node is equal to the phasor sum of currents leaving the node.

In phasor sum, both the magnitude and the angle of the current are added.

Concept:

The characteristics equation is given by 1 + G(s)H(s).

1 + G(s)H(s) = s(s2 + s + 1) + K

s³ + s² + s + K = 0

Calculation:

For stable operation:

Product of inner two terms > Product of outer two terms

1 > K

and K > 0

Hence, the range of K for stable operation is:

0 < K < 1

Concept:

The Laplace transform of general signal tⁿ u(t) is given by:

$L[t^n\space u(t)]={n!\over s^{n+1}}$

Calculation:

The ramp function is represented as t u(t).

n = 1

$L[t\space u(t)]={1!\over s^2}$

$L[t\space u(t)]={1\over s^2}$

Usage of different motors:

• DC series motor is used in traction systems because it develops high starting torque which is required for traction purposes.
• DC series motors produce high starting torque at low speeds and low torque at high speeds. The speed of a DC series motor can be controlled easily and effectively.
• The main use of a squirrel-cage motor in a home HVAC system is that it powers the blower fan. If you have a forced-air heating system, such as a furnace, and/or an air conditioning system, the squirrel-cage motor is the part that turns the fans that blow the heated and cooled air through the ventilation system.
• Synchronous motors are used for power factor improvement.
• Universal Motors are used in table fans, hairdryers, and grinders. They are used in portable drill machines. They are used in polishers, blowers, and kitchen appliances.

The most preferred motor used in water pumping applications is typically the induction motor, specifically the squirrel-cage induction motor. This type of motor is widely used for the following reasons:

Key Features:

• Robustness: Induction motors are very durable and can operate in tough environments with minimal maintenance.
• Efficiency: They offer high efficiency, especially when used for continuous operations like water pumping.
• Cost-effective: Induction motors are generally cheaper and simpler in construction compared to other motor types.
• Reliable operation: They provide reliable performance, which is crucial for applications like water pumping that require consistent operation.
• Availability: Induction motors are available in various power ratings, making them suitable for both domestic and industrial water pumping applications.
  

While other types of motors, such as permanent magnet motors and brushless DC motors (BLDC), can be used in specific water pumping systems (e.g., solar water pumps), the squirrel-cage induction motor remains the most common choice for standard water pumping systems.

Concept:

In a right-angled triangle:

$H=\sqrt{P^2+B^2}$

where, H = Hypotenuse

P = Perpendicular

B = Base

Calculation:

Option 1: $12\ne\sqrt{5^2+8^2}$

Option 2: $12\ne\sqrt{6^2+7^2}$

Option 3: $17=\sqrt{15^2+8^2}$

Option 4:  $8\ne\sqrt{3^2+6^2}$

Hence, option 3 is the correct answer.