Basic Electrical Engineering | PYQs | 2023| beu

 (iv) 6, 15, 15

(iv) RC
Explanation: The time constant (τ \tau τ) of an RC circuit is the product of resistance (R R R) and capacitance (C C C), i.e., τ=R⋅C \tau = R \cdot C τ=R⋅C, representing the time to charge to ~63% of the maximum voltage.

(ii) the source impedance is equal to the load impedance
Explanation: Maximum power transfer occurs when the load impedance matches the source impedance (for complex impedances, magnitude matching), based on the maximum power transfer theorem.

Form Factor
Form factor is the ratio of the RMS (Root Mean Square) value to the average value of an alternating quantity. $$\text{Form Factor} = \frac{\text{RMS Value}}{\text{Average Value}}$$

Peak Factor
Peak factor is the ratio of the maximum (peak) value to the RMS value of an alternating quantity. $$\text{Peak Factor} = \frac{\text{Maximum Value}}{\text{RMS Value}}$$

Kirchhoff’s Voltage Law (KVL)

According to Kirchhoff’s Voltage Law, in any closed circuit or mesh, the algebraic sum of all the EMFs and voltage drops will be zero.

Kirchhoff’s Current Law (KCL)

According to Kirchhoff’s Current Law, the algebraic sum of all the currents meeting at a point or a junction will be zero.

• Hysteresis Loss: Proportional to frequency ($ P_h \propto f $) because it depends on the number of magnetization cycles per second in the core material.
• Eddy Current Loss: Proportional to the square of frequency ($P_e \propto f^2 $) because it depends on the rate of change of magnetic flux, which increases with frequency.

• When a 3-phase AC supply is given to stator windings placed 120° apart,
• It creates three magnetic fields, 120° out of phase.
• Their combination produces a rotating magnetic field in the air gap.
• This RMF rotates at synchronous speed.

• Neutral: Return path for current in AC system.
• Grounding: Connecting system parts to ground for safety (used in live circuits).
• Earthing: Connecting metallic body of equipment to earth to prevent shock.

• Magnetic Flux: $\phi = \frac{\text{MMF}}{\text{Reluctance}} = \frac{N I}{\mathcal{R}}$
• Reluctance $\mathcal{R} = \frac{l}{\mu A}$
  Where μ = permeability, A = area, l = length
• Higher permeability → Lower reluctance → More flux

According to the maximum power transfer theorem, the maximum power flow through load resistor R_L can be achieved when the load resistor equals the thevenin’s equivalent resistance of the circuit.

Mathematical Condition:

R_L = R_th
Where:

$R_{th}$​ = Thevenin (or internal) resistance

$R_L$​ = Load resistance

Maximum Power Transferred

If the source voltage is $V_{th}$​ (Thevenin equivalent voltage), then: $$P_{max} = \frac{V_{th}^2}{4R_{th}}$$

1. Voltage Source

Ideal Voltage Source

• Maintains a constant voltage across its terminals regardless of the load current.
• Has zero internal resistance.
• Example: A theoretical battery with no drop under load.

Practical Voltage Source

• Maintains nearly constant voltage but has some internal resistance.
• Voltage slightly drops when load increases.
• Example: Real batteries, DC generators.

2. Current Source

Ideal Current Source

• Delivers a constant current regardless of the load resistance.
• Has infinite internal resistance.
• Example: Theoretical current source used in circuit analysis.

Practical Current Source

• Tries to deliver constant current but changes slightly with load.
• Has very high internal resistance.
• Example: Current-regulated power supplies.

3. Unilateral Element

• Electrical behavior changes with direction of current or voltage.
• Allows current in only one direction.
• Example: Diode, SCR.

4. Bilateral Element

• Electrical behavior is the same in both directions of current.
• Obeys Ohm’s Law in both directions.
• Example: Resistor, capacitor, inductor (in linear form).

Thevenin’s Theorem states that:
Any linear, bilateral DC network with voltage/current sources and resistors can be replaced by an equivalent circuit consisting of a single voltage source VthV_{th} in series with a resistance RthR_{th}, connected across the load.

Explanation
To apply Thevenin’s Theorem:

1. Remove the load resistor $R_L$ from the circuit.
2. Find the open-circuit voltage across the load terminals. This is the Thevenin voltage VthV_{th}.
3. Replace all independent voltage sources with short circuits and current sources with open circuits. Then find the equivalent resistance across the open terminals. This is RthR_{th}.
4. Draw the Thevenin equivalent circuit: a voltage source $V_{th}$ in series with resistance $R_{th}$, connected to the load $R_L$.

Similarities Between Electric and Magnetic Circuits

Dissimilarities Between Electric and Magnetic Circuits

Applications of Magnetic Circuits

1. Transformers
2. Electric motors
3. Generators
4. Electromagnets (relays, contactors, lifting devices)
5. Inductors used in electronic circuits

Eddy-Current Loss
Eddy currents are circulating currents induced in the core material of electrical machines when exposed to a changing magnetic field. These currents cause power loss in the form of heat, known as eddy-current loss.

Undesirable Effects of Eddy Currents

• Unwanted heat generation in the core
• Decrease in efficiency of machines like transformers and motors
• Possible damage to insulation due to overheating
• Increased energy consumption

Methods to Minimize Eddy-Current Loss

• Use laminated core instead of solid iron
• Each lamination is insulated to restrict the flow of eddy currents
• Use high-resistivity materials like silicon steel in the core

Applications of Eddy Currents

• Induction heating for cooking and industrial processes
• Electromagnetic braking in trains and elevators
• Metal detectors and eddy-current testing for crack detection
• Speedometers and energy meters in electric vehicles

Voltage Regulation Definition:

Voltage regulation of a transformer is the change in secondary voltage from no-load to full-load, expressed as a percentage of full-load voltage (keeping primary voltage constant).

$$
\text{Voltage Regulation (\%)} = \frac{V_{NL} – V_{FL}}{V_{FL}} \times 100
$$

Approximate Voltage Regulation Formula:

When the load is at an angle θ (power factor angle), the voltage regulation is approximately:

$$
\text{Voltage Regulation} = \frac{I R_{eq} \cos\theta \pm I X_{eq} \sin\theta}{V_{FL}} \times 100
$$

(Use + for lagging power factor and − for leading power factor)

(i) Condition for Zero Voltage Regulation:

For zero voltage regulation:

$$
I R_{eq} \cos\theta = I X_{eq} \sin\theta
$$

Dividing both sides by I:

$$
R_{eq} \cos\theta = X_{eq} \sin\theta
$$

So,

$$
\tan\theta = \frac{R_{eq}}{X_{eq}}
$$

This happens at a leading power factor.

(ii) Condition for Maximum Voltage Regulation:

Differentiate the voltage regulation expression with respect to θ and set derivative to zero:

For maximum voltage regulation:

$$
\tan\theta = \frac{X_{eq}}{R_{eq}}
$$

This occurs at a lagging power factor.

Derivation of Induced EMF in a Transformer:

Let,

• $N$ = Number of turns in the winding
• $\phi_m$ = Maximum magnetic flux in the core (Wb)
• $f$ = Frequency of supply (Hz)

According to Faraday’s law of electromagnetic induction:

$$
e(t) = N \frac{d\phi}{dt}
$$

If the flux is sinusoidal,

$$
\phi(t) = \phi_m \sin(\omega t)
$$

Then,

$$
\frac{d\phi}{dt} = \omega \phi_m \cos(\omega t)
$$

So, the instantaneous emf is:

$$
e(t) = N \omega \phi_m \cos(\omega t)
$$

The RMS value of emf is given by:

$$
E = 4.44 f N \phi_m
$$

Where:

• $E$ = RMS value of induced emf (V)
• $f$ = frequency in Hz
• $N$ = number of turns
• $\phi_m$ = maximum magnetic flux in Weber

Construction

A three-phase induction motor consists of several key components that work together to convert electrical energy into mechanical energy. The main parts are:

1. Yoke

• The yoke is the outer frame of the motor, providing mechanical support to internal components.
• Made of cast iron or steel to ensure high magnetic permeability and protect the motor from external damage.
• Provides a low reluctance path for the magnetic field, ensuring efficient operation.

2. Rotor (Armature)

• Contains coils (armature winding) where current flows.
• Rotates within the magnetic field generated by the stator.
• Converts electrical energy into mechanical (rotational) energy.
• Typically made of laminated steel to reduce eddy current losses.

3. Commutator

• Reverses the direction of current flow in the armature winding.
• Ensures unidirectional torque production in the motor.
• Made of copper segments, insulated from each other.
• Wears over time due to brush contact, requiring periodic maintenance.

4. Brushes

• Conduct current between the external circuit and the rotating commutator.
• Usually made of carbon for good conductivity and low wear.
• Require replacement as they wear out over time.
• Minimize friction while maintaining good electrical contact.

5. Field Windings

• Wound around the poles of the stator to create a magnetic field.
• Can be series-wound, shunt-wound, or compound-wound, depending on motor design.
• The strength of the magnetic field determines the torque produced.
• Consumes electrical power to generate the magnetic field.

6. Shaft

• Transmits the mechanical energy generated by the rotor to an external load.
• Made of steel or other durable materials to handle torque.
• Supports rotational movement with minimal deflection.

Principle of Operation

• When 3-phase AC supply is given to the stator winding, it produces a rotating magnetic field (RMF).
• This RMF cuts the rotor conductors and induces an EMF in them (according to Faraday’s law).
• Since the rotor conductors form a closed circuit, current flows.
• The interaction between the rotor current and RMF produces torque (as per Lorentz force law).
• This torque causes the rotor to rotate in the same direction as the RMF.

Slip in an Induction Motor:

• The rotor never rotates at the same speed as the stator’s magnetic field.
• The difference in speed is called slip.

Formula: $$s = \frac{N_s – N_r}{N_s}$$

Where:

• $s$ = slip (unitless or %)
• $N_s$​ = synchronous speed
• $N_r$​ = rotor speed

At full load, slip is usually between 2% to 6%.