GTU 3110005 Basic Electrical Engineering (BEE) Summer 2026 Solved Paper & Solutions

Definitions:

• Ideal Voltage Source: A source that maintains a constant terminal voltage $V_s$ regardless of the load current drawn ($R_{int} = 0$).
• Ideal Current Source: A source that supplies a constant output current $I_s$ regardless of the voltage across its terminals ($R_{int} = \infty$).

Difference Between Ideal and Practical Voltage Source:

1. What is Power Factor?

Power Factor (p.f.) is defined as the cosine of the phase angle ($\phi$) between the applied alternating voltage ($V$) and the resulting alternating current ($I$).

It can also be mathematically expressed as:

1. Ratio of True Power to Apparent Power:

2. Ratio of Resistance to Impedance:

• Significance: A low power factor means the system draws higher line current for the same useful active power, leading to higher $I^2 R$ transmission losses, larger voltage drops, and oversized equipment ratings.

2. Methods of Power Factor Improvement

Most industrial loads (induction motors, welding sets, transformers) are inductive and operate at a lagging power factor. To improve the power factor towards unity, leading reactive power (VAR) must be supplied locally.

1. Static Shunt Capacitors:

• Capacitors connected in parallel with inductive loads draw a leading current ($90^\circ$ ahead of voltage) which neutralizes the lagging reactive component of the load current.
• Advantages: Low maintenance, no moving parts, highly scalable, and cost-effective.

2. Synchronous Condenser:

• An over-excited 3-phase synchronous motor running on no-load acts like a variable capacitor, drawing leading current from the supply mains.
• Advantages: Continuous smooth stepless power factor adjustment; commonly installed in major grid sub-stations.

3. Phase Advancers:

• An AC exciter mounted on the main rotor shaft of slip-ring induction motors to supply magnetizing ampere-turns at slip frequency directly to the rotor circuit, eliminating lagging reactive current from the stator supply.

1. Average Value ($I_{avg}$ or $V_{avg}$)

The average value of an alternating current or voltage is defined as the steady DC current/voltage that transfers across any circuit the exact same amount of electrical charge ($q = \int i \, dt$) as is transferred by that alternating quantity in the same time interval.

• For a symmetrical sinusoidal wave over a half-cycle:

(Over a complete full cycle, symmetrical AC has an average value of zero).

2. Maximum Value (Peak Value / Amplitude $V_m$ or $I_m$)

The maximum instantaneous value attained by an alternating quantity during one complete cycle. It represents the positive or negative crest amplitude of the waveform.

3. Frequency ($f$)

The number of complete cycles executed by an alternating quantity per second.

• Unit: Hertz ($\text{Hz}$) or cycles per second.
• Relation with time period: $f = \frac{1}{T}$.
• In India, the standard generation and distribution frequency is $50\text{ Hz}$.

4. Resonance

Resonance is the operating condition of an AC circuit containing both inductance ($L$) and capacitance ($C$) wherein the inductive reactance ($X_L$) equals the capacitive reactance ($X_C$).

• Under resonance, the net reactive component cancels out ($X_L - X_C = 0$), causing the total circuit impedance to become purely resistive, and the input voltage and current fall into exact phase ($\,\cos\phi = 1$).

5. Form Factor ($k_f$)

Form factor is defined as the numerical ratio of the RMS (Root Mean Square) value of an alternating quantity to its Average value.

• For a standard sinusoidal waveform:

6. Time Period ($T$)

The total time taken in seconds by an alternating quantity to execute one complete cycle of positive and negative variations.

• For $50\text{ Hz}$ supply: $T = \frac{1}{50} = 0.02\text{ s} = 20\text{ ms}$.

7. Phase Sequence

Phase sequence is the chronological order in which the alternating voltages in a polyphase (typically 3-phase) system reach their respective positive maximum values.

• The standard clockwise phase sequence adopted internationally is R–Y–B (Red $\to$ Yellow $\to$ Blue), where Yellow lags Red by $120^\circ$ and Blue lags Yellow by $120^\circ$. Reversing the phase sequence (e.g., to R–B–Y) reverses the direction of rotation of 3-phase induction motors.

Given Data

• Resistance: $R = 10\ \Omega$
• Inductance: $L = 0.1\text{ H}$
• Applied Voltage: $V = 230\text{ V}$
• Frequency: $f = 50\text{ Hz}$

Step 1: Calculate Inductive Reactance ($X_L$)

Step 2: Calculate Circuit Impedance ($Z$)

Step 3: Calculate Circuit Current ($I$)

Summary of Results

A single-phase transformer consists of the following primary and auxiliary components:

1. Primary Electrical & Magnetic Parts

1. Magnetic Core:

• Constructed of thin, high-grade Cold Rolled Grain Oriented (CRGO) silicon steel laminations (0.35 to 0.5 mm thick), insulated by varnish to minimize hysteresis and eddy-current losses. It provides a low-reluctance path for mutual magnetic flux.

2. Primary Winding:

• Copper winding connected to the alternating AC input power supply.

3. Secondary Winding:

• Copper winding from which transformed electric power is delivered to the load.

4. Winding Insulation:

• Dielectric materials (kraft paper, pressboard, varnishes) placed between core and coils, and between primary and secondary windings to prevent dielectric breakdown.

2. Mechanical & Cooling Accessories

5. Transformer Tank: Steel container housing the core-winding assembly submerged in dielectric transformer oil.
6. Transformer Oil: Mineral hydrocarbon oil serving a dual purpose: high dielectric electrical insulation and transfer of heat to cooling fins.
7. Conservator Tank: Cylindrical drum mounted above the tank accommodating the expansion and contraction of oil during thermal loading.
8. Breather (with Silica Gel): Extracts atmospheric moisture from incoming air during breathing cycles; fresh silica gel is blue and turns pale pink when saturated.
9. Buchholz Relay: Gas-actuated safety relay installed on the connecting pipe between tank and conservator to detect internal electrical faults.
10. Bushings (HV & LV): Porcelain insulators providing safe electrical terminals from internal windings to external transmission lines.

1. Construction of Three-Phase Induction Motor

A 3-phase induction motor consists of two main parts:

1. Stator:

• Stator Frame: Cast iron or fabricated steel enclosure providing mechanical support and protection.
• Stator Core: Laminated high-grade silicon steel rings with slots along the inner periphery to reduce eddy current loss.
• Stator Winding: Three-phase balanced winding wound for a specific number of poles ($P$), displaced in space by $120^\circ$ electrical.

2. Rotor:

• Squirrel Cage Rotor: Cylindrical laminated core with copper or aluminum heavy conductor bars embedded in slots, permanently short-circuited at both ends by heavy end-rings. Extremely rugged and maintenance-free.
• Slip-Ring (Wound) Rotor: Carries a balanced 3-phase distributed winding similar to the stator, with terminals connected to three slip rings on the shaft, allowing external resistance for high starting torque.

3. Air Gap: Very small clearance ($0.4\text{ mm}$ to $4\text{ mm}$) to minimize magnetizing current requirements.

2. Working Principle

1. When a 3-phase balanced AC supply is connected to the stator windings, currents displaced by $120^\circ$ electrical produce a Rotating Magnetic Field (RMF) in the air gap.
2. The RMF rotates at synchronous speed:

3. As this magnetic field rotates, it cuts the stationary rotor conductors, inducing an electromotive force (EMF) by Faraday's Law of Electromagnetic Induction:

4. Because the rotor bars form a permanently closed circuit (via end rings), the induced EMF causes heavy rotor currents to circulate.
5. By Lenz's Law, the direction of induced current and resulting electromagnetic torque opposes the cause producing it—the relative motion between the rotating stator field ($N_s$) and the rotor ($N$).
6. Hence, the rotor accelerates in the same direction as the rotating magnetic field, attempting to catch up with $N_s$.

3. Role of Slip ($s$)

Slip is defined as the relative speed between synchronous speed ($N_s$) and actual rotor speed ($N$), expressed as a fraction or percentage of synchronous speed:

• Why Rotor Cannot Reach Synchronous Speed ($N = N_s$):

If the rotor were to reach synchronous speed ($N = N_s$), the relative speed $(N_s - N)$ would become zero ($s = 0$). Rotor conductors would no longer cut any magnetic flux, resulting in zero induced EMF, zero rotor current, and zero torque. The rotor immediately slows down. Thus, an induction motor can never run at synchronous speed.

• Role in Operational Quantities:
• Rotor Frequency: $f_r = s \cdot f$ (at full load, $s \approx 3\% \implies f_r \approx 1.5\text{ Hz}$).
• Rotor Induced EMF: $E_{2r} = s \cdot E_2$.
• Rotor Reactance: $X_{2r} = s \cdot X_2$.
• Torque Generation: Torque is directly proportional to slip ($T \propto s$) in the stable normal operating zone.

Short Note on Plate Earthing:

Plate earthing provides a low-resistance return path for leakage and fault currents into the ground to protect humans and equipment.

Key Specifications:

1. Earthing Plate:

• Copper Plate: Minimum $60\text{ cm} \times 60\text{ cm} \times 3.18\text{ mm}$.
• Galvanized Iron (GI) Plate: Minimum $60\text{ cm} \times 60\text{ cm} \times 6.35\text{ mm}$.

2. Burial Depth: Buried vertically at a minimum depth of $3\text{ meters}$ below ground level into moist soil.
3. Surrounding Bed: Surrounded by alternate layers of powdered charcoal and common salt (NaCl) ($\,\approx 15\text{ cm}$ thick each) to absorb moisture and lower soil resistivity.
4. Watering Pipe: A $19\text{ mm}$ diameter GI pipe with a funnel and wire mesh strainer allows water pouring during summer to keep earth resistance below $5\ \Omega$.

1. Comparison Between Mesh and Nodal Analysis

2. Analysis: Suitability for Circuits with More Voltage Sources

Nodal Analysis is significantly more suitable and efficient for circuits containing more voltage sources (especially when connected between independent nodes and the reference ground):

1. Direct Node Voltage Determination:

• When an independent voltage source is connected between an essential node and the reference ground, that node's voltage is immediately known ($V_k = V_s$), completely eliminating that node's equation from the system of simultaneous equations.

2. Supernode Simplicity:

• When an ideal voltage source is connected between two non-reference nodes, a supernode is formed. The constraint equation ($V_1 - V_2 = V_s$) provides an immediate algebraic substitution, reducing the complexity of KCL matrix inversion.

3. In Contrast with Mesh Analysis:

• While voltage sources easily appear in KVL mesh equations without supermeshes, when the number of nodes is small and voltage sources are tied to ground, Nodal Analysis reduces the problem to only 1 or 2 unknowns, avoiding large matrix inversions.

1. Introduction & Types of ELCB

An Earth Leakage Circuit Breaker (ELCB) is a specialized electrical safety device designed to detect residual earth fault/leakage currents and instantly disconnect the supply to protect humans from fatal electric shocks and prevent electrical fires.
Two main types exist:

1. Voltage-Operated ELCB: Senses hazardous voltage rise on the metal casing of equipment.
2. Current-Operated ELCB (RCCB / Residual Current Circuit Breaker): Standard modern device that senses current imbalance between Phase and Neutral.

2. Construction of Current-Operated ELCB (RCCB)

The main components are:

1. Core Balance Current Transformer (CBCT): A high-permeability circular toroidal magnetic core through which both the Phase conductor and the Neutral conductor pass.
2. Primary Windings: The phase and neutral conductors act as primary coils wound in opposite directions on the toroidal core ($N_1 = N_2$).
3. Secondary / Sensing Winding: A multi-turn search coil wound on the same core, connected directly to a high-speed sensitive electromagnetic relay.
4. Trip Mechanism: Spring-loaded mechanical contacts coupled to the relay that disconnect both phase and neutral simultaneously.
5. Test Circuit: A test pushbutton with a series current-limiting resistor to simulate a fault and verify operational integrity.

3. Working Principle

• Normal Healthy Condition:
• The current leaving the source via the Phase conductor equals the return current via the Neutral conductor ($I_L = I_N$).
• Since the currents flow in opposite directions through the toroidal core, their magnetomotive forces (MMF) cancel each other completely:

• With zero net magnetic flux in the core, no EMF is induced in the secondary sensing winding ($e_s = -N_s \frac{d\Phi_{net}}{dt} = 0$). The trip relay remains de-energized, keeping the contacts firmly closed.

• Earth Fault / Leakage Condition:
• When insulation fails or a human touches an active energized part, a portion of the current leaks to ground ($I_{fault}$ or $I_{leak}$).
• Now, the return neutral current is less than the phase current:

• This net unbalanced current produces a resultant alternating magnetic flux ($\Phi_{net} \ne 0$) inside the toroidal core.
• The alternating flux cuts the secondary sensing coil, inducing an EMF by Faraday’s law.
• This induced EMF drives current through the trip solenoid, energizing the electromagnet, releasing the spring-loaded mechanical latch, and tripping the circuit in less than $30\text{ milliseconds}$ when $I_{fault} \ge 30\text{ mA}$.

Main Parts of a Separately Excited DC Motor:

In a separately excited DC motor, the field winding is energized by an independent, separate DC power source:

1. Yoke (Magnetic Frame): Heavy cast steel outer enclosure providing mechanical protection and low-reluctance magnetic flux path.
2. Field Poles & Pole Shoes: Laminated steel poles that carry the field winding; pole shoes support coils and spread flux uniformly across the air gap.
3. Field Winding: Copper coils excited by an external DC supply to produce the working magnetic flux.
4. Armature Core & Winding: Slotted cylindrical silicon steel laminations carrying insulated copper conductors where electromagnetic torque ($T \propto \Phi I_a$) is developed.
5. Commutator & Carbon Brushes: Wedge-shaped copper segments insulated by mica that reverse current direction in armature coils; spring-loaded carbon brushes collect/supply current.

1. Working Principle of Transformer

A transformer operates on the principle of Faraday’s Law of Electromagnetic Induction, specifically through Mutual Induction between two stationary magnetically-coupled windings:

1. When an alternating voltage $V_1$ is applied to the primary winding, it draws an alternating magnetizing current that sets up an alternating magnetic flux $\Phi$ in the iron core.
2. This common time-varying flux links both the primary and secondary windings through the low-reluctance magnetic core.
3. By Faraday's Law, a self-induced EMF is set up in the primary winding, and a mutually-induced EMF is set up in the secondary winding of frequency identical to the supply.

2. Derivation of EMF Equation of a Transformer

Let:

• $N_1, N_2$ = Number of turns in primary and secondary windings
• $f$ = Supply frequency in $\text{Hz}$
• $\Phi_m$ = Maximum core flux in Webers ($B_m \times A$)

Assuming sinusoidal flux variation:

According to Faraday's Law, instantaneous induced EMF per turn is:

The maximum value of induced EMF per turn is:

The RMS value of induced EMF per turn is:

Multiplying by the number of turns:

3. Turns Ratio ($K$)

The turns ratio is defined as the ratio of the number of turns in the secondary winding to the number of turns in the primary winding:

(Alternatively defined by some standards as $\frac{N_1}{N_2}$; here $K > 1$ denotes a step-up transformer and $K < 1$ denotes a step-down transformer).

1. Derivation of Resonance Condition ($X_L = X_C$)

Consider a series circuit with resistance $R$, inductance $L$, and capacitance $C$ supplied by an AC voltage $v = V_m \sin(\omega t)$.

The total complex impedance of the series circuit is:

The magnitude of impedance is:

The phase angle is:

• Resonance Definition: Electrical resonance occurs when the supply voltage and line current are in exact phase with each other (i.e., power factor $\cos\phi = 1$, or $\phi = 0^\circ$).
• Setting the reactive (imaginary) component to zero:

• The resonant frequency in $\text{Hz}$ is:

2. Why Impedance is Minimum at Resonance

From the impedance magnitude equation:

Since $(X_L - X_C)^2 \ge 0$ for all frequencies, the smallest possible value of $|Z|$ occurs strictly when $(X_L - X_C)^2 = 0$.
At this resonant frequency ($f_0$):

The impedance is purely resistive, meaning the opposing effects of inductive reactance and capacitive reactance completely cancel each other out.

3. Graphical Analysis: Impedance & Current vs Frequency

Analysis of Curves:

1. Below Resonant Frequency ($f < f_0$):

• $X_C = \frac{1}{2\pi f C}$ is very large and $X_L = 2\pi f L$ is small ($X_C > X_L$).
• Circuit is predominantly capacitive; current leads voltage.
• Impedance decreases as frequency increases towards $f_0$.

2. At Resonant Frequency ($f = f_0$):

• $X_L = X_C$, so net reactance is zero.
• Impedance is minimum: $Z = R$.
• Current is maximum: $I_0 = \frac{V}{R}$ (limited only by resistance).
• Power factor is unity ($\cos\phi = 1$).

3. Above Resonant Frequency ($f > f_0$):

• $X_L > X_C$ as inductive reactance grows linearly with frequency.
• Circuit becomes predominantly inductive; current lags voltage.
• Impedance increases, causing circuit current to decline rapidly towards zero.

Construction of Single-Phase Transformer:

A single-phase transformer consists of two primary elements:

1. Magnetic Core:

• Built from thin ($0.35\text{ to }0.5\text{ mm}$) Cold-Rolled Grain-Oriented (CRGO) silicon steel laminations coated with insulating varnish to minimize eddy current and hysteresis losses.
• Vertical segments carrying the windings are called limbs; horizontal bridging sections are called yokes.

2. Windings (Coils):

• Made of high-grade insulated electrolytic copper.
• Low-Voltage (LV) Winding: Placed closest to the core limb to simplify dielectric insulation.
• High-Voltage (HV) Winding: Placed concentrically over the LV winding, separated by an insulating cylinder.
• Categorized as Core-Type (windings surround the limbs) or Shell-Type (magnetic core encloses the central winding limb).

Derivation and Comparison

Let:

• $V_L$ = Line voltage of the balanced 3-phase supply
• $Z_{ph}$ = Impedance per phase of the balanced load ($|Z_{ph}| = Z$, power factor angle = $\phi$)

1. Star ($Y$) Connection

• Phase Voltage: $V_{ph(Y)} = \frac{V_L}{\sqrt{3}}$
• Phase Current: $I_{ph(Y)} = \frac{V_{ph(Y)}}{Z} = \frac{V_L}{\sqrt{3} \, Z}$
• Line Current: In star connection, $I_{L(Y)} = I_{ph(Y)}$:

• Power Consumption:

2. Delta ($\Delta$) Connection

• Phase Voltage: $V_{ph(\Delta)} = V_L$
• Phase Current: $I_{ph(\Delta)} = \frac{V_{ph(\Delta)}}{Z} = \frac{V_L}{Z}$
• Line Current: In delta connection, $I_{L(\Delta)} = \sqrt{3} \, I_{ph(\Delta)}$:

• Power Consumption:

3. Comparison of Line Current and Power

1. Comparison of Line Currents:

The line current drawn by the load in Delta connection is 3 times that drawn in Star connection.

2. Comparison of Power Consumption:

The active power consumed by the load in Delta connection is 3 times that consumed in Star connection.

Circuit Analysis from Diagram (Fig. 1)

• Node A: Potential fixed by 5V DC source: $V_A = 5\text{ V}$.
• Node B: Potential fixed by 10V DC source: $V_B = 10\text{ V}$.
• Central Node C: Connection between $2\ \Omega$ resistor (from A), $4\ \Omega$ resistor (from B), and $6\ \Omega$ load resistor (to ground).
• Top Branch: A $2\ \Omega$ resistor is connected directly between Node A and Node B.
• Important Observation: Since both Node A and Node B are connected directly across ideal voltage sources to ground, the top $2\ \Omega$ resistor is connected directly across the ideal 5V and 10V sources. While it draws current between those two source nodes, it has no direct connection to the central node C. Under superposition, with either source shorted, this top resistor simply connects across that source's nodes to ground without affecting branch currents into Node C!

Step 1: Case 1 — Only 5V Source Active (10V Source Short-Circuited)

1. Deactivate the $10\text{ V}$ source by replacing it with a short circuit to ground ($V_B = 0\text{ V}$).
2. The circuit connected to the 5V source consists of:

• A direct parallel $2\ \Omega$ resistor across the 5V source to ground (top branch).
• A path through the $2\ \Omega$ resistor leading to central Node C.

3. At Node C, two paths go to ground:

• The $6\ \Omega$ resistor to ground.
• The $4\ \Omega$ resistor connected to shorted Node B (ground).

4. Parallel combination at Node C:

5. Total resistance of the path through Node C seen by 5V source:

6. Current entering Node C from Node A:

7. Current through $6\ \Omega$ resistor due to 5V source ($I_1'$):

Using current division between the $6\ \Omega$ and $4\ \Omega$ resistors:

Step 2: Case 2 — Only 10V Source Active (5V Source Short-Circuited)

1. Deactivate the $5\text{ V}$ source by replacing it with a short circuit to ground ($V_A = 0\text{ V}$).
2. The path through Node C from the 10V source consists of:

• The $4\ \Omega$ resistor from Node B to central Node C.

3. At Node C, two paths go to ground:

• The $6\ \Omega$ resistor to ground.
• The $2\ \Omega$ resistor connected to shorted Node A (ground).

4. Parallel combination at Node C:

5. Total resistance of the path through Node C seen by 10V source:

6. Current entering Node C from Node B:

7. Current through $6\ \Omega$ resistor due to 10V source ($I_1''$):

Using current division between the $6\ \Omega$ and $2\ \Omega$ resistors:

Step 3: Total Current by Superposition

By the Superposition Theorem, the total current $I$ through the $6\ \Omega$ resistor is the algebraic sum of the individual currents produced by each source acting alone. Since both $I_1'$ and $I_1''$ flow in the downward direction:

Step 4: Verification via Nodal Analysis

Applying KCL at central Node C:

Multiply throughout by 12:

(Exact match verifies our Superposition solution!)

Final Answer: