GTU 3110005 Basic Electrical Engineering (BEE) Winter 2025 Solved Paper & Solutions

• Voltage rating ($V$) = $230\text{ V}$
• Power rating ($P$) = $2\text{ kW} = 2000\text{ W}$
• Operating duration ($t$) = $12\text{ hours/day}$
• Days in month = $30\text{ days}$
• Tariff rate = $\text{Rs. } 5\text{ per unit (kWh)}$

• 1. Current ($I$):

• 2. Resistance ($R$):

• 3. Daily Energy Consumed:

• 4. Monthly Energy Consumed & Energy Bill:

1. Sinusoidal Wave: Standard smooth periodic AC waveform ($v(t) = V_m \sin(\omega t)$).
2. Square Wave: Jumps abruptly between positive and negative peak levels.
3. Triangular Wave: Constant linear rise and linear fall.
4. Sawtooth Wave: Slow linear rise followed by an instantaneous vertical drop.

• Peak Value ($V_m$): The maximum instantaneous amplitude attained during a half cycle.
• Peak-to-Peak Value ($V_{p-p}$): Total vertical difference between positive and negative peaks:

• RMS Value ($V_{rms}$): The effective heating equivalent of DC voltage:

• Average Value ($V_{avg}$): The mean value over one half-cycle:

1. Mathematical Invariance: The derivative and integral of a sine wave is also a sine wave. It passes through inductors and capacitors without waveform distortion.
2. Easy Generation: Simple rotating coils in a uniform magnetic field naturally generate pure sine waves.
3. Minimal Harmonic Losses: Non-sinusoidal waveforms contain higher-order harmonics causing eddy current losses, motor vibration, and acoustic noise.
4. Efficient Transformation: Essential for smooth operation of transformers and AC electrical machines.

1. De-energize & Lockout-Tagout (LOTO): Always isolate the power supply and lock the main disconnect switch before inspecting or repairing circuits.
2. Use Insulated Tools: Always handle tools with rubber/PVC insulated grips rated for the operating voltage ($1000\text{ V}$ rated).
3. Wear Personal Protective Equipment (PPE): Wear dry rubber gloves, safety goggles, and rubber-soled electrical hazard footwear.
4. Never Work in Wet Conditions: Water drastically reduces skin resistance, increasing shock fatality.
5. Ensure Reliable Earthing: Metallic chassis of all electrical appliances must have low-resistance solid earth bonding.
6. Avoid Socket Overloading: Do not daisy-chain multi-plugs, which causes conductor overheating and electrical fire risks.
7. Discharge Capacitors: High-voltage capacitors can retain fatal electrical charges hours after disconnection; discharge them with a bleeder resistor.

• Lamp 1 ($L_1: 100\text{ V}, 100\text{ W}$):

• Lamp 2 ($L_2: 100\text{ V}, 60\text{ W}$):

• Total series resistance:

• Circuit current drawn:

• Power in $L_1$:

• Power in $L_2$:

• (Check total power: $56.25 + 93.75 = 150\text{ W} = 200\text{ V} \times 0.75\text{ A}$)

In any linear, bilateral network with multiple independent sources, the response (voltage or current) in any branch is the algebraic sum of the responses caused by each independent source acting alone, with all other independent sources deactivated.

1. Non-Linear Circuits: It does not apply to non-linear components (diodes, BJTs, Zener diodes, or non-linear resistors) where $V-I$ characteristics are not straight lines.
2. Power Calculation: It cannot be used to calculate power directly because power is a non-linear quadratic function of current/voltage:

(Total branch current must be calculated first before computing $I^2 R$).

3. Unilateral Elements: It cannot be applied to circuits with unilateral elements (e.g., rectifiers) where conduction depends on current direction.
4. Dependent Sources: Dependent sources cannot be turned off (set to zero); they must remain active while deactivating independent sources.

Any linear active two-terminal network containing voltage/current sources and resistances can be replaced by an equivalent circuit consisting of a single voltage source $V_{th}$ in series with a resistance $R_{th}$.

• Load resistance: $R_L = 2\;\Omega$ across terminals $A-B$.

1. Find $V_{th}$ (Open-Circuit Voltage):

• Disconnect the load resistor $R_L = 2\;\Omega$ from terminals $A$ and $B$.
• Calculate the open-circuit voltage across $A-B$ using nodal analysis or mesh analysis:

2. Find $R_{th}$ (Thevenin Resistance):

• Deactivate all independent sources:
• Replace independent voltage sources with a short-circuit.
• Replace independent current sources with an open-circuit.
• Calculate the equivalent resistance looking back into open terminals $A-B$:

3. Load Current ($I_L$):

• Reconnect the load resistor $R_L = 2\;\Omega$ to the Thevenin equivalent:

• Line Voltage ($V_L$) = $400\text{ V}$
• Phase Impedance ($Z_{ph}$) = $4 + j3\;\Omega$
• Phase Resistance ($R_{ph}$) = $4\;\Omega$
• Phase Reactance ($X_{ph}$) = $3\;\Omega$

• Phase Voltage in Star:

• Magnitude of Phase Impedance:

• Power Factor:

• Phase Current ($I_{ph}$):

(In Star connection, $I_L = I_{ph} = 46.188\text{ A}$)

• Active Power ($P$):

• Reactive Power ($Q$):

In a series RLC circuit, total impedance is:

Resonance occurs at frequency $f_0$ when the inductive reactance equals capacitive reactance:

1. Net Reactance: $X = X_L - X_C = 0$. Circuit is purely resistive.
2. Minimum Impedance: $Z_{\min} = R$.
3. Maximum Current: $I_0 = \frac{V}{R}$ (in phase with applied voltage).
4. Unity Power Factor: $\cos\phi = \frac{R}{Z} = 1$.
5. Voltage Magnification: Voltages across $L$ and $C$ can be much higher than supply voltage: $V_L = V_C = Q \cdot V$ (where $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$ is the Quality factor).

• 1. Sub-resonant region ($f < f_0$):

$X_C > X_L$. Circuit is capacitive. Impedance decreases with $f$; current leads; power factor is leading.

• 2. Resonant frequency ($f = f_0$):

$X_L = X_C$. Impedance reaches global minimum $Z = R$; current peaks at $I_{\max} = V/R$; power factor is exactly $1.0$.

• 3. Super-resonant region ($f > f_0$):

$X_L > X_C$. Circuit is inductive. Impedance increases with $f$; current lags; power factor is lagging.

The B-H curve illustrates the non-linear relationship between Magnetic Flux Density ($B$, in Tesla) and Magnetic Field Intensity ($H = \frac{N I}{l}$, in A/m) for a ferromagnetic material.

1. Initial Region (OA): At low $H$, magnetic domains slowly overcome friction; $B$ increases slowly.
2. Linear / Knee Region (AB): Rapid domain alignment; flux density rises steeply with $H$. Transformers and motors are designed to operate just below the "knee point".
3. Magnetic Saturation Region (BC): Almost all magnetic dipoles are fully aligned with the applied field. Further increases in $H$ yield negligible increases in $B$ (slope approaches $\mu_0$).

When alternating field $H$ is applied, $B$ traces a closed loop showing Hysteresis Loss:

• Retentivity / Residual Magnetism ($B_r$): The residual magnetic flux remaining in the core when magnetizing field $H$ drops to zero.
• Coercivity ($H_c$): The reverse magnetic field strength required to completely demagnetize the core ($B = 0$).
• Hysteresis Loss: Energy dissipated as heat per cycle is directly proportional to the area enclosed by the B-H loop ($W_h = \eta B_{\max}^{1.6} f V$).

A transformer operates on the principle of Mutual Induction governed by Faraday's Laws of Electromagnetic Induction.

• When alternating voltage $V_1$ is applied to the primary winding, it draws magnetizing current establishing alternating magnetic flux $\Phi = \Phi_m \sin(\omega t)$ in the core.
• This flux mutually links the secondary winding, inducing an electromotive force (EMF):

• Voltage ratio:

1. Laminated Magnetic Core:

• High-grade silicon steel laminations ($0.35\text{ mm}$ thick) coated with insulating varnish.
• Function: Provides a low-reluctance path for magnetic flux; lamination suppresses eddy current losses.

2. Primary & Secondary Windings:

• High-conductivity copper or aluminum coils insulated with paper/enamel.
• Function: Primary transfers electrical power into the core; secondary delivers converted voltage to the load.

3. Transformer Tank & Mineral Oil:

• Welded steel enclosure filled with dielectric mineral oil.
• Function: Provides high dielectric electrical insulation between windings and tank; circulates to dissipate heat.

4. Conservator Tank:

• Cylindrical drum mounted above the main tank connected via a pipe.
• Function: Accommodates expansion and contraction of oil due to temperature fluctuations without exposing oil to atmosphere.

5. Breather (Silica Gel):

• Vessel containing cobalt chloride impregnated silica gel crystals.
• Function: Absorbs moisture from incoming air when transformer "breathes" during cooling (blue when dry, pink when saturated).

6. Buchholz Relay:

• Gas-actuated safety relay installed in the pipe connecting main tank and conservator.
• Function: Triggers alarm on minor internal oil decomposition and instantly trips circuit breaker on severe electrical faults.

7. Bushings (Porcelain):

• Function: Insulates high-voltage terminal conductors passing through the grounded metal tank.

An underground cable consists of six concentric layers (from the innermost conductor outward):

Earthing (grounding) is the process of connecting the non-current carrying metallic parts of electrical equipment and the neutral star point of a supply system directly to the general mass of the earth via low-resistance conductors.

1. Protection of Human Life (Shock Prevention): If internal insulation fails, live conductor touches metallic casing. Without earthing, a person touching the case receives a fatal shock. With proper earthing, fault current is safely shunted to ground, clamping casing potential near $0\text{ V}$.
2. Prompt Operation of Protective Devices: Low earth resistance provides an easy path for heavy earth-fault current, instantly tripping MCBs, blowing fuses, or activating RCCBs within milliseconds.
3. Voltage Stabilization in 3-Phase Systems: Grounding the neutral wire keeps neutral potential clamped at zero volts, preventing dangerous phase voltage shifts during unbalanced loads.
4. Lightning & Surge Discharge: Diverts high-voltage atmospheric lightning strikes and inductive switching transients harmlessly to the ground through surge arresters.
5. Prevention of Electrical Fires: Discharges stray leakage currents that could otherwise cause localized arc heating and structural fire hazards.