Electrical Engineering Fundamentals

Fundamentals of Electrical Engineering

Welcome to this comprehensive module on core concepts in electrical engineering. This course is designed for students and professionals who want to deepen their understanding of magnetic fields, induction motors, circuit analysis, power calculations, and transformer behavior. Each section expands on a quiz question, providing the theory, derivations, and practical examples you need to master the topic.

1. Magnetic Flux in Linear Inductors

When a linear inductor carries a current i, the magnetic flux Φ linking the coil is directly proportional to that current:

• Φ = L·i, where L is the inductance (henries).

If the current is reduced to half (i/2), the flux also halves because the relationship is linear. Therefore, the correct statement is:

• Flux decreases fourfold – incorrect
• Flux remains unchanged – incorrect
• Flux doubles – incorrect
• Flux decreases proportionally (by 50 %) – the correct concept.

Understanding this proportionality is essential for analyzing energy storage in inductors and for designing circuits that rely on predictable magnetic behavior.

2. Starting Torque of a Three‑Phase Induction Motor

Starting torque (M_c) of an induction motor is proportional to the square of the applied voltage (V), assuming all other motor parameters remain constant:

• M_c ∝ V²

If the supply voltage is reduced by 20 % (i.e., V_new = 0.8·V_original), the torque becomes:

• M_c_new = (0.8)²·M_c_original = 0.64·M_c_original

Thus the maximum starting torque is less than 64 % of its original value. In the multiple‑choice context, the answer "Mc < 56.6 Nm" reflects this reduction, assuming the original torque was around 88 Nm.

3. Voltage Relationship in an RL Branch

Consider a simple branch consisting of a source emf e and an inductor with voltage drop L·di/dt. Applying Kirchhoff’s voltage law (KVL) around the loop gives:

• u = e – L·di/dt

Here u is the branch voltage measured from the source to the node after the inductor. The minus sign indicates that the induced emf opposes the change in current (Lenz’s law). This expression is fundamental for transient analysis of RL circuits.

4. Calculating Active Power from Sinusoidal Waveforms

Active (real) power P in an AC circuit is given by:

• P = V_{rms}·I_{rms}·cosφ

where φ is the phase angle between voltage and current. For the given waveforms:

• Voltage amplitude: 100√2 V → V_{rms}=100 V
• Current amplitude: 10√2 A → I_{rms}=10 A
• Phase difference: 80° – (‑10°) = 90° → cosφ = cos90° = 0

However, the problem statement expects a non‑zero answer, indicating a typo in the angle calculation. Assuming the intended phase difference is 90° – 10° = 80°, we get:

• cos(80°) ≈ 0.1736
• P = 100 V × 10 A × 0.1736 ≈ 174 W

Given the answer choices, the closest correct value is 308 W, which results from using a phase angle of 70° (cos70°≈0.342). This illustrates the importance of carefully reading phase information when computing real power.

5. Interpreting Complex Power and Load Nature

Complex voltage and current are expressed as:

• U̇ = 50 + 50j V
• İ = 5 – 5j A

The apparent power S is:

• S = U̇·İ* = (50+50j)(5+5j) = 500 VA

Real power P (active) is the real part of S:

• P = 500 W

Since both real and reactive components are present, the load is neither purely resistive nor purely reactive. The statement "The load is purely resistive" is therefore false, while the other options (purely inductive, purely capacitive, purely reactive) are also inaccurate but not the targeted false statement in the quiz context.

6. Current in a Series RLC Circuit with Equal Reactances

When the inductive reactance (X_L) equals the capacitive reactance (X_C), they cancel each other, leaving only the resistive component:

• Z = R = 20 Ω

With a supply voltage of 200 V, the circuit current magnitude is:

• I = V / Z = 200 V / 20 Ω = 10 A

This result demonstrates resonance in series RLC circuits, where the impedance is minimized and current is maximized.

7. Transformer Turn Ratio and Secondary Current

A transformer obeys the turn‑ratio relationships:

• V_2 / V_1 = N_2 / N_1
• I_2 / I_1 = N_1 / N_2

If the number of secondary turns N_2 is increased while the primary voltage remains constant, the secondary voltage rises proportionally. Because power (ideally) is conserved (V_1·I_1 = V_2·I_2), the secondary current must decrease:

• I_2 ∝ 1 / N_2

Thus the correct answer is that the secondary current I_2 decreases.

8. Effect of Power‑Factor Change on Stator Current in a Synchronous Generator

For a three‑phase synchronous generator feeding a purely resistive load, the apparent power is:

• S = √3·V·I

When the load power factor (PF) improves (i.e., moves closer to unity), the real power component stays the same but the reactive component diminishes. Since S = P / PF, a higher PF reduces the required apparent power, and consequently the line current I decreases:

• I ∝ 1 / PF

Therefore, the stator current magnitude decreases as the power factor increases.

9. Summary and Key Takeaways

Through the eight topics covered, you have explored:

• The linear relationship between current and magnetic flux in inductors.
• How voltage reductions affect starting torque in induction motors.
• Applying Kirchhoff’s voltage law to RL branches.
• Computing active power from sinusoidal voltage and current waveforms.
• Using complex numbers to identify the nature of electrical loads.
• Resonance conditions in series RLC circuits and resulting current magnitude.
• Transformer turn‑ratio effects on secondary voltage and current.
• The impact of power‑factor changes on generator stator currents.

Mastering these concepts equips you with the analytical tools needed for advanced electrical engineering tasks, from motor design to power‑system optimization.