Electrical Engineering 2021 Paper I 50 marks Solve

Q6

(a) The magnetic field intensity of a linearly polarized uniform plane wave propagating in the +Y-direction in sea water (ε_r = 80, μ_r = 1, σ = 4 S/m) is H = 0·1 sin (10^10 πt - π/3) a_x A/m. At Y = 0, determine the following: (i) The attenuation constant, intrinsic impedance, the wavelength and skin depth. (ii) The location at which the amplitude of H is 0·01 A/m. (iii) The expression for E(y, t) and H(y, t) at Y = 0·5 (m) as functions of t. (20 marks) (b) A three-phase bridge inverter shown in Figure 6(b) is used to feed a Y-connected resistive load with R = 10 Ω per phase. The dc input to the inverter V_S = 400 V and the output frequency is 50 Hz. If the inverter is operating with 180° conduction mode, (i) compute the rms value of the load current, (ii) compute the rms value of the current in each switching device, (iii) calculate the output power, and (iv) draw the waveforms of phase and line voltages. (20 marks) (c) Let the measurement error of a physical quantity be defined by a random variable X and its density function as follows: f(x) = {K(3-x²) for -1≤x≤1, {0 elsewhere. Determine the value of 'K' and find the probability that a random error in measurement is less than 1/2. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) समुद्र के जल (ε_r = 80, μ_r = 1, σ = 4 S/m) में +Y-दिशा में संचरित एक रेखीय ध्रुवित एकसमान समतल तरंग के चुंबकीय क्षेत्र की तीव्रता H = 0·1 sin (10^10 πt - π/3) a_x A/m है । Y = 0 पर निम्न का मान ज्ञात कीजिए : (i) शीणन स्थिरांक, नैज (इंट्रिंसिक) प्रतिबाधा, तरंगदैर्ध्य तथा सतही गहराई (स्किन डेप्थ) । (ii) वह स्थान जहाँ H का आयाम 0·01 A/m है । (iii) Y = 0·5 (m) पर t के फलन के रूप में E(y, t) तथा H(y, t) के लिए व्यंजक । (20 अंक) (b) चित्र 6(b) में प्रदर्शित एक त्रि-कला सेतु (ब्रिज) प्रतिलोमक (इन्वर्टर), एक Y-संयोजित R = 10 Ω प्रति कला वाले प्रतिरोधीय भार को भारित करने के लिए प्रयुक्त किया जाता है । प्रतिलोमक में dc निवेश V_S = 400 V तथा निगत आवृत्ति 50 Hz है । यदि प्रतिलोमक 180° चालकता विधा में कार्यशील है, तो (i) भार धारा का rms मान ज्ञात कीजिए, (ii) प्रत्येक स्विचिंग युक्ति में धारा का rms मान ज्ञात कीजिए, (iii) निर्गत शक्ति ज्ञात कीजिए, और (iv) कला वोल्टता तथा लाइन वोल्टता के तरंग रूपों को आरेखित कीजिए । (20 अंक) (c) मान लीजिए कि एक भौतिक राशि की मापन त्रुटि एक यादृच्छिक चर X तथा इसका घनत्व फलन निम्नलिखित है : f(x) = {K(3-x²) for -1≤x≤1, {0 अन्यथा । तो, 'K' के मान की गणना कीजिए तथा मापन में यादृच्छिक त्रुटि के 1/2 से कम होने की प्रायिकता ज्ञात कीजिए । (10 अंक)

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Approach

This is a computational problem requiring systematic solution of three distinct parts. Allocate approximately 40% of time to part (a) given its 20 marks and complexity involving good conductor analysis; 40% to part (b) for inverter calculations and waveform sketching; and 20% to part (c) for probability determination. Begin each part with stated assumptions and relevant formulas, proceed through step-by-step calculations with proper units, and conclude with boxed final answers.

Key points expected

  • Part (a): Calculate attenuation constant α, intrinsic impedance η, wavelength λ, and skin depth δ for sea water at given frequency, identifying it as a good conductor (σ/ωε >> 1)
  • Part (a)(ii): Determine propagation distance for amplitude decay from 0.1 A/m to 0.01 A/m using exponential attenuation formula
  • Part (a)(iii): Derive time-domain expressions for E(y,t) and H(y,t) at y=0.5m, accounting for phase shift and attenuation
  • Part (b): Compute RMS load current, device current, output power for 180° conduction mode three-phase bridge inverter with Y-connected resistive load
  • Part (b)(iv): Sketch phase voltages (V_AN, V_BN, V_CN) and line voltages (V_AB, V_BC, V_CA) showing 120° phase displacement and six-step waveform
  • Part (c): Determine normalization constant K by integrating PDF over [-1,1], then calculate P(X < 0.5) through proper integration

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies sea water as good conductor for part (a) with σ/ωε ≈ 18 >> 1; applies proper 180° conduction mode relationships for three-phase inverter in (b); recognizes PDF normalization requirement in (c); uses correct formulas for intrinsic impedance η = √(jωμ/σ) and skin depth δ = 1/αIdentifies some key concepts but makes errors in conductor classification or uses approximate formulas; applies basic inverter relationships but confuses line/phase quantities; understands PDF requires normalization but makes integration errorsTreats sea water as lossless dielectric or uses free space formulas; applies incorrect conduction mode assumptions; fails to recognize PDF normalization requirement or uses invalid probability formulas
Numerical accuracy25%12.5Precise calculations: α ≈ 83.78 Np/m, η ≈ 9.93∠45° Ω, δ ≈ 11.94 mm, λ ≈ 7.5 cm; location y ≈ 27.5 mm for amplitude decay; correct RMS values I_L(rms) = 23.09 A, I_device(rms) = 13.33 A, P_out = 16 kW; K = 3/8, P(X<0.5) = 0.6875Correct methodology with minor calculation errors (within 10%); proper order of magnitude for all quantities; correct final answers for at least two parts with minor slips in thirdOrder-of-magnitude errors in key parameters; confused units (dB vs Np); incorrect RMS calculations; wrong probability values due to integration or arithmetic errors
Diagram quality15%7.5Clear, labeled waveforms for part (b)(iv) showing six-step line voltages (±400V, ±200V, 0) and quasi-square phase voltages with 120° phase displacement; proper time axis scaled to 20ms period; distinct voltage levels marked; clean hand-drawn or sketched appearanceRecognizable waveforms with basic structure correct but missing labels or incorrect voltage levels; phase relationships approximately correct; some confusion between phase and line voltage waveformsMissing diagrams where required; incorrect waveform shapes (e.g., sinusoidal instead of square/step); no distinction between phase and line quantities; unlabeled axes making interpretation impossible
Step-by-step derivation25%12.5Explicit derivation showing: good conductor verification, complex propagation constant γ = α + jβ calculation, separation of real/imaginary parts; inverter voltage relationships V_L-L = V_S for 180° mode, duty cycle analysis for RMS; definite integral setup for K and probability with limits clearly statedShows key intermediate steps but skips some algebraic manipulation; states formulas used but limited derivation of RMS relationships; sets up integrals correctly but shows limited workingJumps to final answers without showing working; no intermediate steps for complex quantities; missing integration steps for probability; simply states RMS formulas without justification
Practical interpretation15%7.5Interprets skin depth relevance for submarine communication/sonar in Indian Ocean context; notes impracticality of EM propagation in sea water at 5 GHz (very high attenuation); discusses 180° mode suitability for motor drives; comments on measurement error distribution being symmetric and boundedBrief mention of practical implications without elaboration; notes high attenuation in sea water or mentions typical inverter applications; recognizes error bounds in part (c)No physical interpretation provided; purely mathematical treatment without context; fails to recognize why 5 GHz propagation in sea water is problematic for practical systems

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