Electrical Engineering 2023 Paper I 50 marks Compulsory Solve

Q5

(a) A 400 V, 50 Hz, 3-phase star-connected cylindrical rotor synchronous motor has synchronous impedance of (0·5 + j 2·5) Ω per phase. It develops a maximum power of 50 kW at rated terminal voltage. Find the excitation voltage, motor current and input power factor under maximum power condition. (10 marks) (b) A half-controlled converter fed from 240 V, 50 Hz single-phase ac source is feeding 1800 W power to a 100 V battery as shown in the figure below. The battery is connected in series with a large inductance and a resistance of 2 Ω. The inductance is large enough to make the load current flat and continuous. Find : (i) the triggering angle of the thyristors, (ii) rms value of fundamental component of converter input current, and (iii) the input power factor in the ac side. (Assume the inductor has a resistance of 1 Ω) (10 marks) (c) A signal x(t) is described as x(t) = (5/2) cos (160 × 10³ πt) + 7 cos (170 × 10³ πt) + (5/2) cos (180 × 10³ πt) Show that this is an Amplitude modulated signal. Find : (i) the ratio Pₛ/Pc where Pₛ is power in side bands and Pc is power in carrier. (ii) the power efficiency in this AM signal. (10 marks) (d) When a transmission line of characteristic impedance 50 Ω is short-circuited at the termination, the voltage minima were found to be 25 cm apart. If the short circuit is replaced by unknown load impedance Z_L Ω, the minima shifted 8 cm towards the load and the standing wave ratio was found to be 4. Calculate the unknown load impedance Z_L. (10 marks) (e) In the series RLC circuit shown in the figure, the capacitor has an initial charge Q_0 = 1 mC and the switch is in position 1 long enough to establish the steady state. Find the transient current which results when the switch is moved from position 1 to 2 at t = 0. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक 400 V, 50 Hz, त्रि-कला तारा-संयोजित वृत्तीय घुर्णक तुल्यकालिक मोटर की तुल्यकालिक प्रतिबाधा (0·5 + j 2·5) Ω प्रति कला है । यह निर्धारित अन्तस्थ बोल्टता पर 50 kW की अधिकतम शक्ति उत्पन्न करती है । अधिकतम शक्ति की स्थिति में उत्तेजना बोल्टता, मोटर धारा तथा निवेश शक्ति गुणांक ज्ञात कीजिए । (10 अंक) (b) 240 V, 50 Hz एकल-कला ए.सी. स्रोत द्वारा पोषित एक अर्ध-नियंत्रित परिवर्तक, एक 100 V की बैटरी को 1800 W शक्ति संभरित करता है, जैसा कि चित्र में प्रदर्शित है । बैटरी श्रेणीक्रम में एक विशाल प्रेरकत्व तथा एक 2 Ω के प्रतिरोध के साथ संयोजित है । प्रेरकत्व भार धारा को सपाट व सतत बनाने के लिए पर्याप्त बड़ा है । ज्ञात कीजिए : (i) थायरिस्टर का उत्त्प्रेरण (ट्रिगर) कोण, (ii) परिवर्तक की निवेश धारा के मूल घटक का rms मान, और (iii) ac की तरफ निवेश शक्ति गुणांक । (मान लीजिए कि प्रेरक में 1 Ω प्रतिरोध है) (10 अंक) (c) एक संकेत x(t), x(t) = (5/2) cos (160 × 10³ πt) + 7 cos (170 × 10³ πt) + (5/2) cos (180 × 10³ πt) द्वारा वर्णित है । दर्शाइए कि यह एक आयाम मॉडुलित संकेत है । ज्ञात कीजिए : (i) Pₛ/Pc का अनुपात, जहाँ Pₛ पार्श्व पट्टिका में शक्ति तथा Pc संवाहक में शक्ति है । (ii) इस AM संकेत में शक्ति दक्षता । (10 अंक) (d) जब 50 Ω लाक्षणिक प्रतिबाधा वाली एक पारेषण लाइन को अन्तस्थ पर लघु-परिपथित किया जाता है, तो बोल्टता के अल्पतमों की आपस में दूरी 25 cm पाई जाती है । यदि लघु परिपथ को एक अज्ञात भार प्रतिबाधा Z_L Ω से प्रतिस्थापित कर दिया जाता है, तो अल्पतम, भार की ओर 8 cm विस्थापित हो जाता है तथा अप्रगामी (स्थिर) तरंग अनुपात 4 पाया जाता है । अज्ञात भार प्रतिबाधा Z_L की गणना कीजिए । (10 अंक) (e) चित्र में प्रदर्शित श्रेणीक्रम RLC परिपथ में संधारित्र का आरम्भिक आवेश Q_0 = 1 mC है तथा स्विच पर्याप्त समय से स्थिति 1 में है ताकि स्थिर अवस्था स्थापित हो सके । t = 0 पर जब स्विच को स्थिति 1 से 2 में खिसकाया जाता है, तो परिणामी क्षणिक धारा ज्ञात कीजिए । (10 अंक)

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Approach

Solve all five numerical sub-parts systematically, allocating approximately 2 minutes per mark (20 minutes each). Begin with clear circuit/phasor diagrams where applicable, show all formulae with proper substitutions, and present final answers with units. For (a) use power-angle characteristics; for (b) analyze half-controlled converter operation; for (c) identify AM components using trigonometric identities; for (d) apply transmission line Smith chart or impedance transformation; for (e) solve second-order RLC transient with initial conditions.

Key points expected

  • (a) Correct application of P_max = (3 V_t E_f)/(X_s) for cylindrical rotor machine with δ = 90°; proper calculation of phase voltage and synchronous reactance
  • (a) Accurate computation of excitation voltage E_f, armature current I_a, and input power factor cos(φ) under maximum power condition
  • (b) Correct voltage balance equation for half-controlled converter: V_dc = (V_m/π)(1 + cos α) - I_a(r_L + r_bat); solving for firing angle α
  • (b) Proper Fourier analysis for fundamental component of input current and displacement factor calculation for power factor
  • (c) Recognition of AM signal structure using cos(A)cos(B) identity: identification of carrier frequency 85 kHz, sidebands at 80 kHz and 90 kHz, modulation index m = 5/7
  • (c) Correct calculation of sideband power ratio P_s/P_c = m²/2 and power efficiency η = m²/(2+m²)
  • (d) Proper use of wavelength λ = 2 × 25 cm = 50 cm; correct application of shift formula: tan(βl) = tan(2π/λ × 8 cm) for finding load impedance from SWR = 4
  • (e) Correct initial condition analysis: steady-state inductor current and capacitor voltage; proper second-order differential equation for RLC circuit with R, L, C values; complete transient solution with damping classification

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies all underlying concepts: power-angle relationship for (a), converter operation modes for (b), AM modulation theory for (c), standing wave phenomena for (d), and second-order transient analysis for (e); no conceptual errors in formula selectionMost concepts identified correctly but minor errors in formula application or missing one key theoretical foundation (e.g., wrong assumption about δ angle or confusion between half-controlled and full-controlled converter)Major conceptual errors such as using cylindrical rotor saliency equations, treating half-controlled as full-controlled, or applying first-order transient analysis to RLC circuit
Numerical accuracy25%12.5All calculations accurate to 2-3 significant figures; correct handling of complex arithmetic for (a), proper radian/degree conversion for (d), accurate Fourier coefficients for (b); final answers with proper unitsMost calculations correct but minor arithmetic errors or unit inconsistencies in 1-2 sub-parts; answers within 10% of correct valueSignificant calculation errors, wrong order of magnitude, or missing units in multiple sub-parts; fundamental misunderstanding of per-phase vs line quantities
Diagram quality15%7.5Clear phasor diagram for (a) showing V_t, E_f, I_a and reactance drop; converter circuit diagram for (b) with proper thyristor/diode labeling; AM spectrum for (c); transmission line setup for (d); RLC circuit for (e) with switch positionsDiagrams present but lacking labels or clarity; missing one required diagram or incomplete representation of circuit elementsNo diagrams or seriously flawed sketches that misrepresent the physical systems; incorrect circuit topology
Step-by-step derivation25%12.5Complete logical flow from given data to final answer for each sub-part; explicit statement of formulae before substitution; clear intermediate steps showing complex number operations, integration for Fourier series, or differential equation solvingDerivation present but skips key steps or assumes intermediate results without proof; some logical gaps in problem-solving sequenceMinimal working shown or 'magical' jumps to answers; no derivation of AM identity or transient solution form; purely numerical plug-in without theory
Practical interpretation15%7.5Interprets results meaningfully: comments on leading/lagging power factor implications for synchronous motor, discusses harmonic effects in converter, explains bandwidth requirements for AM signal, relates SWR to matching importance, discusses damping nature of transientBrief mention of practical significance for 2-3 sub-parts without elaboration; standard concluding statements without specific insightNo interpretation of results; purely mathematical exercise without connection to engineering applications or physical meaning of answers

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