Electrical Engineering 2025 Paper II 50 marks Compulsory Calculate

Q5

(a) Given a second-order linear time-invariant system G(s) with a relative degree of 2. G(s) admits a zero steady-state error for unit step input and steady-state error of 0·1 for unit ramp input under unity feedback configuration. Further, it admits a settling time of 4 seconds for 2% tolerance band in its unit step response under unity feedback. A delay of T seconds is now placed in cascade with G(s). Calculate the value of T in seconds that will make the delayed system oscillate under unity feedback configuration. 10 marks (b) A frequency counter with an accuracy of ±1LSD ±(1×10⁻⁶) is employed to measure frequencies of 100 Hz, 1 MHz and 100 MHz. Calculate the percentage measurement error in each case. What is the effect of time base on error? 10 marks (c) An 11 kV, 50 Hz alternator is connected to a system which has inductance and capacitance per phase of 10 mH and 0·01 µF respectively. Determine (i) the maximum voltage across circuit breaker contacts, (ii) the frequency of transient oscillation, (iii) the average RRRV and (iv) the maximum RRRV. 10 marks (d) Four 50 MVA alternators of 15% reactance each are connected via four 35 MVA reactors each of 10% reactance to a common bus bar. The feeders are connected to the junction of each alternator and its reactor. Determine the rating of each feeder circuit breaker. 10 marks (e) A code is made up of 'dots' and 'dashes'. Assuming that a dash is three times as long as a dot with one-third the probability of occurrence of a dot, calculate— (i) the information in a dot and a dash; (ii) the entropy of the dot-dash code; (iii) the average rate of information, if a dot lasts for 10 ms and this time is allowed between symbols. 10 marks

हिंदी में प्रश्न पढ़ें

(a) एक द्वितीय क्रम (ऑर्डर) का रैखीय समय-अपरिवर्ती तंत्र G(s) दिया गया है, जिसकी सापेक्ष डिग्री 2 है। इकाई पुनर्निवेश विन्यास के तहत G(s) के इकाई पद निवेश (यूनिट स्टेप इनपुट) के लिये स्थायी-दशा त्रुटि शून्य है, तथा इकाई प्रवण (रैप) निवेश के लिये स्थायी-दशा त्रुटि 0·1 है। पुनः यह तंत्र इकाई पुनर्निवेश के तहत, इकाई पद अनुक्रिया (रेस्पांस) में, 2% सहिष्णुता (टॉलरेंस) बैंड में स्थिरण काल 4 सेकंड प्राप्त करता है। अब G(s) के साथ T सेकंड का एक विलम्बन (डिले) सोपानित (कैस्केड) किया जाता है। T के उस मान की गणना सेकंड में कीजिये, जो कि विलम्बित (डिलेड) तंत्र को इकाई पुनर्निवेश विन्यास के तहत दोलित करेगा। 10 अंक (b) एक आवृत्ति गणित्र (फ्रीक्वेंसी काउंटर), जिसकी परिशुद्धता (एक्युरेसी) ±1LSD ±(1×10⁻⁶) है, को 100 Hz, 1 MHz और 100 MHz आवृत्तियों के मापन हेतु लगाया जाता है। प्रत्येक दशा के लिये मापन त्रुटि के प्रतिशत की गणना कीजिये। त्रुटि पर समय आधार (टाइम बेस) का क्या प्रभाव है? 10 अंक (c) एक 11 kV, 50 Hz प्रत्यावर्ती धारा जनित्र (आल्टरनेटर) एक ऐसे तंत्र से संयोजित है जिसका प्रतिफल प्रेरकत्व (इंडक्टेंस) एवं धारिता क्रमशः: 10 mH और 0·01 µF है। निर्धारण कीजिये (i) परिपथ विचोजक (सर्किट ब्रेकर) के संपर्क बिंदुओं के आर-पार अधिकतम वोल्टता, (ii) अल्पकालिक (ट्रांजियेंट) दोलन आवृत्ति, (iii) औसत RRRV और (iv) अधिकतम RRRV। 10 अंक (d) 50 MVA के चार प्रत्यावर्ती धारा जनित्रों को, जिनमें प्रत्येक का प्रतिघात 15% है, 35 MVA के चार रिएक्टरों के साथ एक साझे बस बार में संयोजित किया गया है। प्रत्येक रिएक्टर का प्रतिघात 10% है। प्रत्येक प्रत्यावर्ती धारा जनित्र एवं उसके रिएक्टर की संधि (जंक्शन) पर फीडर संयोजित हैं। प्रत्येक फीडर के परिपथ विचोजक (सर्किट ब्रेकर) की रेटिंग का निर्धारण कीजिये। 10 अंक (e) एक कूट (कोड) 'डॉट्स' एवं 'डैशेज' से निर्मित है। मान लीजिये कि एक डैश, एक डॉट से तीन गुना लम्बा है, तथा इसके उपस्थित होने की प्रायिकता, डॉट के उपस्थित होने की प्रायिकता से एक-तिहाई है। गणना कीजिये— (i) एक डॉट एवं एक डैश में सूचना; (ii) डॉट-डैश कूट की एन्ट्रॉपी; (iii) सूचना की औसत दर, यदि एक डॉट 10 ms तक रहता है और यह समय प्रतीकों के मध्य अनुमत है। 10 अंक

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Directive word: Calculate

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How this answer will be evaluated

Approach

Begin by identifying the directive 'calculate' demands precise numerical solutions across all five sub-parts. Allocate approximately 20% time each: for (a) derive ζ and ωn from steady-state and transient specifications, then apply Nyquist/delay stability criterion; for (b) compute percentage errors using LSD and time-base accuracy formulas; for (c) apply restriking voltage transient analysis for circuit breaker ratings; for (d) calculate fault MVA using symmetrical components for breaker sizing; for (e) apply Shannon entropy and information rate formulas. Present each sub-part with clear problem statement, formula application, substitution, and boxed final answer.

Key points expected

  • Part (a): Extract ζ=0.456 and ωn=2.19 rad/s from given specifications; determine critical delay T=0.458s using phase margin=0 condition for oscillation
  • Part (b): Calculate ±1LSD errors (±1%, ±10⁻⁴%, ±10⁻⁶%) and time-base errors; show percentage errors are 1.000001%, 0.000101%, and 0.000002% respectively; explain inverse relationship between frequency and time-base error contribution
  • Part (c): Compute maximum restriking voltage = 2×11√2/√3 = 17.96 kV; transient frequency = 1/(2π√(LC)) = 15.92 kHz; average RRRV = 4fVmax and maximum RRRV = ωVmax
  • Part (d): Calculate equivalent reactance of parallel alternator-reactor branches; determine fault level at bus; feeder breaker rating = 50 MVA based on alternator contribution to fault
  • Part (e): With P(dot)=0.75, P(dash)=0.25, calculate I(dot)=0.415 bits, I(dash)=2 bits; entropy H=0.811 bits/symbol; average symbol duration=13.33 ms; information rate=60.8 bits/second

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies second-order system type from error constants (Type 1), applies delay stability via characteristic equation 1+G(s)e^(-sT)=0, uses frequency counter error model ±1LSD±(time-base error), recognizes restriking voltage phenomenon in circuit breakers, applies symmetrical components for fault MVA, and correctly uses Shannon information theory with proper probability weightingIdentifies most concepts correctly but confuses Type 0/1/2 systems, applies approximate delay stability criteria, or misinterprets LSD vs. time-base error components; minor errors in fault current distribution or entropy formula applicationFundamental conceptual errors: treats system as Type 0, ignores delay effect on stability, confuses LSD with percentage error only, applies DC fault analysis to AC system, or uses incorrect probability values for entropy calculation
Numerical accuracy20%10All five sub-parts yield precise numerical answers: (a) T≈0.458s, (b) errors 1.000001%, 0.000101%, 0.000002%, (c) Vmax≈17.96kV, f≈15.92kHz, RRRV values correct, (d) breaker rating 50 MVA, (e) entropy 0.811 bits/symbol, rate ≈60.8 bps; maintains 3-4 significant figures throughoutCorrect methodology with minor calculation errors (±5% in final answers) or unit conversion mistakes; one sub-part with significant error but others correctMultiple sub-parts with order-of-magnitude errors, incorrect unit handling (kHz vs Hz, ms vs s), or missing final numerical answers despite correct setup
Diagram quality15%7.5Includes clear block diagrams for (a) unity feedback with delay element, (c) equivalent LC circuit for restriking voltage with breaker contact representation, (d) single-line diagram showing four alternator-reactor-feeder branches to common bus; all diagrams properly labeled with variablesSketches at least two relevant diagrams with basic labeling; may omit (d) system configuration or (c) equivalent circuit; diagrams functional but lacking polishNo diagrams provided where essential for problem understanding, or diagrams drawn without labels making them uninterpretable; missing critical system representations
Step-by-step derivation25%12.5Explicit derivation chain for each part: (a) Kv=10 from ess, ζ from Ts=4/(ζωn), then T from phase condition; (b) error propagation formula with clear LSD and time-base separation; (c) differential equation for LC circuit solution; (d) successive network reduction; (e) entropy definition to final rate formula; all steps logically connectedShows major steps but skips intermediate algebraic manipulations or assumes results without derivation; some sub-parts with complete derivation, others with gapsJumps directly to final formulas without derivation, presents unexplained numerical substitutions, or shows disconnected calculations without logical flow between steps
Practical interpretation20%10Interprets (a) delay margin for digital control implementation; (b) explains why high-frequency measurement needs better time-base stability (relevant to Indian ISRO/DRDO precision applications); (c) relates RRRV to actual 245kV/420kV breaker ratings; (d) connects to Central Electricity Authority fault level guidelines; (e) links to Morse code efficiency and modern data compressionBrief contextual mention for 2-3 sub-parts without elaboration; generic statements about 'practical importance' without specific engineering contextPurely mathematical treatment with no physical interpretation; fails to connect calculated values to real-world equipment ratings, measurement practices, or communication system design

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