All 8 questions from UPSC Civil Services Mains Electrical Engineering
2021 Paper II (400 marks total). Every stem reproduced in full,
with directive-word analysis, marks, word limits, and answer-approach pointers.
8Questions
400Total marks
2021Year
Paper IIPaper
Topics covered
Control systems, power systems, computer programming, measurements, communication systems (1)Power system analysis, control systems, error control coding (1)Power system protection and signal detection (1)Transmission lines and AC bridges (1)Current transformer, Routh stability, Piezo-electric transducer, Source coding, 8085 microprocessor (1)Synchronous machine stability, Nyquist criterion, Spectrum analyzer (1)Protection relays, network analysis, and graph theory (1)Error control coding, computer interfacing, and power system analysis (1)
A
Q1
50MCompulsorysolveControl systems, power systems, computer programming, measurements, communication systems
(a) A system is described by the following state equations :
$$\dot{x}_1 = x_1 + x_2 + 3x_3$$
$$\dot{x}_2 = 2x_1 + 3x_2 + u_1$$
$$\dot{x}_3 = 2x_2 + x_3 + u_2$$
Check the controllability of the system. (10 marks)
(b) A single phase, single line diagram of a power system is shown in figure. Find the sending end voltage and the value of load resistance in p.u. referred to sending end if the voltage across load resistance is 9·8 KV. (10 marks)
(c) Explain the following related to computer programming :
(i) Machine Language (ii) Assembly Language (iii) Compiler
(iv) Interpreter (v) ASCII (10 marks)
(d) A current of (0·5 + 0·3 sinωt – 0·2 sin 2ωt) amps is passed through the circuit shown in figure. Determine the reading of each instrument if ω = 10⁶ rad/sec. (10 marks)
(e) A DPCM system uses a linear predictor with a single tap. The normalized autocorrelation function of the input signal for a lag of one sampling interval is 0·75. The predictor is designed to minimize the prediction error variance. Determine the processing gain attained by the use of this predictor. (10 marks)
हिंदी में पढ़ें
(a) एक तंत्र को निम्नलिखित अवस्था समीकरणों द्वारा वर्णित किया गया है :
$$\dot{x}_1 = x_1 + x_2 + 3x_3$$
$$\dot{x}_2 = 2x_1 + 3x_2 + u_1$$
$$\dot{x}_3 = 2x_2 + x_3 + u_2$$
तंत्र की नियंत्रणीयता की जाँच करें । (10 अंक)
(b) एक एकल कला विद्युत शक्ति प्रणाली को एकल रेखीय आरेख द्वारा चित्र में दर्शाया गया है । यदि भार प्रतिरोध पर वोल्टता 9·8 KV हो तो प्रेषण सिरे की वोल्टता व प्रेषण सिरे के संदर्भ में भार प्रतिरोध का मान प्रति इकाई में ज्ञात करें । (10 अंक)
(c) अभिकलित्र प्रक्रमण के संदर्भ में निम्नलिखित की व्याख्या करें :
(i) मशीन भाषा (ii) समन्वयोजन (असेंबली) भाषा (iii) संकलक (कम्पाइलर)
(iv) भाषांतरक (v) ASCII (10 अंक)
(d) चित्र में दर्शाये गये परिपथ में (0·5 + 0·3 sinωt – 0·2 sin 2ωt) Amp विद्युत धारा का प्रवाह हो रहा है । यदि ω = 10⁶ rad/sec हो तो प्रत्येक मापन के पठन (रीडिंग) का निर्धारण करें । (10 अंक)
(e) एक DPCM तंत्र एक रैखिक प्राक्सूचक को एकल टैप के साथ प्रयोग करता है । एक प्रतिचयन अंतराल पश्चता के लिए सामान्यीकृत स्वतः सह संबंध फलन के निवेश का मान 0·75 है । प्राक्सूचक को, पूर्वानुमान त्रुटि के प्रसरण को निम्नतम करने के लिए अभिकल्प किया गया है । इस प्राक्सूचक के प्रयोग करने से प्राप्त संसाधन लब्धि का निर्धारण करें । (10 अंक)
Answer approach & key points
This is a multi-part technical question requiring precise problem-solving across five distinct domains: control systems, power systems, computer programming, measurements, and communication systems. Allocate approximately 20% time each to parts (a), (b), (d), and (e) which involve numerical calculations, and 20% to part (c) which is descriptive. Begin each part with the relevant governing equations, show systematic derivations, and conclude with boxed final answers for numerical parts.
Part (a): Form correct A and B matrices from state equations, construct controllability matrix Qc = [B AB A²B], compute rank and conclude controllability status
Part (b): Draw equivalent circuit with sending end, line impedance and load; apply KVL/KCL to find sending end voltage and load resistance in p.u. given load voltage of 9.8 kV
Part (c): Define machine language (binary, processor-specific), assembly language (mnemonic-based, one-to-one with machine code), compiler (full translation to object code), interpreter (line-by-line execution), and ASCII (7-bit character encoding standard)
Part (d): Analyze circuit with given current components; calculate instrument readings considering frequency response at ω = 10⁶ rad/sec (ammeter reads RMS, voltmeter responds to specific frequency component)
Part (e): Apply linear prediction theory with single tap; use normalized autocorrelation R(1) = 0.75 to find optimal predictor coefficient, then compute prediction gain as ratio of input variance to prediction error variance
50McalculatePower system analysis, control systems, error control coding
(a) Draw the sequence networks and calculate the load sequence impedances of a load circuit as shown in figure. The load circuit is connected to a balanced three phase supply. The value of z₁, z₂ and zₙ are (4 + j6) Ω, –j45 Ω and j4 Ω. (20 marks)
(b) For the network shown in figure, draw a block diagram representing each circuit element by a block. Use block diagram reduction technique to obtain the transfer function of the network. The voltage $V_i(t)$ is the applied input and the voltage across the capacitor $V_o(t)$ is the output. (20 marks)
(c) A convolutional code is described by
$$g_1 = [1\ 1\ 0],\ g_2 = [1\ 0\ 1],\ g_3 = [1\ 1\ 1].$$
Find the transfer function and the free distance for this code. Also verify whether or not this code is catastrophic. (10 marks)
हिंदी में पढ़ें
(a) चित्र में दर्शाये गये भार परिपथ के अनुक्रम संजालों (सीक्वेंस नेटवर्क्स) को आरेखित करें तथा भार अनुक्रम प्रतिबाधाओं की गणना करें । भार परिपथ को संतुलित तीन कलाओं की आपूर्ति से जोड़ा गया है । परिपथ की प्रतिबाधाओं का मान निम्न प्रकार है :
z₁ = 4 + j6 Ω, z₂ = –j45 Ω, zₙ = j4 Ω. (20 अंक)
(b) चित्र में दर्शाये गये परिपथ के लिए प्रत्येक परिपथ अंश को एक खण्ड से दर्शाते हुए खण्ड आरेखण करें । खण्ड आरेख लघुकरण तकनीक द्वारा संजाल (नेटवर्क) का अंतरण फलन प्राप्त करें । परिपथ की निवेश बोल्टता $V_i(t)$ तथा संधारित्र पर निर्गत बोल्टता $V_o(t)$ है । (20 अंक)
(c) एक संवलक कूट को निम्न प्रकार वर्णित किया गया है :
$$g_1 = [1\ 1\ 0],\ g_2 = [1\ 0\ 1],\ g_3 = [1\ 1\ 1].$$
इस कूट के लिए अंतरण फलन व मुक्त दूरी ज्ञात करें । यह भी सत्यापित करें कि क्या यह कूट आपातपूर्ण (कैटास्ट्रोफिक) है या नहीं । (10 अंक)
Answer approach & key points
Begin with a brief introduction acknowledging the three distinct domains covered: symmetrical components, control systems, and coding theory. For part (a), spend approximately 40% of effort (20 marks) drawing sequence networks and computing Z₀, Z₁, Z₂ with proper handling of the neutral impedance. For part (b), allocate 40% (20 marks) to converting the electrical network to a block diagram, applying reduction rules systematically to obtain Vₒ(s)/Vᵢ(s). For part (c), use remaining 20% (10 marks) to construct the state diagram, derive the transfer function matrix, compute free distance via minimum weight path, and apply Massey-Sain criterion for catastrophic property. Conclude with a summary table of results.
For (a): Correct sequence network diagrams showing positive, negative, and zero sequence connections with proper treatment of neutral impedance (3Zₙ in zero sequence)
For (a): Accurate calculation of Z₁ = Z₂ = z₁ + z₂ = (4+j6) + (-j45) = 4-j39 Ω and Z₀ = z₁ + 3zₙ + z₂ = (4+j6) + j12 + (-j45) = 4-j27 Ω
For (b): Proper block diagram construction with integrators, summers, and gain blocks representing the RLC network dynamics in Laplace domain
For (b): Systematic application of block diagram reduction rules (series, parallel, feedback) to arrive at final transfer function without algebraic errors
For (c): Correct generator matrix G(D) = [1+D, 1+D², 1+D+D²] and state diagram with 4 states (memory m=2)
For (c): Computation of free distance d_free = 5 by finding minimum weight non-zero codeword path through state diagram
For (c): Application of catastrophic code test: checking if GCD of generator polynomials equals D^l, concluding this code is non-catastrophic since GCD(1+D, 1+D², 1+D+D²) = 1
50McalculatePower system protection and signal detection
3.(a) A solidly earthed 400 KV, 3 phase busbar system is connected with two incoming and four outgoing lines (feeders). A differential protection is provided with switchgear of 4000 MVA capacity having the following parameters:
CT secondary resistance = 0.8 Ω
Lead wire resistance = 1.2 Ω
Relay load = 1.0 Ω
CT magnetization current = 0.3 mA/V
Max. full load current in one feeder = 100 A
Voltage setting of over current relay = 100 V
If the O.C. relay in the spill path is set at 1.0 A, find the following:
(a) The maximum 'through fault' current up to which the protection scheme remains stable. (20 marks)
(b) Whether the switchgear is capable to handle maximum through fault current. (20 marks)
(c) The value of minimum internal fault current that can be detected by protection scheme. (20 marks)
(d) The pick-up setting for detecting minimum internal fault current of 90 Amp. (20 marks)
3.(b) Consider a signal detector with an input
r = ±A + n
where +A and −A occur with equal probability and the noise variable n is characterized by the Laplacian pdf shown.
p(n) = (1/√2σ) e^(-|n|√2/σ)
(i) Determine the probability of error as a function of the parameters A and σ. (20 marks)
(ii) Determine the SNR required to achieve an error probability of 10^(-6). (20 marks)
3.(c) A coil of 300 V moving iron voltmeter has a resistance of 500 ohms and an inductance of 0.8 H. The instrument reads correctly at 50 Hz AC supply and takes 100 mA at full scale deflection. What is the percentage error in the instrument reading, when it is connected to 200 V DC supply. (10 marks)
हिंदी में पढ़ें
3.(a) दो आने वाले प्रदायकों व चार जाने वाले प्रदायकों को एक दृढ़ता से भू संपर्कित, त्रिकला, 400 KV बसबार तंत्र से जोड़ा गया है । एक 4000 MVA क्षमता वाले सिचंगियर द्वारा अंतरण संरक्षण (डिफरेंशियल प्रोटेक्शन) प्रदान कराया गया है । प्रणाली के प्राचल निम्न प्रकार हैं :
CT का द्वितीयक प्रतिरोध = 0.8 Ω
चालक तार का प्रतिरोध = 1.2 Ω
रिले भार का प्रतिरोध = 1.0 Ω
CT की चुंबकत्व धारा = 0.3 mA/V
प्रत्येक प्रदायक की अधिकतम पूर्ण भार धारा = 100 A
अधिधारा रिले की बोल्टता का निर्धारण = 100 V
यदि अधिधारा रिले, जो परिपथ में लगाई गई है, को 1.0 पर निर्धारित किया गया है, तो निम्नलिखित तथ्यों का निर्धारण करें ।
(a) अधिकतम शु-फाल्ट धारा का मान, ताकि संरक्षण प्रणाली संतुलित रहे । (20 marks)
(b) क्या संरक्षण प्रणाली अधिकतम शु-फाल्ट धारा को सहन करने में सक्षम है ? (20 marks)
(c) संरक्षण प्रणाली द्वारा संसूचित न्यूनतम अंतरण दोष धारा का मान ज्ञात करें । (20 marks)
(d) न्यूनतम अंतरण दोष धारा 90 A के लिए पिकअप सेटिंग का मान ज्ञात करें । (20 marks)
3.(b) एक संकेत संसूचक का निवेश निम्नप्रकार है
r = ±A + n
+A व −A समप्रायिकता के साथ घटित होता है तथा रवर (नॉयज वेरिएबल) n की विशेषता को लाप्लासियन pdf द्वारा दर्शाया गया है ।
p(n) = (1/√2σ) e^(-|n|√2/σ)
(i) प्राचल A व σ के फलन के रूप में त्रुटि की प्रायिकता का निर्धारण करें । (20 marks)
(ii) 10^(-6) त्रुटि प्रायिकता के लिए आवश्यक SNR ज्ञात करें । (20 marks)
3.(c) एक 300 V चल लोहे वोल्टमापी की कुण्डली का प्रतिरोध 500 ohm व प्रेरकत्व 0.8 H है । मापक यंत्र 50 Hz AC आपूर्ति पर दोषरहित मापन करता है तथा पूर्ण स्केल विचेषण के समय 100 mA धारा ग्रहण करता है । जब यह मापन यंत्र 200 V DC आपूर्ति के साथ जोड़ा जाता है तो प्रतिशत त्रुटि का मान ज्ञात करें । (10 marks)
Answer approach & key points
Calculate the required parameters systematically across all six sub-parts, allocating approximately 35% time to 3(a) parts (differential protection stability, switchgear capability, minimum internal fault, and pickup setting), 35% to 3(b) parts (Laplacian noise error probability and SNR calculation), and 30% to 3(c) (moving iron voltmeter DC error). Begin with clear circuit diagrams for differential protection, show all formulae with proper substitutions, and conclude with practical significance of each result for power system and communication engineering applications.
For 3(a): Calculation of CT knee-point voltage, stability limit using magnetization characteristic, and verification against switchgear fault MVA rating (4000 MVA at 400 kV)
For 3(a): Determination of minimum internal fault current considering relay burden and CT saturation, with correct application of 1.0 A spill path setting
For 3(a): Pick-up setting calculation for 90 A internal fault with proper CT ratio derivation from 100 A feeder current
For 3(b): Derivation of error probability P(e) = 0.5*exp(-A√2/σ) for Laplacian noise using ML detection criterion and integration of conditional densities
For 3(b): SNR calculation (A²/2σ²) for Pe = 10^-6, solving transcendental equation or using numerical approximation
For 3(c): Percentage error calculation comparing AC impedance (R + jωL) at 50 Hz with pure DC resistance, showing frequency-dependent moving iron instrument behavior
4.(a)(i) The configuration of a 400 KV 3 phase line is shown in figure. The radius of each sub-conductor is 2 cm. Calculate the charging mega volt-amperes if line is operating at 50 Hz and has a length of 300 km. (10 marks)
4.(a)(ii) Calculate the most economical overall diameter of insulation of a cable to be operated at 400 KV, 3 phase power system if maximum stress is limited to 100 KV/cm. (10 marks)
4.(b) Derive the conditions of balance of an Anderson's bridge and also draw the phasor diagram of the bridge under balanced condition. Determine the unknown quantities in terms of known parameters and comment on easy convergence of balance of the bridge. (20 marks)
4.(c) The approximate magnitude plot, obtained experimentally, of a nonminimum phase system is shown in figure. Calculate the phase in degrees at w = 3 rad/sec. (10 marks)
हिंदी में पढ़ें
4.(a)(i) एक 400 KV, त्रिकला लाइन का विन्यास चित्र में दर्शाया गया है । प्रत्येक सहिषित चालक की त्रिज्या 2 cm है । यदि लाइन की लम्बाई 300 km हो और 50 Hz पर संचालित हो तो लाइन का आवेशक (चार्जिंग) मेगा वोल्ट-एम्पीयर ज्ञात करें । (10 marks)
4.(a)(ii) 400 KV, त्रिकला शक्ति तंत्र में प्रयोग होने वाले केबल का अति मितव्ययी विद्युत रोधन सहित संपूर्ण व्यास का निर्धारण करें । केबल का सीमान्त अधिकतम रोधक प्रतिबल 100 KV/cm है । (10 marks)
4.(b) ऐण्डरसन सेतु के संतुलन की शर्त को व्युत्पन्न कीजिए व संतुलित अवस्था में कला आरेख (फेजर डायग्राम) बनाइए । ज्ञात प्राचलों के रूप में अज्ञात राशियों का मान ज्ञात करें । सेतु के संतुलन के सुगम अभिसरण पर टिप्पणी कीजिए । (20 marks)
4.(c) एक अनिम्नतम कला तंत्र का अनुमानित परिमाण आलेख प्रयोगात्मक विधि द्वारा प्राप्त किया गया है व जैसे चित्र में दर्शाया गया है । w = 3 rad/sec के लिए कला के मान की अंश में गणना करें । (10 marks)
Answer approach & key points
Solve this multi-part numerical and derivation problem by systematically addressing each sub-question: first calculate charging MVA for the 400 kV transmission line using geometric mean distance and capacitance formulas, then determine optimal cable insulation diameter using maximum stress criterion, followed by complete derivation of Anderson's bridge balance conditions with phasor diagram, and finally compute phase angle from the given Bode plot. Structure as: direct calculations for parts (a)(i)-(ii), rigorous derivation with diagram for part (b), and frequency response analysis for part (c).
Part (a)(i): Calculate geometric mean distance (GMD) from conductor configuration, then capacitance per phase C = 2πε₀/ln(GMD/r), and charging MVA = 3 × Vph² × ωC × length (in appropriate units)
Part (a)(ii): Apply most economical cable design condition where d = 2r = V/(gmax×e) for single-core cable, giving D = d×e, with overall diameter derived from maximum stress limitation
Part (b): Derive balance condition using star-delta transformation or mesh analysis: R1/R2 = R3/R4 + C3/C4 (approximately), with exact relation involving frequency-independent balance for inductance measurement
Part (b): Phasor diagram showing voltage drops across arms with proper phase relationships, indicating quadrature components due to capacitive branch
Part (c): Identify transfer function from asymptotic Bode plot slopes (±20 dB/decade changes), locate corner frequencies, reconstruct poles/zeros including right-half plane zero for non-minimum phase, evaluate phase at ω=3 rad/s
(a) Explain the ratio error and phase angle error of current transformer. 10 marks
(b) The two top rows of a Routh table of a characteristic polynomial is given in the table. Determine the roots of the characteristic equation which lie in the left half s-plane. Complete the remaining rows of the table. 10 marks
(c) A pulse is applied to a piezo-electric transducer for a time T. Prove that in order to keep the undershoot of the response to a value within 5%, the value of time constant should be approximately 20T. 10 marks
(d) A discrete memoryless source (DMS) has five symbols x₁, x₂, x₃, x₄ and x₅ with P(x₁) = 0·4, P(x₂) = 0·19, P(x₃) = 0·16, P(x₄) = 0·15 and P(x₅) = 0·1.
(i) Construct a Shannon Fano code for the source and calculate the efficiency of the code.
(ii) Repeat for Huffman code. Compare the results of (i) and (ii). 10 marks
(e) List the functional classification of 8085 instruction set. Give one example for each class. 10 marks
हिंदी में पढ़ें
(a) विद्युत धारा परिणामित्र की अनुपातिक त्रुटि व कला कोण त्रुटि की व्याख्या करें । 10
(b) एक अभिलक्षण बहुपद (कैरेक्टरिस्टिक पॉलिनोमियल) की रूथ सारणी की सबसे ऊपर की दो पंक्तियाँ नीचे दर्शायी गई हैं । s-तल के अर्ध बाम में स्थित अभिलक्षण समीकरण के मूलों का निर्धारण करें व सारणी की शेष पंक्तियों को पूर्ण करें । 10
(c) एक दाब विद्युत पारातंत्र पर T समय के लिए एक स्पंदन अनुप्रयुक्त किया गया है । सिद्ध करें कि समय स्थिरांक का अनुमानित मान 20T होगा यदि प्रतिक्रिया का अधोचरम (अंडरशूट) मान 5% तक सीमित हो । 10
(d) एक असतत स्मृतिहीन स्रोत (DMS) पांच प्रतीक चिह्न x₁, x₂, x₃, x₄, x₅ हैं जहाँ कि; P(x₁) = 0·4, P(x₂) = 0·19, P(x₃) = 0·16, P(x₄) = 0·15, P(x₅) = 0·1 है ।
(i) इस स्रोत के लिए शैनन फैनो कूट (कोड) का निर्धारण करें व कूट (कोड) की दक्षता की गणना करें ।
(ii) इस स्रोत के लिए हफमैन कूट का भी निर्धारण करें व (i) व (ii) के परिणामों की तुलना करें । 10
(e) 8085 सूक्ष्म संसाधक के अनुदेश समुच्चय को कार्यात्मक वर्गीकरण के अनुसार सूचीबद्ध करें । प्रत्येक वर्ग का एक उदाहरण लिखें । 10
Answer approach & key points
Begin with a brief introduction acknowledging the diverse instrumentation, control systems, and digital electronics topics covered. For part (a), explain ratio and phase angle errors with phasor diagrams and reduction methods; for (b), complete the Routh table systematically and apply stability criteria to count left-half plane roots; for (c), derive the piezo-electric transducer response equation and solve for the time constant condition; for (d)(i)-(ii), construct Shannon-Fano and Huffman codes stepwise, computing efficiencies and comparing optimality; for (e), enumerate 8085 instruction classes with clear examples. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 30% to (d), and 15% to (e), ensuring all derivations and calculations are shown explicitly.
Part (a): Define ratio error (KnIs - Ip)/Ip × 100% and phase angle error (δ) with phasor diagram showing excitation current I0, secondary current Is, and primary current Ip; mention reduction by using high permeability core, stranded conductors, and minimizing secondary burden
Part (b): Complete Routh table using standard recurrence formula b1 = (a1a2 - a0a3)/a1 etc.; determine sign changes in first column to identify right-half plane roots; deduce left-half plane roots count from polynomial degree minus RHP roots
Part (c): Derive piezo-electric transducer as second-order system with transfer function; apply rectangular pulse input of duration T; solve for undershoot condition at t=T+ and set ≤5% to obtain τ ≈ 20T
Part (d)(i): Construct Shannon-Fano code by recursive probability bisection (0.4|0.19,0.16,0.15,0.1 then subdivide); calculate average code length L and entropy H; efficiency η = H/L × 100%
Part (d)(ii): Construct Huffman code by iterative minimum probability combination; compare code lengths, efficiency, and note Huffman achieves optimal prefix code while Shannon-Fano may be suboptimal
Part (e): List 8085 functional classes: Data transfer, Arithmetic, Logical, Branch, Stack I/O & Machine control; provide one specific example each (e.g., MOV A,B, ADD B, ANA B, JMP addr, PUSH B, HLT)
Cross-cutting: Demonstrate awareness of practical CT applications in Indian power grid metering, 8085 relevance in legacy industrial controllers, and source coding in modern communication systems
(a) A synchronous machine is connected to an infinite bus through a transformer and a double circuit line as shown in figure. The infinite bus voltage is V = 1·0 ∠0° p.u. The direct axis transient reactance of the machine is 0·20 p.u., the transformer reactance is 0·10 p.u. and the reactance of each of the transmission lines is 0·4 p.u. all the values are to a base of the rating of the synchronous machine. Initially, the machine is delivering 0·8 p.u. power with a terminal voltage |Vₜ| = 1·05 p.u. The inertia constant H = 5 MJ/MVA. All resistances are neglected. Determine the equation of motion of the machine rotor. 20 marks
(b) State Nyquist stability criterion. Is the feedback system shown in figure in open loop stable ? Determine the closed loop stability of the system using Nyquist stability criterion. Show all the required plots clearly. 20 marks
(c) Write advantages, disadvantages and application of spectrum analyzer. 10 marks
हिंदी में पढ़ें
(a) एक तुल्यकालिक मशीन एक परिणामित्र व द्विपरिपथ लाइन के द्वारा एक अनंत बसबार से जुड़ी है । इस शक्ति तंत्र को चित्र में दर्शाया गया है । अनंत बसबार की बोल्टता V = 1·0 ∠0° p.u. है । मशीन का प्रत्यक्ष अक्ष क्षणिक प्रतिघात 0·20 p.u., परिणामित्र का प्रतिघात 0·10 p.u. व प्रत्येक प्रेषण लाइन का प्रतिघात 0·4 p.u. है । सभी राशियाँ मशीन की रेटिंग के आधार पर प्रति इकाई में परिवर्तित की गई है । प्रारंभ में मशीन अंतर्य बोल्टता |Vₜ| = 1·05 p.u. के साथ 0·8 p.u. शक्ति प्रदान करती है । यदि मशीन का जड़त्व स्थिरांक H = 5 MJ/MVA है तो सभी प्रतिरोधों की उपेक्षा करते हुए मशीन के रोटर की गति समीकरण का निर्धारण करें । 20
(b) नाइक्विस्ट स्थायित्व कसौटी व्यक्त करें । क्या चित्र में दर्शाया गया पुनर्निवेश तंत्र खुले पाश के रूप में स्थिर है ? नाइक्विस्ट स्थायित्व कसौटी का उपयोग करते हुए तंत्र के बंदपाश स्थायित्व का निर्धारण करें । सभी आवश्यक आरेखों को स्पष्ट रूप से दर्शायें । 20
(c) वर्णक्रम (स्पेक्ट्रम) विश्लेषक के लाभ, हानि व उपयोग लिखिए । 10
Answer approach & key points
Begin with part (a) by deriving the swing equation using the given reactances and initial conditions, calculating the equivalent reactance and initial rotor angle. For part (b), state the Nyquist criterion precisely, then construct the Nyquist plot by mapping the s-plane contour to the G(s)H(s) plane, counting encirclements to determine closed-loop stability. Conclude with part (c) by systematically listing advantages, disadvantages, and applications of spectrum analyzers in power system harmonic analysis and communication testing. Allocate approximately 40% time to (a), 40% to (b), and 20% to (c) based on mark distribution.
Part (a): Calculate equivalent reactance X = Xd' + Xt + XL/2 = 0.20 + 0.10 + 0.20 = 0.50 p.u. for parallel lines, then determine initial power angle δ₀ using P = (|E'||V|/X)sinδ₀ with given terminal voltage and power
Part (a): Derive the swing equation M(d²δ/dt²) = Pm - Pe where M = H/(πf₀), obtaining the second-order differential equation of motion with numerical coefficients
Part (b): State Nyquist criterion correctly: Z = P - N where Z is closed-loop RHP poles, P is open-loop RHP poles, N is net clockwise encirclements of (-1,0)
Part (b): Determine open-loop stability by locating poles of G(s)H(s), construct Nyquist contour (D-contour with indentations if poles on imaginary axis), map to GH-plane showing encirclements of critical point
Part (c): Advantages: wide frequency range, real-time display, measurement of harmonic distortion; Disadvantages: limited dynamic range, expensive; Applications: power quality analysis, EMI/EMC testing, vibration analysis in Indian power plants like NTPC installations
50MdiscussProtection relays, network analysis, and graph theory
(a) Discuss the percentage differential Relay with harmonic restraint with the help of diagram and also draw the conceptual representation of it. (20 marks)
(b) A sinusoidal voltage of 10 V amplitude at 100 Hz is applied to a lead network shown in figure. The phase difference between the input voltage $V_i(t)$ and output voltage $V_o(t)$ is 44.43°. If $C = 0.1 \mu F$ and $R_1 = 100 k\Omega$, determine the value of $R_2$ and the magnitude of steady state output voltage. (20 marks)
(c) Consider a connected graph G = (N, A) with N nodes and A arcs, and a weight ωij for each arc (i, j)∈A.
(i) Define minimum weight spanning tree (MST).
(ii) If all arc weights of G are distinct, prove that there exists a unique MST. (10 marks)
हिंदी में पढ़ें
(a) आरेख की सहायता से प्रतिशत विभेदी रिले का संयमित हार्मोनिक्स व्यवस्था समेत वर्णन करें व इसके संकल्पनात्मक प्रतिनिधित्व का आरेख बनाएं | (20 अंक)
(b) चित्र में दर्शाए गए अग्रता संजाल में 10 V आयाम, 100 Hz की ज्यावक्रीय वोल्टता को अनुप्रयुक्त किया गया है | निवेश वोल्टता $V_i(t)$ व निर्गत वोल्टता $V_o(t)$ में 44.43° का कलांतर है | यदि $C = 0.1 \mu F$ व $R_1 = 100 k\Omega$ है तो $R_2$ के मान का निर्धारण करें तथा स्थिर अवस्था में निर्गत वोल्टता के परिमाण की गणना करें | (20 अंक)
(c) एक संयोजित आरेख (ग्राफ) G = (N, A) में N नोड्स, A चाप (आर्क) व प्रत्येक चाप का मान (वेट) ωij है (i, j)∈A |
(i) न्यूनतम वेट स्पार्निंग ट्री (MST) को परिभाषित करें |
(ii) यदि G के सभी चाप का मान (वेट) भिन्न है तो सिद्ध करें कि एक विशिष्ट MST विद्यमान है | (10 अंक)
Answer approach & key points
Begin with a brief introduction on protection systems, then allocate approximately 40% effort to part (a) on percentage differential relays with harmonic restraint, 35% to part (b) on the lead network numerical solution, and 25% to part (c) on MST definition and uniqueness proof. Structure each part with clear headings, present derivations stepwise, and conclude with practical significance of harmonic restraint in transformer protection and MST applications in power system planning.
Part (a): Principle of percentage differential protection with percentage slope characteristic; need for harmonic restraint (2nd harmonic) to prevent maloperation during transformer inrush; block diagram showing CTs, restraint and operating coils, harmonic filter circuit, and trip logic
Part (a): Conceptual representation showing percentage differential characteristic with slope, operating region, and restraint region; explanation of why 15-20% slope is typical for percentage differential
Part (b): Correct transfer function derivation for lead network; phase angle condition tan(φ) = (R1+R2)/(ωCR1R2) or equivalent; solving for R2 using given phase difference of 44.43°
Part (b): Magnitude calculation |Vo/Vi| = ωCR2/√[1+(ωCR2)²] or appropriate expression; final numerical values with proper units (R2 in kΩ, |Vo| in volts)
Part (c)(i): Formal definition of MST as a spanning tree with minimum total weight; connected, acyclic subgraph containing all nodes with N-1 arcs
Part (c)(ii): Proof of uniqueness using cut property or cycle property; argument that with distinct weights, any two MSTs would lead to contradiction via edge exchange
Application context: Mention use of harmonic restraint in Indian power transformers (NTPC, state electricity boards) and MST in optimal transmission network design
50MproveError control coding, computer interfacing, and power system analysis
(a)(i) Prove that the minimum distance of any linear (n, k) block code satisfies dmin ≤ 1 + n – k. (5 marks)
(a)(ii) Show that the minimum Hamming distance of a linear block code is equal to the minimum number of columns of its parity check matrix that are linearly dependent. From this conclude that the minimum Hamming distance of a Hamming code is always equal to 3. (15 marks)
(b) A commercial interface unit uses different names for the handshake lines associated with the transfer of data from the I/O device into the interface unit. The interface input handshake line is labelled STB (strobe), and the interface output handshake line is labelled IBF (input buffer full). A low-level signal on STB loads data from the I/O bus into the interface data register. A high-level signal on IBF indicates that the data item has been accepted by the interface. IBF goes low after an I/O read signal from the CPU when it reads the content of the data register.
(i) Draw the block diagram showing the CPU, the interface, and the I/O device together with the pertinent interconnections among the three units.
(ii) Draw a timing diagram for the handshaking transfer.
(iii) Obtain a sequence of events flowchart for the transfer from the device to the interface and from the interface to the CPU. (20 marks)
(c) For a 3-bus power system, assume
Voltage at bus – 1 : V₁ = (1·05 + j 0) pu,
Voltage at bus – 2 : V₂ = (0·9812 – j 0·0522) pu and
Voltage at bus – 3 : V₃ = (0·999 – j 0·0468) pu.
The line impedances are shown below :
Bus code Impedances (in p.u.)
1 – 2 (0·02 + j 0·04)
1 – 3 (0·01 + j 0·03)
2 – 3 (0·0125 + j 0·025)
Compute Real and Reactive power loss in all the lines and also compute total system loss. (10 marks)
हिंदी में पढ़ें
(a)(i) सिद्ध करें कि किसी रैखिक (n, k) खंड कूट (कोड) की न्यूनतम दूरी का मान dmin ≤ 1 + n – k को संतुष्ट करता है | (5 अंक)
(a)(ii) दर्शाइए कि एक रैखिक खंड कूट की न्यूनतम हैमिंग दूरी इसकी पैरिटी चेक मैट्रिक्स जो कि रेखीय आधारित (आश्रित) है के न्यूनतम स्तंभों की संख्या के बराबर है | उपरोक्त से निष्कर्ष निकालिए कि हैमिंग कूट की न्यूनतम हैमिंग दूरी हमेशा 3 होती है | (15 अंक)
(b) एक वाणिज्य अंतःप्रेष्ट इकाई द्वारा I/O युक्ति से अंतःप्रेष्ट इकाई में डाटा स्थानांतरित करने हेतु संबद्ध हैंडशेक लाइनों के लिए भिन्न नामों का उपयोग होता है | अंतःप्रेष्ट निवेशक हैंडशेक लाइन पर STB (स्ट्रोब) अंकित किया गया है व निर्गत हैंडशेक लाइन पर IBF (इनपुट बफर फुल) अंकित किया गया है | निम्नतर संकेत अवस्था में STB डाटा को I/O बस से अंतःप्रेष्ट डाटा पंजी में भारित (लोड) किया जाता है | STB पर उपस्थित उच्च स्तर संकेत दर्शित करता है कि अंतःप्रेष्ट ने डाटा को ग्रहण कर लिया है | CPU से संकेतों को I/O द्वारा पढ़ने के बाद IBF का मान निम्न हो जाता है, जब यह डाटा पंजी के कंटेंट को पढ़ लेता है |
(i) खंड आलेख की सहायता से CPU, अंतःप्रेष्ट व I/O युक्ति को दर्शाते हुए आरेखण करें | साथ ही तीनों इकाइयों के मध्य उपयुक्त अंतरसंयोजनों को भी आलेख में प्रदर्शित करें |
(ii) हैंडशेकिंग स्थानांतरण के लिए समय-आलेख का आरेखण करें |
(iii) युक्ति से अंतःप्रेष्ट व अंतःप्रेष्ट से CPU में स्थानांतरण की क्रमबद्ध घटनाओं का प्रवाह चार्ट बनायें | (20 अंक)
(c) एक 3-बस शक्ति तंत्र के लिए माने कि
बस न. 1 पर बोल्टता : V₁ = (1·05 + j 0) pu,
बस न. 2 पर बोल्टता : V₂ = (0·9812 – j 0·0522) pu
बस न. 3 पर बोल्टता : V₃ = (0·999 – j 0·0468) pu. है |
लाइनों की प्रतिबाधा निम्नलिखित है :
बस कोड प्रतिबाधा (p.u. में)
1 – 2 (0·02 + j 0·04)
1 – 3 (0·01 + j 0·03)
2 – 3 (0·0125 + j 0·025)
सभी लाइनों में वास्तविक व प्रतिघाती शक्ति की हानि (ह्रास) की गणना करें व तंत्र की संपूर्ण शक्ति हानि की गणना करें | (10 अंक)
Answer approach & key points
Begin with rigorous mathematical proofs for (a)(i)-(ii) establishing the Singleton bound and Hamming code properties using parity check matrix concepts. For (b), construct clear block diagrams, timing diagrams, and flowcharts showing the STB-IBF handshake protocol between CPU, interface, and I/O device. Conclude with systematic power flow calculations for (c), computing complex power injections and line losses using the given 3-bus voltages and impedances. Allocate approximately 40% time to part (a), 40% to part (b), and 20% to part (c) based on mark distribution.
Proof of Singleton bound dmin ≤ 1 + n – k using linear code properties and dimension arguments
Demonstration that dmin equals minimum number of linearly dependent columns in parity check matrix H
Conclusion that Hamming codes have dmin = 3 based on H having distinct non-zero columns
Block diagram showing CPU, interface unit, and I/O device with STB, IBF, data bus, and control signal interconnections
Timing diagram with correct sequence: STB low → data loaded → IBF high → CPU read → IBF low
Flowchart distinguishing device-to-interface transfer (STB-driven) from interface-to-CPU transfer (read-driven)
Calculation of line currents using Iij = (Vi - Vj)/Zij for all three lines in the power system
Computation of complex power loss Sloss = |Iij|² × Zij and separation into real (P) and reactive (Q) components for each line