Q7
(a) A discrete memoryless source generates three independent message symbols m_1, m_2 and m_3 with probabilities 0·9, 0·08 and 0·02 respectively. (i) Design a binary variable length Huffman code. (Assign '0' to the message with highest probability). Determine the average code length, compression ratio and coding efficiency. (ii) Design a binary first-order extension code for this source and find the average code length for the designed code. Determine the compression ratio and efficiency for the extended code. 20 (b) What is the universal relay torque equation? Using this equation, derive the impedance relay, reactance relay and mho relay characteristics. Also draw the operating characteristic and indicate clearly the zones of operation and no operation. 20 (c) A single-phase distributor ABC fed at A is shown in the figure below. The impedances of the sections AB and BC are same and equal to (0·15 + j0·25) Ω. The power factors are lagging with respect to the voltage at the far end. If the voltage at the far end is 240 V, then determine the supply voltage and its phase angle with respect to the far end : 10
हिंदी में प्रश्न पढ़ें
(a) एक असतत स्मृतिहीन स्रोत, तीन स्वतंत्र संदेश प्रतीक-चिह्नों m_1, m_2 एवं m_3 का क्रमशः: 0·9, 0·08 एवं 0·02 प्रायिकताओं के साथ जनन करता है। (i) एक द्वि-आधारी परिवर्ती लम्बाई का हफमैन कूट डिजाइन कीजिए। ('0' उस संदेश को निर्दिष्ट कीजिए, जिसकी प्रायिकता अधिकतम हो।) औसत कूट लम्बाई, संपीड़न अनुपात एवं कोडन दक्षता ज्ञात कीजिए। (ii) इस स्रोत के लिए एक द्वि-आधारी प्रथम-श्रेणी विस्तार कूट अभिकल्पित कीजिए और इस अभिकल्पित कूट के लिए औसत कूट लम्बाई ज्ञात कीजिए। विस्तारित कूट के लिए संपीड़न अनुपात एवं दक्षता निकालिए। 20 (b) यूनिवर्सल रिले बल-आघूर्ण समीकरण क्या होता है? इस समीकरण का उपयोग करते हुए प्रतिबाधा रिले, प्रतिघात रिले एवं म्हो रिले के अभिलक्षणों को व्युत्पन्न कीजिए। इनके प्रचालन अभिलक्षणों को भी आरेखित कीजिए और प्रचालन व अप्रचालन क्षेत्रों (जोनस) को स्पष्टतापूर्वक दिखाइए। 20 (c) एक एकल-कला वितरक ABC, जिसका भरण A सिरे पर है, नीचे चित्र में प्रदर्शित किया गया है। AB एवं BC खंडों (सेक्शन्स) की प्रतिबाधाएँ एकसमान हैं और उसका मान (0·15 + j0·25) Ω के बराबर है। शक्ति गुणक अंतिम सिरे की वोल्टता से पश्चता में हैं। यदि अंतिम सिरे की वोल्टता 240 V है, तो सप्लाई वोल्टता एवं उसका कला कोण, अंतिम सिरे के संदर्भ में, ज्ञात कीजिए : 10
Directive word: Design
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How this answer will be evaluated
Approach
Design the Huffman code and first-order extension code for part (a) with proper probability tree construction, then derive the three relay characteristics from the universal torque equation for part (b) with clear R-X plane diagrams, and finally solve the distributor voltage drop problem for part (c) using phasor analysis. Allocate approximately 40% time to part (a) due to its 20 marks and dual sub-parts, 40% to part (b) for its derivations and diagrams, and 20% to part (c) for the numerical solution. Structure with clear headings for each part, showing all calculations stepwise and labeling diagrams precisely.
Key points expected
- For (a)(i): Construct Huffman code tree assigning '0' to highest probability (m₁), calculate average length L̄ = Σpᵢlᵢ, compression ratio η = H(X)/L̄, and efficiency; correct code: m₁=0, m₂=10, m₃=11 with L̄=1.12 bits/symbol
- For (a)(ii): Form 9-symbol first-order extension with joint probabilities, design Huffman code for extended source, calculate new average length per original symbol, and compare efficiency improvement approaching Shannon limit
- For (b): State universal torque equation T = K₁I² + K₂V² + K₃VIcos(θ-τ) + K₄, derive impedance relay (K₂=K₄=0), reactance relay (K₂=K₄=0, τ=90°), and mho relay (K₁=K₄=0) characteristics with proper algebraic manipulation
- For (b): Draw accurate R-X plane diagrams showing circular/linear characteristics, clearly marking operating zones (inside/on line for operation) and non-operating zones with directional indication for mho relay
- For (c): Calculate load currents from given power and power factors, determine voltage drops in sections AB and BC using IZ drops, apply KVL to find supply voltage magnitude and phase angle with respect to far end C
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 10 | Correctly applies Huffman coding algorithm with proper tree construction for (a), accurately states universal torque equation with all four terms for (b), and properly models the distributor as series impedance circuit with phasor KVL for (c); no conceptual errors in any part | Minor errors in Huffman tree construction or torque equation terms; generally correct approach but some confusion in relay characteristic derivations or distributor modeling | Fundamental misunderstanding of Huffman coding (e.g., fixed-length codes), incorrect universal torque equation, or treats distributor as DC circuit ignoring reactive components |
| Numerical accuracy | 20% | 10 | Precise calculations: (a)(i) L̄=1.12, η=0.985/1.12=87.9%, compression ratio vs fixed 2-bit; (a)(ii) extension L̄≈1.03; (c) correct phasor arithmetic yielding |V_A|≈250-255V with proper angle; all values to 2-3 decimal places | Correct method but arithmetic slips in probability products or complex number operations; final answers slightly off but method traceable | Major calculation errors, incorrect probability handling, wrong complex arithmetic, or missing units; answers without working get no benefit of doubt |
| Diagram quality | 20% | 10 | Clear Huffman tree diagram with probabilities and codes labeled for (a); for (b), three separate R-X plane plots with origin, impedance/reactance/mho circles/lines properly drawn, operating zones shaded and labeled, τ angle marked; distributor schematic for (c) with current directions | Diagrams present but lacking labels, or combined relay characteristics causing confusion; scale not indicated but shapes roughly correct | Missing diagrams, or sketches too rough to interpret; no indication of operating zones; no figure for (c) or incorrect current directions |
| Step-by-step derivation | 20% | 10 | Explicit stepwise: (a) shows probability sorting, tree merging steps, code assignment; (b) derives each relay by substituting K values, simplifies to |Z|<Z_set, X<X_set, |Z-V/2I|<V/2I forms; (c) shows current calculation, drop V_AB=I_AB×Z_AB, V_BC=I_BC×Z_BC, vector addition | Some steps skipped but key equations present; jumps from torque equation to final characteristic without intermediate algebra | Only final answers stated; no derivation of relay characteristics from torque equation; no intermediate voltage drop calculations in (c) |
| Practical interpretation | 20% | 10 | Interprets Huffman efficiency as approaching entropy limit with extension; explains relay applications—impedance for simple distance protection, reactance for ground faults (R-independent), mho for directional comparison (immune to load encroachment); notes voltage drop significance for Indian rural distribution standards | Brief mention of why extension improves efficiency or generic relay purpose; no specific context for distributor voltage regulation | No interpretation of results; purely mathematical treatment without physical meaning; no mention of practical relay selection criteria or distribution system voltage standards |
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