Q6
(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 प्रश्न 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
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How this answer will be evaluated
Approach
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.
Key points expected
- 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
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 10 | Correctly identifies swing equation derivation for (a), states Nyquist criterion with proper encirclement convention for (b), and accurately describes spectrum analyzer working principle for (c); no conceptual confusion between transient and steady-state stability | Minor errors in swing equation form or Nyquist sign convention; understands basic concepts but confuses clockwise/counter-clockwise encirclements or omits M = H/(πf₀) relationship | Fundamental misconceptions: treats synchronous machine as static load, states incorrect Nyquist criterion (e.g., encirclements of origin), or describes oscilloscope instead of spectrum analyzer |
| Numerical accuracy | 20% | 10 | Precise calculation: Xeq = 0.5 p.u., δ₀ ≈ 28.7° or correct value via iterative solution, M = 0.0265 s²/rad, and correct encirclement count N for stability conclusion in (b) | Correct method but arithmetic errors in reactance combination or power angle calculation; correct Nyquist mapping but wrong stability conclusion due to counting error | Major numerical errors: adds reactances incorrectly (e.g., 0.4+0.4 instead of parallel combination), uses degrees instead of radians in swing equation, or completely omits numerical evaluation |
| Diagram quality | 20% | 10 | Clear single-line diagram for (a) showing machine, transformer, double-circuit line with reactances labeled; proper Nyquist plot for (b) with D-contour, GH-mapping, (-1,j0) point marked, and direction arrows; block diagram for feedback system | Diagrams present but missing key labels or arrows; Nyquist plot drawn without showing complete contour or critical point; hand-drawn appearance acceptable but lacks clarity | Missing essential diagrams, no Nyquist plot despite explicit instruction, or diagrams that contradict the textual solution; illegible sketches without axes labels |
| Step-by-step derivation | 20% | 10 | Systematic derivation: from power balance to swing equation in (a); from s-plane to GH-plane mapping with clear justification of each segment in (b); logical structure in (c) with clear categorization | Correct final results but skips intermediate steps; omits derivation of initial conditions or assumes δ₀ without calculation; Nyquist criterion applied without showing mapping process | No derivation shown—only final answers; jumps from given data to swing equation without explaining Pe = (E'V/X)sinδ relationship; states stability conclusion without any plotting or analysis |
| Practical interpretation | 20% | 10 | Interprets swing equation for transient stability assessment in Indian grid context; explains Nyquist criterion's value for systems with unknown open-loop poles; cites specific applications like harmonic analysis in 400kV substations or spectrum analyzer use in BHEL testing | Generic statements about stability importance; mentions standard applications without Indian context; understands practical relevance but lacks specific examples | No practical interpretation provided; treats all parts as purely mathematical exercises; fails to connect spectrum analyzer to electrical engineering practice |
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