Q7
(a) It is given that $\vec{E} = E_m \sin (\omega t - \alpha z) \hat{a}_y$ in free space $\alpha > 0$. (i) Determine $\vec{D}$, $\vec{B}$ and $\vec{H}$. Plot $\vec{E}$ and $\vec{H}$ at $t = 0$. State clearly if any assumption is made. (ii) Show that these $\vec{E}$ and $\vec{H}$ fields constitute a wave travelling in the z-direction. Also demonstrate that the wave speed and E/H depend solely on the properties of free space. Given : $\mu_0 = 4\pi \times 10^{-7}$ H/m, and $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m. (b) (i) A 3-phase, 4-pole, 400 V, 10 kW, 50 Hz slip ring induction motor develops rated output at rated voltage and frequency with its slip ring short-circuited. The maximum torque equal to twice the full load torque, occurs at a slip of 12·5% with zero external resistance in rotor circuit. Neglect stator impedance, stator core and mechanical losses. Determine : I. slip and motor speed at full load torque, and II. starting current in terms of full load current. (10 marks) (ii) An industry has an average electrical load of 600 kW at a p.f. of 0·6 lagging. A synchronous motor with an efficiency of 90% is used to raise the combined p.f. to 0·9 lagging and at the same time supply a mechanical load of 100 kW. Calculate kVA capacity of the synchronous motor and synchronous motor operating power factor. (10 marks) (c) The buck-boost converter has an input voltage of $V_s = 12$ V. The duty cycle $D = 0·25$ and the switching frequency is 20 kHz. The inductance $L = 150$ μH and filter capacitor $C = 250$ μF. The average load current $I_0 = 1·25$ A. Determine : (i) the peak-to-peak ripple in the inductor current, and (ii) the critical values of inductor L and capacitor C for CCM. (10 marks)
हिंदी में प्रश्न पढ़ें
(a) दिया गया है कि $\vec{E} = E_m \sin (\omega t - \alpha z) \hat{a}_y$ और मुक्त अंतराल (फ्री-स्पेस) में $\alpha > 0$ है। (i) $\vec{D}$, $\vec{B}$ और $\vec{H}$ ज्ञात कीजिए। $t = 0$ पर $\vec{E}$ और $\vec{H}$ का अंकन कीजिए। यदि कोई कल्पना है तो स्पष्ट उल्लेख कीजिए। (ii) दर्शाइए कि यह $\vec{E}$ और $\vec{H}$ क्षेत्र, z-दिशा में सरण (गमन) करती हुई एक तरंग निर्मित करते हैं। यह भी दर्शाइए कि तरंग गति और E/H पूर्णतः मुक्त अंतराल के गुणधर्मों पर आधारित है। दिया गया है : $\mu_0 = 4\pi \times 10^{-7}$ H/m और $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m. (b) (i) एक 3-कला, 4-ध्रुव, 400 V, 10 kW, 50 Hz सर्पी बलय प्रेरण मोटर, सर्पी बलय लघुपथित होने की दशा में निर्धारित (रेटेड) वोल्टता और आवृत्ति पर निर्धारित (रेटेड) निर्गत विकसित करती है। अधिकतम बल-आघूर्ण जो निर्धारित पूर्ण भार बल-आघूर्ण से 2 गुना है, 12·5% सर्पण पर उत्पन्न होता है जब कि रोटर परिपथ में बाह्य प्रतिरोध शून्य है। स्टेटर प्रतिबाधा, स्टेटर कोर तथा यांत्रिक क्षतियाँ नगण्य मानते हुए, I. पूर्ण भार बल-आघूर्ण पर सर्पण और मोटर की गति, तथा II. पूर्ण भार धारा के पदों में प्रवर्तन धारा का मान ज्ञात कीजिए। (10 अंक) (ii) एक उद्योग का 0·6 पश्चगामी p.f. पर औसत विद्युत भार 600 kW है। संयुक्त p.f. को 0·9 पश्चगामी तक बढ़ाने के लिए 90% दक्षता वाली एक तुल्यकालिक मोटर प्रयोग की जाती है, जो साथ ही साथ 100 kW यांत्रिक भार भी प्रदाय करती है। तुल्यकालिक मोटर की kVA धारिता और कार्यकारी (ऑपरेटिंग) शक्ति गुणांक की गणना कीजिए। (10 अंक) (c) एक बक-बूस्ट परिवर्तित्र में निवेश वोल्टता $V_s = 12$ V है। कर्म चक्र $D = 0·25$ तथा स्विचन आवृत्ति 20 kHz है। प्रेरक $L = 150$ μH तथा छनक संधारित्र $C = 250$ μF और औसत भार धारा $I_0 = 1·25$ A है, तो (i) प्रेरक धारा में शिखर-से-शिखर झ्रिमिका का मान, और (ii) CCM के लिए प्रेरक L और संधारित्र C के क्रांतिक मानों की गणना कीजिए। (10 अंक)
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How this answer will be evaluated
Approach
This is a multi-part numerical problem requiring systematic solution of electromagnetic wave propagation, induction motor characteristics, power factor correction, and DC-DC converter analysis. Allocate approximately 35% time to part (a) on EM waves (20 marks), 35% to part (b) on machines and power systems (20 marks), and 30% to part (c) on power electronics (10 marks). Begin each sub-part with stated assumptions, proceed with clear derivations, and conclude with numerical answers in proper units.
Key points expected
- Part (a)(i): Correct application of constitutive relations D = ε₀E, B = μ₀H, and Maxwell's equations to derive B = (αEₘ/ω)cos(ωt-αz)âₓ and H = B/μ₀; proper sinusoidal plots of E and H at t=0 showing 90° spatial phase difference
- Part (a)(ii): Demonstration that (ωt-αz) = constant implies phase velocity vₚ = ω/α = 1/√(μ₀ε₀) = c ≈ 3×10⁸ m/s; proof that |E|/|H| = √(μ₀/ε₀) = η₀ ≈ 377Ω (intrinsic impedance of free space)
- Part (b)(i): Using torque-slip relation T ∝ sR₂/(R₂²+(sX₂)²), determination of full-load slip s_FL = 0.05 (5%) giving speed = 1425 rpm; starting current ratio I_st/I_FL = 2.5 using equivalent circuit with neglected stator impedance
- Part (b)(ii): Calculation of reactive power compensation where original load has 800 kVAR lagging; synchronous motor must draw 260.4 kVAR leading to achieve 0.9 lagging combined p.f.; resulting motor kVA = 370.3 kVA at 0.27 leading p.f.
- Part (c)(i): Application of buck-boost inductor current ripple formula ΔI_L = DV_s/(Lf) = 0.25×12/(150×10⁻⁶×20×10³) = 1 A peak-to-peak
- Part (c)(ii): Critical inductance L_crit = D(1-D)²R/(2f) = 0.25×0.75²×9.6/(2×20×10³) = 33.75 μH; critical capacitance C_crit = (1-D)/(8Lf²×(ΔV₀/V₀)) for specified ripple criterion
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 12 | Correctly identifies wave propagation direction from phase (ωt-αz); applies proper torque-slip characteristics for induction motor; understands synchronous motor as reactive power compensator; correctly identifies buck-boost converter voltage polarity inversion and CCM/DCM boundary conditions | Minor errors in identifying wave characteristics or torque-slip relationship; partially correct understanding of power factor correction but confuses leading/lagging operation; basic understanding of converter operation but incorrect polarity awareness | Fundamental misunderstanding of Maxwell's equations application; incorrect torque-slip formula; treats synchronous motor as induction motor; fails to recognize buck-boost converter characteristics |
| Numerical accuracy | 20% | 12 | All calculations precise: wave speed exactly 3×10⁸ m/s, intrinsic impedance 120π ≈ 377Ω; slip exactly 0.05, speed 1425 rpm; motor kVA 370.3 at 0.27 leading p.f.; ripple 1.00 A, L_crit 33.75 μH; proper unit handling throughout | Minor calculation errors (e.g., 1420 rpm instead of 1425 rpm, or 376Ω vs 377Ω); correct methodology but arithmetic slips; acceptable unit conversions with occasional omissions | Major calculation errors; incorrect formulas leading to wrong orders of magnitude; missing or inconsistent units; failure to use given ε₀, μ₀ values |
| Diagram quality | 20% | 12 | Clear plots for (a)(i): E as sine wave and H as cosine wave versus z at t=0, properly labeled axes, amplitude ratio shown; phasor diagrams for (b) showing power triangles; converter circuit diagram with current/voltage waveforms for (c) | Basic sketches present but lacking proper labels or scale; rough indication of phase relationships; circuit diagrams correct but waveforms missing or poorly drawn | Missing required plots; incorrect waveforms (e.g., E and H in phase); no diagrams despite explicit 'plot' instruction; illegible or irrelevant sketches |
| Step-by-step derivation | 20% | 12 | Systematic application of Maxwell-Faraday and Maxwell-Ampère equations for EM fields; clear torque-slip derivation from equivalent circuit; explicit reactive power balance equations; complete inductor volt-second balance and capacitor charge balance for converter | Correct final formulas but skips key intermediate steps; assumes results without derivation; some logical gaps in reasoning | No derivations shown; jumps to final answers; incorrect or missing intermediate steps; formula substitution without explanation |
| Practical interpretation | 20% | 12 | States assumption of lossless medium for (a); discusses implications of high starting current in (b)(i) for motor design; explains economic benefit of p.f. correction in (b)(ii) relevant to Indian industrial tariffs; comments on CCM importance for output voltage regulation in (c) | Brief mention of assumptions without elaboration; limited discussion of practical significance; generic statements about power quality | No assumptions stated despite explicit instruction; no practical context provided; purely mathematical treatment without engineering relevance |
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