Q7
(a) (i) Draw the neat and properly labelled output voltage waveform of a three-phase, phase-controlled rectifier having firing angle α. Also derive the relationship for average output voltage in terms of line voltage V_LL and firing angle α. (10 marks) (ii) A three-phase full-wave controlled rectifier is being operated from a star-connected, 415 V, 50 Hz supply. This rectifier is feeding a constant current load of 15 kW. It is required to obtain an average output voltage of 80% of maximum possible output voltage. Find the firing angle, r.m.s. value of line current and input power factor. Assume devices are ideal. (10 marks) (b) (i) Show that the maximum power that a synchronous generator can supply when connected to constant voltage, constant frequency busbars increases with the excitation. (10 marks) (ii) An 11 kV, 3-phase, star-connected turbo-alternator delivers 250 A at unity power factor when running on constant voltage and frequency busbars. If the excitation is increased so that the delivered current rises to 300 A, find the power factor at which now machine works and percentage increase in the induced e.m.f., assuming a constant steam supply and unchanged efficiency. The armature resistance is 0·5 Ω per phase and the synchronous reactance is 10 Ω per phase. (10 marks) (c) A medium has infinite conductivity for z ≤ 0, ε_r = 7 and μ_r = 18, and σ = 0 for z > 0. The electric field for z > 0 is given as $\vec{E} = 10\cos(3 \times 10^8 t - 15x)\hat{z}$, as shown below. Determine the surface charge density and surface current density at location (3, 4, 0) at t = 0·8 ns. Given, $\mu_0 = 4\pi \times 10^{-7}$ H/m, $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m : (10 marks)
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(a) (i) एक त्रिकला, कला-नियंत्रित दिष्कारी, जिसका फायरन कोण α है, का स्वच्छ एवं यथायोग्य चिह्नित निर्गत बोल्टता तरंगरूप आरेखित कीजिए। लाइन बोल्टता V_LL और फायरन कोण α के सापेक्ष औसत निर्गत बोल्टता के लिए संबंध भी व्युत्पन्न कीजिए। (10 अंक) (ii) एक त्रिकला पूर्ण-तरंग नियंत्रित दिष्कारी एक तारा-संयोजित, 415 V, 50 Hz प्रदाय द्वारा संचालित है। यह दिष्कारी एक 15 kW के स्थिर धारा भार को पोषित करता है। अधिकतम संभव निर्गत बोल्टता का 80% औसत निर्गत बोल्टता प्राप्त करना वांछित है। फायरन कोण, लाइन धारा का r.m.s. मान और निवेश शक्ति गुणांक का मान ज्ञात कीजिए। मान लीजिए कि उपकरण आदर्श हैं। (10 अंक) (b) (i) दिखाइए कि अधिकतम शक्ति, जो स्थिर बोल्टता, स्थिर आवृत्ति बसबार पर संयोजित एक तुल्यकालिक जनित्र प्रदान कर सकता है, उतेजन के साथ बढ़ती है। (10 अंक) (ii) एक 11 kV, 3-कला, तारा-संयोजित टर्बो-प्रत्यावर्तित्र जब स्थिर बोल्टता और आवृत्ति के बसबार पर क्रियाशील है, इकाई शक्ति गुणांक पर 250 A देता है। यदि उतेजन को बढ़ा दिया जाता है ताकि प्रदत धारा 300 A तक बढ़ जाए, तो शक्ति गुणांक, जिस पर अब यंत्र काम करता है, और प्रेरित e.m.f. में प्रतिशत वृद्धि को स्थिर भार प्रदाय व अपरिवर्तित दक्षता मानते हुए ज्ञात कीजिए। आर्मेचर प्रतिरोध 0·5 Ω प्रति कला तथा तुल्यकालिक प्रतिघात 10 Ω प्रति कला है। (10 अंक) (c) z ≤ 0, ε_r = 7 और μ_r = 18 होने पर एक माध्यम की चालकता अनंत है और z > 0 के लिए σ = 0 है। z > 0 के लिए विद्युत क्षेत्र $\vec{E} = 10\cos(3 \times 10^8 t - 15x)\hat{z}$ है, जैसा कि नीचे प्रदर्शित है। स्थान (3, 4, 0) पर t = 0·8 ns पर सतह आवेश घनत्व और सतह धारा घनत्व ज्ञात कीजिए। दिया गया है, $\mu_0 = 4\pi \times 10^{-7}$ H/m, $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m : (10 अंक)
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How this answer will be evaluated
Approach
Solve this multi-part numerical problem by allocating approximately 25% time to each of parts (a)(i), (a)(ii), (b)(ii), and (c), with part (b)(i) requiring brief theoretical proof. Begin with clear diagrams and derivations for the rectifier waveform, then proceed systematically through calculations for firing angles, power factors, and electromagnetic boundary conditions, concluding with physical interpretations of each result.
Key points expected
- For (a)(i): Correct three-phase bridge rectifier output waveform with 6-pulse ripple, proper labeling of firing angle α, conduction intervals, and phase voltages; derivation of V_avg = (3√3/π)V_LL cos(α) for continuous conduction
- For (a)(ii): Calculation of firing angle α = cos⁻¹(0.8) = 36.87°, RMS line current = 20.82 A, and input power factor = 0.8 lagging using proper relationships for constant current load
- For (b)(i): Proof that P_max = EV/X_s increases with excitation E, using power-angle characteristics and showing ∂P_max/∂E > 0 for constant V and X_s
- For (b)(ii): Calculation of new power factor = 0.833 lagging, percentage increase in induced EMF = 19.6%, using power balance with constant steam input and phasor diagrams
- For (c): Application of boundary conditions at z=0 for perfect conductor; surface charge density ρ_s = 83.14 nC/m² and surface current density J_s = -0.424 ŷ A/m at (3,4,0) using wave impedance and propagation constants
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 20% | 6 | Correctly identifies 3-phase full-wave rectifier as 6-pulse bridge; applies proper synchronous machine power-angle theory with constant power constraint; uses correct electromagnetic boundary conditions for perfect conductor (tangential E=0, normal B=0) with proper wave impedance η = √(μ/ε) | Minor errors in rectifier configuration identification or confuses leading/lagging power factors; incomplete boundary condition application or incorrect wave impedance formula | Fundamental misconceptions such as treating 3-phase half-wave as full-wave, ignoring constant power constraint in (b), or applying free-space conditions instead of material medium properties |
| Numerical accuracy | 20% | 6 | Precise calculations: α=36.87°, I_rms=20.82A, PF=0.8 for (a)(ii); new PF=0.833, 19.6% EMF increase for (b)(ii); ρ_s=83.14 nC/m², J_s=-0.424 A/m for (c) with correct unit handling and significant figures | Correct methodology with minor arithmetic errors (±5% deviation) or unit conversion mistakes (degrees vs radians, kV vs V); correct final formulas with substitution errors | Order-of-magnitude errors, incorrect formula substitutions, or missing critical steps like √3 factor for line quantities; no numerical answers despite setup |
| Diagram quality | 20% | 6 | Neat 3-phase output waveform showing 6-pulses per cycle, clear α marking from natural commutation points, phase voltage references, conduction intervals labeled (T1-T6), and phasor diagram for synchronous machine showing E, V, I, and δ angles | Recognizable waveform but missing labels for conduction intervals or unclear α demarcation; phasor diagram present but missing key angles or current direction | Unrecognizable sketches, missing waveform entirely, or diagrams that contradict written derivations; no phasor diagram for synchronous machine analysis |
| Step-by-step derivation | 20% | 6 | Complete integration for V_avg showing 6 intervals of (π/3) width with proper limits; explicit P_max derivation using dP/dδ=0; clear boundary condition derivation from Maxwell's equations with tangential H discontinuity leading to J_s | Jumps key steps in integration or uses standard formula without derivation; states P_max condition without proof; applies boundary conditions without showing field component analysis | No derivations, only final formulas stated; or incorrect integration limits, missing the 6-pulse nature; confuses power transfer equations with motor operation |
| Practical interpretation | 20% | 6 | Explains why α>0 reduces output voltage and power factor in HVDC/Indian railway traction contexts; discusses stability implications of operating near P_max with increased excitation; relates surface currents to shielding effectiveness in electromagnetic compatibility for Indian telecom infrastructure | Brief mention of applications without specific context; generic statements about power electronics or synchronous machines without linking to calculated values | No physical interpretation provided; or completely irrelevant applications showing misunderstanding of the technologies involved |
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