Electrical Engineering 2024 Paper I 50 marks Compulsory Solve

Q5

(a) A uniform plane wave travels in vacuum along +y direction. The electric field of the wave at some instant is given as $\vec{E} = 4\hat{x} + 3\hat{z}$. Find the vector magnetic field $\vec{H}$. (Given, $\mu_0 = 4\pi \times 10^{-7}$ H/m, $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m) (10 marks) (b) The maximum efficiency of a 200 kVA, 3300/600 V, 50 Hz, single-phase transformer is 98% and occurs at 75% full load and unity power factor. If the leakage impedance is 10%, find the voltage regulation at full load and power factor 0.8 lagging. (10 marks) (c) A diode circuit with an L-C load is shown in the figure, with the capacitor having an initial voltage $V_C(t=0) = 120$ V, capacitance $C = 12$ μF and inductance $L = 48$ μH. If switch S is closed at $t = 0$ s, then find the following: (i) Peak value of current $i$ (ii) Conduction time of the diode (10 marks) (d) How can linear pre-emphasis and de-emphasis filters be employed to improve the performance of an FM system? Is the improvement in output SNR dependent on both the frequency responses of the pre-emphasis filter and the de-emphasis filter? (10 marks) (e) A transmission line is 25 m long. It has characteristic impedance Z₀ = 40 Ω and operates at 2 MHz. The line is terminated with a load of Z_L = (50 + j30) Ω. If the wave velocity is u = 0.8c (with c = 3×10⁸ m/s) on the line, determine (i) the reflection coefficient and (ii) the input impedance. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक एकसमान समतल तरंग निर्वात में +y दिशा में चल रही है। तरंग का विद्युत क्षेत्र किसी समय पर $\vec{E} = 4\hat{x} + 3\hat{z}$ द्वारा प्रदर्शित है। सदिश चुंबकीय क्षेत्र $\vec{H}$ निकालिए। (दिया है, $\mu_0 = 4\pi \times 10^{-7}$ H/m, $\varepsilon_0 = \frac{1}{36\pi} \times 10^{-9}$ F/m) (10 अंक) (b) एक 200 kVA, 3300/600 V, 50 Hz, एकल कला परिणामित्र की अधिकतम दक्षता 98% है एवं पूर्ण भार के 75% भार तथा इकाई शक्ति गुणांक पर प्राप्त होती है। यदि क्षरण प्रतिबाधा 10% हो, तो पूर्ण भार एवं 0.8 पश्चगामी शक्ति गुणांक पर वोल्टता नियमन ज्ञात कीजिए। (10 अंक) (c) दर्शाए गए डायोड एवं L-C भार संयुक्त परिपथ में संधारित्र की प्रारंभिक वोल्टता $V_C(t=0) = 120$ V, धारिता का मान $C = 12$ μF एवं प्रेरकत्व का मान $L = 48$ μH है। यदि स्विच S को समय $t = 0$ s पर बंद किया जाए, तो निम्नलिखित ज्ञात कीजिए: (i) धारा $i$ का शिखर मान (ii) डायोड का चालन समय (10 अंक) (d) रैखिक पूर्व-प्रबलन और विप्रबलन छक्कों (फिल्टरों) को एक FM तंत्र का प्रदर्शन उन्नत करने के लिए कैसे नियोजित किया जा सकता है? क्या निगत S/N अनुपात में उन्नयन पूर्व-प्रबलन छक्क और विप्रबलन छक्क दोनों की आवृत्ति प्रतिक्रियाओं पर निर्भर है? (10 अंक) (e) एक प्रेषण लाइन 25 m लम्बी है। इसकी लाक्षणिक प्रतिबाधा Z₀ = 40 Ω है और यह 2 MHz पर कार्य करती है। लाइन एक भार Z_L = (50 + j30) Ω पर समाप्त होती है। यदि लाइन पर तरंग वेग u = 0.8c है (जहाँ c = 3×10⁸ m/s है), तो (i) परावर्तन गुणांक और (ii) निवेश प्रतिबाधा ज्ञात कीजिए। (10 अंक)

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Approach

Solve all five sub-parts systematically, allocating approximately 20% time to each part since marks are equal. Begin with clear statement of given data and required unknowns for each sub-part. Present derivations step-by-step with proper units, then substitute numerical values. For part (c), sketch the L-C circuit diagram showing diode, switch, inductor and capacitor with initial polarity. Conclude each part with boxed final answers and brief physical interpretation.

Key points expected

  • Part (a): Apply Poynting vector relation; use η₀ = √(μ₀/ε₀) = 120π Ω; determine H = (1/η₀)(âₓ × E) with propagation in +y direction giving H = (3/120π)âₓ - (4/120π)â_z A/m
  • Part (b): Calculate core loss and copper loss at maximum efficiency condition; use P_cu = x²P_cu,FL to find full-load copper loss; apply voltage regulation formula with leakage impedance to find % regulation ≈ 6.5%
  • Part (c): Analyze underdamped RLC circuit; derive i(t) = (V_C/ω_dL)e^(-αt)sin(ω_d t); find peak current I_peak = V_C√(C/L) ≈ 60 A; conduction time = π/ω_d ≈ 48 μs until current returns to zero
  • Part (d): Explain pre-emphasis boosts high frequencies before modulation matching FM noise triangle; de-emphasis attenuates highs after demodulation; SNR improvement depends only on de-emphasis filter matching noise spectrum, not pre-emphasis
  • Part (e): Calculate reflection coefficient Γ = (Z_L - Z₀)/(Z_L + Z₀) = 0.35∠56.3°; find electrical length βl = 2πf/u = 0.418 rad; apply transmission line equation for input impedance Z_in = Z₀(Z_L + jZ₀tanβl)/(Z₀ + jZ_Ltanβl)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies all underlying principles: Maxwell's equations for (a), transformer efficiency conditions for (b), second-order RLC circuit analysis for (c), FM noise pre-emphasis/de-emphasis theory for (d), and transmission line theory for (e); no conceptual errors in any sub-partIdentifies most principles correctly but has minor errors in one sub-part, such as wrong wave impedance direction in (a) or incorrect efficiency condition application in (b)Major conceptual errors in multiple sub-parts, such as confusing Poynting vector direction, applying DC transformer analysis, or using lumped circuit analysis for transmission lines
Numerical accuracy20%10All calculations accurate to 3 significant figures; correct handling of complex arithmetic in (e); proper unit conversions (μF, μH to SI); final answers match expected values within reasonable precisionMost calculations correct but arithmetic errors in 1-2 sub-parts, such as sign errors in complex reflection coefficient or incorrect angular frequency calculation in (c)Multiple calculation errors, wrong formula substitutions, or missing units; answers orders of magnitude incorrect due to power-of-ten errors
Diagram quality15%7.5Clear circuit diagram for part (c) showing switch S, diode D, inductor L, capacitor C with polarity marking for initial voltage; phasor diagrams for (b) and (e) if helpful; properly labeled with all component valuesBasic circuit diagram present for (c) but missing initial voltage polarity or component labels; no diagrams for other parts where they could aid clarityNo diagram for (c) despite explicit figure reference; messy or incorrect circuit topology showing series instead of parallel L-C
Step-by-step derivation25%12.5Complete derivations showing: wave impedance calculation and cross-product for (a); loss separation and regulation formula development for (b); differential equation setup, characteristic roots, and solution for (c); clear mathematical progression for (d) and (e)Most steps shown but skips key intermediate steps like root calculation for underdamped case in (c) or direct substitution without showing regulation formulaFinal answers only with no derivation; or incorrect formulas stated without justification; jumps from given data to answer without logical flow
Practical interpretation20%10Physical interpretation for each result: explains orthogonal E-H relationship in (a); discusses significance of maximum efficiency loading for Indian distribution transformers in (b); interprets current waveform and diode conduction in (c); relates FM improvement to broadcast standards in (d); discusses impedance matching implications in (e)Brief interpretation for 2-3 sub-parts but misses practical significance; generic statements without specific contextPurely mathematical treatment with no physical insight; no discussion of engineering relevance or real-world implications

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