(a) The plane wave E = 30 cos (ωt – z) aₓ V/m in air normally hits a lossless medium (μ = μ₀, ε = 4ε₀) at z = 0. (i) Find reflection coefficient (Γ), transmission coefficient (τ), standing wave ratio (S). (ii) Calculate the reflected electric and mag

Question

(a) The plane wave E = 30 cos (ωt – z) aₓ V/m in air normally hits a lossless medium (μ = μ₀, ε = 4ε₀) at z = 0. (i) Find reflection coefficient (Γ), transmission coefficient (τ), standing wave ratio (S). (ii) Calculate the reflected electric and magnetic fields. 20 marks

(b) Determine the value of inductance L, capacitance C and duty cycle of a buck regulator shown in Figure 6(b). The input voltage is 16 V, output voltage is 4 V, and ripple voltage (peak-to-peak) is 30 mV. The regulator is operating at 20 kHz switching frequency and peak-to-peak ripple current in inductance is 0·75 A. 20 marks

(c) Using the superposition theorem find the voltage 'V' across the 5 Ω resistance in the circuit as shown in Figure 6(c). 10 marks

UPSC Answer Check · Accepted Answer

Solve this multi-part numerical problem by allocating time proportionally to marks: approximately 40% for part (a) on wave reflection, 40% for part (b) on buck converter design, and 20% for part (c) on superposition theorem. Begin each part with clear identification of given parameters, apply relevant formulas with proper unit handling, show all intermediate calculations, and conclude with boxed final answers. For parts (a) and (b), include neatly labeled diagrams showing wave propagation and converter topology respectively.
- Part (a)(i): Calculate intrinsic impedances η₁ = 377Ω (air) and η₂ = 188.5Ω (dielectric), then Γ = (η₂-η₁)/(η₂+η₁) = -0.333, τ = 1+Γ = 0.667, and S = (1+|Γ|)/(1-|Γ|) = 2
- Part (a)(ii): Derive reflected field Eᵣ = -10 cos(ωt+z) aₓ V/m and Hᵣ = Eᵣ/η₁ = -26.53 cos(ωt+z) aᵧ mA/m; transmitted field Eₜ = 20 cos(ωt-2z) aₓ V/m with wavenumber change
- Part (b): Calculate duty cycle D = Vₒ/Vᵢₙ = 4/16 = 0.25; inductance L = Vₒ(1-D)/(fₛ×ΔIₗ) = 0.5 mH; capacitance C = ΔVₒ/(8L×fₛ²×ΔVₒ) or using C = (1-D)/(8L×fₛ²×(ΔVₒ/Vₒ)) yielding approximately 31.25 μF
- Part (c): Apply superposition by considering 10V source alone (with 5A open) then 5A source alone (with 10V short), calculate contributions through 2Ω and 3Ω resistances, sum to find V = 10V contribution + 5A contribution across 5Ω
- Verify boundary conditions at z=0 for part (a): tangential E and H continuity; verify ripple current and voltage specifications are met in part (b) design

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies wave impedance formulas for lossless media, applies buck converter CCM equations, and recognizes superposition applicability; distinguishes between intrinsic impedance and load impedance concepts; correctly interprets 'peak-to-peak' versus peak ripple specifications	Uses correct basic formulas but confuses η = √(μ/ε) with transmission line impedance, or applies buck-boost equations instead of buck; minor errors in identifying series/parallel combinations for superposition	Fundamental conceptual errors such as using same wavenumber in both media, treating buck converter as boost, or applying superposition to nonlinear circuits; omits critical concepts like boundary condition matching
Numerical accuracy	25%	12.5	All calculations precise to 3 significant figures: Γ = -1/3 or -0.333, τ = 2/3 or 0.667, S = 2 exactly; L = 0.5 mH, C ≈ 31.25 μF, D = 0.25; superposition yields exact voltage with proper sign conventions; unit conversions handled flawlessly	Correct methodology but arithmetic slips in final decimals (e.g., Γ = -0.3, S = 2.1); inductance calculation correct but capacitance formula error using wrong ripple voltage relation; magnitude correct but sign errors in superposition	Order-of-magnitude errors (mH vs μH), incorrect duty cycle calculation (using Vᵢₙ/Vₒ), or algebraic errors in solving simultaneous equations; final answers without units or with inconsistent units
Diagram quality	15%	7.5	Clear diagram for (a) showing incident, reflected, transmitted waves with propagation directions and field orientations; labeled buck converter schematic with switch, diode, L, C, and load; circuit diagram for (c) with current directions marked for each superposition case; proper labeling of all components and variables	Basic sketches present but missing critical labels (e.g., no aₓ, aᵧ, aᵤ directions in wave diagram); converter diagram omits diode or shows incorrect switch position; circuit diagram lacks node labels or current direction arrows	No diagrams despite visual nature of problems, or highly inaccurate sketches (e.g., parallel instead of series inductor in buck converter); messy, unlabeled figures that hinder rather than aid understanding
Step-by-step derivation	25%	12.5	Systematic derivation: for (a) explicit η calculation → Γ → τ → S with formula statements; for (b) clear sequential calculation of D → L → C with justification of each formula choice; for (c) separate complete analyses for each source with intermediate voltage calculations clearly shown; all steps logically connected	Correct final answers but skips key intermediate steps (e.g., jumps from given to Γ without showing η calculation); some formulas stated without derivation or justification; superposition steps combined confusingly rather than separated	Disorganized 'answer-only' approach with no visible methodology; incorrect formula substitutions without explanation; no indication of how boundary conditions or circuit theorems were applied
Practical interpretation	15%	7.5	Interprets S = 2 as moderate mismatch requiring impedance matching for RF applications; discusses practical inductor selection (core material, saturation) and capacitor ESR effects for 20kHz SMPS; notes superposition as essential for linear circuit analysis in power system protection studies; relates to Indian standards IS 61689 for medical equipment or BIS specifications	Brief mention of practical relevance without elaboration (e.g., 'used in antennas' or 'used in power supplies'); generic statements about importance without specific application context	No physical interpretation offered; treats problems as purely mathematical exercises without connecting to electromagnetic interference, power electronics applications, or circuit analysis practice

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