(a) The figure shows a compensator network, where R₁ = 3 MΩ, R₂ = 1 MΩ, C = 1 μF. Vᵢ(t) and Vₒ(t) are the input voltage and output voltage respectively. Determine the attenuation in dB provided by this network at very high frequencies : 10 marks

(b)

Question

(a) The figure shows a compensator network, where R₁ = 3 MΩ, R₂ = 1 MΩ, C = 1 μF. Vᵢ(t) and Vₒ(t) are the input voltage and output voltage respectively. Determine the attenuation in dB provided by this network at very high frequencies : 10 marks

(b) A resistive strain gauge, with a gauge factor 2·2, is cemented on a rectangular steel bar with the elastic modulus, E = 205×10⁶ kN/m². The width and thickness of the steel bar is 3·5 cm and 0·55 cm respectively. An axial force of 12 kN is applied. If the nominal resistance of the strain gauge is 100 Ω, determine the change in resistance of the strain gauge. 10 marks

(c) A three-phase, 50 Hz, 415 V supply delivers 250 kW power at power factor of 0·8 lagging. The line power factor is desired to be improved to 0·9 lagging by installing shunt capacitors. Calculate the capacitance if they are connected in delta. 10 marks

(d) Binary data is transmitted over additive white Gaussian noise (AWGN) channel at a bit rate of 5 kilobits/sec. The single-sided power spectral density for the channel is 10⁻⁷ W/Hz. Non-coherent orthogonal binary FSK with higher frequency signalling tone of 1 MHz is used. The bit energy, E_b = 2×10⁻⁶ J. Determine the minimum required bandwidth and average bit error probability. 10 marks

(e) Consider a three-phase, Δ-Y connected, 30 MVA, 33/11 kV transformer with differential relay protection. If the CT ratios are 500 : 5A on the primary side and 2000 : 5A on the secondary side, compute the relay current setting for faults drawing up to 200% of rated transformer current. 10 marks

UPSC Answer Check · Accepted Answer

This is a multi-part numerical problem requiring precise calculations across five distinct electrical engineering domains. Begin by identifying the circuit topology for part (a) as a lag compensator, then systematically solve each sub-part: (a) derive transfer function and evaluate at ω→∞ for attenuation; (b) apply stress-strain-resistance relationship for strain gauges; (c) use reactive power compensation formulas for delta-connected capacitors; (d) apply non-coherent FSK bandwidth and error probability formulas; (e) perform CT ratio matching for differential protection. Allocate approximately 2-2.5 minutes per mark, with roughly equal time distribution across all five 10-mark parts. Present each solution with clear circuit diagrams where applicable, state all formulas before substitution, and conclude with physically meaningful interpretations.
- Part (a): Identify the network as a lag compensator, derive Vₒ/Vᵢ = (R₂ + 1/sC)/(R₁ + R₂ + 1/sC), evaluate at s→jω where ω→∞ to get attenuation = 20log₁₀(R₂/(R₁+R₂)) = -12 dB
- Part (b): Calculate axial stress σ = F/A = 12×10³/(3.5×10⁻² × 0.55×10⁻²), strain ε = σ/E, then ΔR/R = GF × ε, yielding ΔR ≈ 0.067 Ω
- Part (c): Calculate initial and final reactive power, Qc = P(tanφ₁ - tanφ₂) = 250×10³×(0.75-0.484) = 66.5 kVAR, then CΔ = Qc/(3×2πf×V²) ≈ 409 μF per phase
- Part (d): For non-coherent orthogonal FSK, bandwidth B = 2R_b + |f₂-f₁| ≈ 2×5 kHz + separation, with E_b/N₀ = 20, giving P_e = ½exp(-E_b/2N₀) ≈ 2.3×10⁻³
- Part (e): Calculate rated currents I₁ = 30×10⁶/(√3×33×10³) ≈ 524.9 A, I₂ = 30×10⁶/(√3×11×10³) ≈ 1574.6 A, match CT secondary currents (5A vs 5A) through proper connection, set relay at 200% with appropriate bias characteristic

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies all five circuit/equipment types: lag compensator in (a), metallic strain gauge in (b), shunt power factor correction in (c), non-coherent FSK in (d), and percentage differential protection in (e); applies appropriate fundamental laws (KVL, Hooke's law, power triangle, matched filter detection, CT polarity rules) without conceptual errors	Correctly identifies 3-4 of the five systems with minor conceptual gaps; may confuse lag vs lead compensator, or coherent vs non-coherent detection, or star-delta CT connection implications	Major misidentification of 2+ systems; treats (a) as simple voltage divider ignoring frequency dependence, or (d) as coherent FSK, or (e) as overcurrent protection; fundamental conceptual errors in applying basic laws
Numerical accuracy	20%	10	All five calculations yield precise answers with correct significant figures: (a) exactly -12.04 dB or -12 dB, (b) ΔR ≈ 0.066-0.067 Ω, (c) C ≈ 408-410 μF, (d) B ≈ 15 kHz (minimum orthogonal spacing) and P_e ≈ 2.3×10⁻³, (e) relay setting 10A with proper percentage slope; unit conversions (MΩ, μF, kN/m²) handled flawlessly	3-4 correct numerical answers with minor arithmetic errors or unit conversion mistakes; may use 10⁶ vs 10³ incorrectly in (b) elastic modulus, or forget √3 in three-phase calculations, or E_b/N₀ ratio errors by factor of 2	Major numerical errors in 2+ parts; order-of-magnitude mistakes, completely wrong formulas substituted, or missing unit conversions leading to absurd results (e.g., capacitance in Farads, bandwidth in MHz)
Diagram quality	15%	7.5	Clear labeled diagrams for (a) showing R₁, R₂, C with input/output nodes; (b) strain gauge orientation on bar with force direction; (c) delta-connected capacitor bank with phase/line voltage markings; (d) FSK frequency allocation showing orthogonality condition; (e) CT connections showing polarity dots and relay coil; all diagrams use standard IEEE/IEC symbols	Adequate sketches for 3-4 parts with minor labeling omissions; may miss polarity dots in (e), or frequency separation illustration in (d), or strain gauge orientation in (b); acceptable but not publication-quality	Missing or seriously flawed diagrams for 2+ parts; incorrect symbols, no labels, or diagrams that contradict the described problem; hand-drawn without ruler or proper proportions
Step-by-step derivation	25%	12.5	Each sub-part shows complete logical flow: (a) impedance approach → transfer function → limit evaluation; (b) area → stress → strain → resistance change; (c) power triangle → reactive power → per-phase calculation → capacitance; (d) E_b/N₀ derivation → bandwidth formula → error probability integral; (e) rated current → CT secondary → mismatch analysis → percentage setting; all algebraic steps explicit with no jumps	Clear derivation for 3-4 parts with some steps condensed or assumed; may skip intermediate algebraic manipulation, or combine steps in (c) power calculations, or omit E_b/N₀ explicit calculation in (d); generally followable but with gaps	Missing derivations or 'answer-only' responses for 2+ parts; jumps directly to final formula without justification; no showing of limit evaluation in (a), or stress-strain relationship in (b), or CT ratio matching logic in (e)
Practical interpretation	20%	10	Meaningful physical interpretation for each result: (a) explains why lag compensator attenuates at high frequencies (capacitor shorts); (b) comments on small ΔR requiring Wheatstone bridge measurement; (c) notes delta connection advantage (lower capacitance per phase) for Indian 415V industrial systems; (d) relates P_e to practical BER requirements and bandwidth efficiency; (e) discusses relay stability during magnetizing inrush and through-fault conditions with percentage bias necessity	Brief physical comments for 3-4 parts; generic statements about 'improving power factor' or 'reducing errors' without specific connection to calculated values; misses practical measurement challenges in (b) or protection coordination in (e)	No physical interpretation or entirely generic statements; fails to explain why results matter, or makes incorrect physical claims (e.g., 'higher attenuation is better' without context, or ignores inrush problem in transformer protection)

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