UPSC Electrical Engineering 2023 — Paper II

Solve all five sub-parts systematically with equal time allocation (~20% each) since all carry equal marks. Begin with the control system problem (a) requiring steady-state error analysis and overshoot calculation, then proceed sequentially through circuit breaker ratings (b), PCM sampling and bit rate (c), 8085 instruction explanations (d), and transmission line capacitance (e). Present each solution with clear problem identification, relevant formulas, step-by-step working, and final boxed answers.

• Part (a): Apply final value theorem to find K from steady-state value 0.8, then determine damping ratio ζ and natural frequency ωn, finally calculate percentage overshoot using Mp = exp(-πζ/√(1-ζ²)) × 100
• Part (b): Identify rated normal current = 2500 A, breaking current = 2500 A (symmetrical), making current = 2.55 × breaking current (IEC standard), short-time rating = 2500 A for 3 seconds, MVA rating = √3 × 33 kV × breaking current
• Part (c): Apply Nyquist criterion for minimum sampling rate = 2 × 15 kHz = 30 kHz, actual rate = 1.2 × 30 kHz = 36 kHz, bits per sample = log₂(1024) = 10, bit rate = 36 kHz × 10 = 360 kbps
• Part (d): Explain ANA M (AND accumulator with memory), XRA M (XOR accumulator with memory), CMC (complement carry flag), STC (set carry flag), RRC (rotate accumulator right through carry) with flag effects
• Part (e): Calculate capacitance per phase using C = 2πε₀/ln(D/r) per km, where D = 5 m, r = 1.5 cm, then multiply by 400 km; convert to μF or nF with proper units

Solve this multi-part problem by allocating approximately 40% time to part (a) root locus analysis, 40% to part (b) Schering bridge derivation, and 20% to part (c) delta modulation calculations. Begin with sketching the root locus using standard rules, then derive balance equations for the Schering bridge with clear circuit diagram, and conclude with slope overload calculations for the DM system. Ensure all numerical answers are boxed and diagrams are neatly labeled.

• Part (a)(i): Correct identification of poles and zeros, asymptote angles and centroid, breakaway/break-in points, and complete root locus sketch with proper direction arrows
• Part (a)(ii): Application of magnitude condition at s=0 to find K value using |G(s)H(s)|=1
• Part (a)(iii): Use of Routh-Hurwitz criterion or jω-axis crossing method to determine stable range of K
• Part (b): Accurate connection diagram of Schering bridge with four arms labeled, balance equations derived from Z1Z4=Z2Z3, and final expressions Cx=C3R1/R2 and tanδ=ωC4R4
• Part (c): Calculation of maximum amplitude A_max = (Δ·fs)/(2πfm) for slope overload avoidance, and maximum power P_max = A_max²/2

Begin by deriving state-space representation for part (a) using the given block diagram, then apply Kalman's controllability and observability tests. For part (b), contrast full and partial decoding with memory mapping examples, then illustrate DAA with a concrete BCD addition like 38+25. Conclude with part (c) by calculating cable parameters using standard formulae for cylindrical geometry. Allocate approximately 40% time to (a), 20% to (b)(i), 20% to (b)(ii), and 20% to (c) based on mark distribution.

• Part (a): Correct state equations ẋ = Ax + Bu and output y = Cx + Du derived from block diagram; controllability matrix [B AB] and observability matrix [C; CA] computed with rank determination
• Part (b)(i): Clear distinction between full decoding (all address lines decoded, unique addresses) and partial decoding (some lines unused, address overlap); advantages (hardware simplicity vs. complete utilization) and disadvantages (address ambiguity vs. complexity) stated
• Part (b)(ii): DAA instruction operation explained: addition first, then adjustment if lower nibble >9 or AC=1, and if upper nibble >9 or CY=1; concrete example like 88H + 88H = 10H (with carry) then DAA gives 76H with CY=1
• Part (c): Insulation resistance R = ρln(r₂/r₁)/(2πL) calculated correctly; capacitance C = 2πε₀εᵣL/ln(r₂/r₁) computed; maximum electric stress E_max = V/(r₁·ln(r₂/r₁)) at conductor surface determined
• Part (c): Proper unit conversions (cm to m, km to m) and final values with correct units (MΩ, μF, kV/cm or MV/m) presented

Solve this multi-part numerical and descriptive problem by allocating approximately 40% time to part (a) coding theory (20 marks), 40% to part (b) transmission line calculations (20 marks), and 20% to part (c) 8085 interrupt instructions (10 marks). Begin with systematic matrix construction for (a), proceed to ABCD parameter calculations for (b), and conclude with concise instruction descriptions for (c). Present all derivations stepwise with clear intermediate results.

• For (a)(i): Construct 4×8 generator matrix G = [I₄|P] and 4×8 parity check matrix H = [Pᵀ|I₄] from given parity equations, verifying GHᵀ = 0
• For (a)(ii): Determine minimum Hamming distance dₘᵢₙ = 3 from H matrix columns, hence error detection capability = 2 errors, correction capability = 1 error
• For (a)(iii): Compute syndrome S = rHᵀ, identify error position from syndrome pattern, and correct received word 10100100 to transmitted codeword
• For (b)(i): Apply ABCD parameters with Vᵣ = 132∠0° kV, Iᵣ = 40×10⁶/(√3×132×10³×0.8) ∠-36.87° A to find Vₛ = AVᵣ + BIᵣ and voltage regulation = (|Vᵣₙₗ| - |Vᵣբₗ|)/|Vᵣբₗ| × 100%
• For (b)(ii): Use Vₛ = 132∠30° kV, Vᵣ = 132∠0° kV in transmission equation to solve for current and hence complex power S = 3VᵣIᵣ* and power factor
• For (c): Explain EI (Enable Interrupts), DI (Disable Interrupts), SIM (Set Interrupt Mask), and RIM (Read Interrupt Mask) instructions with their specific roles in 8085 interrupt control structure

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

This is a multi-part problem-solving question requiring systematic calculation and derivation across power systems, microprocessors, and control systems. Allocate approximately 20 minutes (40%) to part (a) covering economic load dispatch and Ybus formation, 15 minutes (30%) to part (b) on DMA operations and 8085 assembly programming, and 10 minutes (20%) to part (c) for critical damping calculation. Begin each sub-part with the governing equation, show complete step-by-step working, and verify boundary conditions and physical feasibility of results.

• For (a)(i): Apply equal incremental cost criterion (dC₁/dP_g1 = dC₂/dP_g2 = λ) with P_g1 + P_g2 = 650 MW, check generator limits, and resolve if limits violated
• For (a)(ii): Convert impedances to admittances (y = 1/z), identify bus connections, and construct Ybus matrix with diagonal elements as sum of connected admittances and off-diagonals as negative of connecting branch admittances
• For (b)(i): Describe HOLD, HLDA, RESET, and address/data bus pins; explain DMA handshake sequence: peripheral requests → CPU grants hold → DMA controller takes bus → transfer → release
• For (b)(ii): Use PUSH PSW to save flags on stack, POP into register pair, complement accumulator, then restore; or use direct flag manipulation via alternate register operations
• For (c): Derive second-order characteristic equation from RLC circuit, set damping ratio ζ = 1 for critical damping, solve for R = 2√(L/C) with proper unit conversion from μH and nF to ohms
• Cross-check: Verify (a)(i) solution satisfies both power balance and generator limits; confirm (c) result yields repeated real poles

Design the Huffman code and first-order extension code for part (a) with proper probability tree construction, then derive the three relay characteristics from the universal torque equation for part (b) with clear R-X plane diagrams, and finally solve the distributor voltage drop problem for part (c) using phasor analysis. Allocate approximately 40% time to part (a) due to its 20 marks and dual sub-parts, 40% to part (b) for its derivations and diagrams, and 20% to part (c) for the numerical solution. Structure with clear headings for each part, showing all calculations stepwise and labeling diagrams precisely.

• For (a)(i): Construct Huffman code tree assigning '0' to highest probability (m₁), calculate average length L̄ = Σpᵢlᵢ, compression ratio η = H(X)/L̄, and efficiency; correct code: m₁=0, m₂=10, m₃=11 with L̄=1.12 bits/symbol
• For (a)(ii): Form 9-symbol first-order extension with joint probabilities, design Huffman code for extended source, calculate new average length per original symbol, and compare efficiency improvement approaching Shannon limit
• For (b): State universal torque equation T = K₁I² + K₂V² + K₃VIcos(θ-τ) + K₄, derive impedance relay (K₂=K₄=0), reactance relay (K₂=K₄=0, τ=90°), and mho relay (K₁=K₄=0) characteristics with proper algebraic manipulation
• For (b): Draw accurate R-X plane diagrams showing circular/linear characteristics, clearly marking operating zones (inside/on line for operation) and non-operating zones with directional indication for mho relay
• For (c): Calculate load currents from given power and power factors, determine voltage drops in sections AB and BC using IZ drops, apply KVL to find supply voltage magnitude and phase angle with respect to far end C

Solve all three parts systematically, allocating approximately 40% time to part (a) given its 20 marks and complexity involving polar plots and stability analysis; 35% to part (b) for shunt calculations and current distribution; and 25% to part (c) for fault level calculations. Begin with clear problem statements, show all derivations with proper units, and conclude with physical interpretations of results.

• For (a)(i): Calculate magnitude and phase of G(jω) at key frequencies (ω→0, ω→∞, and phase crossover), identify type number (n=1) and starting angle (-90°), sketch polar plot showing encirclement pattern
• For (a)(ii): Apply Nyquist stability criterion correctly, count encirclements of (-1+j0) point, determine N and P, conclude closed-loop stability with Z = P - N = 0
• For (a)(iii): Calculate gain margin as 1/|G(jω_pc)| where phase crossover frequency ω_pc satisfies ∠G(jω_pc) = -180°, express in dB
• For (b): Calculate shunt resistances R_sh1 = (1×0.2)/(20-0.2) and R_sh2 = (2×0.25)/(20-0.25), then find equivalent resistances of extended-range ammeters, solve current divider for 15A total current
• For (c): Convert short-circuit capacities to equivalent reactances (X1 = 11²/1500, X2 = 11²/1000), combine with interconnector reactance (0.7Ω), calculate new fault levels at each bus using Thevenin equivalent
• For (c): Recognize that interconnected system modifies fault levels due to mutual contribution through tie-line, calculate S_sc_new at each station considering parallel contribution from other station