UPSC Electrical Engineering 2023

Solve each sub-part systematically, allocating approximately 20% time to each 10-mark section. Begin with clear circuit diagrams for parts (a), (d), and (e), then apply standard network theorems, Laplace transform techniques, machine equations, and electromagnetic principles. Present derivations stepwise with final boxed answers for numerical quantities.

• Part (a): Correct application of Norton's theorem to coupled circuits with proper handling of mutual inductance; calculation of short-circuit current and equivalent impedance; final current through 5Ω resistor
• Part (b): Application of periodic Laplace transform formula F(s) = (1/(1-e^(-sT)))∫[0 to T]f(t)e^(-st)dt for both waveforms; correct identification of period and piecewise functions
• Part (c): Calculation of cable impedance from 20% voltage drop condition; blocked rotor current using standstill impedance; starting current and power factor with cable impedance in series
• Part (d): DC analysis for operating point; small-signal model for CE amplifier; calculation of lower corner frequency f_L = 1/(2π(R_S + r_π)C_C) with proper input resistance reflection
• Part (e): Derivation of motional EMF (ε = B₀lu); power delivered to resistance (P = ε²/R = B₀²l²u²/R); mechanical power input (F = B₀²l²u/R, P_mech = Fu); equality proof for energy conservation

Solve this multi-part numerical problem by allocating approximately 40% time to part (a) nodal analysis, 40% to part (b) DSP system analysis, and 20% to part (c) differential pair sensitivity analysis. Begin each part with clear circuit/system identification, show complete mathematical working with proper notation, and conclude with verified numerical answers. For (b), explicitly state ROC for causality; for (c), use small-signal approximation and symmetry arguments.

• Part (a): Correct identification of reference node and formulation of nodal equations using KCL; proper handling of conductances and current sources in the network
• Part (a): Accurate solution of simultaneous equations yielding specific voltage values across 0.5 Ω and 2.5 Ω resistors with proper units
• Part (b)(i): Derivation of H(z) = 1/(1 - 0.75z⁻¹ + 0.125z⁻²) with correct ROC |z| > 0.5 for causal system; proper factorization of denominator
• Part (b)(ii): Partial fraction expansion and inversion to obtain h[n] = [2(0.5)ⁿ - (0.25)ⁿ]u[n] showing recognition of distinct real poles at 0.5 and 0.25
• Part (b)(iii): Convolution of h[n] with unit step or multiplication by z/(z-1) in z-domain to derive s[n] = [4 - 4(0.5)ⁿ + (0.25)ⁿ/3]u[n] with steady-state value 8/3
• Part (c): Application of half-circuit concept and symmetry; for (i) ΔV_X = ΔV_Y = ΔV (common-mode), for (ii) ΔV_X = -ΔV_Y = -ΔI·R_C/2 (differential), for (iii) Δ(V_X-V_Y) = -I_EE·ΔR/2 showing CMRR degradation

Solve this multi-part numerical problem by allocating time proportionally: spend ~40% on part (a) covering shift register truth table, timing diagram and MUX implementation; ~40% on part (b) for Fourier transform derivation with spectrum sketch and convolution integral evaluation; and ~20% on part (c) for Op-Amp resistor design. Begin each sub-part with stated assumptions, show complete derivations with intermediate steps, and conclude with boxed final answers.

• For (a)(i): Complete truth table showing CLR̄=0 resets all outputs to 0; CLR̄=1, Load=0 enables shift right; CLR̄=1, Load=1 enables parallel load from X₃X₂X₁X₀; timing diagram showing Q outputs tracking 0101 with proper clock edge triggering
• For (a)(ii): Implement 4-variable function using 4:1 MUX with A,B as select lines, deriving C,D-based data inputs using K-map or Boolean algebra for minterms 3,4,5,6,7,9,10,12,14,15
• For (b)(I): Derive Fourier transform of triangular pulse X(ω) = (a/2)sinc²(ωa/4π) or equivalent form using integration by parts or known transform pairs
• For (b)(II): Sketch amplitude spectrum showing |X(ω)| as sinc² function with nulls at ω = ±4π/a, ±8π/a, etc., and peak value a/2 at ω=0
• For (b)(ii): Evaluate convolution integral for L⁻¹[F₁(s)F₂(s)] = cos(t) - e⁻ᵗ + sin(t) - t·cos(t) or simplified form using convolution theorem with proper limits
• For (c): Design inverting summer with v₀ = -R_f(v₁/R₁ + v₂/R₂), yielding R_f = 4kΩ, R₁ = 4kΩ, R₂ = 1kΩ satisfying I_f ≤ 1mA at v₀ = 4V

Solve this multi-part numerical and design problem by allocating approximately 40% of effort to part (a) given its 20 marks, 40% to part (b) for its 20 marks, and 20% to part (c) for its 10 marks. Begin with clear circuit diagrams for each part, then apply BJT current equations for (a), use K-map simplification with the given constraint for (b), and apply two-port network parameter conversions for (c). Conclude with verification of results and practical significance.

• Part (a): Apply Ebers-Moll model with V_A = ∞ (neglect Early effect), use I_C = βI_B/(1+β/β) relationship, and recognize that at edge of saturation V_CE(sat) ≈ 0.2V with I_C/I_B < β
• Part (a): Calculate reverse saturation current I_S using I_C = I_S exp(V_BE/V_T) for both operating conditions (active mode and edge of saturation)
• Part (b): Construct 4-variable K-map with SW1, SW2, SW3, SW4 as inputs, identify minterms where two or more switches are closed (at least two 1s in input combination)
• Part (b): Apply the constraint that SW1 and SW4 cannot close simultaneously (don't care or forbidden condition), simplify using K-map grouping for minimal SOP or POS realization
• Part (c): Calculate Z-parameters from given experimental data using V1 = z11 I1 + z12 I2 and V2 = z21 I1 + z22 I2, then convert to Y-parameters using matrix inversion
• Part (c): Determine missing data in Experiments 3 and 4 by substituting known Z-parameters into the network equations

Solve all five numerical sub-parts systematically, allocating approximately 2 minutes per mark (20 minutes each). Begin with clear circuit/phasor diagrams where applicable, show all formulae with proper substitutions, and present final answers with units. For (a) use power-angle characteristics; for (b) analyze half-controlled converter operation; for (c) identify AM components using trigonometric identities; for (d) apply transmission line Smith chart or impedance transformation; for (e) solve second-order RLC transient with initial conditions.

• (a) Correct application of P_max = (3 V_t E_f)/(X_s) for cylindrical rotor machine with δ = 90°; proper calculation of phase voltage and synchronous reactance
• (a) Accurate computation of excitation voltage E_f, armature current I_a, and input power factor cos(φ) under maximum power condition
• (b) Correct voltage balance equation for half-controlled converter: V_dc = (V_m/π)(1 + cos α) - I_a(r_L + r_bat); solving for firing angle α
• (b) Proper Fourier analysis for fundamental component of input current and displacement factor calculation for power factor
• (c) Recognition of AM signal structure using cos(A)cos(B) identity: identification of carrier frequency 85 kHz, sidebands at 80 kHz and 90 kHz, modulation index m = 5/7
• (c) Correct calculation of sideband power ratio P_s/P_c = m²/2 and power efficiency η = m²/(2+m²)
• (d) Proper use of wavelength λ = 2 × 25 cm = 50 cm; correct application of shift formula: tan(βl) = tan(2π/λ × 8 cm) for finding load impedance from SWR = 4
• (e) Correct initial condition analysis: steady-state inductor current and capacitor voltage; proper second-order differential equation for RLC circuit with R, L, C values; complete transient solution with damping classification

Solve this multi-part numerical problem by allocating approximately 40% time to part (a) [20 marks], 40% to part (b) [20 marks], and 20% to part (c) [10 marks]. Begin each part with the relevant governing equations, show complete step-by-step calculations with proper units, and conclude with physically meaningful results. For part (a), establish rated conditions first; for part (b), use transmission line theory for lossy dielectrics; for part (c), analyze the discontinuous conduction mode of the rectifier-fed motor.

• Part (a): Correct determination of rated armature current using no-load data and back EMF relationships; calculation of full-load torque using power balance and speed-torque characteristics
• Part (a): Analysis of field fault condition where flux collapses and armature current rises to 5× rated, with speed determined from torque balance under weakened field
• Part (b): Calculation of complex intrinsic impedance η = √(μ/ε) and propagation constant γ = jω√(με) for the lossy absorber material with εᵣ = μᵣ = 6–j6
• Part (b): Application of transmission line theory for normal incidence on conductor-backed absorber, using reflection coefficient and attenuation to achieve 30 dB reduction
• Part (c): Analysis of half-controlled single-phase bridge with discontinuous conduction, computing average output voltage considering 105° firing angle and 30° extinction angle
• Part (c): Determination of back EMF from terminal voltage and armature resistance drop, then speed calculation using constant flux assumption

This is a multi-part numerical problem requiring precise calculations across three distinct domains: synchronous machines (40% weight, 20 marks), probability theory (40% weight, 20 marks), and power electronics (20% weight, 10 marks). Begin with part (a) by drawing the phasor diagram for synchronous condenser operation, clearly showing Vt, Ef, and Ia with 90° phase relationship; proceed to parts (b)(i)-(ii) using moment generating functions or convolution for Poisson proof, and limit analysis (n→∞, p→0, np=λ) for Binomial-Poisson relationship; conclude with part (c) by analyzing the RLC circuit transient response, identifying the damping condition and solving the second-order differential equation for current extrema and conduction angle.

• Part (a): Correct phasor diagram for synchronous condenser with Ia leading Vt by 90°, calculation of phase current from Q = √3 VL IL, and excitation voltage Ef = Vt + IaZs using complex arithmetic
• Part (b)(i): Application of convolution or MGF to prove sum of independent Poisson variables is Poisson with parameter λ₁+λ₂, showing P(Y=k) = Σ P(X₁=i)P(X₂=k-i)
• Part (b)(ii): Rigorous limit derivation showing lim(n→∞) C(n,k)p^k(1-p)^(n-k) = (λ^k e^-λ)/k! with substitution p=λ/n and appropriate limit theorems
• Part (c)(i)-(iii): Correct circuit analysis assuming series RLC with pre-charged capacitor, formulation of differential equation, identification of underdamped/overdamped condition, and calculation of current maximum, its time instant, and total conduction time until current returns to zero
• Proper handling of per-phase vs line quantities in part (a), and clear statement of assumptions for thyristor circuit in part (c) including initial inductor current and capacitor voltage

Calculate the required quantities systematically across all four sub-parts. For part (a), spend approximately 50% of time (20 marks) on VVVF induction motor analysis including torque-slip characteristics and voltage-frequency relationships. For part (b)(i)-(ii), allocate 25% of time (10+10 marks) on VSB power calculations and noise figure analysis with attenuator temperature. For part (c), use remaining 25% of time (10 marks) on unbalanced transformer loading with delta-star connection. Present clear per-phase equivalent circuits, modulation spectra diagrams, and transformer connection diagrams where applicable.

• Part (a): Calculate Thevenin equivalent parameters (V_TH, R_TH, X_TH) for induction motor; determine breakdown torque at 50 Hz using torque-slip relation; apply VVVF control maintaining V/f constant with voltage boost for low frequency; find required voltage at 5 Hz and corresponding speed for rated torque
• Part (a): Recognize that for constant torque operation, slip frequency must remain constant; calculate synchronous speed at 5 Hz (150 rpm) and rotor speed at rated torque condition
• Part (b)(i): Calculate carrier power (300 W), sideband powers (m²P_c/4 each), total VSB power with 15% vestigial component; compare with DSB power (P_c + m²P_c/2) to find percentage saving
• Part (b)(ii): Apply Friis formula for cascaded systems; relate attenuator noise figure to physical temperature using F = 1 + (L-1)T/T_0; solve for attenuator temperature given overall NF ≤ 3.5 dB with 3 dB loss
• Part (c): Determine transformer turns ratio (11000/231); calculate secondary line-neutral voltage (231/√3 = 133.4 V) and line-line voltage (231 V); analyze unbalanced loading with 120 kVA balanced load plus 40 kW single-phase load; apply symmetrical components or direct phase analysis for delta primary currents

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