(a) (i) For the circuit shown in Figure 2(a)(i), initial current through the coil is zero. The switch is closed at time t = 0. Find the time domain expression of current flowing through the coil for t ≥ 0. How long will it take for the coil current t

Question

(a) (i) For the circuit shown in Figure 2(a)(i), initial current through the coil is zero. The switch is closed at time t = 0. Find the time domain expression of current flowing through the coil for t ≥ 0. How long will it take for the coil current to reach 95% of its final value and what is the final value of this current? (10 marks)
(ii) Obtain the Thevenin's equivalent of the network shown in Figure 2(a)(ii), across the terminals XY. (10 marks)
(b) (i) Determine the Inverse Laplace Transform of the following function: F(s) = (s³ + 7s² + 14s + 11)/(s³ + 6s² + 11s + 6) (6 marks)
(ii) Find the initial and final value of the current whose Laplace Transform is given below: I(s) = 0.32/[s(s² + 2.42s + 0.672)] (4 marks)
(iii) Solve the following differential equation: d²i/dt² + di/dt = t² + 2t. Given that i(0-) = 4 and (di/dt)₀₋ = -2. (10 marks)
(c) A single phase AC bridge rectifier as shown in Figure 2(c) is operating at firing delay angle α = 45°. The thyristor T₃ gets damaged and behaves as an open circuit. Calculate the value of load resistance R if load current is 3·1556 A. (10 marks)

UPSC Answer Check · Accepted Answer

Solve this multi-part numerical problem by allocating time proportionally to marks: approximately 40% on part (a) [20 marks], 40% on part (b) [20 marks], and 20% on part (c) [10 marks]. Begin with clear circuit diagrams for (a)(i), (a)(ii) and (c), then proceed with systematic mathematical derivations. For transient analysis in (a)(i), establish the differential equation and apply initial conditions. For Thevenin's equivalent in (a)(ii), show open-circuit voltage and short-circuit current calculations. In part (b), demonstrate partial fraction expansion for (i), apply Initial and Final Value Theorems correctly for (ii), and use Laplace transforms to solve the differential equation in (iii). Conclude with practical interpretations of time constants, steady-state values, and fault implications in power electronics.
- For (a)(i): Derive i(t) = (V/R)(1 - e^(-t/τ)) for RL circuit, calculate time constant τ = L/R, determine 95% settling time as t = 3τ, and state final value I_final = V/R
- For (a)(ii): Calculate Thevenin voltage V_TH using mesh/nodal analysis and Thevenin resistance R_TH by deactivating independent sources, presenting equivalent circuit with values
- For (b)(i): Perform polynomial long division to make F(s) proper, factor denominator (s+1)(s+2)(s+3), apply partial fraction expansion, and obtain inverse Laplace transform
- For (b)(ii): Verify applicability of Initial Value Theorem (proper rational function) and Final Value Theorem (poles in LHP), then apply limits as s→∞ and s→0
- For (b)(iii): Take Laplace transform of differential equation, apply initial conditions i(0-) = 4 and i'(0-) = -2, solve for I(s), decompose by partial fractions, and invert to get i(t)
- For (c): Analyze single-phase bridge rectifier with T₃ open (fault condition), determine conduction pattern with only two thyristors firing, derive average output voltage V_o = (V_m/π)(1+cosα) for half-wave equivalent, and solve for R = V_o/I_o

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly applies RL transient theory for (a)(i), Thevenin's theorem with proper source transformation for (a)(ii), partial fraction decomposition and ILT techniques for (b)(i)-(iii), and fault analysis of bridge rectifier with modified conduction angle for (c); no conceptual errors in network theorems or Laplace properties	Minor errors in applying theorems (e.g., incorrect sign convention in KVL, confusion between Thevenin and Norton, partial fraction arithmetic mistakes) or incomplete justification for IVT/FVT applicability; rectifier analysis shows understanding but incorrect voltage integration limits	Fundamental misconceptions such as treating RL circuit as RC, applying Thevenin's theorem incorrectly (e.g., keeping dependent sources active while finding R_TH), using Final Value Theorem when poles exist in RHP, or analyzing bridge rectifier as normal operation ignoring T₃ fault
Numerical accuracy	20%	10	All calculations precise: correct time constant and 95% settling time in (a)(i), accurate V_TH and R_TH values in (a)(ii), exact partial fraction coefficients and final time-domain expressions in (b), correct initial value i(0+) = 0 and final value i(∞) = 0.476 A in (b)(ii), and R ≈ 50Ω in (c) with proper handling of α = 45°	Computational errors in 10-20% of calculations (e.g., arithmetic mistakes in partial fractions, incorrect pole identification, time constant calculation errors, or final resistance value off by 10-15%); method correct but execution flawed	Major numerical errors (>30% deviation), incorrect formula substitution (e.g., using ωt instead of α in rectifier), wrong unit conversions, or answers without any supporting calculations; demonstrates inability to handle complex algebra
Diagram quality	15%	7.5	Clear, labeled circuit diagrams for (a)(i) showing switch, source, R and L with reference directions; (a)(ii) with terminals X-Y explicitly marked and Thevenin equivalent drawn; (c) showing bridge configuration with thyristors T₁-T₄, AC source, and load R with correct firing sequence indication; all component values marked	Diagrams present but incomplete labeling (missing current directions, polarity marks, or terminal labels), hand-drawn sketches acceptable but lacking neatness, or Thevenin equivalent shown without original circuit; rectifier diagram misses firing angle indication	Missing essential diagrams, unrecognizable sketches, or failure to show Thevenin equivalent circuit; no attempt to illustrate the fault condition in (c) or incorrect circuit topology (e.g., half-wave instead of bridge)
Step-by-step derivation	30%	15	Systematic progression: KVL setup → characteristic equation → complete solution with transient and steady-state parts for (a)(i); sequential calculation of V_OC, I_SC, then R_TH = V_OC/I_SC for (a)(ii); explicit polynomial division, factorization, partial fraction setup with coefficient comparison, and term-by-term inversion for (b)(i); rigorous limit evaluation with theorem verification for (b)(ii); complete Laplace transformation, algebraic manipulation, and inverse transformation for (b)(iii); modified voltage waveform integration with proper limits for (c)	Derivations present but with skipped steps (e.g., jumping from differential equation to solution without showing characteristic roots, omitting partial fraction coefficient calculations, or presenting final answers without intermediate working); some logical gaps but recoverable	Disorganized or missing derivations, no shown work for partial fractions, direct statement of answers without methodology, or incorrect mathematical procedures (e.g., algebraic errors in solving simultaneous equations, wrong integration limits)
Practical interpretation	15%	7.5	Interprets 95% settling time as 3τ for relay/inductor sizing applications; explains Thevenin equivalent utility for load matching and maximum power transfer; discusses physical significance of initial/final values in circuit transients; analyzes T₃ fault impact on output voltage ripple, reduced average voltage, and potential damage to remaining thyristors due to increased current stress—relevant to Indian railway traction and industrial drive protection systems	Brief mention of practical relevance without elaboration (e.g., stating 'used in filters' without context, or 'fault affects output' without quantifying impact); generic statements about Laplace transform utility without circuit-specific interpretation	No physical interpretation provided, purely mathematical treatment, or incorrect practical conclusions (e.g., claiming fault improves rectifier performance, misunderstanding time constant significance, or irrelevant applications cited)

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