(a) A 400 V, 3-phase, 50 Hz, star-connected synchronous motor has per phase synchronous impedance Zₛ = (0·5 + j 3·5) Ω. It is required to operate the motor as synchronous condenser to deliver 100 kVAr at rated voltage and no load. Find the motor curr

Question

(a) A 400 V, 3-phase, 50 Hz, star-connected synchronous motor has per phase synchronous impedance Zₛ = (0·5 + j 3·5) Ω. It is required to operate the motor as synchronous condenser to deliver 100 kVAr at rated voltage and no load. Find the motor current and excitation voltage under this condition. (Assume zero motor input power at no load) (20 marks)

(b) (i) Two statistically independent Poisson random variables X₁ and X₂ with respective parameters λ₁ and λ₂ are added to form Y = X₁ + X₂. Show that the random variable Y is Poisson distributed with parameter (λ₁ + λ₂). (10 marks)
(ii) Derive the relationship between Binomial and Poisson random variables when Binomial distribution becomes equal to the Poisson distribution. (10 marks)

(c) The circuit given in the figure below is in steady state initially before the thyristor is triggered. The thyristor is triggered at t = 0. Calculate (10 marks)
(i) the maximum current the thyristor will carry.
(ii) the instant of carrying maximum current by the thyristor.
(iii) the conduction time of the thyristor.
(Assume zero latching and holding current for the thyristor)

UPSC Answer Check · Accepted Answer

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

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Demonstrates flawless conceptual grasp: for (a) recognizes synchronous condenser operates at zero power factor leading with Ia perpendicular to Vt; for (b)(i) correctly identifies independence requirement and uses appropriate proof technique; for (b)(ii) rigorously establishes the limiting conditions; for (c) correctly classifies the RLC response and applies thyristor switching principles	Shows generally correct concepts with minor errors: phasor direction may be confused in (a), proof method in (b) is valid but incomplete, limiting process in (b)(ii) is heuristic rather than rigorous, circuit analysis in (c) uses correct approach but wrong initial conditions	Fundamental conceptual errors: treats synchronous condenser as motor with mechanical load in (a), attempts invalid proof for Poisson sum in (b), fails to recognize Binomial-Poisson relationship as limit in (b)(ii), or analyzes thyristor circuit as simple resistive load in (c)
Numerical accuracy	25%	12.5	All numerical results accurate to appropriate significant figures: part (a) gives correct Ia = 144.34 A (phase) or 144.34/√3 A (line) and Ef magnitude/angle; part (c) yields precise values for Imax, tmax, and tcond with correct unit handling and proper complex arithmetic throughout	Most calculations correct but with arithmetic slips: minor errors in complex number manipulation in (a), correct approach in (c) but wrong characteristic roots or integration constants, final answers within 10% of correct values	Serious numerical errors: incorrect conversion between line and phase quantities in (a), fundamental errors in differential equation solution in (c), or answers that are dimensionally inconsistent or orders of magnitude wrong
Diagram quality	15%	7.5	Clear, labeled diagrams essential for full marks: part (a) shows complete phasor diagram with Vt reference, Ia leading by 90°, Ef, and drop IaZs forming right triangle; part (c) shows the complete thyristor-RLC circuit with initial conditions marked, current waveform sketch showing firing at t=0, peak, and extinction	Diagrams present but incomplete: phasor diagram in (a) lacks proper angle markings or scale, circuit in (c) shown but without initial conditions or current waveform sketch, labels missing or unclear	Diagrams absent or seriously flawed: no phasor diagram in (a) despite explicit need for visualizing 90° phase relationship, no circuit diagram in (c), or diagrams that misrepresent the physical system
Step-by-step derivation	25%	12.5	Systematic, complete derivations with no skipped steps: part (a) explicitly writes Q = 3VtIa = 100 kVAR, solves for Ia, then Ef = Vt + jIaXs + IaRa; part (b)(i) shows full convolution sum or MGF product; part (b)(ii) writes Binomial PMF, substitutes p=λ/n, applies Stirling's approximation or limit definition of e; part (c) sets up KVL, solves characteristic equation, applies initial conditions, finds di/dt=0 for maximum	Derivations mostly complete but with gaps: skips algebraic manipulation in complex arithmetic, omits intermediate steps in limit proof, or assumes solution form in (c) without justification; key results obtained but reasoning not fully transparent	Derivations fragmented or absent: jumps directly to formulas without setup, uses unjustified shortcuts, or presents only final answers with no derivation; in (b), states results without proof; in (c), gives numerical answers without showing differential equation solution
Practical interpretation	15%	7.5	Insightful practical connections: for (a), explains synchronous condenser application in Indian grid for reactive power support (e.g., at HVDC terminals or wind farms); for (b), notes Poisson approximation utility in rare event modeling like power system faults; for (c), discusses thyristor turn-off requirements and practical snubber design implications	Basic practical awareness mentioned: generic statement about power factor correction in (a), standard textbook remark on Poisson approximation in (b), or simple comment on thyristor switching in (c) without specific engineering context	No practical interpretation provided: purely mathematical/numerical answer with no connection to engineering applications, or incorrect practical statements (e.g., claiming synchronous condenser consumes real power, or confusing thyristor turn-off with transistor switching)

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