(a) A three-phase, 5 kW, 440 V, 6 pole, star connected synchronous motor having negligible stator resistance and synchronous reactance of 6 Ω is operated at 0·8 rated power factor lagging. Calculate the following :

(i) Torque angle at full load

(ii

Question

(a) A three-phase, 5 kW, 440 V, 6 pole, star connected synchronous motor having negligible stator resistance and synchronous reactance of 6 Ω is operated at 0·8 rated power factor lagging. Calculate the following :

(i) Torque angle at full load

(ii) Pull-out torque

(iii) Armature current and power factor at half the rated torque (20 marks)

(b) X and Y are two independent random variables with probability density functions given by

f_X(x) = {1/4  for  -2 ≤ x ≤ 2
         {0    otherwise

and

f_Y(y) = {A e^{-3y}  for  0 ≤ y < ∞
         {0          otherwise .

(i) Determine A.

(ii) Determine the probability density function of Z = 3X + 4Y. (20 marks)

(c) Evaluate both sides of Stokes theorem for the field H = (2ρz a_ρ + 3z sin φ a_φ − 4ρ cos φ a_z) A/m and for the open surface defined by z = 1, 0 < ρ < 2m, 0° < φ < 45°. (10 marks)

UPSC Answer Check · Accepted Answer

Calculate the required quantities for all three parts systematically. For part (a), apply the cylindrical rotor synchronous motor power equations and phasor diagram; allocate ~40% time (20 marks). For part (b), use probability normalization for A, then convolution/characteristic function for Z = 3X + 4Y; allocate ~40% time (20 marks). For part (c), verify Stokes theorem by computing both line integral and surface integral in cylindrical coordinates; allocate ~20% time (10 marks). Present clear final answers with units.
- Part (a)(i): Correct application of power equation P = (3VE_f/X_s)sinδ to find torque angle δ at 0.8 pf lagging, with proper phasor diagram construction
- Part (a)(ii): Calculation of pull-out torque using T_max = (3VE_f)/(X_s·ω_s) with synchronous speed ω_s = 4πf/P rad/s
- Part (a)(iii): Determination of new excitation voltage E_f at half torque, then solving for armature current and power factor using modified power equation
- Part (b)(i): Normalization of f_Y(y) to find A = 3, verifying ∫f_Y(y)dy = 1 over 0 to ∞
- Part (b)(ii): Derivation of PDF of Z = 3X + 4Y using convolution of transformed variables or characteristic functions, with correct limits for -6 ≤ 3X ≤ 6 and 0 ≤ 4Y < ∞
- Part (c): Verification of Stokes theorem with correct curl computation in cylindrical coordinates, proper surface integral over z=1, 0<ρ<2, 0°<φ<45°, and matching line integral around the boundary contour

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	25%	12.5	Correctly identifies synchronous motor power-angle relationship, applies probability axioms for normalization, and understands Stokes theorem as ∮H·dl = ∫(∇×H)·dS with proper coordinate transformations	Uses correct basic formulas but makes errors in applying lagging pf convention, confuses independence with identical distributions, or misidentifies surface orientation in Stokes theorem	Fundamental misconceptions: treats synchronous motor as induction motor, fails to normalize PDF, or confuses Stokes theorem with Gauss's divergence theorem
Numerical accuracy	25%	12.5	All calculations precise to 2-3 decimal places: δ ≈ 23.5°, T_pull-out ≈ 53.8 Nm, I_a ≈ 4.2 A at 0.95 pf, A = 3 exactly, correct piecewise PDF for Z with proper breakpoints, both sides of Stokes theorem matching to <1% error	Minor arithmetic errors in one part (e.g., wrong synchronous speed calculation, sign errors in convolution limits, or numerical integration mistakes) but methodologically sound	Major calculation errors: wrong phase voltage (uses 440V instead of 440/√3), incorrect characteristic function transformation, or order-of-magnitude errors in final answers
Diagram quality	15%	7.5	Clear phasor diagram for (a) showing V, I_a, E_f, and δ with proper 36.87° lagging angle; neat sketch of surface and contour for (c) with labeled cylindrical coordinates and boundary orientation	Basic phasor diagram present but missing angle labels or scale; surface sketch for (c) lacks clear indication of open surface vs closed boundary	No diagrams where essential, or completely wrong diagrams (e.g., induction motor equivalent circuit instead of synchronous motor phasor diagram)
Step-by-step derivation	25%	12.5	Systematic derivation: for (a) shows power equation → E_f calculation → δ → T_max → iterative solution for half torque; for (b) shows normalization → characteristic functions → inverse transform; for (c) explicit curl computation and parameterization of boundary curves	Correct final formulas shown but skips key intermediate steps (e.g., jumps from power to δ without showing E_f calculation, or states convolution result without integration steps)	Only final answers with no derivation, or logically disconnected steps that don't lead to the claimed answer
Practical interpretation	10%	5	Interprets torque angle significance for stability margin, explains why Z's PDF has asymmetric triangular/exponential hybrid shape, and relates Stokes verification to electromagnetic field continuity in cylindrical geometries relevant to motor design	Brief mention of physical meaning without elaboration, or generic statements about 'checking the theorem'	Purely mathematical treatment with no physical interpretation, or incorrect physical interpretation (e.g., claiming leading pf is better for synchronous motors without justification)

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