(a) (i) What do you mean by grading of cables? What are the methods of grading?
(ii) Derive the condition for minimum value of gradient at the surface of the conductor.
(iii) Determine the economic overall diameter of a single-core cable metal sheath

Question

(a) (i) What do you mean by grading of cables? What are the methods of grading?
(ii) Derive the condition for minimum value of gradient at the surface of the conductor.
(iii) Determine the economic overall diameter of a single-core cable metal sheathed for a working voltage of 75 kV, if the dielectric strength of the insulating material is 60 kV/cm.

(b) A 400 V, 50 Hz, 6-pole, 960 r.p.m., Y-connected induction motor has the following parameters per phase referred to stator :
r₁ = 0·4 Ω; r₂' = 0·2 Ω; x₁ = x₂' = 1·5 Ω; Xₘ = 30 Ω

The motor is controlled by a variable frequency inverter at a constant flux of rated value for operation below synchronous speed, while in super-synchronous operation region flux is weakened by keeping voltage constant at rated value. Assume straight line for torque vs. slip characteristics for slip s < sₘ (motor region) and s > sₘ' (generator region). The connected load on the shaft is constant torque type.

Calculate the inverter frequency and current drawn by the stator when torque on the shaft is half-rated while motoring at 500 r.p.m.

(c) Why is the waveshape of magnetizing current of a transformer non-linear? Explain the phenomenon of in-rush magnetizing current and derive its expression in terms of α, the angle of the voltage sinusoid at t = 0 and Φᵣ, the residual core flux at t = 0.

Use the graph sheet to show non-linearity of current from the assumed Φ-i diagram of magnetic core of the transformer.

UPSC Answer Check · Accepted Answer

This is a multi-part numerical and theoretical problem requiring systematic solving. Begin with part (a) on cable grading: define grading, derive the condition for minimum gradient using calculus on the electric field expression, then calculate economic diameter using the optimal radius ratio condition. For part (b), apply variable frequency control principles: first determine rated torque and slip, then use constant V/f control below synchronous speed to find inverter frequency at 500 rpm, and calculate stator current using the equivalent circuit with modified frequency parameters. For part (c), explain core saturation physics, derive the inrush current expression from flux balance equation considering residual flux and switching angle, and sketch the Φ-i curve showing non-linear magnetization. Allocate approximately 35% time to (a), 40% to (b), and 25% to (c) based on computational complexity.
- Part (a)(i): Definition of cable grading as equalizing dielectric stress; identification of capacitance grading (using multiple dielectrics) and intersheath grading (using intermediate conducting layers) methods
- Part (a)(ii): Derivation showing E_max is minimized when ln(R/r) = 1, i.e., R/r = e ≈ 2.718, giving optimal conductor-to-sheath radius ratio
- Part (a)(iii): Calculation of economic diameter using g_max = V/(r·ln(R/r)) with ln(R/r)=1, yielding overall diameter ≈ 4.06 cm for 75 kV working voltage
- Part (b): Determination of rated torque from synchronous speed (1000 rpm) and rated slip; application of constant flux (V/f) control to find inverter frequency ≈ 25 Hz at 500 rpm; calculation of stator current ≈ 12-14 A using modified equivalent circuit parameters
- Part (c): Explanation of non-linear magnetizing current due to saturation in B-H curve; derivation of inrush current i = (V_m/ωL)(cosα - cos(ωt+α)) + Φ_r/L showing DC offset; graph showing peaked waveform with harmonic content

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	2	Precise definitions of capacitance vs intersheath grading; correct identification of constant V/f control for part (b) and flux-weakening regions; accurate physical explanation of core saturation causing non-linear magnetization; correct interpretation of economic cable design	Basic definitions present but confusion between grading methods; partially correct control strategy with errors in identifying operating regions; generic explanation of saturation without linking to B-H curve non-linearity	Misidentification of grading methods; fundamental errors in variable frequency control principles; incorrect physical explanation of transformer magnetization or complete omission of saturation effects
Numerical accuracy	20%	2	Correct calculation of economic diameter ≈ 4.06 cm using optimal ratio R/r = e; accurate inverter frequency of 25 Hz and stator current calculation within ±5% tolerance; proper handling of per-phase equivalent circuit with frequency scaling	Correct approach but arithmetic errors in final values; correct frequency calculation but errors in current magnitude due to incorrect impedance calculation; partial credit for setting up correct equations	Major errors in applying optimal condition for cable; incorrect synchronous speed or slip calculations; failure to scale reactances with frequency; order-of-magnitude errors in final answers
Diagram quality	20%	2	Clear Φ-i characteristic showing linear region, knee point, and saturation with corresponding non-linear current waveform; properly labeled cable cross-section showing grading; equivalent circuit diagram for induction motor with parameter values marked	Diagrams present but missing key features like saturation region labeling or proper axes; rough sketches without clear indication of non-linearity; motor equivalent circuit without referred values indicated	Missing required Φ-i diagram; incorrect cable diagram showing wrong geometry; no diagrams despite explicit instruction to use graph sheet; completely unlabeled sketches
Step-by-step derivation	20%	2	Rigorous derivation of dE/dr = 0 condition leading to ln(R/r) = 1; complete flux linkage equation setup with integration for inrush current showing transient and steady-state components; systematic equivalent circuit analysis with Thevenin or direct approach	Key steps shown but gaps in mathematical rigor; missing intermediate steps in differentiation or integration; correct final expressions but unclear derivation path	Final formulas stated without derivation; missing critical steps like boundary condition application for inrush current; no mathematical working, only numerical substitution
Practical interpretation	20%	2	Insight on 765 kV Indian transmission cables using grading; discussion of inverter-fed drives in Indian Railways traction; explanation of inrush current mitigation using pre-insertion resistors in power transformers; relevance of economic design to XLPE cable standards	Brief mention of practical applications without specific Indian context; generic statements about motor drives or transformer protection without elaboration	No practical context provided; purely theoretical treatment ignoring engineering significance; failure to mention why economic diameter matters for material cost vs dielectric stress trade-off

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