(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

(a)(i) Grading of cables means controlling the dielectric stress in the insulation so that the maximum electric stress at the conductor surface is reduced and the insulation is used economically. The main methods are:
Capacitance grading: two or more dielectric layers of different permittivities are used concentrically.
Intersheath grading: one or more metallic intersheaths are placed at suitable potentials inside the dielectric, each layer being designed for the same maximum stress.

(a)(ii) Let conductor radius = r, sheath radius = R, and voltage between conductor and sheath = V. For a single-core cable, electric stress at radius x is
g(x) = V/(x ln(R/r)).
At the conductor surface, x = r:
gₘₐₓ = V/(r ln(R/r)).
For fixed R and V, minimise gₘₐₓ by maximising y = r ln(R/r).
dy/dr = ln(R/r) - 1.
Set dy/dr = 0: ln(R/r) = 1, so R/r = e.
Since d²y/dr² = -1/r < 0, this gives the minimum surface gradient.
Thus the condition is R = e r.

(a)(iii) For economic design, the surface gradient equals the dielectric strength gₛ. Using R = e r,
V = gₛ r ln(R/r) = gₛ r.
Given V = 75 kV and gₛ = 60 kV/cm,
r = 75/60 = 1.25 cm.
R = e r = 2.71828 × 1.25 = 3.3979 cm.
Overall diameter = 2R = 6.7957 cm.
Overall diameter ≈ 6.80 cm.

(b) At 50 Hz, synchronous speed:
Nₛ = 120f/p = 120 × 50/6 = 1000 rpm.
Rated slip: s_N = (1000 - 960)/1000 = 0.04.
Rated slip frequency: fₛ_N = s_N f = 0.04 × 50 = 2 Hz.
For constant flux below base speed, torque is proportional to slip frequency. Hence for half-rated torque,
fₛ = 1 Hz.
At N = 500 rpm motoring,
Nₛ' = N + (120/p)fₛ = 500 + (120/6) × 1 = 520 rpm.
Inverter frequency:
f' = Nₛ' p/120 = 520 × 6/120 = 26 Hz.
Inverter frequency = 26 Hz.

Slip at this point:
s = (520 - 500)/520 = 0.0384615.
Rated phase voltage = 400/√3 = 230.94 V.
At constant flux, V' = 230.94 × (26/50) = 120.09 V.
Reactances at 26 Hz:
x₁ = x₂' = 1.5 × 26/50 = 0.78 Ω,
Xₘ = 30 × 26/50 = 15.6 Ω.
Rotor branch: r₂'/s = 0.2/0.0384615 = 5.2 Ω.
Using the per-phase equivalent circuit,
Z₂ = 5.2 + j0.78 Ω, Zₘ = j15.6 Ω.
Parallel combination:
Zₚ = (j15.6)(5.2 + j0.78)/(j15.6 + 5.2 + j0.78)
= 4.285 + j2.103 Ω.
Total impedance:
Z = 0.4 + j0.78 + 4.285 + j2.103 = 4.685 + j2.883 Ω.
|Z| = √(4.685² + 2.883²) = 5.501 Ω.
Stator current:
I₁ = V'/|Z| = 120.09/5.501 = 21.83 A.
Stator current ≈ 21.8 A (line = phase).

(c) The magnetizing current waveshape is non-linear because the core B-H or Φ-i relation is non-linear. For a sinusoidal voltage, flux is almost sinusoidal, but the current required to produce that flux is not linear in flux; saturation makes the current peaked and rich in odd harmonics, especially third harmonic.

In-rush magnetizing current occurs when a transformer is switched on with residual core flux Φᵣ. Let the applied voltage be v(t) = Vₘ sin(ωt + α). By Faraday’s law,
Vₘ sin(ωt + α) = N dΦ/dt.
Integrating from t = 0, with Φ(0) = Φᵣ:
Φ(t) = Φᵣ + (Vₘ/(Nω))[cos α - cos(ωt + α)].
Let Φₘ = Vₘ/(Nω). Then
Φ(t) = Φᵣ + Φₘ cos α - Φₘ cos(ωt + α).
The term Φᵣ + Φₘ cos α is a dc offset. The worst case occurs at α = 0, giving
Φₘₐₓ = Φᵣ + 2Φₘ.
This drives the core deep into saturation, producing a large unidirectional magnetizing current pulse that decays with the primary time constant. The current is
iₘ(t) = f[Φ(t)] = f[Φᵣ + Φₘ(cos α - cos(ωt + α))],
where f denotes the non-linear Φ-i characteristic.

Graph description: plot Φ on the y-axis and i on the x-axis. The curve is almost linear up to the knee and then bends sharply, showing saturation. When the flux waveform with dc offset is projected onto this curve, the resulting current is non-sinusoidal with narrow, high peaks.

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