(a) A hollow cylinder made of insulating material surrounded by a thin metal sheath has inner and outer diameters of 4 cm and 8…

(a) Safe operating voltage of a conductor in a composite dielectric (20 marks)

Geometry. The conductor has radius r₁ = 1 cm. It lies in air out to the inner surface of the insulating cylinder at r₂ = 2 cm (inner diameter 4 cm). The cylinder has ε_r = 4 and extends to r₃ = 4 cm (outer diameter 8 cm), where the thin earthed metal sheath lies. So there are two coaxial dielectric layers in series: air (1 to 2 cm) and the insulator (2 to 4 cm).

Field in each layer. For a charge λ per unit length on the conductor, Gauss's law gives

• in air: E(r) = λ/(2πε₀r), which is greatest at the conductor surface, r = r₁;
• in the insulator: E(r) = λ/(2πε₀ε_r r).

The maximum gradient in air is therefore g_max = λ/(2πε₀r₁), so that λ/(2πε₀) = g_max r₁.

Voltage. The peak voltage between the conductor and the sheath is the integral of the field:

V = ∫E dr = (λ/2πε₀)[ln(r₂/r₁) + (1/ε_r) ln(r₃/r₂)] = g_max r₁ [ln(r₂/r₁) + (1/ε_r) ln(r₃/r₂)].

Numbers. The permissible peak gradient in air is g_max = 33 kV/cm.

V_peak = 33 × 1 × [ln 2 + (1/4) ln 2] = 33 × [0.6931 + 0.1733] = 33 × 0.8664 = 28.59 kV.

RMS value. V_rms = V_peak/√2 = 28.59/1.4142 = 20.2 kV.

Check on the insulator. At r = 2 cm the gradient in air is 33 × 1/2 = 16.5 kV/cm, and just inside the insulator it is 16.5/4 = 4.1 kV/cm. The stress in the insulator is far below that at the conductor, so the limiting stress is at the surface of the conductor in air, as assumed. The safe rms operating voltage of the conductor is about 20.2 kV.

(b)(i) Capacity of an ideal AWGN channel of infinite bandwidth (10 marks)

The noise has a two-sided power spectral density η/2, so the noise power in a bandwidth B is N = (η/2)(2B) = ηB. The Shannon-Hartley capacity is

C = B log₂(1 + S/(ηB)) bits/s.

Put x = S/(ηB), so that B = S/(ηx). Then

C = (S/η)(1/x) log₂(1 + x) = (S/η)(1/ln 2)[ln(1 + x)/x].

As B → ∞, x → 0, and ln(1 + x) = x − x²/2 + x³/3 − … so ln(1 + x)/x → 1. Hence

C_∞ = lim(B→∞) C = (1/ln 2)(S/η) = 1.44 S/η bits/s, since 1/ln 2 = log₂e = 1.4427.

Interpretation. Capacity rises with B, but only up to this limit, because the noise power ηB grows in proportion to B. With S = E_bC, the limit gives E_b/η ≥ ln 2 = 0.693, that is −1.6 dB, the Shannon limit for error-free communication.

(b)(ii) Error probability of coherent binary FSK (10 marks)

Data. R_b = 2.5 × 10⁶ b/s, so T_b = 1/R_b = 0.4 μs. The noise has a two-sided power spectral density N₀/2 = 10⁻²⁰ W/Hz, so N₀ = 2 × 10⁻²⁰ W/Hz. This is the convention used in part (i).

Energy per bit. The received sinusoid has amplitude A = 1 μV across R = 50 Ω, so the power is A²/(2R) = (10⁻⁶)²/(2 × 50) = 10⁻¹⁴ W, and

E_b = (A²/2R)T_b = 10⁻¹⁴ × 0.4 × 10⁻⁶ = 4 × 10⁻²¹ J.

Error probability. For coherent detection of binary FSK (orthogonal signals),

P_e = Q(√(E_b/N₀)) = ½ erfc(√(E_b/2N₀)).

E_b/N₀ = 4 × 10⁻²¹/(2 × 10⁻²⁰) = 0.2, so

P_e = Q(√0.2) = Q(0.447) ≈ 0.327.

Comment. An error probability of about 0.33 means the link is almost unusable, because E_b/N₀ = 0.2 (−7 dB) is very low; an E_b/N₀ of about 13.5 dB is needed for P_e = 10⁻⁶. (If 10⁻²⁰ W/Hz were read as N₀ itself, E_b/N₀ = 0.4 and P_e = Q(0.632) = 0.264.)

(c) Abnormal running conditions of a generator and their protection (10 marks)

Abnormal running conditions are not internal short circuits. They are usually slow to develop, so the protection gives an alarm or trips after a time delay, and it is set to keep the generator within its thermal and mechanical limits.

• Overloading (thermal overload): heats the stator winding. It is guarded by thermal relays or temperature detectors (RTDs) in the slots, with an alarm and then a delayed trip.
• Unbalanced loading (negative-sequence current): the negative-sequence field rotates against the rotor at twice the synchronous speed and induces double-frequency currents that overheat the rotor surface. A negative-sequence relay with an inverse-time I₂²t = K characteristic gives the protection.
• Loss of excitation (field failure): the machine runs as an induction generator, drawing reactive power from the grid and overheating the rotor, and may lose synchronism. It is protected by an offset mho (impedance) relay, or an under-excitation or field-failure relay, which trips after a short delay.
• Loss of synchronism (pole slipping): causes large swings in current and torque. An out-of-step (impedance-swing) relay detects it and trips.
• Motoring (loss of prime mover): the generator takes power from the grid, which can overheat the turbine blades or damage the engine. A reverse-power (directional power) relay trips after a time delay.
• Overspeed: follows a sudden loss of load and endangers the rotor. The governor limits it, and a centrifugal overspeed trip (or overfrequency relay) shuts the prime mover down.
• Overvoltage and over-fluxing (high V/Hz): caused by loss of load or an AVR fault. An overvoltage relay and a volts-per-hertz relay protect the insulation and core.
• Stator or rotor cooling failure, bearing failure and vibration: temperature and vibration detectors give an alarm and trip.
• Rotor earth fault: the first earth fault does not cause damage, but a second one would short-circuit part of the winding, so a field earth-fault relay raises an alarm or trips.