22-Elec-A6 Power Systems and Machines · May 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exam 22-Elec-A6 Power Systems and Machines (May 2018, code 16-Elec-A6). Closed book; one double-sided aid sheet and an approved calculator permitted. Five questions of equal value constitute a complete paper — all five are worked here as a study resource. All AC quantities are RMS; three-phase voltages are line-to-line and power is total real power unless noted.
Reference texts. S. J. Chapman, Electric Machinery Fundamentals, 5th ed. (magnetic circuits Ch. 1; transformers Ch. 2; synchronous machines Ch. 4; induction machines Ch. 7). J. D. Glover et al., Power System Analysis and Design, 6th ed. (three-phase power and power-factor correction, Ch. 2). IEEE Std 519 (harmonic limits).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
(a) Motor on a delta supply with one leg open. Yes — the motor keeps running. Opening one leg of the three-phase delta source leaves the two remaining sources in an open-delta (V-V) connection, which still develops all three line-to-line voltages and therefore a genuine (if unbalanced) three-phase supply. The motor continues to rotate because the two live phases still produce a rotating field, but the available capacity falls to \(1/\sqrt{3}\approx57.7\%\) of the closed-delta rating and the currents become unbalanced, so the machine must be de-rated to avoid overheating. (A different reading is that one of the three supply lines feeding the motor opens. That is single-phasing: only one line-to-line voltage reaches the motor. A motor that is already running may keep turning with reduced torque, but it draws heavy overcurrent and overheats. It cannot start from rest, so it needs overload protection to trip it.)
(b) Generator droop. Droop is the deliberate, near-linear fall of a generator's frequency (speed) as its real-power output increases, set by the prime-mover governor: \(f = f_{nl} - m\,P\), where \(f_{nl}\) is the no-load frequency and \(m\) the droop slope. It is quantified as a percentage, \(\text{droop}\%=\dfrac{f_{nl}-f_{fl}}{f_{fl}}\times100\%\) (no-load to full-load), typically 3–5%. Droop lets several generators share load stably in parallel: because each unit's frequency-vs-power line has a downward slope, they settle at one common system frequency with the load divided inversely to their slopes (a stiffer, lower-droop machine takes the larger share of any load change).
(c) Is a synchronous motor self-starting? No. At standstill the stator produces a field rotating at synchronous speed while the DC-excited rotor is stationary; the torque between them reverses every half-cycle and averages to zero, so the heavy rotor cannot lock in and accelerate on its own. Synchronous motors are therefore started by another means — amortisseur (damper) windings so the machine starts as an induction motor and then pulls into step, a pony motor, or a variable-frequency drive that ramps the supply frequency up from near zero.
(d) Five specifications for selecting a power transformer. (1) kVA (apparent-power) rating; (2) rated primary/secondary voltages and turns ratio; (3) rated frequency (50/60 Hz); (4) per-unit (percent) impedance, which fixes voltage regulation and fault current; (5) winding connection and vector group (e.g. Dyn11). Other valid choices: cooling class (ONAN/ONAF), insulation/temperature class and BIL, efficiency, and whether tap-changing is required.
(e) Low-power-factor penalty. A low power factor means the customer draws far more current (and apparent power, kVA) than the real power (kW) actually billed as energy. That extra current loads the utility's generators, transformers and lines, raises \(I^2R\) losses and voltage drop, and consumes system capacity that must be built and maintained but earns no energy revenue. The penalty (billing on kVA or kVA-demand, or a direct p.f. surcharge) recovers those costs and gives the customer an incentive to install power-factor correction.
(f) Volts-per-hertz check on the distribution transformer.
Given. Transformer rated 480 V, 60 Hz on the LV winding; proposed operation 415 V, 50 Hz. Find. whether the core flux stays within its rated value. Core flux obeys \(\Phi_{max}\propto V/f\), so the safe condition is \(V/f \le (V/f)_{rated}\): $$\left(\frac{V}{f}\right)_{rated}=\frac{480}{60}=8.0\ \text{V/Hz},\qquad \left(\frac{V}{f}\right)_{new}=\frac{415}{50}=8.3\ \text{V/Hz}.$$ Since \(8.3 \gt 8.0\), the flux would rise by \(8.3/8.0-1=3.75\%\), pushing the core into saturation with excessive magnetizing current, heating and noise. No — it is not safe. The limiting factor is flux, not kVA: although 15 kVA is within the 18 kVA rating, at 50 Hz the voltage must be held to \(480\times(50/60)=400\) V or below.
(g) Poor induction-motor efficiency at high slip. Of the air-gap power \(P_{ag}\) crossing to the rotor, a fraction equal to the slip is dissipated as rotor copper loss (\(P_{rcl}=s\,P_{ag}\)) and only the remainder \((1-s)P_{ag}\) becomes mechanical power. The rotor-circuit efficiency is therefore at most \((1-s)\): at high slip a large share of the input is burned as \(I^2R\) heat in the rotor resistance rather than converted to shaft power, so efficiency is intrinsically low (e.g. at \(s=0.5\) the rotor conversion efficiency cannot exceed 50%).
(h) Harmonics — three causes and effects. (1) Power-electronic / non-linear loads (rectifiers, VFDs, switched-mode supplies) draw non-sinusoidal current, injecting 5th, 7th, 11th… harmonics → overheating of transformers and neutrals, and interference. (2) Transformer / reactor core saturation produces odd (esp. 3rd) harmonics in the magnetizing current → triplen currents circulating in delta windings and neutrals. (3) Arcing loads (arc furnaces, discharge lighting, welders) generate broadband harmonics → flicker and resonance. General effects: extra losses and heating, resonance with power-factor capacitors, nuisance relay/breaker operation, and distorted voltage affecting other customers (IEEE Std 519 sets limits).
(i) Why the core is laminated, and three causes of core damage. The core is built from thin, individually insulated steel laminations to interrupt the eddy currents that the alternating flux induces in the iron. Eddy-current loss varies with the square of the lamination thickness, so thin sheets sharply reduce circulating currents and the associated \(I^2R\) heating. Three causes of core damage: (1) over-fluxing / over-voltage (or under-frequency operation) driving the core into saturation and overheating; (2) inter-lamination insulation breakdown from ageing, moisture or overheating, which lets eddy currents bridge laminations; (3) mechanical vibration and clamping-force loss that loosens the laminations and abrades their insulation. (DC magnetization and mechanical shock during transport are further causes.)
