22-Elec-B8 Power Electronics and Drives · Undated paper
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Examinations, May 2019 — 16-Elec-B8 Power Electronics and Drives. Three hours; open book; any non-communicating calculator, whose make and model must be written on the first inside sheet of the work book. The paper is in two parts and the candidate must attempt all parts: Part 1-A is ten five-point short-answer items, Part 1-B is ten five-point multiple-choice items each requiring a written explanation, and Part 2 is four thirty-point problems. Page-1 Note 4 states that the maximum total score is 220 points and that 150 points is a full mark (100 per cent), so there are 70 points of built-in bonus. Note 5 warns that an answer without its working scores nothing. All a.c. voltages and currents below are rms unless stated otherwise, and three-phase voltages are line-to-line. Every part of every question is solved below, in the exam's own order, with the six sections mapping to Part 1-A, Part 1-B and PROBLEMs 1 to 4.
Reference texts.
Two printing anomalies in the paper itself, both worth a line in the answer book. First, Part 1-B Question 9 repeats Question 5 word for word (“AC voltage controllers convert…”) — ten of the fifty Part 1-B points are the same item asked twice. Both are answered below, the second with the quantitative detail the first does not need. Second, the arithmetic of the cover page does not close in the candidate's favour by accident: 50 + 50 + 120 = 220 points are on offer against a full mark of 150, so a candidate should attempt every part and let the surplus absorb the inevitable slips rather than budgeting time as if 100 points were the target.
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.
Ten independent five-point items. Each is answered at the depth a five-point answer earns in a three-hour open-book paper: the definition or mechanism, the governing relation where one exists, and the practical consequence.
Harmonics are the sinusoidal components of a distorted but periodic voltage or current whose frequencies are integer multiples of the power-system fundamental — 60 Hz in Canada, so the 5th harmonic is 300 Hz and the 7th is 420 Hz. Any periodic waveform admits a Fourier series $$v(t)=V_0+\sum_{n=1}^{\infty}\sqrt{2}\,V_n\sin(n\omega_1 t+\theta_n),$$ and the harmonic content is summarised by the total harmonic distortion $\mathrm{THD}=\sqrt{\sum_{n\ge 2}V_n^{2}}\big/V_1$.
Harmonics are created by non-linear loads — diode and thyristor rectifiers, a.c. voltage controllers, arc furnaces, saturated transformers and switch-mode supplies — which draw non-sinusoidal current from a sinusoidal supply; that current then produces harmonic voltage drops across the system impedance and distorts the bus voltage for every other customer. The practical consequences are extra copper and iron loss (skin effect makes the loss rise faster than $I^{2}R$ at fundamental frequency), overheated neutral conductors because triplen harmonics add arithmetically in a four-wire system, resonance between line inductance and power-factor-correction capacitors, torque pulsations in motors, and mis-operation of protection and metering. In Canada, IEEE Std 519 is the usual benchmark for the allowable voltage and current distortion at the point of common coupling.
At constant applied rms voltage the peak core flux follows directly from Faraday's law in the transformer form $V=4.44\,f\,N\,\phi_{m}$, so
$$\phi_{m}\;\propto\;\frac{V}{f}\qquad\Longrightarrow\qquad \frac{B_{m,60}}{B_{m,50}}=\frac{50}{60}=0.833 .$$Raising the frequency therefore reduces the peak flux density by about 17 per cent. The two core-loss mechanisms respond differently. Hysteresis loss follows Steinmetz, $P_{h}=k_{h}fB_{m}^{n}$ with $n\approx 1.6$; substituting $B_{m}\propto 1/f$ gives $P_{h}\propto f^{\,1-n}=f^{-0.6}$, so
$$\frac{P_{h,60}}{P_{h,50}}=\left(\frac{60}{50}\right)^{-0.6}=0.896 .$$Eddy-current loss follows $P_{e}=k_{e}f^{2}B_{m}^{2}$; with $B_{m}\propto 1/f$ the two frequency factors cancel exactly and $P_{e}$ is unchanged. The net effect is that total core loss falls by roughly ten per cent of its hysteresis share, the magnetising current falls with the flux, and the transformer runs cooler and quieter — a 50 Hz unit is safe on 60 Hz at rated voltage. The reverse move is the dangerous one: a 60 Hz transformer on 50 Hz sees 20 per cent more flux, drives the core into saturation, and the magnetising current and core loss both climb sharply. Note also that leakage reactance rises with frequency, so per-unit impedance and voltage regulation change slightly.
Pulse width modulation (PWM) controls the average value of a switched waveform by varying the width of the switching pulses while the switching period is held constant, rather than by varying their amplitude. Over one switching period $T_{s}$ the mean output is $\bar{v}_{o}=\delta V_{dc}$ with duty ratio $\delta=t_{on}/T_{s}$, so the average tracks any reference that changes slowly compared with $T_{s}$.
In the sinusoidal PWM used for inverters, a sinusoidal modulating signal at the wanted output frequency $f_{1}$ is compared with a triangular carrier at $f_{c}$; the comparator output gates the switches. Two ratios describe the modulator: the amplitude modulation index $m_{a}=\hat{V}_{ref}/\hat{V}_{carrier}$, which sets the fundamental output amplitude linearly ($\hat{V}_{o1}=m_{a}V_{dc}/2$ for $m_{a}\le 1$), and the frequency modulation ratio $m_{f}=f_{c}/f_{1}$, which determines where the harmonics land. The virtue of PWM is precisely that: the significant harmonics are pushed up to sidebands clustered about $m_{f}$ and its multiples, far from the fundamental, so a small filter (or the load's own inductance) removes them. The costs are switching loss proportional to $f_{c}$ and steep $dv/dt$ at the machine terminals.
A d.c. link converter is a two-stage, back-to-back arrangement: a rectifier converts the fixed-voltage fixed-frequency a.c. supply to d.c., an energy-storage element forms the link, and an inverter converts that d.c. back to a.c. at whatever voltage and frequency the load requires. The link is what makes the scheme work: because the capacitor (voltage-source link) or series reactor (current-source link) decouples the two converters, the output frequency is set entirely by the inverter's switching pattern and bears no relation to the supply frequency. That is the key difference from a cycloconverter, whose output frequency is limited to roughly one third of the input frequency.
In the common voltage-source form the link capacitor holds a stiff d.c. bus, the inverter is PWM-controlled, and a constant-voltage-per-hertz law $V_{1}/f_{1}\approx\text{constant}$ keeps the machine flux at rated value while speed is varied. A diode front end gives one-way power flow and needs a braking chopper for regeneration; an active front end (a second PWM converter) gives bidirectional flow and near-unity input displacement factor. D.c. link converters are the backbone of adjustable-speed drives, wind-turbine converters, traction drives and HVDC transmission.
A multilevel inverter synthesises its output from more than the two levels of a conventional bridge by switching among several d.c. voltage levels, so the pole voltage climbs to the peak in a staircase rather than in one step. The three standard topologies are the diode-clamped (neutral-point-clamped) inverter, the flying-capacitor inverter, and the cascaded H-bridge fed from isolated d.c. sources. With $n$ levels per phase the line-to-line waveform has $2n-1$ steps.
Three benefits follow directly from the staircase. First, the waveform is much closer to a sinusoid, so total harmonic distortion falls sharply and the output filter shrinks or disappears. Second, each switching transition is only $V_{dc}/(n-1)$, so $dv/dt$ and the associated common-mode currents, bearing currents and EMI are all reduced. Third — and this is why medium-voltage drives use them — each device only ever blocks $V_{dc}/(n-1)$, so a 4.16 kV drive can be built from ordinary 1.7 kV IGBTs without series connection and its attendant static and dynamic voltage-sharing problems. The price is a larger device count, more gate drives, and the need to balance the capacitor voltages actively.
A snubber is a small auxiliary network placed around a switching device to shape its switching trajectory and hold it inside the safe operating area. Three variants cover most practice: an R–C snubber across a thyristor or diode, an R–C–D turn-off snubber across a transistor, and a small series inductor (a turn-on snubber) in the switch leg.
Its functions are: to limit the rate of rise of off-state voltage so a thyristor does not turn on spuriously through its junction capacitance (the classical $dv/dt$ triggering, since the displacement current $C_{j}\,dv/dt$ can exceed the gate trigger current); to limit $di/dt$ at turn-on so the conducting area of the device spreads before the full current arrives; to absorb the energy trapped in stray wiring inductance at turn-off, which would otherwise appear as a voltage spike $L_{\sigma}\,di/dt$ across the device; to damp the ringing between stray inductance and device capacitance and so reduce conducted EMI; and to take reverse-recovery energy off the diode. A snubber transfers switching loss from the semiconductor to a resistor, where it is easier to remove, but it does not eliminate that loss — which is why regenerative and lossless snubbers are used at high switching frequency.
The power MOSFET is a majority-carrier device: conduction is by electrons drifting through an induced channel, with no injected minority-carrier plasma to be swept out at turn-off. There is therefore no current tail and no stored-charge recovery time, and switching times of tens of nanoseconds are routine, which keeps the switching loss $\tfrac{1}{2}V I (t_{on}+t_{off})f_{s}$ tolerable even at hundreds of kilohertz. The gate is a capacitor, so the device is voltage controlled and needs only charging energy $Q_{g}V_{gs}$ — a simple, cheap driver — whereas a BJT needs continuous base current.
The on-state resistance has a positive temperature coefficient, so a hot device sheds current to its cooler neighbours; MOSFETs therefore parallel readily and cannot suffer the current-crowding second breakdown of a bipolar transistor. The restriction to low voltage is fundamental: the drift-region resistance scales roughly as $R_{DS(on)}\propto V_{BD}^{2.5}$, so conduction loss $I^{2}R_{DS(on)}$ becomes prohibitive above a few hundred volts and the IGBT, whose conductivity modulation gives an almost voltage-independent forward drop, takes over. At low voltage the opposite holds — a milliohm-class $R_{DS(on)}$ beats an IGBT's one-to-two-volt saturation drop outright.
All of these are shunt-connected, electronically controlled sources of reactive power. The classical SVC is a thyristor-controlled reactor in parallel with thyristor-switched capacitors: the firing angle sets the effective inductive susceptance, and the combination gives a continuously variable $Q$ between a capacitive and an inductive limit. The STATCOM (or SVG) is a voltage-source converter tied to the bus through a coupling reactance $X$, and its reactive output follows from the two-bus relation
$$Q=\frac{V_{bus}\left(V_{bus}-V_{conv}\right)}{X}\;,$$so simply raising the converter's fundamental output voltage above the bus voltage makes it supply reactive power, and lowering it makes the converter absorb reactive power — no capacitor bank is switched at all.
Power-factor improvement follows because the compensator supplies the load's reactive demand locally, so the line carries only the real component: the source current falls from $S/(\sqrt{3}V)$ towards $P/(\sqrt{3}V)$, line loss falls as the square of that ratio, and released capacity becomes available. Bus-voltage control follows because injecting $Q$ raises the local voltage through the system's $X\,\Delta Q/V$ sensitivity. The STATCOM's decisive advantage over a capacitor bank or an SVC is that its current limit is set by the converter rating, not by $V^{2}$: during a voltage sag, when support is most needed, a capacitor bank's output collapses while the STATCOM still delivers rated current. Response is within one or two cycles, which is why STATCOMs are used for flicker compensation on arc furnaces and for transient-stability support.
A variable autotransformer (a Variac-type tapped or brush-swept winding) does produce an adjustable a.c. voltage, but it is a poor drive controller for six reasons. It is bulky, heavy and expensive at motor ratings, because the magnetic circuit must carry the full flux at supply frequency. Its adjustment is mechanical — a moving brush or a tap changer — so it is slow, wears, arcs, needs maintenance, and cannot be placed in a fast closed speed loop; tapped versions give only discrete steps. It provides no galvanic isolation in the autotransformer connection, so a fault couples the supply straight to the load.
More fundamentally, it varies voltage only, not frequency. For an induction motor, torque at a given slip varies as $T\propto V^{2}$, so reducing voltage lowers the torque-speed curve and the machine finds a new operating point at higher slip; since rotor copper loss is $sP_{ag}$, the lost power is dissipated in the rotor and efficiency collapses. Only fan and pump loads, whose torque falls as speed squared, tolerate this at all, and even then the speed range is narrow and the breakdown torque margin shrinks as $V^{2}$. Constant flux requires constant $V/f$, which an autotransformer cannot deliver because it cannot change frequency. Finally it offers no starting-current limit beyond the voltage reduction itself and no regenerative braking. A PWM voltage-source inverter drive solves every one of these at lower mass and cost.
A bidirectional (full-wave) a.c. voltage controller places two thyristors in antiparallel — or a single TRIAC — in series between a fixed a.c. supply and the load, so that one device carries the positive half cycle and the other the negative half cycle. Each device is gated at a delay angle $\alpha$ measured from its own voltage zero crossing and turns off naturally when its current reaches zero, at the extinction angle $\beta$. For a purely resistive load $\beta=180^\circ$ and the output is a symmetrically chopped sinusoid; for an inductive load the stored energy keeps the current flowing past the voltage zero, so $\beta>180^\circ$ and the conduction angle is $\gamma=\beta-\alpha$, exactly the behaviour analysed in PROBLEM 2 below.
The rms output voltage is obtained by integrating over the conducting window,
$$V_{o,rms}=V_{s}\sqrt{\frac{1}{\pi}\left[\left(\beta-\alpha\right) -\frac{\sin 2\beta-\sin 2\alpha}{2}\right]},$$and falls monotonically from $V_{s}$ at $\alpha=0$ towards zero as $\alpha$ approaches $180^\circ$ (resistive load). Because the supply frequency is untouched, the converter is a fixed-a.c. to variable-a.c. stage. Its two control modes are phase-angle control, used for lighting dimmers, resistance heating and induction motor soft starters, and integral-cycle (on–off) control, used where the load is thermally slow. The drawbacks are severe input current harmonics and a displacement factor that worsens as $\alpha$ increases.
