6.3 Power Electronics & Transmission

Rectifiers and inverters

Power electronics uses switching devices to convert between AC and DC. A rectifier converts AC to DC; an inverter converts DC to AC; a DC-DC converter (buck/boost) changes DC voltage level; a cycloconverter changes AC frequency. Common rectifier circuits:

• Half-wave: one diode passes a single half-cycle; output is pulsating DC with high ripple. Average V_dc = V_m/pi.
• Full-wave bridge: four diodes use both half-cycles. Average V_dc = 2·V_m/pi.
• Center-tapped full-wave: two diodes plus a center-tapped transformer achieve full-wave output.

Worked example: rectifier output

A full-wave bridge is fed a sinusoid of peak V_m = 170 V (i.e., 120 V_rms).

• V_dc = 2·V_m/pi = 2·170/3.1416 = 108 V.
• A half-wave rectifier on the same source gives V_m/pi = 170/3.1416 = 54 V — exactly half. A filter capacitor across the load reduces ripple toward the peak value.

Switching devices: diodes, thyristors, transistors

Many FE power-electronics questions hinge on conduction conditions rather than calculation, so know each device's behavior cold.

A diode is uncontrolled — it conducts whenever forward-biased. A thyristor (silicon-controlled rectifier, SCR) is a controlled latching switch: it stays off until a gate trigger, then conducts until its current drops near zero (line commutation). A transistor is fully controlled — gate drive both turns it on and off, which is why inverters use IGBTs/MOSFETs. In a phase-controlled rectifier, delaying the SCR firing angle alpha reduces the average DC output (V_dc proportional to (1+cos alpha)/2 for a full-wave controlled bridge).

Transmission line models

A transmission line is modeled by lumped elements whose complexity scales with length:

• Short line (under ~80 km / 50 mi): series resistance R and inductance L only; shunt capacitance neglected.
• Medium line (~80 to 250 km): nominal-pi or nominal-T adds shunt capacitance, usually split half at each end (nominal-pi).
• Long line (over ~250 km): distributed-parameter model with hyperbolic functions of the propagation constant gamma = sqrt(z·y).

The series impedance Z = R + jX_L causes voltage drop and I^2R loss; the shunt capacitance can raise receiving-end voltage above sending-end at light load — the Ferranti effect. Picking the right model for the stated length is the most common exam decision here. Surge impedance Z_0 = sqrt(L/C) sets the line's natural loading (SIL); loading below SIL leaves net capacitive (voltage-rising) behavior.

Voltage regulation and protection

Voltage regulation measures how far the output sags from no load to full load:

VR (%) = (V_no-load - V_full-load) / V_full-load × 100

Lower regulation means a stiffer source; the same formula covers transformers and lines.

Worked example: voltage regulation

A feeder reads 7200 V at no load and 6840 V at full load: VR = (7200 - 6840)/6840 × 100 = 360/6840 × 100 = 5.26%. The denominator is the full-load value — using no-load gives a wrong 5.0%.

Protection keeps faults from harming equipment and people. Fuses melt to interrupt overcurrent; circuit breakers are resettable interrupters; protective relays sense abnormal current, voltage, or impedance and command breakers to trip. Good design uses selective coordination so the device nearest the fault clears first, isolating the smallest section and keeping the rest energized. Grounding limits touch voltage and provides a low-impedance fault-current return path. Symmetrical-component analysis (positive/negative/zero sequence) underlies unbalanced-fault calculations.

Converter families and fault basics

Four converter types cover most power-electronics questions, and naming the conversion direction is half the battle:

A buck converter steps voltage down (V_out = D·V_in for duty cycle D); a boost steps it up (V_out = V_in/(1-D)). These ideal duty-cycle relations are quick Handbook lookups.

Fault basics: a short-circuit fault current is set by the system source voltage divided by the impedance to the fault, I_fault = V/Z. With impedances in per-unit, the three-phase fault MVA at a bus is roughly S_base/Z_pu. The interrupting rating of a breaker must exceed the available fault current. The most common fault is a single-line-to-ground fault; a bolted three-phase fault is usually the most severe for equipment rating.

Coordinating relays and breakers so the nearest device trips first keeps an isolated fault from cascading into a wider outage.

Per-unit method: before adding impedances, refer every quantity to one common base by pu = actual/base, with V_base, S_base chosen and I_base = S_base/(sqrt(3)·V_base_LL) and Z_base = V_base^2/S_base derived from them. The big payoff is that an ideal transformer's turns ratio disappears when each side uses its own voltage base, so a multi-voltage network collapses into a single per-unit impedance. 0 pu) by that total to get pu fault current, then multiply by base current to recover amperes.

A common distractor swaps the per-unit base or forgets to refer all impedances to a single common base before adding them.