Power System Analysis
Power system analysis
Use per-unit analysis to remove transformer voltage levels. Choose S_base and V_base, then I_base = S_base/(√3 V_base) and Z_base = V_base²/S_base for three-phase studies, with V_base always line-to-line. When changing bases, Z_new = Z_old (S_new/S_old)(V_old/V_new)². A common trap is applying the voltage ratio backwards or using line-to-neutral voltage with three-phase MVA.
Load-flow problems distinguish slack, PV, and PQ buses. The slack bus fixes voltage magnitude and angle; PV buses fix real power and voltage magnitude; PQ buses fix real and reactive power. Newton-Raphson converges quickly but needs a good Jacobian; Gauss-Seidel is simpler but slower. For faults, build sequence networks: positive, negative, and zero. A balanced three-phase fault uses only the positive-sequence network, while single line-to-ground faults involve all three, with zero-sequence impedance strongly dependent on transformer grounding.
Stability questions distinguish steady-state, transient, and voltage stability. The equal-area criterion shows that faster fault clearing increases transient stability margin. Do not confuse a load-flow solution with a stability limit.
Worked example: a generator has Z = j0.10 pu on a 50 MVA, 13.8 kV base. On 100 MVA and 13.8 kV, Z_new = j0.10 × (100/50) = j0.20 pu. If the fault current base is 100 MVA/(√3 × 13.8 kV) = 4184 A, a 0.20 pu fault gives about 20,900 A. Rule of thumb: doubling the MVA base doubles the per-unit impedance of a fixed physical impedance.
Protection
Protection
Protection begins with dependability, selectivity, speed, and security. Primary protection clears faults inside its zone; backup protection covers adjacent zones after a time delay. Overcurrent relays use pickup and time-current curves, and coordination requires a time margin between downstream and upstream devices. Directional overcurrent uses a polarizing quantity to avoid tripping for reverse faults. Distance relays measure impedance and use zones: zone 1 is typically set to about 80% of the protected line, zone 2 reaches beyond the remote terminal with delay, and zone 3 provides backup. Differential protection compares currents at zone boundaries and is inherently selective, making it common for transformers, buses, and generators.
Traps include CT saturation during faults, transformer magnetizing inrush causing false differential operation, incorrect CT ratios or polarity, and forgetting that a broken-delta or zero-sequence filter needs grounded-system voltage. Ground-fault protection must account for system grounding; an ungrounded system may use sensitive zero-sequence voltage or residual current.
Worked example: a feeder CT has a 400:5 ratio and the relay pickup is 4 A secondary. Primary pickup is 4 × (400/5) = 320 A. Rule of thumb: always convert relay settings to primary amperes before comparing with load or fault current, and keep the coordination time interval at roughly 0.2–0.4 s for electromechanical relays.
Power Electronics
Power electronics
Power-electronics questions hinge on device states, waveforms, and average versus rms values. A diode conducts when forward-biased; a thyristor needs a gate pulse and turns off when current falls below holding current or is commutated. In controlled rectifiers, firing angle α delays conduction and reduces average output; a fully controlled bridge can invert when α > 90° if the load supports current. Choppers control DC voltage by duty ratio D, with buck V_o = D V_in and boost V_o = V_in/(1 − D) in continuous conduction. Inverters synthesize AC from DC; PWM controls magnitude and harmonic content, and a six-pulse bridge produces harmonics at orders 6k ± 1, with the fifth and seventh dominant.
Traps: using average voltage where rms is needed for power, ignoring discontinuous conduction, assuming ideal commutation, and forgetting freewheeling diodes or capacitive filters. For a single-phase full-wave controlled rectifier with resistive load, V_avg = (2V_m/π)cosα.
Worked example: 120 V rms input, V_m = 170 V, α = 30°. V_avg = (2 × 170/π) × cos30° = 93.7 V. Rule of thumb: increasing α reduces the average DC output of a rectifier but increases harmonic distortion and reactive power demand.