PE Electrical Power cheat sheet

Per-unit conversion: Z_pu = Z_actual / Z_base, where Z_base = V_base^2 / S_base; the per-unit impedance of a transformer is identical whether computed from either winding's base quantities.

Balanced three-phase relations: S_3φ = √3 · V_LL · I_L; P = √3 · V_LL · I_L · cos θ; Q = √3 · V_LL · I_L · sin θ; wye V_L = √3 V_ph, delta I_L = √3 I_ph.

Thevenin equivalent: V_th is the open-circuit terminal voltage, Z_th = V_oc / I_sc; maximum real power transfers to a load when R_L = |Z_th|.

Series resonance occurs at f_r = 1/(2π√(LC)); at resonance X_L = X_C, the impedance is purely resistive, and voltage across L and C can greatly exceed the source voltage (Q magnification).

Two-wattmeter method on a balanced three-phase load: P_total = W1 + W2, Q_total = √3 (W1 − W2); a negative reading on one meter indicates power factor below 0.5.

Instrument accuracy: percent error = (measured − true)/true × 100; accuracy class defines the maximum error as a percentage of full-scale (or reading) deflection.

A current transformer secondary must never be open-circuited while primary current flows; the collapsing core flux induces dangerous voltage, so short the secondary before removing burden.

Skin and proximity effects raise the effective AC resistance of conductors above the DC value at higher frequency, larger cross-section, and closer spacing.

Synchronous speed: n_s = 120 f / P rpm; a 60 Hz, 4-pole machine runs at 1800 rpm, and an induction rotor always turns slower than n_s in motoring.

Slip: s = (n_s − n_r)/n_s; rotor frequency f_r = s · f; rotor-induced EMF and reactance scale directly with slip.

Induction motor developed torque is approximately proportional to the square of applied terminal voltage, so reduced-voltage starting sharply lowers starting torque while lowering inrush current.

Round-rotor synchronous machine per phase: E_f = V_t + j X_s I_a; real power P ≈ (E_f V_t / X_s) sin δ, maximum at δ = 90° (steady-state stability limit).

DC machine: E_a = K Φ ω (generated EMF proportional to flux and speed); electromagnetic torque T = K Φ I_a; separately excited and shunt fields behave differently under load.

Transformer: turns ratio a = N1/N2 = V1/V2; impedance referred to the primary = a² × (secondary impedance); current ratio is inverse to the voltage ratio.

Transformer losses: core (hysteresis + eddy current) loss is essentially constant at rated voltage and frequency; copper loss varies as I² and is found from the short-circuit test; core loss from the open-circuit test.

Machine efficiency η = P_out / P_in; losses appear as heat, and temperature rise at a given load depends on the cooling method (self, forced, or totally enclosed).

Surge (characteristic) impedance of a lossless line Z_c = √(L/C), roughly 400 Ω for overhead lines in air; underground cables have far lower values because of high capacitance.

Voltage regulation = (|V_no-load| − |V_full-load|)/|V_full-load| × 100, evaluated at the receiving end for a specified load power factor.

Ferranti effect: on a long, lightly loaded or open line the receiving-end voltage exceeds the sending-end voltage because charging current through the series inductance adds voltage.

Bundled conductors (two, three, or four subconductors per phase) reduce series inductive reactance, reduce corona and audible noise, and raise the effective GMR of the phase.

Corona inception depends on conductor surface voltage gradient, diameter, and spacing; larger conductors, bundled phases, and higher humidity/altitude effects change the critical voltage.

Lossless line power transfer: P ≈ (V_s V_r / X) sin δ; the angle δ limits transfer, and series compensation reduces X to raise the steady-state limit.

Transposition of phase conductors at regular intervals balances the mutual inductances so that the three phases see approximately equal impedance over the line length.

Distribution grounding practice determines step and touch potentials; a ground grid is designed so that mesh and step voltages stay below tolerable body-current limits during a fault.

Symmetrical components: an unbalanced three-phase set resolves into positive, negative, and zero sequences; operator a = 1∠120°, with 1 + a + a² = 0.

Zero-sequence current requires a grounded return path; it circulates in grounded-wye windings and grounded networks but cannot pass an ungrounded delta or an ungrounded neutral.

Per-unit base change: Z_pu,new = Z_pu,old × (V_base,old / V_base,new)² × (S_base,new / S_base,old).

Single line-to-ground fault: I_a1 = V_f / (Z1 + Z2 + Z0 + 3Z_n), and the three sequence networks are connected in series for this fault type.

Swing equation: (2H/ω_s) d²δ/dt² = P_m − P_e (per unit); accelerating power produces rotor angle excursion, and H (inertia constant) sets the response speed.

Equal-area criterion: transient stability holds when the available decelerating area equals or exceeds the accelerating area; critical clearing time is when the two areas just balance.

Load flow: Newton–Raphson converges quadratically but needs a Jacobian each iteration; Gauss–Seidel is simple and slow; both solve the Y_bus power balance equations.

Three-phase fault at a bus: I_f = V_th / (√3 |Z_th|) using per-phase Thevenin impedance, with prefault voltage typically assumed 1.0 pu.

Distance relay zone 1 reaches approximately 80–85% of the protected line with no intentional delay so that it underreaches the remote bus; zone 2 reaches roughly 120–150% with a coordinating time delay.

Differential protection compares currents entering and leaving the protected zone; balanced through-fault current produces no operating signal, while an internal fault produces a differential current.

Overcurrent coordination: the device closest to the fault clears first; time-current curves, pickup multiples, and the coordination time interval set the grading margin between upstream and downstream relays.

Generator protection functions include 87G (differential), 51V (voltage-restrained overcurrent), 40 (loss of field), 32 (reverse power), 46 (negative sequence), and 59/27 (over/under voltage).

Directional overcurrent elements need a polarizing quantity (voltage, current, or memory) to establish a reference so that they respond only to faults in the intended direction.

Current transformer saturation distorts secondary current, delays relay pickup, and can cause overreach or miscoordination; DC offset in the primary current worsens saturation.

Transformer differential protection must compensate for the phase shift and ratio of wye-delta windings, and use harmonic restraint (often second harmonic) to avoid tripping on magnetizing inrush.

Breaker failure protection trips all surrounding breakers after a short timer if current through the faulted breaker persists after a trip command.

Single-phase full-controlled rectifier: V_dc = (2 V_m / π) cos α for continuous conduction; output becomes negative for α > 90°, permitting inversion.

Three-phase full bridge (six-pulse) rectifier: V_dc = (3√3 V_m / π) cos α ≈ 1.35 V_LL cos α, with V_m the peak phase voltage.

Buck chopper: V_o = D · V_in; boost chopper: V_o = V_in / (1 − D); both relations assume continuous inductor conduction and ideal switches.

Line-commutated converter: V_d = V_d0 cos α − (3/π) X_c I_d, where the second term is the commutation (reactance) voltage drop.

Twelve-pulse operation uses two six-pulse bridges fed by delta and wye secondaries; the 30° shift cancels the 5th and 7th harmonic currents on the AC side.

PWM inverters control both magnitude and frequency; the modulation index sets fundamental output, and higher switching frequency pushes harmonics to higher orders for easier filtering.

Thyristors turn off only at natural current zero (line commutation), while IGBTs and GTOs allow forced commutation; snubbers limit dv/dt and di/dt and dissipate switching energy.

Rectifier input power factor degrades as firing angle increases; displacement factor ≈ cos α, and total power factor also includes current harmonic distortion.