PE Electrical cheat sheet

S = P + jQ; |S|^2 = P^2 + Q^2; power factor = P/|S| = cos θ, where θ is the angle by which current lags or leads voltage.

Balanced three-phase: P = √3 V_LL I_L cos θ; Q = √3 V_LL I_L sin θ; S = √3 V_LL I_L.

Per-unit: Z_pu = Z/Z_base; for three-phase, Z_base = V_LL,base^2/S_base and I_base = S_base/(√3 V_LL,base).

Capacitor reactive power for power-factor correction: Q_c = P(tan θ_old − tan θ_new).

Efficiency: η = P_out/P_in; total loss = P_in − P_out; for motors P_in = √3 V_LL I_L cos θ.

Load factor = average load/peak load; demand factor = maximum demand/connected load; both are dimensionless.

Energy cost: kWh = kW × hours; cost = kWh × rate.

Ideal transformer: V1/V2 = N1/N2, I1/I2 = N2/N1, and S1 = S2.

KCL: algebraic sum of currents at a node is zero. KVL: algebraic sum of voltages around a closed loop is zero.

Series resistances add; parallel conductances add: 1/R_eq = Σ(1/R_i); two resistors: R_eq = R1R2/(R1+R2).

Thevenin equivalent: V_th = open-circuit voltage; R_th = V_oc/I_sc, with independent sources set to zero for resistance calculation.

Maximum power transfer occurs at R_L = R_th, giving P_max = V_th^2/(4R_th).

Impedance: Z_R = R; Z_L = jωL; Z_C = −j/(ωC); admittance Y = 1/Z.

Series resonance: f0 = 1/(2π√(LC)); at resonance X_L = X_C and the impedance is minimum and resistive.

First-order transient: τ = RC for RC circuits, τ = L/R for RL circuits; response reaches 63.2% of final value in one τ.

Sinusoidal RMS: V_rms = V_peak/√2; full-wave rectified average = 2V_peak/π.

AC power: P = V_rms I_rms cos θ; Q = V_rms I_rms sin θ; complex power S = V I*.

Synchronous speed: n_s = 120f/P rpm, where P is poles; electrical radian frequency ω_e = (P/2)ω_m.

Induction motor slip: s = (n_s − n_r)/n_s; rotor frequency f_r = s f_s; at standstill s = 1.

Induction motor developed torque: T = P_ag/ω_s at small slip; torque is proportional to slip near no load and to V^2.

Transformer EMF: E_rms = 4.44 f N Φ_max; core flux is set by applied voltage and frequency.

DC machine: E_a = KΦω; separately excited shunt generator terminal voltage V_t = E_a − I_aR_a.

Lorentz force on a conductor: F = B I l sin θ; induced voltage e = B l v for perpendicular motion.

Mechanical power: P = Tω; in SI, T (N·m) = P (W)/ω (rad/s).

Synchronous generator: frequency f = nP/120; voltage regulation = (V_no-load − V_full-load)/V_full-load × 100%.

Short line: V_s = V_r + I Z; series impedance Z = R + jX; shunt capacitance neglected.

Medium line: nominal π or T model uses ABCD parameters; sending-end V_s = A V_r + B I_r.

Voltage regulation = (|V_r,no-load| − |V_r,full-load|)/|V_r,full-load| × 100%.

Three-phase line loss = 3 I^2 R = (P^2 + Q^2)R/V_LL^2 for a balanced load.

Surge impedance Z_c = √(L/C); surge impedance loading SIL = V_LL^2/Z_c.

Ferranti effect: receiving-end voltage rises above sending-end voltage on light or open-circuit lines because charging current aids voltage.

Corona depends on conductor surface gradient, air density, and spacing; bundle conductors reduce gradient and corona loss.

Skin effect increases AC resistance with frequency and conductor size; steel-cored aluminum conductors are common for overhead lines.

Per-unit fault current: I_f,pu = V_th,pu/(Z_th,pu + Z_f,pu); actual current = I_f,pu × I_base.

Three-phase short-circuit MVA = V_LL^2/Z_th(Ω) = S_base/Z_th,pu on the same base.

Symmetrical components: phase quantities transform to positive, negative, and zero sequences using a = 1∠120°.

Load-flow bus types: slack (V, θ fixed), PV (P, V fixed), PQ (P, Q fixed).

Transient stability: equal-area criterion; critical clearing time increases with inertia constant H.

Swing equation: (2H/ω_s)(d^2δ/dt^2) = P_m − P_e in per unit.

Protection: overcurrent for radial, distance for transmission lines, differential for transformers and buses.

Grounding: touch and step voltages depend on soil resistivity and grid geometry; ground resistance R ≈ ρ/(2πr) for a hemisphere.