Power Systems and Electrical Machines: cram pack

Balanced three-phase systems: in a Y (star) connection the line-to-line voltage is √3 times the phase voltage, V_L = √3·V_ph, and the line current equals the phase current. In a Δ (delta) connection the line-to-line voltage equals the phase voltage and the line current is √3 times the phase current, I_L = √3·I_ph. The total power is three times the power of one phase: P = 3·V_ph·I_ph·cos φ = √3·V_L·I_L·cos φ.

Balanced three-phase systems: in a Y (star) connection the line-to-line voltage is √3 times the phase voltage, V_L = √3·V_ph, and the line current equals the phase current. In a Δ (delta) connection the line-to-line voltage equals the phase voltage and the line current is √3 times the phase current, I_L = √3·I_ph. The total power is three times the power of one phase: P = 3·V_ph·I_ph·cos φ = √3·V_L·I_L·cos φ. Power factor correction: capacitors supply reactive power locally, Q_C = P·(tan φ₁ − tan φ₂). Each of the three capacitors supplies Q_C/3 = V²·ω·C, where V is the voltage across that capacitor: V_L in a Δ bank, V_L/√3 in a Y bank.

Ideal transformer: the voltages are in the ratio of the turns and the currents in the inverse ratio: V₁/V₂ = N₁/N₂ = a and I₁/I₂ = N₂/N₁ = 1/a, so the power in equals the power out, V₁·I₁ = V₂·I₂. A load impedance Z on the secondary looks like a²·Z from the primary.

Transformer losses: the core loss (hysteresis and eddy currents) depends on the voltage and is the same at any load. The copper loss in the windings goes with the current squared, so at a fraction x of rated load it is x²·P_cu,fl. Efficiency η = P_out/(P_out + P_core + P_cu), with P_out = x·S·pf. The efficiency is highest at the load where the copper loss equals the core loss.

AC power: apparent power S = V·I (VA) splits into real power P = S·cos φ (W), which does useful work, and reactive power Q = S·sin φ (var), which moves back and forth between source and load. The power factor is cos φ. A capacitor in parallel supplies reactive power locally, so the line current drops for the same real power.

AC power: apparent power S = V·I (VA) splits into real power P = S·cos φ (W), which does useful work, and reactive power Q = S·sin φ (var), which moves back and forth between source and load. The power factor is cos φ. A capacitor in parallel supplies reactive power locally, so the line current drops for the same real power.

AC circuits with phasors: resistors, inductors and capacitors have impedances R, jωL and 1/(jωC), with ω = 2πf. In series, impedances add like resistances, but as complex numbers. Rms current and voltage follow Ohm’s law with |Z|.

AC circuits with phasors: resistors, inductors and capacitors have impedances R, jωL and 1/(jωC), with ω = 2πf. In series, impedances add like resistances, but as complex numbers. Rms current and voltage follow Ohm’s law with |Z|.

DC motor: the armature is a resistance R_a in series with the back-EMF E, so V = E + I_a·R_a. The back-EMF grows with speed, E = k·ω, and the torque with current, T = k·I_a, with the same motor constant k (V·s/rad = N·m/A) in SI units. Convert speeds with ω = 2π·n/60. The developed power is E·I_a = T·ω; the input power is V·I_a, and the copper loss I_a²R_a and the rotational loss make the difference. The speed falls in a straight line with torque, ω = V/k − R_a·T/k², from the no-load speed V/k to zero at the stall torque k·V/R_a. At standstill E = 0, so the starting current V/R_a is large unless a resistor limits it.

DC motor: the armature is a resistance R_a in series with the back-EMF E, so V = E + I_a·R_a. The back-EMF grows with speed, E = k·ω, and the torque with current, T = k·I_a, with the same motor constant k (V·s/rad = N·m/A) in SI units. Convert speeds with ω = 2π·n/60. The developed power is E·I_a = T·ω; the input power is V·I_a, and the copper loss I_a²R_a and the rotational loss make the difference. The speed falls in a straight line with torque, ω = V/k − R_a·T/k², from the no-load speed V/k to zero at the stall torque k·V/R_a. At standstill E = 0, so the starting current V/R_a is large unless a resistor limits it.