ACTIVE POWER

  P  =  U × I              (DC)
  P  =  U × I × cos φ      (AC single-phase)
  P  =  √3 × U × I × cos φ (AC three-phase)

  Variables :
    U       voltage (V) — line-to-line in three-phase
    I       line current (A)
    cos φ   power factor (0 to 1)
    √3      ≈ 1.732  (line-to-line / phase-to-neutral voltage ratio)

  Rationale : P is the energy actually consumed by the load (in W).
  It is the one that converts into heat, motion, light — the only
  power that shows up on the electricity bill (in kWh).


REACTIVE POWER

  Q  =  U × I × sin φ      (AC single-phase)
  Q  =  √3 × U × I × sin φ (AC three-phase)

  Variables :
    Q       reactive power (var)
    sin φ   = √(1 − cos²φ)

  Rationale : Q is the energy that continuously oscillates between
  the source and the inductors/capacitors of the load. It performs
  no useful work, but it flows through cables and transformers —
  hence the penalties applied by utilities beyond a certain cos φ.


APPARENT POWER

  S  =  U × I              (AC single-phase)
  S  =  √3 × U × I         (AC three-phase)

  Variables :
    S   apparent power (VA)

  Rationale : S is what the network must physically supply. It is the
  quantity that sizes cables, transformers and alternators. A
  transformer is rated in kVA (S), not in kW (P).


POWER TRIANGLE

  S²  =  P² + Q²
  cos φ  =  P / S

  Rationale : fundamental Pythagorean relation between the three
  powers. cos φ measures what fraction of S is actually useful.
  Examples : cos φ = 1 → resistive load (heater, halogen lamp) ;
  cos φ = 0.85 → typical motor ; cos φ = 0.6 → heavily inductive load.