Complex power and power triangle

Complex power (S) is the complex quantity whose real part is the active power P and whose imaginary part is the reactive power Q:

S = V I* = P + jQ (volt-ampere, VA)

Here V is the voltage phasor and I* is the conjugate of the current phasor. Its magnitude |S| = VI is the apparent power.

R-L series circuit

Take V = V∠0° and Z = R + jX_L = Z∠φ. Then I = V/Z = I∠−φ (lagging).

S = V I* = V∠0° × I∠+φ = VI∠φ = VI cos φ + j VI sin φ

• Active power P = VI cos φ = I²R (watt)
• Reactive power Q = VI sin φ = I²X_L (VAR), positive for an inductive load
• Apparent power S = VI = I²Z (VA)

S = P + jQ, |S| = √(P² + Q²), cos φ = P/S

Power triangle

          /|
     S   / |
  (VA)  /  | Q = VI sin φ (VAR)
       /   |
      / φ  |
     /_____|
      P = VI cos φ (W)

It is obtained by multiplying the sides of the impedance triangle (R, X_L, Z) by I². For an inductive load, Q is drawn upward (lagging). For a capacitive load, Q is drawn downward (leading).