Complex power and sign conventions

Complex power is the phasor product of voltage and the conjugate of current. Its real part is the active power and its imaginary part is the reactive power:

$$S = VI^* = |V||I|\angle(\theta_v - \theta_i) = P + jQ$$

• $P = |V||I|\cos\phi$ (W), the average power converted to work or heat.
• $Q = |V||I|\sin\phi$ (var), the power that oscillates between the source and the fields of L and C.
• $|S| = \sqrt{P^2+Q^2}$ (VA), the apparent power. Equipment is rated in VA.
• $\phi = \theta_v - \theta_i$ is the power-factor angle. Q is positive for a lagging (inductive) current.

Sign conventions

Load (motor) convention: the current enters the positive terminal. $S = VI^*$ is then the power absorbed.

• P > 0: the element absorbs real power (resistor, motor).
• Q > 0: it absorbs reactive power (inductor, lagging load).
• Q < 0: it supplies reactive power (capacitor).

Source (generator) convention: the current leaves the positive terminal. $S = VI^*$ is then the power delivered.

• P > 0: the source delivers real power (generator action). P < 0 means it absorbs P (motor action).
• Q > 0: it delivers reactive power (over-excited machine). Q < 0 means it absorbs Q.

 Load convention          Source convention
   I -->  +                 +  --> I
   +------o                 o------+
   | Load |  S absorbed     | Src  |  S delivered
   +------o                 o------+
          -                 -

Example: $V = 100\angle0^\circ$ V and the current leaving the + terminal is $I = 5\angle-30^\circ$ A. Then $S = 500\angle30^\circ = 433 + j250$ VA, so the source delivers 433 W and 250 var.