Complex power and its significance

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

$$S = VI^* = |V||I|\angle\phi = P + jQ$$

where:

• $P = |V||I|\cos\phi$ (W)
• $Q = |V||I|\sin\phi$ (var)
• $|S| = |V||I|$ (VA) is the apparent power
• $\phi = \theta_v - \theta_i$

For a three-phase system, $S_{3\phi} = 3V_{ph}I_{ph}^* = \sqrt3 V_LI_L\angle\phi$.

Power triangle

        S (VA)
       /|
      / |  Q (var)
     /phi|
    /____|
      P (W)

Significance in circuit and power system analysis

• One complex number gives both $P$ and $Q$, together with their signs, so power flow is described completely.
• Sign convention: With $S = VI^*$, an inductive load absorbs $+Q$ and a capacitor absorbs $-Q$ (that is, it supplies Q). A source delivers power when $S$, computed with current leaving its + terminal, is positive.
• Conservation: The sum of complex power over all elements is zero (Tellegen's theorem), so $\sum P$ and $\sum Q$ each balance. This is the basis of load flow equations.
• It gives the power factor directly, $\cos\phi = P/|S|$, and so the size of the compensation needed.
• Equipment such as generators, transformers and cables is rated in VA (that is, $|S|$), because heating depends on $|I|$.
• With $S = I^2Z = |V|^2/Z^*$ it gives the losses and the reactive drop in lines directly.

Example: If $V = 230\angle0^\circ$ V and $I = 10\angle-36.87^\circ$ A, then $S = 2300\angle36.87^\circ = 1840 + j1380$ VA. The load absorbs 1840 W and 1380 var at a power factor of 0.8 lagging.