Complex power, impedance and power triangles

Complex power is $S = VI^* = P + jQ$. Its real part P (W) is the active power, its imaginary part Q (var) is the reactive power, and its magnitude $|S|$ (VA) is the apparent power. The angle of S is the power-factor angle $\phi = \theta_v - \theta_i$.

Symmetry with the impedance triangle

For a load $Z = R + jX = |Z|\angle\phi$ carrying current I, $V = IZ$. So

$$S = VI^* = (IZ)I^* = |I|^2Z = |I|^2R + j|I|^2X$$

So $P = |I|^2R$, $Q = |I|^2X$ and $|S| = |I|^2|Z|$. Every side of the impedance triangle is multiplied by the same real number $|I|^2$. The power triangle is therefore similar to the impedance triangle:

 Impedance triangle          Power triangle
        /|                        /|
   |Z| / | X              |S|    / | Q = |I|^2 X
      /  |                      /  |
     /phi|                     /phi|
    +----+                    +----+
      R                     P = |I|^2 R

Significance:

• The two triangles have the same angle. The power factor of a load can be found either from its impedance or from its powers.
• Inductive X (+) gives +Q (vars absorbed). Capacitive X (−) gives −Q (vars supplied). Both triangles flip below the axis together.
• This symmetry holds only because S is defined as $VI^*$. With $V^*I$, Q would be $-|I|^2X$ and the power triangle would be the mirror image of the impedance triangle.

Example: $Z = 6 + j8\ \Omega$ with $|I| = 10$ A gives $S = 600 + j800$ VA. Both triangles have angle 53.13° and pf 0.6 lagging.