Complex Power

Complex power is a complex number with the real part equal to real power and the imaginary part equal to the reactive power.

$$\begin{array}{rl}S& =\vec{V}{\vec{I}}^{\ast }\\ & =Z{I}^2\\ & =P+jQ\end{array}$$

where ${\vec{I}}^{\ast }$ is the complex conjugate of $\vec{I}$.

Then, we have:

$$\begin{array}{r}P=\text{Re}(S)\\ Q=\text{Im}(S)\end{array}$$

We can also write:

$$\begin{array}{rl}S& =V{I}^{\ast }\\ & =IZ{I}^{\ast }\\ & =Z{I}^2\\ & =V{\left(\frac{V}{Z}\right)}^{\ast }\\ & =\boxed{\frac{V^2}{Z^{\ast }}}\end{array}$$