
The current and voltage phasor transformation is defined as

Where $|V|e^{j\theta _v}$ and $|I|e^{j\theta _i}$ are phasors of voltage and current, and usually denoted with a tilde $\tilde {}$ over capital letters $\tilde {V}$, $\tilde {I}$.

The real part of a complex number can also be found as $\Re \{z\}=\frac {1}{2}(z +z^*)$, so the above two equations can be re-written as

Power is defined as a product of voltage and current.

If we replace voltage and current in Equation eq:power in the time domain with Equations eq:phasorV and eq:phasorI we get

Multiplying the terms above, and rearanging, we get:

We can again apply equation $\Re \{z\}=\frac {1}{2}(z +z^*)$ to simplify the above equation to

This can also be re-written as

p(t) above is instantaneous power, $S=\tilde {V}\tilde {I}^*=|V| |I| e^{j(\theta _v-\theta _i)}$ is complex power. Complex power has real and reactive parts S=P+jQ. The first part of the equation represents the average real power P delivered to the load $P=\frac {1}{2}\Re \{\tilde {V}\tilde {I}^*\}$, and the second part represents the fluctuating power. We are usually interested in the average real power P delivered to the load.

To find the real power delivered to the load, one would take the real part of the complex power. If we know that the impedance of the load is $Z=R+jX$, the voltage is $\tilde {V} = Z \tilde {I}$ and we remember that $\tilde {I} \tilde {I}^* = |I|^2$ then the real power is