Sinusoidal waveforms expressed in the time domain.

\begin{aligned} v(t) &= V_{peak}cos(\omega t + \theta_{v}) \\ i(t) &= I_{peak}cos(\omega t + \theta_{i}) \end{aligned}

Using Euler’s formula:

e^{i \theta} = cos(\theta) + i sin(\theta)

Sinusoidal waveforms described in the time domain can be converted to phasors:

\begin{aligned} \vec{v} &= V_{peak}e^{i \theta_{v}}= V_{peak} \angle \theta_{v} = V_{peak}cos(\theta_{v}) + jV_{peak}sin(\theta_{v}) \\ \vec{i} &= I_{peak}e^{i \theta_{i}} = I_{peak} \angle \theta_{i} = I_{peak}cos(\theta_{i}) + jI_{peak}sin(\theta_{i}) \end{aligned}

By working in phasors, engineers simplify difficult computations into complex number operations.

One limitation with phasor operations is that they must have one identical frequency. When working with multiple frequencies, each frequency needs to be computed individually and then superimposed afterwards.