# Phase Difference

In AC circuits, voltage and current waveform have their respective phase shifts. The difference in phase shift of these two waveforms is often referred to as phase difference:

\phi = \theta_{v} – \theta_{i} = \phi_{v} – \phi_{i}

When the phase difference is positive, it is said that the current waveform lags the voltage waveform.

When the phase difference is negative, it is said that the current waveform leads the voltage waveform.

During circuit analysis, it is bothersome to keep track of two phase angles. Furthermore, the relative phase difference between waveforms is the only thing that matters. The absolute value of the phase shift for each waveform is not important.

To simplify computations, the voltage waveform is typically selected as a reference point and all other waveforms have phase shifts relative to the reference.

Thus:

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

can be redefined as

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