# MyOpenMath/Complex phasors

- For more help with phasors visit Phasor algebra

## Time average of the product of two signals

editHere we multiply two signals with the same angular frequency but with different phases:

- ,

where denotes complex conjugate. For example,

Define , and make the algebra easier to follow by defining two phases:

- and, .

Note that is the product of two binomials, which yield four terms:

When the two binomials are multiplied we obtain four terms. We group them according to whether they involve the sum or difference between the two phases, and , because whether it is a sum or difference affects the time-dependence as follows:

These terms can be grouped into real and imaginary parts, expressed in terms of the sine and cosine functions:

With ac circuits it is customary to average over one period, , defined by the expression .^{[1]} Using the overbar notation to denote this time average, we have:

## Footnotes

edit- ↑ We may also average over an integral number of periods. Or, with minimal error, we may simply average over any interval of time much greater than T.