MyOpenMath/Complex phasors

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Time average of the product of two signals[edit | edit source]

Here 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:

Graphs of current i, voltage v, and power p for an ac circuit with a phase shift between current and voltage.

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 | edit source]

  1. 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.