MyOpenMath/Complex phasors

Time average of the product of two signalsEdit

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:

 

FootnotesEdit

  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.