Phasor
المؤلف:
Krantz, S. G
المصدر:
"Polar Form of a Complex Number." §1.2.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser,
الجزء والصفحة:
pp. 8-10
24-10-2018
1174
Phasor
The representation, beloved of engineers and physicists, of a complex number in terms of a complex exponential
 |
(1)
|
where i (called j by engineers) is the imaginary number and the complex modulus and complex argument (also called phase) are
Here, 
(sometimes also denoted 
) is called the complex argument or the phase. It corresponds to the counterclockwise angle from the positive real axis, i.e., the value of 
such that 
and 
. The special kind of inverse tangent used here takes into account the quadrant in which 
lies and is returned by the FORTRAN command ATAN2(Y,X) and the Wolfram Language function ArcTan[x, y], and is often restricted to the range 
. In the degenerate case when 
,
![phi=<span style=]() {-1/2pi if y<0; undefined if y=0; 1/2pi if y>0. " src="http://mathworld.wolfram.com/images/equations/Phasor/NumberedEquation2.gif" style="height:78px; width:153px" /> |
(4)
|
It is trivially true that
![sum_(i)R[psi_i]=R[sum_(i)psi_i].](http://mathworld.wolfram.com/images/equations/Phasor/NumberedEquation3.gif) |
(5)
|
Now consider a scalar function 
. Then
where 
is the complex conjugate. Look at the time averages of each term,
Therefore,
 |
(19)
|
Consider now two scalar functions
Then
In general,
 |
(28)
|
REFERENCES:
Krantz, S. G. "Polar Form of a Complex Number." §1.2.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 8-10, 1999.
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