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Phasor


The representation, beloved of engineers and physicists, of a complex number in terms of a complex exponential

 x+iy=|z|e^(iphi),
(1)

where i (called j by engineers) is the imaginary number and the complex modulus and complex argument (also called phase) are

|z|=sqrt(x^2+y^2)
(2)
phi=tan^(-1)(y/x).
(3)

Here, phi (sometimes also denoted theta) is called the complex argument or the phase. It corresponds to the counterclockwise angle from the positive real axis, i.e., the value of phi such that x=|z|cosphi and y=|z|sinphi. The special kind of inverse tangent used here takes into account the quadrant in which z 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 -pi<theta<=pi. In the degenerate case when x=0,

 phi={-1/2pi   if y<0; undefined   if y=0; 1/2pi   if y>0.
(4)

It is trivially true that

 sum_(i)R[psi_i]=R[sum_(i)psi_i].
(5)

Now consider a scalar function psi=psi_0e^(iphi). Then

I=[R(psi)]^2
(6)
=[1/2(psi+psi^_)]^2
(7)
=1/4(psi+psi^_)^2
(8)
=1/4(psi^2+2psipsi^_+psi^_^2),
(9)

where psi^_ is the complex conjugate. Look at the time averages of each term,

<psi^2>=<psi_0^2e^(2iphi)>
(10)
=psi_0^2<e^(2iphi)>
(11)
=0
(12)
<psipsi^_>=<psi_0e^(iphi)psi_0e^(-iphi)>
(13)
=psi_0^2
(14)
=|psi|^2
(15)
<psi^_^2>=<psi_0^2e^(-2iphi)>
(16)
=psi_0^2<e^(-2iphi)>
(17)
=0.
(18)

Therefore,

 <I>=1/2|psi|^2.
(19)

Consider now two scalar functions

psi_1=psi_(1,0)e^(i(kr_1+phi_1))
(20)
psi_2=psi_(2,0)e^(i(kr_2+phi_2)).
(21)

Then

I=[R(psi_1)+R(psi_2)]^2
(22)
=1/4[(psi_1+psi^__1)+(psi_2+psi^__2)]^2
(23)
=1/4[(psi_1+psi^__1)^2+(psi_2+psi^__2)^2+2(psi_1psi_2+psi_1psi^__2+psi^__1psi_2+psi^__1psi^__2)]
(24)
<I>=1/4[2psi_1psi^__1+2psi_2psi^__2+2psi_1psi^__2+2psi^__1psi_2]
(25)
=1/2[psi_1(psi^__1+psi^__2)+psi_2(psi^__1+psi^__2)]
(26)
=1/2(psi_1+psi_2)(psi^__1+psi^__2)=1/2|psi_1+psi_2|^2.
(27)

In general,

 <I>=1/2|sum_(i=1)^npsi_i|^2.
(28)

See also

Affix, Cis, Complex Argument, Complex Modulus, Complex Multiplication, Complex Number, Exponential Function, Inverse Tangent, Phase, Vector Magnitude

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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.

Referenced on Wolfram|Alpha

Phasor

Cite this as:

Weisstein, Eric W. "Phasor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Phasor.html

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