C

#### Chris Carlen

- Jan 1, 1970

- 0

This is just a reflection on some mathematical semantics and an interest

in having a neat notation for calculating the average power dissipated

in an AC circuit from rectangular forms of the voltage and current

phasors. It arose out of my deciding to review in detail my

understanding of AC power, so might be of some interest to students of

EE. There is nothing here which is not found in any basic circuits

text, but just my particular description of it.

To get started, my definition of a phasor would go like this:

"A phasor is a complex number that carries the information about the

magnitude and phase of a sinusoidal time varying function of fixed

frequency."

Thus, any old complex number is not a phasor, unless it is a complex

number arising from the phasor transform applied to a sinusoidal time

varying function of fixed frequency.

"Furthermore, a phasor is not a vector, though a phasor may be

graphically represented as a vector in the complex plane."

"Nor do phasors or the vectors representing them "rotate" in the complex

plane. There is nothing in the phasor transform that leads to the real

and imaginary components of the phasor being time dependent functions.

Indeed it's very point is to remove the time variation aspect from the

voltage and current quantities. Thus, the components of the phasors are

simply constants. It is only a demonstrative tool to illustrate how

IF the vector is rotated in time, that for the case of starting with a

cosine time domain function (and applying the respective phasor

transform), then the real component of the phasor traces out the

instantaneous magnitude of the time varying quantity. Likewise for the

case of starting with a sine time domain function (and applying the

respective phasor transform), then the imaginary component of the phasor

traces out the instantaneous magnitude of the time varying quantity."

Average power dissipated in a linear AC circuit driven by a fixed

frequency sinusoidal source can be determined by the dot product of the

vectors representing the phasors for

V(t) = Vm cos(wt+phi_V)

I(t) = Im cos(wt+phi_I)

Let's use the notation ~V and ~I to represent the phasor transforms of

V(t) and I(t) respectively.

But since ~V and ~I are phasors we can't notate or even speak of their

dot product as we would with vectors. Ie., the notation

~V . ~I

doesn't mean the dot product of the vectors, since the quantities ~V and

~I are phasors, not vectors.

Instead we would have to write out something ugly like:

Pave = [ Re{~V} Re{~I} + Im{~V} Im{~I} ]/2

which works, but isn't a concise mathematical notation such as the

simple dot between two vectors.

Obviously, if we have the phasors in polar form, then the calculation is

a more straightforward:

Pave = 0.5 Vm Im cos( phi_V - phi_I )

which of course is the definition of the dot product of the vectors

representing the phasors.

Additionally, there is the defnition of complex power as

S = P + jQ , where

P = 0.5 Vm Im cos( phi_V - phi_I ) = Pave

Q = 0.5 Vm Im sin( phi_V - phi_I ) = reactive power

In which case average power may be neatly expressed as:

Pave = Re{S}

And since S can be shown to be:

S = 1/2 ~V ~I* then

Pave = Re{ 1/2 ~V ~I* }

So perhaps that is it, huh? The real part of the product of a phasor

with the conjugate of another gives the same effect as the dot product

of the vectors representing those phasors.

Good day!

--

_______________________________________________________________________

Christopher R. Carlen

Principal Laser/Optical Technologist

Sandia National Laboratories CA USA

[email protected]

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