C
Chris Carlen
Greetings:
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]
NOTE, delete texts: "RemoveThis" and "BOGUS" from email address to reply.
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]
NOTE, delete texts: "RemoveThis" and "BOGUS" from email address to reply.
