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Phasors vs. vectors and power calculations

Discussion in 'Electronic Basics' started by Chris Carlen, Mar 10, 2005.

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  1. Chris Carlen

    Chris Carlen Guest

    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

    NOTE, delete texts: "RemoveThis" and "BOGUS" from email address to reply.
     
  2. Don Pearce

    Don Pearce Guest

    I think that I would have to say a phasor is a rotating vector.

    d

    Pearce Consulting
    http://www.pearce.uk.com
     

  3. Then I would have to say: A phasor is a
    stationary vector used to conveniently
    refer to a rotating vector.
     
  4. krw

    krw Guest

    Either this, or a weapon of mass destruction.
     
  5. And the coordinate plane it is drawn on and the viewer of that plane
    are rotating at the same frequency. ;-)
     
  6. Chris Carlen

    Chris Carlen Guest

    The phasor transform is found by taking only the non time-dependent
    factors of the argument to the real operator in:

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

    =Re{Vm [cos(wt+phi_V) + j sin(wt+phi_V)]}

    =Re{Vm exp[j(wt+phi_V)]}

    =Re{Vm exp(jwt) exp(j phi_V)}

    Thus:

    ~V = Vm exp(j phi_V)

    ~V = Vm cos phi_V + j sin phi_V

    There is nothing time dependent here to allow for any conception of
    rotation. The phasor is a constant complex number. How can a constant
    rotate? The only difference between it and an ordinary constant complex
    number, such as an impedance, is mathematically no difference at all,
    but contextually that it is derived from the phasor transform of a time
    varying function.

    That's why I said it is a semantics issue. Rotating the vector that
    graphically represents the phasor in the complex plane is a
    demonstrative tool. But it remains that there is nothing about the
    mathematical representation of the phasor that allows for rotation.


    One last thing. A vector represents a magnitude and a direction in a
    coordinate system. It is based upon orthonormal basis vectors such as
    what we might consider to be r_hat and j_hat in the complex plane,
    analagous to x_hat, y_hat in the XY plane.

    It is the definition of the basis vectors that gives mathematical
    consistency to a statement such as:

    A_vec . B_vec = (A_x x_hat + A_y y_hat).(B_x x_hat + B_y y_hat)
    = A_x B_x x_hat.x_hat + 2 A_x B_y x_hat.y_hat
    + A_y B_y y_hat.y_hat
    = A_x B_x + A_y B_y

    since x_hat.x_hat=1, y_hat.y_hat=1, and x_hat.y_hat=0

    But there are no basis vectors in a complex number or phasor. Thus, the
    phasor doesn't contain information about "direction" in an orthonormal
    coordinate system. That is why I carefully say that a phasor may be
    *graphically represented* as a vector in a complex coordinate plane
    characterized by orthonormal basis vectors r_hat and j_hat such that

    r_hat.r_hat=1 and
    j_hat.j_hat=1 and
    r_hat.j_hat=0

    So that we can represent the phasors as a vector by letting:

    V_vec = Re{~V} r_hat + Im{~V} j_hat
    I_vec = Re{~I} r_hat + Im{~I} j_hat

    Then the real average power:

    Pave = 0.5 V_vec.I_vec works.

    But the expression:

    ~V.~I is meaningless.


    Good day!


    --
    _______________________________________________________________________
    Christopher R. Carlen
    Principal Laser/Optical Technologist
    Sandia National Laboratories CA USA

    NOTE, delete texts: "RemoveThis" and "BOGUS" from email address to reply.
     
  7. Don Pearce

    Don Pearce Guest

    OK - I'll try again. A phasor is a representation of the instantaneous
    angular difference between subject and reference vectors, with the
    absolute magnitude of the subject vector.

    d

    Pearce Consulting
    http://www.pearce.uk.com
     
  8. Luhan Monat

    Luhan Monat Guest

    Something is rotating here; I seem to be getting dizzy.
     
  9. Terry Given

    Terry Given Guest

    what about exp(jwt) that you carefully dropped off? that is the rotating
    bit.
     
  10. Chris Carlen

    Chris Carlen Guest


    Terry, dropping the exp(jwt) is the *definition* of the phasor transform.

    For references, try here:

    Ulaby, Fawwaz. "Fundamentals of Applied Electromagnetics 2001 media
    ed." pg. 25

    And here:

    Nilsson, James W. and Susan Riedel. "Electric Circuits 6th ed." pg. 418


    And there must be a pile of other texts which explain this definition.


    Good day!


    --
    _______________________________________________________________________
    Christopher R. Carlen
    Principal Laser/Optical Technologist
    Sandia National Laboratories CA USA

    NOTE, delete texts: "RemoveThis" and "BOGUS" from email address to reply.
     
  11. Terry Given

    Terry Given Guest

    Indeed. If machines are your thing, Retter, Krause, Kron, Vas etc. have
    written plenty on the subject.

    I think this is a semantic issue. Phasors are, by definition, rotating
    vectors. The trick with phasor transforms is to rotate synchronously
    with some reference phasor, so all synchronous phasors appear
    stationary, but with angular displacement.

    Mathematically this is achieved by dropping the exp(jwt) term.

    If you build a synchronously rotating reference frame controller for
    electrical machinery, this "phasor transform" is normally implemented as
    follows:

    1) Choose a phasor on which to orient your rotating reference frame - eg
    AC line voltage. By definition this transformed phasor is purely real
    (or purely complex).

    2) Calculate the transformed real and imaginary parts by taking the
    original phasor and multiplying by exp(j*theta) [in a 3-phase system we
    typically assume some symmetry and perform a 3-phase to 2-phase
    transformation, resulting in a single complex phasor.

    3) use the imaginary (or real) part of the transformed phasor as the
    feedback signal for a PI controller to calculate the actual w. The
    setpoint for this PI controller will be zero, again by definition

    4) the output of the PI controller is angular speed, w. Usually the PI
    controller has the desired w (eg 2pi*50Hz) added to its output, so it
    only corrects the phase error, but its not strictly necessary - the PI
    controller will eventually get the right w.

    5) integrate the estimated w, giving theta = wt

    6) use this theta in the phasor transform of step 2.

    Ive implemented quite a few of these, and it works very well, for a wide
    range of PI gains. The magnitude of the error (ie |Eq|) can be used to
    detect synchronisation. Once synchronised, all them pesky rotating
    things suddenly look DC. If your are controlling an induction motor, the
    rotor vectors all spin at the slip speed.

    And of course its quite feasible to transform the synchronous PI
    controller into a stationary reference frame, it just becomes a resonant
    controller (DG Holmes, P Mattiavelli etc have covered these in detail)
    but thats boring power electronics control stuff.

    Interestingly enough, the electrical model for a machine in the
    stationary reference frame implicitly has time-varying inductances
    (hardly surprising, flux linkage being a function of rotor angle),
    making the analysis a right bastard. But through a phasor
    transformation, the time-varying nature of the inductance "disappears",
    making the resultant analysis straightforward. Its been around in power
    systems engineering since the 1920s, often referred to as Clarke (or is
    it Park, one is the 3:2 phase transform, the other is the
    stationary-rotating) transforms.

    This is another form of feedback linearisation based nonlinear control BTW

    Conversely whenever I analyse a 50Hz AC circuit, I implicitly use phasor
    transforms by simply ignoring the rotating aspect, and pretending the
    relevant vectors are all stationary.

    Cheers
    Terry
     
  12. Fred Abse

    Fred Abse Guest


    Are we allowed to Park here?

    :)
     
  13. Terry Given

    Terry Given Guest

    <Snort> LOL

    Cheers
    Terry
     
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