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Phasor representation of AC voltage and current

  1. Nov 10, 2007 #1

    Astronuc

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    Staff: Mentor

    Phasor representation of AC voltage and current.

    [tex]I\,=\,5\angle{0^o}\,=\,5\,+\,j0\,A[/tex]

    [tex]V\,=\,100\angle{30^o}\,=\,86.6\,+\,j50\,V[/tex]


    in general

    [tex]V\,=\,A\angle{\theta^o}\,=\,A cos{\theta}\,+\,jA sin{\theta}\,V[/tex]

    and similarly for I


    It is assumed that the angular frequency [itex]\omega[/itex] is the same throughout the system, and it is assumed that the Voltage and Current are RMS values.

    For the above phasor values, the voltage and current are:

    v(t) = 141.4 cos ([itex]\omega[/itex]t + 30°)

    and

    i(t) = 7.07 cos [itex]\omega[/itex]t
     
  2. jcsd
  3. Nov 14, 2007 #2

    Astronuc

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    Staff: Mentor

    AC Power in Phasor Form

    [tex]p(t)\,=\,[V_{max}\,cos(\omega{t}+\theta)] \times [I_{max}\,cos(\omega{t}+\phi)][/tex]

    becomes

    [tex]p(t)\,=\,\frac{V_{max}I_{max}}{2}[cos(\theta-\phi)\,+\,cos(2\omega{t}+\theta+\phi)][/tex]

    The average power is

    [tex] P\,=\,V_{rms}I_{rms}\,cos(\theta-\phi)[/tex]


    In phasor notation,

    [tex]v\,=\,V_{rms}\angle\theta[/tex]

    [tex]i\,=\,I_{rms}\angle\phi[/tex]

    but

    [tex]P\,\neq\,V_{rms}I_{rms}\angle(\theta+\phi)[/tex]

    Instead

    [tex]P\,=\,Re\{VI^*\}[/tex]

    and

    [tex]V\,I^*\,=\,(V_{rms}\angle\theta)\times(I_{rms}\angle-\phi)[/tex]

    [tex]\,=\,V_{rms}I_{rms}\angle(\theta-\phi)[/tex]

    The real part of power is given by

    [tex]P\,=\,V_{rms}I_{rms}cos(\theta-\phi)[/tex]

    and the reactive or imaginary part of power is

    [tex]Q\,=\,V_{rms}I_{rms}sin(\theta-\phi)[/tex]

    and the quantity [itex]cos(\theta-\phi)[/itex] is known as the power factor.

    The apparent power, S, expressed as volt-amperes (VA) is given by

    S (volt-amps) = P (Watts) + jQ (volt-amps-reactive) = VI*

    |S|2 = |P|2 + |Q|2 = Vrms2 Irms2

    PF = |P|/|S|

    VAR is commonly used as a unit for "volt-amperes-reactive"

    Some useful background on AC power and phasors.

    http://hyperphysics.phy-astr.gsu.edu/hbase/electric/phase.html

    http://www.physclips.unsw.edu.au/jw/AC.html

    http://www.walter-fendt.de/ph11e/accircuit.htm
     
    Last edited: Nov 15, 2007
  4. Jun 11, 2008 #3
    phasor representation

    so Phasor representation of an AC voltage is what magnitude? RMS
     
  5. Jun 11, 2008 #4

    rbj

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    you might want to explicitly relate Vmax to Vrms and similar for the currents. in fact, Astronuc, i might define the sinusoids as

    [tex] v(t) \triangleq V_{max} cos(\omega t + \theta) = \sqrt{2} V_{rms} cos(\omega t + \theta) [/tex]

    and

    [tex] i(t) \triangleq I_{max} cos(\omega t + \phi) = \sqrt{2} I_{rms} cos(\omega t + \phi) [/tex]

    and then crank out the instantaneous and mean power as you did.

    i dunno. just a suggestion.
     
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