Phasor representation of AC voltage and current

Astronuc

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

Astronuc

[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 = V_rms² I_rms²

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...ric/phase.html

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

http://www.walter-fendt.de/ph11e/accircuit.htm

infowarrior

so Phasor representation of an AC voltage is what magnitude? RMS

rbj

you might want to explicitly relate V_max to V_rms 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.