Eric Forgy Phasors

Phasors

Voltage and current is given by

v(t) = Re\left(V e^{j\omega t}\right)

i(t) = Re\left(I e^{j\omega t}\right)

where $V$ and $I$ are complex constants.

Instantaneous Power

\begin{aligned} p(t) &= v(t) i(t) \\ &= Re\left(V e^{j\omega t}\right) Re\left(I e^{j\omega t}\right) \\ &= \frac 1 4 \left(V e^{j\omega t} + V^* e^{-j\omega t}\right) \left(I e^{j\omega t} + I^* e^{-j\omega t}\right) \\ &= \frac 1 4 \left(V I e^{j2\omega t} + V^* I^* e^{-j2\omega t}\right) + \frac 1 4 \left(V I^* + V^* I\right) \\ &= \frac 1 2 Re\left(V I e^{j2\omega t}\right) + \frac 1 2 Re\left(V I^*\right) \end{aligned}

Average (or Real) Power

\begin{aligned} P &= \langle p(t)\rangle \\ &= \frac 1 2 Re\left(V I^*\right) \end{aligned}

Reactive Power

Q = \frac 1 2 Im\left(V I^*\right)

Complex Power

S = \frac 1 2 V I^* = P + j Q

Created on April 3, 2010 at 19:36:20 by Eric Forgy