Eric Forgy
Phasors

Phasors

Voltage and current is given by

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

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

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

Instantaneous Power

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

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

Reactive Power

$$Q=\frac{1}{2}\mathrm{Im}\left(V{I}^{*}\right)$$Q = \frac 1 2 Im\left(V I^*\right)

Complex Power

$$S=\frac{1}{2}V{I}^{*}=P+jQ$$S = \frac 1 2 V I^* = P + j Q